**Molecular Mean-Field Theory of Ionic Solutions: A Poisson-Nernst-Planck-Bikerman Model**

**Jinn-Liang Liu 1,\* and Bob Eisenberg 2,3**


Received: 16 April 2020; Accepted: 12 May 2020; Published: 14 May 2020

**Abstract:** We have developed a molecular mean-field theory—fourth-order Poisson–Nernst–Planck– Bikerman theory—for modeling ionic and water flows in biological ion channels by treating ions and water molecules of any volume and shape with interstitial voids, polarization of water, and ion-ion and ion-water correlations. The theory can also be used to study thermodynamic and electrokinetic properties of electrolyte solutions in batteries, fuel cells, nanopores, porous media including cement, geothermal brines, the oceanic system, etc. The theory can compute electric and steric energies from all atoms in a protein and all ions and water molecules in a channel pore while keeping electrolyte solutions in the extra- and intracellular baths as a continuum dielectric medium with complex properties that mimic experimental data. The theory has been verified with experiments and molecular dynamics data from the gramicidin A channel, L-type calcium channel, potassium channel, and sodium/calcium exchanger with real structures from the Protein Data Bank. It was also verified with the experimental or Monte Carlo data of electric double-layer differential capacitance and ion activities in aqueous electrolyte solutions. We give an in-depth review of the literature about the most novel properties of the theory, namely Fermi distributions of water and ions as classical particles with excluded volumes and dynamic correlations that depend on salt concentration, composition, temperature, pressure, far-field boundary conditions etc. in a complex and complicated way as reported in a wide range of experiments. The dynamic correlations are self-consistent output functions from a fourth-order differential operator that describes ion-ion and ion-water correlations, the dielectric response (permittivity) of ionic solutions, and the polarization of water molecules with a single correlation length parameter.

**Keywords:** bioelectricity; electrochemistry; thermodynamics; electrokinetics; molecular mean-field theory; Boltzmann and Fermi distributions; Poisson–Boltzmann; Poisson–Fermi; Poisson–Bikerman; Nernst–Planck; steric and correlation effects; ion channels; ion activity; double-layer capacitance; nanofluidics

#### **1. Introduction**

Water and ions give life. Their electrostatic and kinetic interactions play essential roles in biological and chemical systems such as DNA, proteins, ion channels, cell membranes, physiology, nanopores, supercapacitors, lithium dendrite growth, porous media, corrosion, geothermal brines, environmental applications, and the oceanic system [1–34]. Poisson, Boltzmann, Nernst, and Planck laid the foundations of classical electrostatic and kinetic theories of ions in 1813–1890 [35–39]. Gouy [40] and Chapman [41] formulated the Poisson–Boltzmann (PB) equation in 1910 and 1913, respectively [9]. Bikerman proposed a modified PB equation in 1942 for binary ionic liquids to account for **different-sized** ions with **voids** [42]. Eisenberg puns PNP for Poisson-Nernst-Planck and Positive-Negative-Positive semiconductor transistors to emphasize nonequilibrium flows of

ions through ion channels as life's transistors [43]. Ions in classical PB and PNP theories are treated as volumeless point charges like the 'ions' of semiconductors, namely holes and electrons in semiconductor electronics [44–55]. Water molecules are treated as a dielectric medium (constant) without volumes either. However, advanced technologies in ion channel experiments [56,57] and material science [33,34,58] have raised many challenges for classical continuum theories to describe molecular mechanisms of ions and water (or solvents) with specific size effects in these systems at nano or atomic scale [9,12,14,16–19,30,33].

There is another important property that classical continuum theories fail to describe, namely short-range **ion-ion** or **ion-water correlations** in ion channels [8,9], charge-induced thickening and density oscillations near highly charged surfaces [14], correlation-induced charge inversion on macroions (DNA, actin, lipid membranes, colloidal particles) [59], the phase structure of plasma and polar fluids [60], colloidal charge renormalization [60], etc. Several other properties related to correlations such as the dielectric response of electrolytes solutions and the polarization of water in various conditions or external fields are usually modeled differently from the correlation perspective [61–63].

We have recently developed a molecular mean-field theory called—Poisson-Nernst-Planck-Bikerman (PNPB) theory—that can describe the size, correlation, **dielectric**, and **polarization** effects of ions and water in aqueous electrolytes at equilibrium or nonequilibrium all within a unified framework [64–76]. Water and ions in this theory can have *different shapes* and *volumes* necessarily with intermolecular voids. The theory generalizes and unifies the second-order Poisson–Bikerman equation [42] of binary ionic liquids for different-sized ions with identical steric energies [72] and the **fourth-order** differential permittivity operator in Santangelo's model of one component plasma [77] or in the Bazant, Storey, and Kornyshev theory of general nonlocal permittivity for equal-sized ions in ionic liquids [78].

Ion-ion and ion-water correlations are modeled by the permittivity operator with a correlation length that depends on the diameter of ions or water and the valence of ions of interest [78]. The fourth-order operator yields a permittivity as an *output function* of spatial variables, salt concentration, and hydration shell structure including water diameter from solving the PNPB model and thus describes the **dehydration** of ions from bath to channel pore or from bulk to charged wall, the polarization of water, and the change of permittivities of electrolyte solutions at different locations in response to different configurations and conditions. Water densities also change with configurations and conditions.

The fourth-order operator introduces correlations into the mean-field equations so they can deal more realistically with real systems in which the correlations are of the greatest importance. A remark should be made here that simulations containing only particles do not automatically deal with correlations better than mean-field theories with fourth-order operators like this. It is not at all clear that simulations widely done in biophysics actually compute correlations well. Indeed, it is difficult to see how simulations that use conventions to approximate the electric field, and periodic boundary conditions to approximate macroscopic systems could deal with correlations correctly. The dearth of direct checks of the role of periodic boundary conditions, and of the accuracy of the conventional treatment of electrostatics, does little to assuage these concerns. The detailed direct checks found necessary in computational electronics are not easily found in simulations of ions in electrolyte solutions (see Chap. 6, particularly Figures 6.34–35 of [55] for some details that are found to be necessary in the simulations of computational electronics).

It is important to reiterate the obvious. Our model includes water as a molecule and depends on the hydration structure around ions. Our model uses partial differential equations (PDEs) to describe these essentially discrete properties of ionic solutions, and uses the physical parameters of individual atoms and water molecules, NOT just their mean-field description. This use of PDEs to describe inherently discrete processes is hardly new: most of probability theory [79,80] and the entire theory of wave equations, including the wave equation of the electron called the Schrödinger equation [81], treat discrete processes the same way, using PDEs that measure (in probability theory) the underlying discrete system, or represent it exactly as the discrete solutions of a continuum PDE (e.g., the Schrödinger equation describing a hydrogen atom).

The most important contribution of our work is to include water as discrete molecules by using **Fermi distributions** [82] of classical particles having excluded volumes with interstitial voids. We show that the treatment of water as finite size molecules requires, as a matter of mathematics, not physics, the existence of voids. This is demonstrated by mathematics and simple ways to compute the voids and their role are presented. These Fermi-like distributions yield **saturation** of all particles (ions and water) even under mathematically infinite large external fields. These distributions also satisfy **mass conservation** in the region of interest such as channel pores, which classical theories fail to describe as well. This Fermi distribution of classical particles obeying volume exclusion is reminiscent of the Fermi distribution of identical particles obeying the Pauli exclusion principle [83] in quantum mechanics.

We also introduce a new concept of distance-dependent potential between non-bonded particles for different-sized particles similar to the electric potential for different-charged ions and name it the **steric potential**. The void distribution function describes the van der Waals potential [84] of paired particles [85,86] in the system in a mean-field sense. The steric potential can be written as a distribution function of voids, emphasizing the crucial role of voids in our theory. The specific sizes of particles and the distance-dependent steric potential allow us to calculate steric energies at the **atomic scale**. Using Coulomb's law allows calculation of electric energies at the atomic scale as well. Therefore, our theory applies to biological or chemical systems with explicit atomic structures, as well as classical mean-field representations of bulk solutions, for example. We have shown that solving the PNPB model in different continuum and molecular domains yields *self-consistent* electric and steric potentials in many examples of biological ion channels or chemical systems in [64–76]. The theory is also *consistent* with classical theories in the sense that its model converges to the corresponding classical one when the volume of all particles and the correlation length tend to zero, i.e., steric and correlation effects vanish asymptotically to classical cases.

In this review article, we explain the above bold-face terms in detail and compare them with those of earlier theories in a precise but limited way. The precision means that we display explicitly, to the best of our ability, the significant differences between analogous concepts in our theory and previous treatments. It is obviously impossible to do complete comparisons in this vast and formidable field. No doubt we are ignorant of significant relevant papers. We apologize to those inadvertently slighted and ask them to help us remedy our oversight. The remaining of this article consists as follows.

Section 2 describes the physical meaning of Fermi distributions and the steric potential of ions and water with excluded volumes. We also explain the differences between Fermi and Boltzmann distributions in the context of statistical thermodynamics.

Section 3 unifies Fermi distributions and correlations into the simple and concise 4th-order Poisson–Bikerman (4PBik) equation. The simplicity refers to the correlation length being the only empirical parameter in the equation. The conciseness means that the fourth-order differential operator can describe the complex and correlated properties of ion-ion and ion-water interactions, polarization of water, and dielectric response of electrolytes solutions all in a single model setting.

Section 4 presents a Gibbs free energy functional for the 4PBik equation. We show that minimization of the functional yields the equation and Fermi distributions that reduce to Boltzmann distributions when the volumes of particles vanish in limiting case. This functional is critical to explain a major shortcoming of earlier modified PB models that cannot yield Boltzmann distributions in the limit. These models are thus not consistent with classical theories and may poorly estimate steric energies and other physical properties due to their coarse approximation of size effects.

Section 5 generalizes the 4PBik equation to the PNPB model to describe flow dynamics of ions and water in the system subject to external fields. The most important feature in this section is the introduction of the steric potential to the classical Nernst–Planck equation. Electric and steric potentials describe the dynamic **charge/space competition** between ions and water. We also show that the PNPB model reduces to the 4PBik equation at equilibrium.

Section 6 presents a generalized Debye-Hückel theory from the 4PBik equation for thermodynamic modeling. The theory yields an equation of state that analytically models ion activities in all types of binary and multi-component electrolyte solutions over wide ranges of concentration, temperature, and pressure. It is also useful to study the size, correlation, dielectric, and polarization effects in a clear comparison with those ignoring these effects.

Section 7 discusses numerical methods for solving the PNPB model that is highly nonlinear and complex when coupled with the electrical field generated by protein charges in ion channels, for example. It is very challenging to numerically solve the model with tolerable accuracy in 3D protein structures that generate extremely large electric field, e.g., 0.1 V in 1 Angstrom, in parts of the molecule of great biological importance where crowded charges directly control biological function, in the same sense that a gas pedal controls the speed of a car.

Section 8 demonstrates the usefulness of the PNPB theory for a wide range of biological and chemical systems, where the steric and correlation effects are of importance. We choose a few examples of these systems, namely electric double layers, ion activities, and biological ion channels.

Section 9 summarizes this review with some concluding remarks.

#### **2. Fermi Distributions and Steric Potential**

The total volume of an aqueous electrolyte system with *K* species of ions in a solvent domain Ω*<sup>s</sup>* is

$$V = \sum\_{i=1}^{K+1} v\_i N\_i + V\_{K+2\nu} \tag{1}$$

where *K* + 1 and *K* + 2 denote water and voids, respectively, *v<sup>i</sup>* is the volume of each species *i* particle, *N<sup>i</sup>* is the total number of species *i* particles, and *VK*+<sup>2</sup> is the total volume of all the voids [68]. The volume of each particle *v<sup>i</sup>* will play a central role in our analysis, as well that the limit *v<sup>i</sup>* goes to zero. This limit defines the solution of point particles of classical PB and PNP theory. We must include the voids as a separate species if we treat ions and water having volumes in a model. This necessity can be proven by mathematics (see below). It is also apparent to any who try to compute a model of this type with molecular water, as it was to us [68].

Dividing the volume Equation (1) in bulk conditions by *V*, we get the bulk volume fraction of voids

$$
\Gamma^B = 1 - \sum\_{i=1}^{K+1} v\_i \mathbb{C}\_i^B = \frac{V\_{K+2}}{V} \,\prime \tag{2}
$$

where *C B <sup>i</sup>* = *Ni V* are bulk concentrations. If the system is spatially inhomogeneous with variable electric or steric fields, as in realistic systems, the constants *C B i* then change to functions *Ci*(**r**) and so does Γ *B* to a void volume function

$$\Gamma(\mathbf{r}) = 1 - \sum\_{i=1}^{K+1} v\_i \mathbb{C}\_i(\mathbf{r}). \tag{3}$$

We define the concentrations of particles (i.e., the distribution functions of the number density) in Ω*<sup>s</sup>* [72] as

$$\mathcal{C}\_{i}(\mathbf{r}) = \mathcal{C}\_{i}^{B} \exp\left(-\beta\_{i}\phi(\mathbf{r}) + \frac{v\_{i}}{v\_{0}}\mathcal{S}^{trc}(\mathbf{r})\right), \; \mathcal{S}^{trc}(\mathbf{r}) = \ln \frac{\Gamma(\mathbf{r})}{\Gamma^{B}},\tag{4}$$

where *φ*(**r**) is an electric potential, *S trc*(**r**) is called a *steric potential*, *β<sup>i</sup>* = *qi*/*kBT* with *q<sup>i</sup>* being the charge on species *i* particles and *qK*+<sup>1</sup> = 0, *k<sup>B</sup>* is the Boltzmann constant, *T* is an absolute temperature, and *v*<sup>0</sup> = ∑ *K*+1 *i*=1 *vi* /(*K* + 1) is an average volume. The following inequalities

$$\begin{split} \mathsf{C}\_{i}(\mathbf{r}) &= \ \mathsf{C}\_{i}^{B} \exp\left(-\beta\_{i}\phi(\mathbf{r})\right) \left[\frac{\Gamma(\mathbf{r})}{\Gamma^{B}}\right]^{v\_{i}/v\_{0}} = \mathsf{a}\_{i} \left[1 - \sum\_{j=1}^{K+1} v\_{j} \mathsf{C}\_{j}(\mathbf{r})\right]^{v\_{i}/v\_{0}} \\ &= \ \mathsf{a}\_{i} \left[1 - v\_{i} \mathsf{C}\_{i}(\mathbf{r}) - \sum\_{j=1, j\neq i}^{K+1} v\_{j} \mathsf{C}\_{j}(\mathbf{r})\right]^{v\_{i}/v\_{0}} < \mathsf{a}\_{i} \left[1 - v\_{i} \mathsf{C}\_{i}(\mathbf{r})\right]^{v\_{i}/v\_{0}} \\ &\leq \ \mathsf{a}\_{i} \left[1 - \frac{v\_{i}^{2}}{v\_{0}} \mathsf{C}\_{i}(\mathbf{r})\right] \text{ if } v\_{i}/v\_{0} \leq 1, \text{by Bernoulli's inequality.} \end{split} \tag{5}$$

$$\begin{aligned} \mathbf{C}\_{i}(\mathbf{r}) &< \ \ a\_{i} \left[ 1 - v\_{i} \mathbf{C}\_{i}(\mathbf{r}) \right]^{v\_{i}/v\_{0}} = a\_{i} \left[ 1 - v\_{i} \mathbf{C}\_{i}(\mathbf{r}) \right]^{\gamma} \left[ 1 - v\_{i} \mathbf{C}\_{i}(\mathbf{r}) \right]^{v\_{i}/v\_{0} - \gamma} \\ &< \ \ a\_{i} \left[ 1 - v\_{i} \mathbf{C}\_{i}(\mathbf{r}) \right] \left[ 1 - \left( v\_{i}/v\_{0} - \gamma \right) v\_{i} \mathbf{C}\_{i}(\mathbf{r}) \right] \\ &< \ \ a\_{i} \left[ 1 - v\_{i} \mathbf{C}\_{i}(\mathbf{r}) \right] \text{ if } v\_{i}/v\_{0} > 1, \end{aligned} \tag{6}$$

imply that the distributions are of Fermi-like type [87]

$$\mathcal{C}\_{i}(\mathbf{r}) \quad < \lim\_{a\_{i} \to \infty} \frac{a\_{i}}{1 + a\_{i}v\_{i}^{2}/v\_{0}} < \frac{v\_{0}}{v\_{i}^{2}} \text{ if } v\_{i}/v\_{0} \le 1 \tag{7}$$

$$\mathcal{C}\_{i}(\mathbf{r}) \quad < \lim\_{\mathfrak{a}\_{i} \to \infty} \frac{\mathfrak{a}\_{i}}{1 + \mathfrak{a}\_{i} v\_{i}} < \frac{1}{v\_{i}} \text{ if } v\_{i}/v\_{0} > 1,\tag{8}$$

i.e., *Ci*(**r**) cannot exceed the maximum value 1/*v* 2 *i* or 1/*v<sup>i</sup>* for any arbitrary (or even infinite) potential *φ*(**r**) at any location **r** in the domain Ω*<sup>s</sup>* , where *i* = 1, · · ·, *K* + 1, *α<sup>i</sup>* = *C B i* exp (−*βiφ*(**r**)) / Γ *B vi*/*v*<sup>0</sup> > 0, 0 < *vi*/*v*<sup>0</sup> − *γ* < 1, and *γ* ≥ 1.

The classical Boltzmann distribution appears if all particles are treated as volumeless points, i.e., *v<sup>i</sup>* = 0 and Γ(**r**) = Γ <sup>B</sup> = 1. The classical Boltzmann distribution may produce an infinite concentration *Ci*(**r**) → ∞ in crowded conditions when −*βiφ*(**r**) → ∞, close to charged surfaces for example, which is physically impossible [64–66]. This is a major, even crippling deficiency of PB theory for modeling a system with strong local electric fields or interactions. The difficulty in the application of classical Boltzmann distributions to saturating systems has been avoided in the physiological literature (apparently starting with Hodgkin, Huxley, and Katz [88]) by redefining the Boltzmann distribution to deal with systems that can only exist in two states. This redefinition has been vital to physiological research and is used in hundreds of papers [89,90], but confusion results when the physiologists' saturating two-state Boltzmann is not kept distinct from the unsaturating Boltzmann distribution of statistical mechanics [91].

It should be clearly understood that as beautiful as is Hodgkin's derivation it begs the question of what physics creates and maintains two states. Indeed, it is not clear how one can define the word state in a usefully unique way in a protein of enormous molecular weight with motions covering the scale from femtoseconds to seconds.

The steric potential *S trc*(**r**) in Equation (4) first introduced in [64] is an entropic measure of crowding or emptiness of particles at **r**. If *φ*(**r**) = 0 and *Ci*(**r**) = *C B i* then *S trc*(**r**) = 0. The factor *<sup>v</sup>i*/*v*<sup>0</sup> shows that the steric energy <sup>−</sup>*v<sup>i</sup> v*0 *S trc*(**r**)*kBT* of a type *i* particle at **r** depends not only on the steric potential *S trc*(**r**) but also on its volume *v<sup>i</sup>* similar to the electric energy *βiφ*(**r**)*kBT* depending on both *φ*(**r**) and *q<sup>i</sup>* [72]. The steric potential *S trc*(**r**) is especially relevant to determining selectivity of specific ions by certain biological ion channels [65,66,68,70,72].

In this mean-field Fermi distribution, it is impossible for a volume *v<sup>i</sup>* to be completely filled with particles, i.e., it is impossible to have *viCi*(**r**) = 1 (and thus Γ(**r**) = 0) since that would make *S trc*(**r**) = <sup>−</sup><sup>∞</sup> and hence *<sup>C</sup>i*(**r**) = 0, a contradiction. Therefore, *we must include the voids as a separate species if we treat ions and water having volumes in a model* for which *Ci*(**r**) < 1/*v<sup>i</sup>* and Γ(**r**) 6= 0 for all *i* = 1, · · ·, *K* + 1 and **r** ∈ Ω*<sup>s</sup>* . This is a critical property distinguishing our theory from others that do not consider water as a molecule with volume and so do not have to consider voids. We shall elaborate this property in Section 4.

Our theory is consistent with the classical theory of van der Waals in molecular physics, which describes nonbond interactions between any pair of atoms as a distance-dependent potential such as the Lennard-Jones (L-J) potential that cannot have zero distance between the pair [85,86]. Indeed, the steric potential *S trc*(**r**) can be written as a function of the volume of all molecular species (of course, including water as well as ions). Classical extensions of van der Waals theories often use this variable, but seem not to mention the existence or importance of voids.

The steric potential *S trc*(**r**) lumps all van der Waals potential energies of paired particles in a mean-field sense. It is an approximation of L-J potentials that describe local variations of L-J distances (and thus empty voids) between any pair of particles. L-J potentials are highly oscillatory and extremely expensive and unstable to compute numerically [66]. Calculations that involve L-J potentials [92–98], or even truncated versions of L-J potentials [99–101] must be extensively checked to be sure that results do not depend on irrelevant parameters. Any description that uses L-J potentials has a serious problem specifying the combining rule. The details of the combining rule *directly* change predictions of effects of different ions (selectivity) and so predictions depend on the reliability of data that determines the combining rule and its parameters.

*The steric potential does not require combining rules*. Since we consider specific sizes of ions and water with voids, the steric potential is valid on the *atomic scale* of L-J potentials. It is also *consistent* with that on the *macroscopic scale* of continuum models as shown in Sections 6 and 8.

To our surprise during the writing of this article, we found Equation (2) in Bikerman's 1942 paper [42] is *exactly* the *same* as Equation (4) for a special case of binary ionic liquids with the identical steric energies of different-sized ions, i.e., the factor *vi*/*v*<sup>0</sup> = 1 in (4). The steric potential in Equation (4) is however not explicitly expressed in Bikerman's paper. Therefore, Bikerman's concentration function is a Fermi distribution, a generic term used in statistical mechanics. We do NOT use exactly the Fermi distribution as Fermi derived in 1926 for identical particles now called fermions in quantum mechanics. So it is both more precise and historically correct to use the name "Poisson–Bikerman" equation for finite-sized ions as a generalization of the Poisson–Boltzmann equation for volumeless ions in electrochemical and bioelectric systems.

As noted by Bazant et al. in their review paper [14], Bikerman's paper has been poorly cited in the literature until recently. In our intensive and extensive study of the literature since 2013 [64], we have never found any paper specifically using Bikerman's formula as Equation (4), although of course there may be an instance we have not found. We thus now change the term "Poisson-Fermi" used in our earlier papers to "Poisson-Bikerman" in honor of Bikerman's brilliant work. We present here mathematical as well as physical justifications of a very general treatment of different-sized ions and water molecules in the mean-field framework based on Bikerman's pioneer work.

#### **3. Fourth-Order Poisson-Bikerman Equation and Correlations**

Electrolytes have been treated mostly in the tradition of physical chemistry of isolated systems that proved so remarkably successful in understanding the properties of ideal gases in atomic detail, long before the theory of partial differential equations, let alone numerical computing was developed. Most applications of ionic solutions however involve systems that are not at all isolated. Rather, most practical systems include electrodes to deliver current and control potential, and reservoirs to manipulate the concentrations and types of ions in the solution. Indeed, all biology occurs in ionic solutions and nearly all of biology involves large flows. It is necessary then to extend classical approaches so they deal with external electric fields and other boundary conditions and allow flow so the theory can give useful results that are applicable to most actual systems.

When the electrolyte system in Ω*<sup>s</sup>* is subject to external fields such as applied voltages, surface charges, and concentration gradients on the boundary *∂*Ω*<sup>s</sup>* , the electric field **E**(**r**) of the system, the displacement field **D**(**r**) of free ions, and the polarization field **P**(**r**) of water are generated at all **r** in Ω*<sup>s</sup>* . In Maxwell's theory [102,103], these fields form a constitutive relation

$$\mathbf{D}(\mathbf{r}) = \epsilon\_0 \mathbf{E}(\mathbf{r}) + \mathbf{P}(\mathbf{r}) \tag{9}$$

and the displacement field satisfies

$$\nabla \cdot \mathbf{D}(\mathbf{r}) = \rho\_{ion}(\mathbf{r}) = \sum\_{i=1}^{K} q\_i \mathbb{C}\_i(\mathbf{r}), \tag{10}$$

where *e*<sup>0</sup> is the vacuum permittivity, *ρion*(**r**) is the charge density of ions, and *Ci*(**r**) are the concentrations defined in (4). See [104] for a modern formulation of Maxwell's theory applicable wherever the Bohm version of quantum mechanics applies [105,106].

The electric field **E**(**r**) is thus screened by water (Bjerrum screening) and ions (Debye screening) in a correlated manner that is usually characterized by a correlation length *l<sup>c</sup>* [77,78,107]. The screened force between two charges in ionic solutions (at **r** and **r** 0 in Ω*s*) has been studied extensively in classical field theory and is often described by the van der Waals potential kernel [71,72,84,107,108]

$$\mathcal{W}(\mathbf{r} - \mathbf{r}') = \frac{e^{-|\mathbf{r} - \mathbf{r}'| / l\_c}}{|\mathbf{r} - \mathbf{r}'| / l\_c} \tag{11}$$

that satisfies the Laplace-Poisson equation [108]

$$-\Delta \mathcal{W}(\mathbf{r} - \mathbf{r}') + \frac{1}{l\_c^2} \mathcal{W}(\mathbf{r} - \mathbf{r}') = \delta(\mathbf{r} - \mathbf{r}'), \ \mathbf{r}, \mathbf{r}' \in \mathbb{R}^3 \tag{12}$$

in the whole space *R* 3 , where <sup>∆</sup> <sup>=</sup> ∇ · ∇ <sup>=</sup> <sup>∇</sup><sup>2</sup> is the Laplace operator with respect to **r** and *δ*(**r** − **r** 0 ) is the Dirac delta function at **r** 0 .

The potential *φ*e(**r**) defined in

$$\mathbf{D}(\mathbf{r}) = -\epsilon\_s \nabla \tilde{\phi}(\mathbf{r}) \tag{13}$$

describes an electric potential of free ions [72,107] that are correlated only by the mean electric field according to the Poisson equation

$$-\varepsilon\_{s} \Delta \tilde{\phi}(\mathbf{r}) = \rho\_{\rm ion}(\mathbf{r}),\tag{14}$$

a second-order partial differential equation, where *e<sup>s</sup>* = *ewe*<sup>0</sup> and *e<sup>w</sup>* is the dielectric constant of water. This potential does not account for correlation energies between individual ions or between ion and polarized water in high field or crowded conditions under which the size and valence of ions and the polarization of water play significant roles [17,65–68,77,78,107].

The correlations implicit in Maxwell's equations are of the mean-field and can be summarized by the statement that current is conserved perfectly and universally on all scales that the Maxwell equations are valid, where current includes the term *e*<sup>0</sup> *∂***E**(**r**,*t*) *∂t* . This term allows the Maxwell equations to describe the propagation of light through a vacuum, and it allows charge to be relativistically invariant, i.e., independent of velocity unlike mass, length, and time all of which vary dramatically as velocities approach the speed of light [104,106].

We introduce the *correlated* electric potential

$$\boldsymbol{\phi}(\mathbf{r}) = \int\_{R^3} \frac{1}{l\_c^2} \mathcal{W}(\mathbf{r} - \mathbf{r}') \tilde{\boldsymbol{\phi}}(\mathbf{r}') d\mathbf{r}' \tag{15}$$

in [72] as a convolution of the displacement potential *φ*e(**r** 0 ) with *W*(**r** − **r** 0 ) to deal with the correlation and polarization effects in electrolyte solutions. However, it would be too expensive to calculate *φ*(**r**) using (15). Multiplying (12) by *φ*e(**r** 0 ) and then integrating over *R* <sup>3</sup> with respect to **r** 0 [71], we obtain

$$-l\_c^2 \Delta \phi(\mathbf{r}) + \phi(\mathbf{r}) = \tilde{\phi}(\mathbf{r}) \tag{16}$$

a Laplace-Poisson equation [107,108] that satisfies (15) in the whole unbounded space *R* <sup>3</sup> with the boundary conditions *φ*(**r**) = *φ*e(**r**) = 0 at infinity. From (14) and (16), we obtain the *4th-order* Poisson–Bikerman equation

$$\left[\epsilon\_{\rm s} \left[l\_{\rm c}^{2}\Delta - 1\right] \Delta\phi(\mathbf{r}) = \rho\_{\rm ion}(\mathbf{r}), \; \mathbf{r} \in \Omega\_{\rm s} \tag{17}$$

a PDE that is an approximation of (16) in a bounded domain Ω*<sup>s</sup>* ⊂ *R* <sup>3</sup> with suitable boundary conditions (see below) of *φ*(**r**) on *∂*Ω*<sup>s</sup>* . We can thus use (9) to find the polarization field

$$\mathbf{P}(\mathbf{r}) = \epsilon\_s l\_c^2 \nabla(\Delta \phi(\mathbf{r})) - (\epsilon\_w - 1)\epsilon\_0 \nabla \phi(\mathbf{r}) \tag{18}$$

with **E**(**r**) = −∇*φ*(**r**). If *l<sup>c</sup>* = 0, we recover the standard Poisson Equation (14) and the standard polarization **P** = *e*0(*e<sup>w</sup>* − 1)**E** with the electric susceptibility *e<sup>w</sup>* − 1 (and thus the dielectric constant *ew*) if water is treated as a time independent, isotropic, and linear dielectric medium [103]. In this case, the field relation **D** = *ewe*0**E** with the scalar constant permittivity *ese*<sup>0</sup> is an approximation of the exact relation (9) due to the simplification of the dielectric responses of the medium material to the electric field **E** [109–111].

The exponential van der Waals potential *W*(**r** − **r** 0 ) = *<sup>e</sup>* <sup>−</sup>|**r**−**<sup>r</sup>** 0 |/*lc* |**r**−**r** 0 |/*l<sup>c</sup>* [84] is called the Yukawa [112] potential in [71,72] and usually in physics, which is an anachronism [108,113]. Van der Waals derived this potential in his theory of capillarity based on the proposition that the intermolecular potential of liquids and gases is shorter-ranged, but much stronger than Coulomb's electric potential [108]. Ornstein and Zernike (OZ) introduced short- (direct) and long-ranged (indirect) correlation functions in their critical point theory [114]. There are three important properties of the van der Waals potential: (i) it satisfies the Laplace-Poisson Equation (12), (ii) it generates the same functional form for shortand long-ranged correlations in the OZ theory, and (iii) it solves van der Waals's problem for the intermolecular potential [108].

Therefore, the potential *φ*(**r**) in (15) includes *correlation* energies of *ion-ion* and *ion-water* interactions in *short* as well as *long* ranges in our system. The *correlation length l<sup>c</sup>* can be derived from the OZ equation, see Equation (13) in [108], but the derivation is not very useful. The correlation length becomes an unknown functional of *ρion*(**r**) in (10) and the OZ direct correlation function, and is hence usually chosen as an empirical parameter to fit experimental, molecular dynamics (MD), or Monte Carlo (MC) data [14,64–73,75,77,78,107]. It seems clear that it would be useful to have a theory that showed the dependence of correlation length on ion composition and concentration, and other parameters.

There are several approaches to fourth-order Poisson-Boltzmann equations for modeling ion-ion and ion-water correlations from different perspectives of physics [71,77,78,115,116]. In [77], a decomposed kernel acts on a charge density of counterions in a binary liquid without volumes and water (ion-ion correlations) in contrast to the potential *φ*e(**r**) in (15) that is generated by different-sized ions and water with voids in (14) (ion-ion and ion-water correlations in a multi-component aqueous electrolyte). The kernel consists of short-range (of van der Waals type) and long-range components from a decomposition of Coulomb's interactions. In [78], the kernel is a general nonlocal kernel that acts on a charge density of equal-sized ions in a binary liquid without water (ion-ion correlations). The kernel is a series expansion of the gradient operator ∇ and thus can yield not only a fourth-order PB but even higher-order PDEs. The fourth-order PB is the first-order approximation of the energy expansion that converges only with small wavenumbers *k* in Fourier frequency domain for the dielectric response of ionic liquids [78].

Derived from the framework of nonlocal electrostatics for modeling the dielectric properties of water in [107], the kernel acting on *φ*e(**r**) in [71] (ion-ion and ion-water correlations) consists of a van der Waals function and the Dirac delta function that correspond to the limiting cases *k* = 0 and *k* = ∞, respectively. In [115], a system of three PDEs derived from electrostatics and thermodynamic pressure has electric potential and concentration gradients of equal-sized cations and anions in a binary fluid as three unknown functions. Linearization and simplification of the nonlinear system can yield a linear fourth-order PB (ion-ion correlations). In [116], the fourth-order PB is derived from a free energy functional that models ion-ion correlations in a binary liquid using volume-fraction functions of equal-sized cations and anions with two additional parameters associated with the interaction energies of these two functions and their gradients.

The *dielectric operator e<sup>s</sup> l* 2 *<sup>c</sup>*∆ − 1 in (17) describes changes in dielectric response of water with salt concentrations (ion-water correlations), ion-ion correlations, and water polarizations all via the mean-field charge density function *ρion*(**r**) provided that we can solve (4) and (17) for a consistent potential function *φ*(**r**). Therefore, the operator (a mapping) depends not only on ion and water concentrations (*C B i* for all arbitrary species *i* = 1, · · ·, *K* + 1 of particles with any arbitrary shapes and volumes) but also on the location **r** and the voids at **r**. The operator thus produces a *dielectric function* <sup>b</sup>*e*(**r**,*<sup>C</sup> B i* ) as an *output* from the solution *φ*(**r**) that satisfies the 4PBik (17) that saturates as a function of concentration (4), as we shall repeatedly emphasize. This dielectric function <sup>b</sup>*e*(**r**,*<sup>C</sup> B i* ) is not an additional model for <sup>e</sup>*e*(**r**), <sup>e</sup>*e*(*k*), or <sup>e</sup>*e*(*<sup>C</sup> B i* ) as it often is in other models in the literature [62,63,117–127]. Here the dielectric function is an output, as we have stated.

The 4PBik Equation (17) with (4) is a very general model using only one extra parameter *l<sup>c</sup>* in the fourth-order operator to include many physical properties ignored by the classical Poisson-Boltzmann equation. We shall illustrate these properties of our model in Section 8.

#### **4. Generalized Gibbs Free Energy Functional**

To generalize the Gibbs free energy functional for Boltzmann distributions that satisfy the classical Poisson–Boltzmann equation [3,128,129], we introduce a functional in [72] for saturating Fermi distributions (4) that satisfy the 4th-order Poisson–Bikerman Equation (17)

$$F(\mathbf{C}) = F\_{\text{el}}(\mathbf{C}) + F\_{\text{en}}(\mathbf{C}),\tag{19}$$

$$F\_{\rm el}(\mathbf{C}) = \frac{1}{2} \int\_{\Omega\_{\rm s}} \rho\_{\rm ion}(\mathbf{r}) L^{-1} \rho\_{\rm ion}(\mathbf{r}) d\mathbf{r},\tag{20}$$

$$F\_{\rm en}(\mathbf{C}) = k\_B T \int\_{\Omega\_{\rm s}} \left\{ \sum\_{i=1}^{K+1} \mathbb{C}\_i(\mathbf{r}) \left( \ln \frac{\mathbb{C}\_i(\mathbf{r})}{\mathbb{C}\_i^B} - 1 \right) + \frac{\Gamma(\mathbf{r})}{v\_0} \left( \ln \frac{\Gamma(\mathbf{r})}{\Gamma^B} - 1 \right) \right\} d\mathbf{r},\tag{21}$$

where *Fel*(**C**) is an electrostatic functional, *Fen*(**C**) is an entropy functional, **C** = (*C*1(**r**), *C*2(**r**), · · ·, *CK*+1(**r**)), and *L* −1 is the inverse of the self-adjoint positive linear operator *L* = *e<sup>s</sup> l* 2 *<sup>c</sup>*∆ − 1 ∆ [71] in (17), i.e., *Lφ*(**r**) = *ρion*(**r**). **C** is a 'concentration vector' that specifies the number density, i.e., concentration of each species in the ionic solution, including water. **C** plays a central role in any theory of ionic solutions because it specifies the main property of a solution, namely its composition.

Taking the variations of *F*(**C**) at *Ci*(**r**), we have

$$\frac{\delta F(\mathbf{C})}{\delta \mathbf{C}\_{i}} = \int\_{\Omega\_{k}} \left\{ k\_{B}T \left[ \ln \frac{\mathbf{C}\_{i}(\mathbf{r})}{\mathbf{C}\_{i}^{B}} - \frac{v\_{i}}{v\_{0}} \ln \frac{\Gamma(\mathbf{r})}{\Gamma^{B}} \right] + \frac{1}{2} \left( q\_{i}L^{-1}\rho\_{ion}(\mathbf{r}) + \rho\_{ion}(\mathbf{r})L^{-1}q\_{i} \right) \right\} d\mathbf{r}\_{\prime}$$

$$\frac{1}{2} \left( q\_{i}L^{-1}\rho\_{ion}(\mathbf{r}) + \rho\_{ion}(\mathbf{r})L^{-1}q\_{i} \right) = q\_{i}\phi(\mathbf{r}),$$

$$\frac{\delta F(\mathbf{C})}{\delta \mathbf{C}\_{i}} = 0 \Rightarrow k\_{B}T \left[ \ln \frac{\mathbf{C}\_{i}(\mathbf{r})}{\mathbf{C}\_{i}^{B}} - \frac{v\_{i}}{v\_{0}} \ln \frac{\Gamma(\mathbf{r})}{\Gamma^{B}} \right] + q\_{i}\phi(\mathbf{r}) = 0\tag{22}$$

that yields the saturating Fermi distributions in (4) for all *i* = 1, · · ·, *K* + 1. Moreover, we have

$$\frac{\delta^2 F(\mathbf{C})}{\delta \mathbf{C}\_i^2} = \int\_{\Omega\_\mathbf{s}} \left\{ k\_\mathbf{B} T \left[ \frac{1}{\mathbf{C}\_i(\mathbf{r})} + \frac{v\_i^2}{v\_0} \frac{\Gamma^B}{\Gamma(\mathbf{r})} \right] + q\_i^2 L^{-1} \mathbf{C}\_i \right\} d\mathbf{r} > 0 \tag{23}$$

implying that the saturating Fermi distribution vector **C** is a unique minimizer of the functional *F*(**C**).

The Gibbs-Bikerman free energy functional *F*(**C**) has two important properties. First, its electrostatic part *Fel*(**C**) is defined in terms of the composition vector **C** only. It depends only on concentrations and nothing else. If an electrostatic functional *F*e *el*(*φ*e(**r**)) is defined in terms of ∇*φ*e(**r**) 2 for the PB equation [64,78,124,130–137], the corresponding concentration vector **C**e and the potential *φ*e(**r**) do *not* minimize the corresponding functional *F*e(**C**e, *φ*e(**r**)) [128,129], i.e., *F*e is not a Gibbs free energy functional [3,128]. Second, the limit of its entropic part

$$\lim\_{\varepsilon\_l \to 0} F\_{\varepsilon n}(\mathbf{C}) = k\_B T \int\_{\Omega\_\varepsilon} \sum\_{i=1}^{K+1} \mathbb{C}\_i^0(\mathbf{r}) \left( \ln \frac{\mathbb{C}\_i^0(\mathbf{r})}{\mathbb{C}\_i^B} - 1 \right) d\mathbf{r} \tag{24}$$

exists (*Fen* converges) when the volume *v<sup>i</sup>* tends to zero for all *i* = 1, · · ·, *K* + 1. This implies that all ionic species have Boltzmann distributions *C* 0 *i* (**r**) = *C B i* exp (−*βiφ*(**r**)), *i* = 1, · · ·, *K*, the water concentration *C* 0 *K*+1 (**r**) = *C B K*+1 is a constant, and the void fraction Γ(**r**) = Γ *<sup>B</sup>* = 1 since all particles are volumeless in PB theory. Therefore, the 4PBik model (4) and (17) is physically and mathematically *consistent* with the classical PB model in the limiting case when we ignore the steric (*v<sup>i</sup>* = 0) and correlation (*l<sup>c</sup>* = 0) effects.

There are many shortcomings of the lattice approach [138] frequently used to account for steric effects in lattice-based PB models [14,61,64,78,124,129,133–136,139–141]. For example, (i) it assumes equal-sized ions and thus cannot distinguish non-uniform particles as in (1), (ii) its effective ion size needs to be unrealistically large to fit data [14], (iii) its correction over Boltzmann's point charge approach appears only at high surface charges [125], (iv) its pressure term diverges very weakly (is greatly underestimated) at close packing [142], and (v) its entropy functional may diverge as the volume of ions tends to zero, i.e., the corresponding *lattice-based PB* model is *not* physically and mathematically *consistent* with the classical PB model in the limiting case [66].

The importance of the restriction in Point (i) is hard to overstate. Almost all the interesting properties of ionic solutions arise because of their selectivity (as it is called in biology) or specificity between species, and those different properties arise in large measure because of the different diameters of the ions. The equal diameter case is dull and degenerate.

Point (v) is a critical problem that is closely related to Points (ii)–(iv). The divergence is obvious for an entropy term *F*e*en* in Equation (2) in [133] as

$$\lim\_{v \to 0} \tilde{F}\_{\text{en}} = \lim\_{v \to 0} \sum\_{i=1}^{K} \tilde{\mathcal{C}}\_{i}(\mathbf{r}) \ln \left( v \tilde{\mathcal{C}}\_{i}(\mathbf{r}) \right) = -\infty \tag{25}$$

which also appears in [61,64,78,124,129,133–136,139–141]. It is impossible to derive Boltzmann distributions *C*e *<sup>i</sup>*(**r**) = *C B i* exp −*βiφ*e(**r**) from *F*e*en* as *v* → 0 without extra assumptions, see (2.6) in [129], for example. In fact, the assumption (2.6), i.e., *vC*e *<sup>i</sup>*(**r**) > 0, actually forbids us from taking *v* to the limit zero.

Our derivation of *Fen*(**C**) does not employ any lattice models but simply uses the exact volume Equation (1). Our theory should not be classified then as a lattice model as sometimes is the case, at least in informal discussions. The void function Γ(**r**) is an analytical generalization of the void fraction 1 − Φ in (20) in [14] with all volume parameters *v<sup>i</sup>* (including the bulk fraction Γ *B* ) being physical instead of empirical as Φ. The excess chemical potential in [14] is −*kBT* ln(1 − Φ) whereas ours is *Fen*(**C**) in (21).

These expressions are different in important respects. Our model is not a lattice-based model because its differences are crucial both mathematically and physically. Indeed, the lattice-based model is in a certain sense internally inconsistent with classical statistical mechanics since a fundamental result of classical statistical mechanics *vC*e *<sup>i</sup>*(**r**) > 0 prevents the model from satisfying the classical imperative of the Boltzmann distribution in the limit of zero *v*.

The Langmuir-type distribution

$$\mathcal{C}\_{i}(\mathbf{x}) = \frac{\mathcal{C}\_{i}^{\mathcal{B}} \exp \left( -\beta\_{i} \phi(\mathbf{x}) \right)}{1 + \sum\_{j=1}^{K} \frac{\mathcal{C}\_{j}^{\mathcal{B}}}{\mathcal{C}\_{j}^{\max}} \left( \exp \left( -\beta\_{j} \phi(\mathbf{x}) \right) - 1 \right)} \tag{26}$$

of different-sized ions (without water) proposed in [125] also reduces to a Boltzmann distribution as *v<sup>j</sup>* → 0, ∀*j*, where *C* max *<sup>j</sup>* = *p*/*v<sup>j</sup>* and *p* ≤ 1 is a packing parameter. This distribution saturates and thus is of Fermi type, i.e., *Ci*(*x*) ≤ *C* max *i* and *viCi*(*x*) ≤ 1. The entropy term <sup>−</sup> ln 1 + ∑ *K j*=1 *C B j C* max *j* exp −*βjφ*(*x*) − 1 does not involve voids so it is different from the *S trc*(**r**) in (4). Our distribution in (4) does not need any packing parameters and satisfies *viCi*(**r**) < 1.

#### **5. Poisson-Nernst-Planck-Bikerman Model of Saturating Phenomena**

For nonequilibrium systems, we can also generalize the classical Poisson-Nernst-Planck model [38,39,43,143,144] to the Poisson-Nernst-Planck–Bikerman model by coupling the flux density equation

$$\frac{\partial \mathbf{C}\_{i}(\mathbf{r},t)}{\partial t} = -\nabla \cdot \mathbf{J}\_{i}(\mathbf{r},t), \ \mathbf{r} \in \Omega\_{\text{s}} \tag{27}$$

of each particle species *i* = 1, · · ·, *K* + 1 (including water) to the 4PBik Equation (17), where the flux density is defined as

$$\mathbf{J}\_{i}(\mathbf{r},t) = -D\_{i}\left[\nabla\mathbf{C}\_{i}(\mathbf{r},t) + \beta\_{i}\mathbf{C}\_{i}(\mathbf{r},t)\nabla\phi(\mathbf{r},t) - \frac{v\_{i}}{v\_{0}}\mathbf{C}\_{i}(\mathbf{r},t)\nabla S^{\text{trc}}(\mathbf{r},t)\right],\tag{28}$$

*Di* is the diffusion coefficient, and the time variable *t* is added to describe the dynamics of electric *φ*(**r**, *t*) and steric *S trc*(**r**, *t*) potentials.

The flux Equation (27) is called the Nernst-Planck-Bikerman equation because the steric potential *S trc*(**r**, *t*) is introduced into the classical NP equation so it can deal with saturating phenomena including those that arise from the unequal volumes of ions and the finite volume of molecular water. The PNPB model can be extended to include hydrodynamic kinetic and potential energies in the variational treatment of energy processes (i.e., EnVarA) by Hamilton's least action and Rayleigh's dissipation principles [145,146]. We shall however consider this as a topic for future work.

At equilibrium, the net flow of each particle species is a zero vector, i.e., **J***i*(**r**) = **0** (in a steady state), which implies that

$$
\nabla \mathbf{C}\_{i}(\mathbf{r}) + \beta\_{i} \mathbf{C}\_{i}(\mathbf{r}) \nabla \phi(\mathbf{r}) - \frac{v\_{i}}{v\_{0}} \mathbf{C}\_{i}(\mathbf{r}) \nabla S^{trc}(\mathbf{r}) = \mathbf{0},
$$

$$
\nabla \left[ \mathbf{C}\_{i}(\mathbf{r}) \exp(\beta\_{i} \phi(\mathbf{r}) - \frac{v\_{i}}{v\_{0}} S^{trc}(\mathbf{r})) \right] = \mathbf{0},
$$

$$
\mathbf{C}\_{i}(\mathbf{r}) \exp(\beta\_{i} \phi(\mathbf{r}) - \frac{v\_{i}}{v\_{0}} S^{trc}(\mathbf{r})) = \mathbf{c}\_{i\prime}.
\tag{29}
$$

where the constant *c<sup>i</sup>* = *C B i* for *φ*(**r**) = *S trc*(**r**) = 0. Therefore, (29) = (4), i.e., the NPB Equation (27) reduces to the saturating Fermi distribution (4) as the classical NP equation reduces to the Boltzmann distribution at equilibrium.

The gradient of the steric potential ∇*S trc*(**r**, *t*) in (28) represents an entropic force of vacancies exerted on particles. The negative sign in −*Ci*(**r**, *t*)∇*S trc*(**r**, *<sup>t</sup>*) means that the steric force <sup>∇</sup>*<sup>S</sup> trc*(**r**, *t*) is in the opposite direction to the diffusion force ∇*Ci*(**r**, *t*).

Larger *S trc*(**r**, *t*) = ln <sup>Γ</sup>(**r**,*t*) Γ *B* implies lower pressure because the ions occupy more space (less crowded) as implied by the numerator Γ(**r**, *t*). The larger the *S trc*(**r**, *t*) the lower pressure at the location **r**, the more the entropic force (the higher pressure) pushes particles to **r** from neighboring locations. The steric force is the opposite of the diffusion force ∇*Ci*(**r**, *t*) that pushes particles away from **r** if the concentration at **r** is larger than that at neighboring locations.

Moreover, the Nernst-Einstein relationship between diffusion and mobility [9] implies that the steric flux *D<sup>i</sup> vi v*0 *Ci*(**r**, *t*)∇*S trc*(**r**, *t*) is greater if the particle is more mobile. The Nernst-Einstein relationship is generalized to

$$
\mu\_i = \upsilon\_i q\_i D\_i / (\upsilon\_0 k\_B T)\_\prime \tag{30}
$$

where the mobility coefficient *µ<sup>i</sup>* of an ion depends on its size *v<sup>i</sup>* in addition to its charge *q<sup>i</sup>* . The mobility coefficient of water is *µK*+<sup>1</sup> = *vK*+1*DK*+1/(*v*0*kBT*). The drift term in (28) is thus −*DiβiCi*(**r**, *t*)∇*φ*(**r**, *t*) = −*µi*(*v*0/*vi*)*Ci*(**r**, *t*)∇*φ*(**r**, *t*).

Therefore, the gradients of electric and steric potentials (∇*φ*(**r**, *t*) and ∇*S trc*(**r**, *t*)) describe the *charge/space competition* mechanism of particles in a crowded region within a mean-field framework. Since *S trc*(**r**, *t*) describes the dynamics of void movements, the dynamic crowdedness (pressure) of the flow system can also be quantified. A large amount of experimental data exists concerning the dependence of diffusion coefficient on the concentration and size of solutes. Comparing our model with this data is an important topic of future work.

The motion of water molecules, i.e., the *osmosis* of water [147,148] is directly controlled by the steric potential in our model and their distributions are expressed by *CK*+1(**r**, *t*) = *C B K*+1 exp *vK*+1*S trc*(**r**, *t*)/*v*<sup>0</sup> . Nevertheless, this motion is still implicitly changed by the electric potential *φ*(**r**, *t*) via the correlated motion of ions described by other *Cj*(**r**, *t*) in the void fraction function Γ(**r**, *t*) and hence in the charge density *ρion*(**r**, *t*) in (17).

In summary, the PNPB model accounts for (i) the *steric* (*pressure*) effect of ions and water molecules, (ii) the *correlation* effect of crowded ions, (iii) the *screening* (*polarization*) effect of polar water, and (iv) the *charge*/*space competition* effect of ions and water molecules of different sizes and valences. These effects are all closely related to the interstitial voids between particles and described by two additional terms, namely the *correlation length* and the *steric potential*. The steric potential is most naturally written as a function of the volume of voids, but it can also be written as a function of the total volume of all molecules, including water and ions.

#### **6. Generalized Debye-Hückel Theory**

Thermodynamic modeling is of fundamental importance in the study of chemical and biological systems [1,6,9,11–13,16,32]. Since Debye and Hückel (DH) proposed their theory in 1923 [149] and Hückel extended it to include Born energy effects in 1925 [150], a great variety of extended DH models (equations of state) have been developed for modeling aqueous or mixed-solvent solutions over wide ranges of composition, temperature, and pressure [6,19,151–155]. Despite these intense efforts, robust thermodynamic modeling of electrolyte solutions still presents a difficult challenge for extended DH models due to an enormous number of parameters that need to be adjusted carefully and often subjectively [19,152–154,156].

It is indeed a frustrating despair (the word *frustration* on p. 11 in [16] and the word *despair* on p. 301 in [1]) that about *22,000* parameters [19] need to be extracted from the available experimental data for one temperature for combinatorial solutions of the most important 28 cations and 16 anions in salt chemistry by the Pitzer model [6], which is the most widely used DH model with unmatched precision for modeling electrolyte solutions [153]. The JESS (joint expert speciation system) is the world's largest system of thermodynamic information relating to electrolytes, reactions in aqueous media, and hydrocarbon phase equilibria [157]. The total number of Pitzer's fitting parameters in JESS is *95* [158].

By contrast, we propose in [75,76] a generalized Debye-Hückel theory from the 4PBik Equation (17) to include (i) steric effects, (ii) correlation effects, (iii) Born solvation energy, and (iv) ion hydration [159–166] that are missing in the original DH theory. The generalized theory can be used to calculate ion activities in all types of binary and multi-component solutions over wide ranges of concentration, temperature, and pressure with only *3* fitting parameters [69,73,75,76].

We briefly outline the derivation of a generalized DH equation of state and refer to [76] for more details. The activity coefficient *γ<sup>i</sup>* of an ion of species *i* in an aqueous electrolyte solution with a total of *K* species of ions describes deviation of the chemical potential of the ion from ideality (*γ<sup>i</sup>* = 1) [11]. The excess chemical potential *µ ex <sup>i</sup>* = *kBT* ln *γ<sup>i</sup>* can be calculated by [69,167]

$$
\mu\_i^{ex} = \frac{1}{2} q\_i \phi(\mathbf{0}) - \frac{1}{2} q\_i \phi^0(\mathbf{0}) \, \tag{31}
$$

where *q<sup>i</sup>* is the charge of the hydrated ion (also denoted by *i*), *φ*(**r**) is a reaction potential [167] function of spatial variable **r** in the domain Ω = Ω*<sup>i</sup>* ∪ Ω*sh* ∪ Ω*<sup>s</sup>* shown in Figure 1, Ω*<sup>i</sup>* is the spherical domain occupied by the ion *i*, Ω*sh* is the hydration shell domain of the ion, Ω*<sup>s</sup>* is the rest of solvent domain, **0** denotes the center (set to the origin) of the ion, and *φ* 0 (**r**) is a potential function when the solvent domain Ω*<sup>s</sup>* does not contain any ions at all with pure water only, i.e., when the solution is ideal. The radii of Ω*<sup>i</sup>* and the outer boundary of Ω*sh* are denoted by *R Born i* (ionic cavity radius [160]) and *R sh i* , respectively.

**Figure 1.** The model domain Ω is partitioned into the ion domain Ω*<sup>i</sup>* (with radius *R Born i* ), the hydration shell domain Ω*sh* (with radius *R sh i* ), and the remaining solvent domain Ω*s*.

The potential function *φ*(**r**) can be found by solving the 4PBik Equation (17) and the Laplace equation [69,73]

$$
\Delta\phi(\mathbf{r}) = 0 \text{ in } \Omega\_{\mathbf{i}} \cup \Omega\_{\text{sh}\prime} \tag{32}
$$

where *e<sup>s</sup>* is defined in Ω*sh* ∪ Ω*<sup>s</sup>* , the correlation length *l<sup>c</sup>* = √ *lBlD*/48 is a density-density correlation length independent of specific ionic radius [168], *l<sup>B</sup>* and *l<sup>D</sup>* are the Bjerrum and Debye lengths, respectively, the concentration *C<sup>k</sup>* (**r**) function (4) is defined in Ω for all *k* = 1, · · ·, *K* + 1 in molarity (M), and *v<sup>k</sup>* = 4*πa* 3 *k* /3 with radius *a<sup>k</sup>* . Since the steric potential takes particle volumes and voids into account, the shell volume *Vsh* of the shell domain Ω*sh* can be determined by the steric potential *S trc sh* = *v*0 *vw* ln *<sup>O</sup><sup>w</sup> i VshC B K*+1 <sup>=</sup> ln *<sup>V</sup>sh*−*vwO<sup>w</sup> i Vsh*Γ *B* [69,73], where the occupant (coordination) number *O<sup>w</sup> i* of water molecules is given by experimental data [166]. The shell radius *R sh i* is thus determined and depends

not only on *O<sup>w</sup> i* but also on the bulk void fraction Γ *B* , namely *on all salt and water bulk concentrations* (*C B k* ) [69,73].

We reduce the complexity of higher-order approximations, and make them easier to implement by transforming the fourth-order PDE (17) to the following two second-order PDEs [64]

$$\left(l\_c^2 \Delta - 1\right)\psi(\mathbf{r}) = \rho\_{ion}(\mathbf{r}) \text{ in } \Omega\_{\mathbf{s}\prime} \tag{33}$$

$$
\epsilon\_{\rm s} \Delta \phi(\mathbf{r}) = \psi(\mathbf{r}) \text{ in } \Omega\_{\rm s} \tag{34}
$$

where the extra unknown function *ψ*(**r**) is a density-like function as seen from (33) by setting *l<sup>c</sup>* = 0. The boundary and interface conditions for *φ*(**r**) and *ψ*(**r**) in (32)–(34) are [64]

$$
\phi(\mathbf{r}) = \psi(\mathbf{r}) = 0 \text{ on } \partial\Omega\_{\mathbf{s}} \backslash \partial\Omega\_{\mathbf{sl}\prime} \tag{35}
$$

$$\psi(\mathbf{r}) = -\rho\_s(\mathbf{r}) \text{ on } \partial\Omega\_{\text{sh}} \cap \partial\Omega\_{\text{s}\prime} \tag{36}$$

$$\left[\phi(\mathbf{r})\right] = 0 \text{ on } \partial\Omega\_i \cup \left(\partial\Omega\_{\text{sh}} \cap \partial\Omega\_{\text{s}}\right), \tag{37}$$

$$\left[\nabla\phi(\mathbf{r})\cdot\mathbf{n}\right] = 0 \text{ on } \partial\Omega\_{\text{sl}} \cap \partial\Omega\_{\text{s}}.\tag{38}$$

$$\mathbb{E}\left[\boldsymbol{\varepsilon}(\mathbf{r})\nabla\phi(\mathbf{r})\cdot\mathbf{n}\right] = \boldsymbol{\varepsilon}\_{l}\nabla\phi^\*(\mathbf{r})\cdot\mathbf{n}\text{ on }\partial\Omega\_{l\prime}\tag{39}$$

where *∂* denotes the boundary of a domain, the jump function [*φ*(**r**)] = lim**r***sh*→**<sup>r</sup>** *φ*(**r***sh*) − lim**r***i*→**<sup>r</sup>** *φ*(**r***i*) at **r** ∈ *∂*Ω*<sup>i</sup>* with **r***sh* ∈ Ω*sh* and **r***<sup>i</sup>* ∈ Ω*<sup>i</sup>* , *e*(**r**) = *e<sup>s</sup>* in Ω*sh* and *e*(**r**) = *eione*<sup>0</sup> in Ω*<sup>i</sup>* , *eion* is a dielectric constant in Ω*<sup>i</sup>* , **n** is an outward normal unit vector at **r** ∈ *∂*Ω*<sup>i</sup>* , and *φ* ∗ (**r**) = *qi*/(4*πe<sup>i</sup>* |**r** − **0**|). Equation (32) avoids large errors in a direct approximation of the delta function *δ*(**r** − **0**) in the singular charge *qiδ*(**r** − **0**) of the solvated ion at the origin **0** by transforming the singular charge to the Green's function *φ* ∗ (**r**) on *∂*Ω*<sup>i</sup>* in (39) as an approximation source of the electric field produced by the solvated ion [169,170].

For simplicity, we consider a general binary (*K* = 2) electrolyte C*z*2A*z*<sup>1</sup> with the valences of the cation C*z*1<sup>+</sup> and anion A*z*2<sup>−</sup> being *z*<sup>1</sup> and *z*2, respectively. The first-order Taylor approximation of the charge density functional *ρion*(*φ*(**r**)) in (17) with respect to the electric potential *φ*(**r**) yields

$$\left[\rho\_{\rm ion}(\phi(\mathbf{r})) \approx \frac{-\mathcal{C}\_1^B q\_1}{k\_B T} \left[ (q\_1 - q\_2) - \Lambda q\_1 \right] \phi(\mathbf{r}) \right. \tag{40}$$

where Λ = *C B* 1 (*v*<sup>1</sup> − *v*2) 2 / - Γ *<sup>B</sup>v*<sup>0</sup> + *v* 2 1*C B* <sup>1</sup> + *v* 2 2*C B* <sup>2</sup> + *v* 2 3*C B* 3 which is a quantity corresponding to a linearization of the steric potential *S trc*(**r**) [76]. Consequently, we obtain a *generalized Debye length*

$$l\_{D4PBik} = \left(\frac{\epsilon\_s k\_B T}{C\_1^B ((1-\Lambda)q\_1^2 - q\_1 q\_2)}\right)^{1/2} \tag{41}$$

that reduces to the original Debye length *l<sup>D</sup>* [11] if *v*<sup>1</sup> = *v*<sup>2</sup> 6= 0 (two ionic species with equal radius and thus Λ = 0) or *v*<sup>1</sup> = *v*<sup>2</sup> = *v*<sup>3</sup> = 0 (all particles treated as volumeless points in standard texts for PB [11]). The nonlinear value of Λ 6= 0 for *v*<sup>1</sup> = *v*<sup>2</sup> 6= 0 can be obtained by Newton's method [76].

Equation (33) is a second-order PDE that requires two boundary conditions like (35) and (36) for a unique solution *<sup>ψ</sup>*(**r**). Since *<sup>ψ</sup>*(**r**) = *<sup>e</sup>s*∇2*φ*(**r**) = <sup>−</sup>*ρ*(**r**) <sup>≈</sup> *<sup>e</sup>s<sup>κ</sup>* <sup>2</sup>*φ*(**r**) if *l<sup>c</sup>* = 0, Equation (36) is a simplified (approximate) boundary condition for *ψ*(**r**) on *∂*Ω*sh* ∩ *∂*Ω*<sup>s</sup>* without involving higher-order derivatives of *ψ*(**r**) (or the third-order derivative of *φ*(**r**)). The approximations in (36) and (40) do not significantly affect our generalized DH model's ability to fit activity data. However, these assumptions should be carefully scrutinized in other applications such as highly charged surfaces. Bazant et al. have recently developed more consistent and general boundary conditions for their fourth-order model by enforcing continuity of the Maxwell stress at a charged interface [171,172].

In [76], we analytically solve the linear 4PBik PDEs (32), (33), and (34) with (40) in a similar way as Debye and Hückel solved the linear PB equation for a spherically symmetric system. However, the spherical domain shown in Figure 1 and the boundary and interface conditions in (35)–(39) are different from those of the standard method for the linear PB equation in physical chemistry texts [11]. The analysis consists of the following steps: (i) The nonlinear term *ρion*(**r**) in (33) is linearized to the linear term −*esφ*/*l* 2 *<sup>D</sup>*4*PBik* in (40) as that of Debye and Hückel. (ii) The linear PDEs corresponding to (33) and (34) are then formulated into a system of eigenvalue problems with eigenfunctions (*φ*(**r**), *ψ*(**r**)) and eigenvalues (*λ*1, *λ*2), where the general solution of *φ*(**r**) is equal to that of Debye and Hückel in the solvent domain Ω*<sup>s</sup>* (not the entire domain) when *l<sup>c</sup>* = *v*<sup>1</sup> = *v*<sup>2</sup> = *v*<sup>3</sup> = 0. (iii) A unique pair of eigenfunctions *φ* <sup>4</sup>*PBik*(**r**), *ψ* <sup>4</sup>*PBik*(**r**) is found under conditions (35)–(39), where *φ* <sup>4</sup>*PBik*(**r**) is equal to that of Debye and Hückel in Ω*<sup>s</sup>* when *l<sup>c</sup>* = *v*<sup>1</sup> = *v*<sup>2</sup> = *v*<sup>3</sup> = 0.

The analytical potential function that we found [76] is

$$\boldsymbol{\phi}^{\text{4PBIik}}(\boldsymbol{r}) = \begin{cases} \frac{q\_{\boldsymbol{i}}}{4\pi\varepsilon\_{s}R\_{i}^{\text{Rur}}} + \frac{q\_{\boldsymbol{i}}}{4\pi\varepsilon\_{s}R\_{i}^{\text{R}l}} \left(\Theta - 1\right) \text{ in } \Omega\_{\boldsymbol{i}}\\ \frac{q\_{\boldsymbol{i}}}{4\pi\varepsilon\_{s}r} + \frac{q\_{\boldsymbol{i}}}{4\pi\varepsilon\_{s}R\_{i}^{\text{R}l}} \left(\Theta - 1\right) \text{ in } \Omega\_{\boldsymbol{s}l}\\ \frac{q\_{\boldsymbol{i}}}{4\pi\varepsilon\_{s}r} \left[\frac{\lambda\_{1}^{2}e^{-\sqrt{\lambda\_{2}}\left(r - R\_{i}^{\text{R}l}\right)} - \lambda\_{2}^{2}e^{-\sqrt{\lambda\_{1}}\left(r - R\_{i}^{\text{R}l}\right)}}{\lambda\_{1}^{2}\left(\sqrt{\lambda\_{2}}R\_{i}^{\text{R}l} + 1\right) - \lambda\_{2}^{2}\left(\sqrt{\lambda\_{1}}R\_{i}^{\text{R}l} + 1\right)}\right] \text{ in } \Omega\_{\boldsymbol{s}r} \end{cases} \tag{42}$$

where

$$\Theta = \frac{\lambda\_1^2 - \lambda\_2^2}{\lambda\_1^2 \left(\sqrt{\lambda\_2} R\_i^{sh} + 1\right) - \lambda\_2^2 \left(\sqrt{\lambda\_1} R\_i^{sh} + 1\right)},\tag{43}$$

*r* = |**r**|, *λ*<sup>1</sup> = 1 − q 1 − 4*l* 2 *<sup>c</sup>* /*l* 2 *D*4*PBik* / 2*l* 2 *c* , and *λ*<sup>2</sup> = 1 + q 1 − 4*l* 2 *<sup>c</sup>* /*l* 2 *D*4*PBik* / 2*l* 2 *c* . Please note that lim*lc*→<sup>0</sup> *<sup>λ</sup>*<sup>1</sup> = 1/*<sup>l</sup>* 2 *<sup>D</sup>*4*PBik*, lim*lc*→<sup>0</sup> *<sup>λ</sup>*<sup>2</sup> = <sup>∞</sup>, and lim*lc*→<sup>0</sup> <sup>Θ</sup> = lim*<sup>C</sup> B* <sup>1</sup> <sup>→</sup><sup>0</sup> <sup>Θ</sup> <sup>=</sup> lim*lD*4*PBik*→<sup>∞</sup> <sup>Θ</sup> <sup>=</sup> 1 [76]. The linearized 4PBik potential *φ* <sup>4</sup>*PBik*(*r*) reduces to the linearized PB potential *φ PB*(*r*) = *qi e* <sup>−</sup>*r*/*l<sup>D</sup>* /(4*πesr*) as in standard texts (e.g., Equation (7.46) in [11]) by taking lim*lc*→<sup>0</sup> *<sup>φ</sup>* <sup>4</sup>*PBik*(*r*) with *v<sup>k</sup>* = 0 for all *k*, *R sh <sup>i</sup>* = 0, and *r* > 0 [76].

As discussed in [173], since the solvation free energy of an ion *i* varies with salt concentrations, the Born energy *q* 2 *i* 1 *ew* − 1 /8*πe*0*R* 0 *i* in pure water (i.e., *C B <sup>i</sup>* = 0) with a constant Born radius *R* 0 *i* should change to depend on *C B <sup>i</sup>* ≥ 0. Equivalently, the Born radius *R Born i* in (42) is variable and we can model it from *R* 0 *i* by a simple formula [69,73]

$$R\_i^{Born} = \theta R\_i^0 \, \, \, \theta = 1 + a\_1^i \left(\overline{\mathbf{C}}\_i^{\mathbb{B}}\right)^{1/2} + a\_2^i \overline{\mathbf{C}}\_i^{\mathbb{B}} + a\_3^i \left(\overline{\mathbf{C}}\_i^{\mathbb{B}}\right)^{3/2} \, \, \, \tag{44}$$

where *C B <sup>i</sup>* = *C B i* /M is a dimensionless bulk concentration and *α i* 1 , *α i* 2 , and *α i* 3 are parameters for modifying the experimental Born radius *R* 0 *i* to fit experimental activity coefficient *γ<sup>i</sup>* that changes with the bulk concentration *C B i* of the ion. The Born radii *R* 0 *i* given below are from [173] obtained from the experimental hydration Helmholtz free energies of those ions given in [12]. The three parameters in (44) have physical or mathematical meanings unlike many parameters in the Pitzer model [19,153,156]. The first parameter *α i* 1 adjusts *R* 0 *i* and accounts for the real thickness of the ionic atmosphere (Debye length), which is proportional to the square root of the ionic strength in the DH theory [11]. The second *α i* 2 and third *α i* 3 parameters are adjustments in the next orders of approximation beyond the DH treatment of ionic atmosphere [73].

The potential value *φ* 0 (**0**) = lim*<sup>C</sup> B* <sup>1</sup> →0 *φ* <sup>4</sup>*PBik*(**0**) = *qi*/ 4*πesR* 0 *i* by lim*<sup>C</sup> B* <sup>1</sup> <sup>→</sup><sup>0</sup> <sup>Θ</sup> <sup>=</sup> <sup>1</sup> and lim*<sup>C</sup> B* <sup>1</sup> →0 *R Born <sup>i</sup>* = *R* 0 *i* . From (31) and (42), we thus have a generalized activity coefficient *γ* 4*PBik i* in

$$\ln \gamma\_i^{4PBik} = \frac{q\_i^2}{8\pi \varepsilon\_s k\_B T} \left( \frac{1}{R\_i^{Born}} - \frac{1}{R\_i^0} + \frac{\Theta - 1}{R\_i^{sh}} \right),\tag{45}$$

which satisfies the DH limiting law, i.e., *γ* 4*PBik <sup>i</sup>* = *γ DH <sup>i</sup>* = 1 for infinite dilute (ideal) solutions as *C B <sup>i</sup>* → 0. The generalized activity coefficient *γ* 4*PBik i* reduces to the classical *γ DH i* proposed by Debye and Hückel in 1923 [149], namely

$$\ln \gamma\_i^{DH} = \frac{-q\_i^2}{8\pi \epsilon\_s k\_B T l\_D (1 + R\_i^{sh}/l\_D)} \tag{46}$$

if *R Born <sup>i</sup>* = *R* 0 *i* (without considering Born energy effects), *R sh <sup>i</sup>* = *R<sup>i</sup>* (an effective ionic radius [149]), *lD*4*PBik* = *l<sup>D</sup>* (no steric effect), and *l<sup>c</sup>* = 0 (no correlation effect). The reduction shown in [76] is by taking the limit of the last term in (45) as *l<sup>c</sup>* → 0, i.e., lim*lc*→<sup>0</sup> Θ−1 *R sh* = <sup>−</sup><sup>1</sup> *Ri*+*l<sup>D</sup>* .

*i* Hückel soon realized that the DH formula (46) failed to fit experimental data at high ionic strengths and modified it in 1925 [150] by adding one more parameter *η*<sup>1</sup> to become (see Equation (7.115) in [11])

$$\ln \gamma\_i^{DHB} = \frac{-q\_i^2}{8\pi \epsilon\_s k\_B T l\_D (1 + \eta\_0 \sqrt{I})} + \eta\_1 I\_\prime \tag{47}$$

where *η*<sup>0</sup> (an approximation of *R sh i* ) and *η*<sup>1</sup> account for the distance of closest approach to the ion *i* and the salting-out effect (an approximation of the Born energy), respectively [11], where *I* = <sup>1</sup> <sup>2</sup> ∑*<sup>i</sup> C B i z* 2 *i* is the ionic strength of the solution. Consequently, a variety of extended DH models *γ DHBx i* [153,174] in the form similar to

$$\ln \gamma\_i^{DHBx} = \frac{-q\_i^2}{8\pi \varepsilon\_s k\_B T l\_D (1 + \eta\_0 \sqrt{I})} + \sum\_{k \neq 0} \eta\_k I^k \tag{48}$$

have been proposed in the literature to express other thermodynamic properties such as temperature and pressure by a power expansion of *I* with more and more parameters *η<sup>k</sup>* that can increase combinatorially with various composition, temperature, and pressure to a frustrating amount [1,16,19]. Please note that *η<sup>k</sup>* may also depend on ionic strength *I* in a complicated way, see e.g., Equation (2) in [153]. Many expressions of those parameters are rather long and tedious and do not have clear physical meaning [19,153,156]. The Davies equation [175] is a special form of (47) with a linear term in *I*.

The *R Born i* term in (45) *differs significantly* from the last term in (48) as they are the *inverse* of each other in terms of *I* and parameters, i.e., *I*, *α*1, *α*2, and *α*<sup>3</sup> are in the denominator in (45) whereas *I* and *η<sup>k</sup>* are in the numerator in (48). This implies that *γ* 4*PBik i* and *γ DHBx i* vary oppositely with *I*. Consequently, the values of *α*1, *α*2, and *α*<sup>3</sup> are totally different from those of *η<sup>k</sup>* when we use *γ* 4*PBik i* and *γ DHBx i* to fit experimental activity coefficients with *I* varying from low to high values [76]. This may explain why the empirical nature of extended DH models requires a great deal of effort to extract parameters (without physical hints) from existent thermodynamic databases by regression analysis [19,153].

#### **7. Numerical Methods**

Numerical simulations are indispensable to study chemical, physical, and mathematical properties of biological and chemical systems in realistic applications, especially with experimental details at atomic scale such as ion channels in the Protein Data Bank (PDB) [57]. Continuum PDE models have substantial advantages over Monte Carlo, Brownian dynamics (BD), or molecular dynamics in physical insights and computational efficiency that are of great importance in studying a range of conditions and concentrations especially for large nonequilibrium or inhomogeneous systems, as are present in experiments and in life itself [10,17,21,95,121,176–185].

The literature on numerical methods for solving PB and PNP models is vast [64,68,74]. We summarize here some important features of the methods proposed in [64,68,74] for Poisson-Bikerman and Poisson-Nernst-Planck-Bikerman models, which may be useful for workers in numerical analysis and coding practice. Since PNPB including 4PBik is highly nonlinear and the

geometry of protein structures is very complex, we emphasize two different types of methods, namely nonlinear iterative methods and discretization methods for these two problems as follows.

#### *7.1. Nonlinear Iterative Methods*

For the PNPB system of *K* + 1 NP equations in (27), Laplace Equation (32), and two 4PBik equations in (33) and (34), the total number of second-order PDEs that we need to solve is *K* + 4. These PDEs are coupled together and highly nonlinear except (32). Numerically solving this kind of nonlinear systems is not straightforward [64,68,74]. We use the following algorithm to explain essential procedures for solving the steady-state PNPB system, where Ω*<sup>m</sup>* denotes the biomolecular domain that contains a total of *Q* fixed atomic charges *q<sup>j</sup>* located at **r***<sup>j</sup>* in a channel protein as shown in Figure 2L for the gramicidin A channel downloaded from PDB with *Q* = 554, for example, *∂*Ω*<sup>m</sup>* denotes the molecular surface of the protein and the membrane lipids through which the protein crosses as shown in Figure 2R, and Ω*<sup>s</sup>* is the solvent domain consisting of the channel pore and the extracellular and intracellular baths for mobile ions and water molecules.

**Figure 2. Left (L)**: Top view of the gramicidin A channel. **Right (R)**: A cross section of 3D simulation domain for the channel placed in a rectangular box, where Ω*m* is the biomolecular domain consisting of the channel protein and the membrane and Ω*s* is the solvent domain consisting of the channel pore and the baths.

*Nonlinear Iterative Algorithm [68]:*


Linearizing the nonlinear 4PBik (17) yields two second-order linear 4PBik1 and 4PBik2 in Steps 4 and 5 that differ from the nonlinear (33) and (34). Newton's iterative Steps 4–6 for solving 4PBik1 and 4PBik2 dictates convergence that also depends on various mappings from an old solution *φ Old* to a new solution *φ New*. This algorithm uses two relaxation and three continuation mappings for which we need to carefully tune two relaxation parameters *ω*4*PBik* and *ωPNPB* and three continuation parameters *λ<sup>c</sup>* (related to correlation effects), *λ<sup>s</sup>* (steric effects), and ∆*V* (incremental voltage for applied voltage). For example, the parameter *λ<sup>s</sup>* in Γ *Old*(**r**) = <sup>1</sup> <sup>−</sup> <sup>∑</sup> *K*+1 *j*=1 *λsvjC Old j* (**r**) can be chosen as *λ<sup>s</sup>* = *k*∆*λ*, *k* = 0, 1, 2, · · ·, 1 ∆*λ* , an incremental continuation from 0 (no steric effects) to 1 (fully steric effects) with a tuning stepping length ∆*λ*. The algorithm can fail to converge if we choose ∆*λ* = 1 (without continuation) for some simulation cases, since we may have Γ *Old*(**r**) < 0 resulting in numerically *undefined Strc*(**r**) = ln <sup>Γ</sup> *Old*(**r**) Γ *<sup>B</sup>* at some **r** where the potential *φ Old*(**r**) is large.

#### *7.2. Discretization Methods*

All PDEs in Steps 1, 2, 4, 5, 8, and 9 are of Poisson type −∇2*φ*(**r**) = *<sup>f</sup>*(**r**). We use the central finite difference (FD) method [64]

$$\frac{-\phi\_{i-1,j,k} + 2\phi\_{i\bar{j}k} - \phi\_{i+1,j,k}}{\Delta x^2} + \frac{-\phi\_{i,j-1,k} + 2\phi\_{i\bar{j}k} - \phi\_{i,j+1,k}}{\Delta y^2} + \frac{-\phi\_{i,j,k-1} + 2\phi\_{i\bar{j}k} - \phi\_{i,j,k+1}}{\Delta z^2} = f\_{i\bar{j}k} \tag{49}$$

to discretize it at all grid points **r***ijk* = (*x<sup>i</sup>* , *y<sup>j</sup>* , *z<sup>k</sup>* ) in a domain, where *φijk* ≈ *φ*(*x<sup>i</sup>* , *y<sup>j</sup>* , *z<sup>k</sup>* ), *f ijk* = *f*(*x<sup>i</sup>* , *y<sup>j</sup>* , *z<sup>k</sup>* ), and ∆*x*, ∆*y*, and ∆*z* are mesh sizes on the three axes from a uniform partition ∆*x* = ∆*y* = ∆*z* = *h*. The domains in Steps 1 and 2 are Ω*<sup>m</sup>* and Ω*<sup>s</sup>* , respectively. The discretization leads to a sparse matrix system *A* −→*φ* = −→*<sup>f</sup>* with the compressed bandwidth of the matrix *<sup>A</sup>* being 7, where the matrix size can be millions for sufficiently small *h* to obtain sufficiently accurate *φijk*.

The matrix system consists of four subsystems, two by the FD method (49) in Ω*<sup>m</sup>* and Ω*<sup>s</sup>* , one by another method (see below) to discretize the jump condition in Step 2 on the interface *∂*Ω*<sup>m</sup>* between Ω*<sup>s</sup>* and Ω*m*, and one by imposing boundary conditions on *∂*Ω. We need to solve the matrix system in the entire domain Ω = Ω*<sup>m</sup>* ∪ Ω*<sup>s</sup>* .

The convergence order of (49) is *O*(*h* 2 ) (optimal) in maximum error norm for sufficiently smooth function *φ*(**r**). However, this optimal order can be easily degraded to *O*(*h* 0.37) [186], for example, by geometric singularities if the jump condition is not properly treated. In [64], we propose the interface method

$$\frac{-\varepsilon\_{i-\frac{3}{2}}\phi\_{i-2} + \left(\varepsilon\_{i-\frac{3}{2}} + (1 - A\_1)\,\varepsilon\_{i-\frac{1}{2}}^{-}\right)\phi\_{i-1} - A\_2\varepsilon\_{i-\frac{1}{2}}^{-}\phi\_i}{\Delta x^2} = f\_{i-1} + \frac{\varepsilon\_{i-\frac{1}{2}}^{-}A\_0}{\Delta x^2} \tag{50}$$

$$\frac{-B\_1 \varepsilon\_{i-\frac{1}{2}}^+ \phi\_{i-1} + \left( (1 - B\_2)\varepsilon\_{i-\frac{1}{2}}^+ + \varepsilon\_{i+\frac{1}{2}} \right) \phi\_i - \varepsilon\_{i+\frac{1}{2}} \phi\_{i+1}}{\Delta x^2} = f\_i + \frac{\varepsilon\_{i-\frac{1}{2}}^+ B\_0}{\Delta x^2} \tag{51}$$

where

$$A\_{1} = \frac{-\left(\mathfrak{e}\_{m} - \mathfrak{e}\_{\mathrm{s}}\right)}{\mathfrak{e}\_{m} + \mathfrak{e}\_{\mathrm{s}}},\ A\_{2} = \frac{2\mathfrak{e}\_{m}}{\mathfrak{e}\_{m} + \mathfrak{e}\_{\mathrm{s}}},\ A\_{0} = \frac{-2\mathfrak{e}\_{m}\left[\mathfrak{e}\right] - \Delta x\left[\mathfrak{e}\phi'\right]}{\mathfrak{e}\_{m} + \mathfrak{e}\_{\mathrm{s}}},$$

$$B\_{1} = \frac{2\mathfrak{e}\_{\mathrm{s}}}{\mathfrak{e}\_{m} + \mathfrak{e}\_{\mathrm{s}}},\ B\_{2} = \frac{\mathfrak{e}\_{m} - \mathfrak{e}\_{\mathrm{s}}}{\mathfrak{e}\_{m} + \mathfrak{e}\_{\mathrm{s}}},\ B\_{0} = \frac{2\mathfrak{e}\_{\mathrm{s}}\left[\mathfrak{e}\right] - \Delta x\left[\mathfrak{e}\phi'\right]}{\mathfrak{e}\_{m} + \mathfrak{e}\_{\mathrm{s}}},$$

to discretize the 1D Poisson equation <sup>−</sup> *<sup>d</sup> dx e*(*x*) *dφ*(*x*) *dx* = *f*(*x*) at every jump position *γ* ∈ *∂*Ω*<sup>m</sup>* that is at the middle of its two neighboring grid points, i.e., *<sup>x</sup>i*−<sup>1</sup> <sup>&</sup>lt; *<sup>γ</sup>* <sup>=</sup> *<sup>x</sup>i*<sup>−</sup> <sup>1</sup> 2 < *x<sup>i</sup>* , where *<sup>x</sup>i*<sup>−</sup> <sup>1</sup> 2 = (*xi*−<sup>1</sup> + *xi*)/2 and *xi*−<sup>1</sup> and *x<sup>i</sup>* belong to different domains Ω*<sup>s</sup>* and Ω*m*. The corresponding cases in *y*- and *z*-axis follow obviously in a similar way. This method yields *optimal* convergence [64].

Since the matrix system is usually very large in 3D simulations and we need to repeatedly solve such systems updated by nonlinear iterations as shown in the above algorithm, linear iterative methods such as the bi-conjugate gradient stabilized (bi-CG) method are used to solve the matrix system [74]. We propose two parallel algorithms (one for bi-CG and the other for nonlinear iterations) in [74] and show that parallel algorithms on GPU (graphic processing unit) over sequential algorithms on CPU (central processing unit) can achieve 22.8× and 16.9× speedups for the linear solver time and total runtime, respectively.

Discretization of Nernst–Planck equations in Step 7 is different from (49) because the standard FD method

$$\frac{\mathbf{C}\_{i+1} - \mathbf{C}\_{i}}{\Delta x} = \frac{\mathbf{C}\_{i+1} + \mathbf{C}\_{i}}{2} \left( -\beta \frac{\Delta \phi\_{i}}{\Delta x} + \frac{\Delta \mathbf{S}\_{i}^{tr}}{\Delta x} \right) \tag{52}$$

for the zero flux (*J*(*x*) = −*D*(*x*) *dC*(*x*) *dx* + *βC*(*x*) *dφ*(*x*) *dx* <sup>−</sup> *<sup>v</sup> v*0 *C*(*x*) *dStrc* (*x*) *dx* = 0) can easily yield

$$\mathbf{C}\_{i+1} - \mathbf{C}\_{i} > \mathbf{C}\_{i+1} + \mathbf{C}\_{i} \tag{53}$$

and thereby a negative (*unphysical*) concentration *C<sup>i</sup>* < 0 at *x<sup>i</sup>* if

$$\frac{1}{2} \left( -\beta \Delta \phi\_i + \Delta S\_i^{trc} \right) > 1,\tag{54}$$

where ∆*φi*−<sup>1</sup> = *φ<sup>i</sup>* − *φi*−<sup>1</sup> , *φ<sup>i</sup>* ≈ *φ*(*xi*) etc. Therefore, it is crucial to check whether the *generalized Scharfetter–Gummel* (SG) condition [68]

$$-\beta \Delta \phi\_i + \Delta S\_i^{\text{tr}} \le 2 \tag{55}$$

is satisfied by any discretization method in implementation. This condition generalizes the well-known SG stability condition in semiconductor device simulations [187,188] to include the steric potential function *S trc*(**r**).

We extend the classical SG method [187] of the flux *J*(*x*) in [68] to

$$J\_{i+\frac{1}{2}} = \frac{-D}{\Delta \mathbf{x}} \left[ \mathcal{B}(-t\_i)\mathcal{C}\_{i+1} - \mathcal{B}(t\_i)\mathcal{C}\_i \right] \tag{56}$$

where *t<sup>i</sup>* = *β*∆*φ<sup>i</sup>* − ∆*S trc i* and *B*(*t*) = *<sup>t</sup> e <sup>t</sup>*−<sup>1</sup> is the Bernoulli function [188]. Equation (56), an exponential fitting scheme, satisfies (55) and is derived from assuming that the flux *J*, the local electric field <sup>−</sup>*d<sup>φ</sup> dx* , and the local steric field *dStrc dx* are all constant in the sufficiently small subinterval (*x<sup>i</sup>* , *xi*+1), i.e.,

$$\frac{d}{D} = \frac{-d\mathbb{C}(\mathbf{x})}{d\mathbf{x}} - k\mathbb{C}(\mathbf{x}), \text{ for all } \mathbf{x} \in (\mathfrak{x}\_{i\prime}, \mathfrak{x}\_{i+1}), \tag{57}$$

where *k* = *β dφ dx* <sup>−</sup> *dStrc dx* . Solving this ordinary differential equation (ODE) with a boundary condition *C<sup>i</sup>* or *Ci*+<sup>1</sup> yields the well-known Goldman-Hodgkin-Katz flux equation in ion channels [9], which is exactly the same as that in (56) but with the subinterval (*x<sup>i</sup>* , *xi*+1) being replaced by the height of the entire box in Figure 2R.

The generalized Scharfetter-Gummel method for Nernst-Planck equations is thus

$$\begin{split} \frac{dI(\mathbf{x}\_{i})}{d\mathbf{x}} &\approx \quad \frac{I\_{i+\frac{1}{2}} - I\_{i-\frac{1}{2}}}{\Delta \mathbf{x}} = \frac{b\_{i-1}\mathbf{C}\_{i-1} + b\_{i}\mathbf{C}\_{i} + b\_{i+1}\mathbf{C}\_{i+1}}{\Delta \mathbf{x}^{2}} = 0 \\ \quad J\_{i-\frac{1}{2}} &= \quad \frac{-D}{\Delta \mathbf{x}} \left[ B(-t\_{i-1})\mathbf{C}\_{i} - B(t\_{i-1})\mathbf{C}\_{i-1} \right] \\ \quad J\_{i+\frac{1}{2}} &= \quad \frac{-D}{\Delta \mathbf{x}} \left[ B(-t\_{i})\mathbf{C}\_{i+1} - B(t\_{i})\mathbf{C}\_{i} \right] \\ \quad t\_{i} &= \quad \beta \Delta \phi\_{i} - \Delta S\_{i}^{\text{tr}}, \mathcal{B}(t) = \frac{t}{e^{t} - 1} \\ \quad b\_{i-1} &= \quad -\mathcal{B}(t\_{i-1}), b\_{i} = \mathcal{B}(-t\_{i-1}) + \mathcal{B}(t\_{i}), b\_{i+1} = -\mathcal{B}(-t\_{i}). \end{split} \tag{58}$$

The SG method is *optimal* in the sense that it integrates the ODE (57) *exactly* at *every* grid point with a suitable boundary condition [189]. Therefore, the SG method can resolve sharp layers very accurately [189] and hence needs few grid points to obtain tolerable approximations when compared with the primitive FD method. Moreover, the exact solution of (57) for the concentration function *C*(*x*) yields an exact flux *J*(*x*). Consequently, the SG method is *current preserving*, which is particularly important in nonequilibrium systems, where the current is possibly the most relevant physical property of interest [190].

It is difficult to overstate the importance of the current preserving feature and it must be emphasized for workers coming from fluid mechanics that preserving current has a significance quite beyond the preserving of flux in uncharged systems. Indeed, conservation of current (defined as Maxwell did to include the displacement current of the vacuum *e*<sup>0</sup> *∂***E**(**r**,*t*) *∂t* ) is an unavoidable consequence, nearly a restatement of the Maxwell equations themselves [104,106]. The electric field is so strong that the tiniest error in preserving current, i.e., the tiniest deviation from Maxwell's equations, produces huge effects. The third paragraph of Feynman's lectures on electrodynamics makes this point unforgettable [191]. Thus, the consequences of a seemingly small error in preserving the flow of charge are dramatically larger than the consequences of the same error in preserving the flux of mass.

We have developed a C++ code for solving 4PBik and PNPB models on laptop and highperformance (with GPU) computers. For solving a 4PBik problem with a matrix system of size 4,096,000, for example, the code requires about 300 MB memory to store the compressed matrix system with double precision. It took about 2 min and 47 s on a laptop computer equipped with 1.3 GHz Intel CPU and 2 GB RAM to solve the linear system once by the successive overrelaxation method with an error tolerance of 10−<sup>6</sup> [64].

#### **8. Applications**

We have used the saturating Poisson-Nernst-Planck-Bikerman theory to study ion activities, electric double layers, and biological ion channels in the past. The theory accounts for the steric effect of ions and water molecules, the effects of ion-ion and ion-water correlations, the screening and polarization effects of polar water, and the charge/space competition effect of ions and water molecules of different sizes and valences. These effects are all closely related to the dielectric operator in (17) and the steric potential in (4) that works on both macroscopic and atomic scales. We now illustrate these properties in the following three areas using mostly experimental data to verify the theory.

#### *8.1. Ion Activities*

The curves in Figure 3 obtained by the generalized Debye-Hückel Formula (45) [75] fit well to the experimental data by Wilczek–Vera et al. [192] for single-ion activities in 8 1:1 electrolytes. There are only three fitting parameters in the formula, namely *α i* 1 , *α i* 2 and *α i* 3 , which we reiterate have specific physical meaning as parameters of the water shell around ions. The values of the parameters are given in Table 1 from which we observe that *R Born i* deviates from *R* 0 *i* slightly. For example, *R Born* Cl− /*R* 0 Cl<sup>−</sup> = 1.007∼1.044 (not shown) for Figure 3a with [LiCl] = 0∼2.5 M, since the cavity radius *R Born* Cl− is an atomic measure from the infinite singularity *δ*(**r** − **0**) at the origin, i.e., *φ* <sup>4</sup>*PBik*(*r*) and thus *γ* 4*PBik i* are very sensitive to *R Born i* . On the other hand, *γ* 4*PBik i* is not very sensitive to *R sh i* (*R sh* Cl<sup>−</sup> = 5.123∼5.083 Å), i.e., the fixed choice of *O*<sup>w</sup> *<sup>i</sup>* = 18 (an experimental value in [166]) for all curves is not critical but reasonable [69]. The error between the estimated *O*<sup>w</sup> *i* and its unknown true value can always be compensated by small adjustments of *R Born i* . The values of other symbols are *a*Li<sup>+</sup> = 0.6 Å, *a*Na<sup>+</sup> = 0.95 Å, *a*<sup>K</sup> <sup>+</sup> = 1.33 Å, *a*<sup>F</sup> <sup>−</sup> = 1.36 Å, *a*Cl<sup>−</sup> = 1.81 Å, *a*Br<sup>−</sup> = 1.95, *a*H2<sup>O</sup> = 1.4 Å, *R* 0 Li<sup>+</sup> = 1.3 Å, *R* 0 Na<sup>+</sup> = 1.618 Å, *R* 0 K <sup>+</sup> = 1.95 Å, *R* 0 F <sup>−</sup> = 1.6 Å, *R* 0 Cl<sup>−</sup> = 2.266, *<sup>R</sup>* 0 Br<sup>−</sup> = 2.47 Å [173], *e<sup>w</sup>* = 78.45, *eion* = 1, *T* = 298.15 K, where *a<sup>i</sup>* is the (Pauling) radius of type *i* particle (ion) [173].

Table 1 also shows the significant order of these parameters, i.e., *α i* 1  > *α i* 2  > *α i* 3 in general cases. Please note that the three sets of the values of *α* Na<sup>+</sup> 1 , *α* Na<sup>+</sup> 2 , and *α* Na<sup>+</sup> 3 for the same Na<sup>+</sup> in three different salts NaCl, NaBr, and NaF are different because the diameters of the anions are different. Figure 4 shows single-ion activities in 6 2:1 electrolytes by experiments [192] and 4PBik, where the significant order (not shown) of three fitting parameters is similar to that in Table 1.


**Table 1.** Values of *α i* 1 , *α i* 2 , *α i* 3 in (44).

**Figure 3.** Single-ion activity coefficients of (**a**) LiCl (**b**) LiBr (**c**) NaF (**d**) NaCl (**e**) NaBr (**f**) KF (**g**) KCl (**h**) KBr electrolytes. Comparison of 4PBik results (curves) with experimental data (symbols) [192] on *i* = C <sup>+</sup> (cation) and A<sup>−</sup> (anion) activity coefficients *γ<sup>i</sup>* in various [CA] from 0 to 1.6 M.

**Figure 4.** Single-ion activity coefficients of (**a**) MgCl<sup>2</sup> (**b**) MgBr<sup>2</sup> (**c**) CaCl<sup>2</sup> (**d**) CaBr<sup>2</sup> (**e**) BaCl<sup>2</sup> (**f**) BaBr<sup>2</sup> electrolytes. Comparison of 4PBik results (curves) with experimental data (symbols) [192] on *i* = C 2+ (cation) and A<sup>−</sup> (anion) activity coefficients *γ<sup>i</sup>* in various [CA2] from 0 to 1.5 M.

The electric potential and other physical properties of ionic activity can be studied in detail according to the partitioned domain in Figure 1 characterized by *R Born i* and *R sh i* . For example, we observe from Figure 5 that the electric potential (*φ* 4*PBik* Br<sup>−</sup> (0) = −2.4744 *kBT*/*e*) and the Born radius (*R Born* Br<sup>−</sup> (2 M) = 2.0637 Å) generated by Br<sup>−</sup> at [LiBr] = 2 M are significantly different from that (*φ* 4*PBik* Br<sup>−</sup> (0) = −0.6860 *kBT*/*e*, *R Born* Br<sup>−</sup> (2 M) = 4.2578 Å) at [KBr] = 2 M. The only difference between these two solutions is the size of cations, i.e., the size of different positive ions significantly changes the activity of the same negative ion at high concentrations. The difference between *φ* 4*PBik* Li<sup>+</sup> (0) and *φ* 4*PBik* K <sup>+</sup> (0) is due to the sizes of Li<sup>+</sup> and K<sup>+</sup> not Br<sup>−</sup> as it is the same for both solutions.

**Figure 5.** Electric potential *φ* <sup>4</sup>*PBik*(*r*) profiles by (42) near the solvated ions Li<sup>+</sup> and Br<sup>−</sup> at [LiBr] = 2 M, and K<sup>+</sup> and Br<sup>−</sup> at [KBr] = 2 M, where *r* is the distance from the center of the respective ion.

This example clearly shows the atomic properties of 4PBik theory in the ion Ω*<sup>i</sup>* and shell Ω*sh* domains and the continuum properties in the solvent domain Ω*<sup>s</sup>* . The Born radius *R Born i* in (42) determined by (44) changes with (i) *ion-water* interactions in Ω*<sup>i</sup>* ∪ Ω*sh* and (ii) *ion-ion* interactions in Ω*<sup>i</sup>* ∪ Ω*<sup>s</sup>* via *φ* <sup>4</sup>*PBik*(*r*) in (42) that is self-consistently determined by the interface conditions in (35)–(39) and by (iii) *multi-salt* [73,76] concentrations in Ω*<sup>s</sup>* , (iv) the *screening* effects of water in Ω*sh* and ions and water in Ω*<sup>s</sup>* , (v) the *polarization* effect of water in Ω*<sup>s</sup>* , (vi) the *correlation* effect between ions in Ω*<sup>s</sup>* , (vii) the *steric* effects of all ions and water in the entire domain Ω = Ω*<sup>i</sup>* ∪ Ω*sh* ∪ Ω*<sup>s</sup>* , (viii) *temperatures* [73,76], and (ix) *pressures* [73,76]. The generalized Debye-Hückel formula (45) includes all these 9 physical properties with only 3 fitting parameters. However, we look forward to the day when we can derive the three fitting parameters for particular types of ions, from independently determined experimental data.

#### *8.2. Electric Double Layers*

We consider a charged surface in contact with a 0.1 M 1:4 aqueous electrolyte, where the charge density is *σ* = 1*e*/(50 Å<sup>2</sup> ), the radius of both cations and anions is *a* = 4.65 Å (in contrast to an edge length of 7.5 Å of cubical ions in [133]), and *e<sup>s</sup>* = 80 [72]. The multivalent ions represent large polyanions adsorbed onto a charged Langmuir monolayer in experiments [133]. We solve (33) and (34) using (49) in the rectangular box Ω = Ω*<sup>s</sup>* = n (*x*, *<sup>y</sup>*, *<sup>z</sup>*) : 0 <sup>≤</sup> *<sup>x</sup>* <sup>≤</sup> 40, <sup>−</sup> <sup>5</sup> <sup>≤</sup> *<sup>y</sup>* <sup>≤</sup> 5, <sup>−</sup> <sup>5</sup> <sup>≤</sup> *<sup>z</sup>* <sup>≤</sup> 5 Å<sup>o</sup> such that *<sup>φ</sup>*(**r**) <sup>≈</sup> <sup>0</sup> within the accuracy to <sup>10</sup>−<sup>4</sup> near and on the surface *<sup>x</sup>* <sup>=</sup> <sup>40</sup> Å. The boundary conditions on the surface and its adjacent four planes are −*es*∇*φ* · **n** = *σ* with **n** = h−1, 0, 0i and −*es*∇*φ* · **n** = 0 with **n** defined similarly, respectively.

The classical PB model (with *a* = *a*H2<sup>O</sup> = *l<sup>c</sup>* = 0, i.e., no size, void, and correlation effects) produces unphysically high concentrations of anions (A4−) near the surface as shown by the dashed curve in Figure 6L. The dotted curve in Figure 6L is similar to that of the modified PB in [133] and is obtained by the 4PBik model with *l<sup>c</sup>* = 0 (no correlations), *VK*+<sup>2</sup> = 0 (no voids), and *a*H2<sup>O</sup> = 0 (water is volumeless as in [133] and hence Γ *<sup>B</sup>* <sup>=</sup> <sup>1</sup> <sup>−</sup> <sup>∑</sup> *K i*=1 *viC B i* is the bulk water volume fraction). The voids (*VK*+<sup>2</sup> 6= 0) and water molecules (*a*H2<sup>O</sup> 6= 0) have slight effects on anion concentration (because of saturation) and electric potential (because water and voids have no charges) profiles as shown by the thin solid curves in Figure 6L,R, respectively, when compared with the dotted curves. However, ion-ion correlations (with *l<sup>c</sup>* = 1.6*a* [78]) have significant effects on ion distributions as shown by the thick solid and dash-dotted curves in Figure 6L, where the saturation layer is on the order of ionic radius *a* and the *overscreening* layer [78] (*C*A4<sup>−</sup> (*x*) ≈ 0 < *C B* <sup>A</sup>4<sup>−</sup> = 0.1 M) of excess co-ions (*C*<sup>C</sup> <sup>+</sup> (*x*) > *C B* C <sup>+</sup> = 0.4 M) is about 18 Å in thickness.

The *saturation layer* is an *output* (not an imposed condition) of our model unlike a Stern layer [193] imposed by most EDL models to account for size effects near charge surfaces [194–196]. The electric potentials *φ*(0) = 5.6 *kBT*/*e* at *x* = 0 and *φ*(11.5) = −1.97 *kBT*/*e* in Figure 6R obtained by 4PBik with voids and correlations deviate dramatically from those by previous models for nearly 100% at *x* = 0 (in the saturation layer) and 70% at *x* = 11.5 Å (in the screening layer) when compared with the maximum potential *φ*(0) = 2.82 *kBT*/*e* of previous models. The 4PBik potential depth *φ*(11.5) = −1.97 *kBT*/*e* of the overscreening layer is very sensitive the size *a* of ions and tends to zero as *a* → 0.

**Figure 6. Left (L)**: Concentration profiles of anions *C*<sup>A</sup> <sup>4</sup><sup>−</sup> (*x*) and cations *C*<sup>C</sup> <sup>+</sup> (*x*) obtained by various models in a C4A electrolyte solution with the charge density *σ* = 1*e*/(50 Å<sup>2</sup> ) at *x* = 0. **Right (R)**: Electric potential profiles *φ*(*x*).

#### *8.3. Biological Ion Channels*

Biological ion channels are a particularly appropriate test of a model of concentrated ionic solutions.

The data available for tens to hundreds of different types of channels and transporters is breathtaking: it is often accurate to a few per cent (because signal to noise ratios are so large and biological variation hardly exists for channels of known amino acid sequence, which means nearly every channel presently). The data is always nonequilibrium, i.e., current voltage relations in a wide range of solutions of different composition and concentration, or (limiting zero voltage) conductance in those solutions. Indeed, many of the channels do not function if concentrations are equal on both sides and the electrical potential is zero. They are said to inactivate.

The data is often available for single channels recorded individually in patch clamp or bilayer configuration. Data is available for a range of divalent (usually calcium ion) concentrations because calcium concentration is often a controller of channel, transporter, and biological activity in the same sense that a gas pedal is the controller of the speed of a car. The structure of the ion channel or transporter is often known in breathtaking detail. The word 'breathtaking' is appropriate because similar structures are rarely if ever known of strictly physical systems. The structure and the structure of the permanent and polarization charge of the channel protein (that forms the pore through which ions move) can be modified by standard methods of site directed mutagenesis, for example that are available in 'kit' form usable by most molecular biology laboratories. Thus, models can be tested from atomic detail to single-channel function to ensemble function to cellular and physiological function, even to the ultimate biological function (like the rate of the heartbeat). Few other systems allow experimental measurement at each level of the hierarchy linking the atomic composition of genes (that encode the channel's amino acid composition), to the atomic structure of the channel, right to the function of the cell. The hierarchy here reaches from 10−<sup>11</sup> to 10−<sup>5</sup> m. When the channel controls the biological function of an organ like the heart, the hierarchy reaches to <sup>2</sup> · <sup>10</sup>−<sup>1</sup> m, in humans for example.

The biological significance of ion channels is hard to exaggerate since they play a role in organisms analogous to the role of transistors in computers. They are the device that execute most of the physical controls of current and ion movement that are then combined in a hierarchy of structures to make biological cells, tissues, and organisms, if not populations of organisms.

From a physical point of view, ion channels provide a particularly crowded environment in which the effects of the steric potential (crowding in more traditional language) and electrical potential can combine to produce striking characteristics of selectivity and rectification. Theories that do not deal explicitly with ion channel data, i.e., that do not predict current voltage relations from known structures, seem to us to be begging central PHYSICAL questions that might falsify their approach. In fact, as a matter of history it is a fact that we learned how to construct our model of bulk solutions from our earlier work on ion channels.

#### 8.3.1. Gramicidin A Channel

We use the gramicidin A (GA) channel in Figure 2L to illustrate the full Poisson–Nernst–Planck–Bikerman theory consisting of Equations (4), (27), (28), (32)–(34), and conditions (35)–(39) with—steric, correlation, polarization, dielectric, charge/space competition, and nonequilibrium effects—at steady state using the algorithm and methods in Section 7 to perform numerical simulations. The union domain Ω*<sup>i</sup>* ∪ Ω*sh* in Figure 1 is replaced by the biomolecular domain Ω*<sup>m</sup>* in Figure 2R.

Figure 7L shows I-V curves obtained by PNPB and compared with experimental data (symbols) by Cole et al. [197] with bath K<sup>+</sup> and Cl<sup>−</sup> concentrations *C <sup>B</sup>* = 0.1, 0.2, 0.5, 1, 2 M and membrane potentials ∆*V* = 0, 50, 100, 150, 200 mV. The PNPB currents in pico ampere (pA) were obtained with *θ* = 1/4.7 in the pore diffusion coefficients *θD<sup>i</sup>* from (30) for all particle species. The reduction

parameter *θ* has been used in all previous PNP papers and is necessary for continuum is comparable to MD, BD, or experimental data [198]. The values of other model parameters are listed in Table I in [68].

**Figure 7. Left (L)**: A comparison of PNPB (curves) and experimental [197] (symbols) I-V results with bath K<sup>+</sup> and Cl<sup>−</sup> concentrations *C* <sup>B</sup> = 0.1, 0.2, 0.5, 1, 2 M and membrane potentials ∆*V* = 0, 50, 100, 150, 200 mV. **Right (R)**: Averaged steric potential *S trc*(**r**) profiles at each cross section along the pore axis with *C <sup>B</sup>* = 0.1, 0.2, 0.5, 1, 2 M and ∆*V* = 200 mV. The same averaging method applies to the following profiles.

We summarize the novel results of PNPB in [68] when compared with those of earlier continuum models for ion channels: (i) The pore *diffusion* parameter *θ* = 1/4.7 agrees with the range 1/3 to 1/10 obtained by many MD simulations of various channel models [199–201] indicating that the steric (Figure 7R), correlation, dehydration (Figure 8L), and dielectric (Figure 8R) properties have made PNPB simulations closer (realistic) to MD than previous PNP for which *θ* differs from MD values by an order to several orders of magnitude [200]. (ii) Figures 7R and 8L,R, which are all absent in earlier work, show that these properties *correlate* to each other and *vary* with salt concentration and protein charges in a *self-consistent* way by PNPB. (iii) The steric potential profiles in Figure 7R clearly illustrate the *charge/space competition* between ions and water under dynamic and variable conditions. For example, the global minimum value in Figure 7R at <sup>b</sup>*<sup>r</sup>* <sup>=</sup> 13.1 on the channel axis, where the channel protein is most negatively charged, is *S trc*(b*r*) = ln <sup>Γ</sup>(b*r*) Γ *<sup>B</sup>* <sup>=</sup> <sup>−</sup>0.485 yielding <sup>Γ</sup>(b*r*)/<sup>Γ</sup> *<sup>B</sup>* = 0.616. Namely it is 38.4% more crowded at <sup>b</sup>*<sup>r</sup>* than in the bath and mainly occupied by K<sup>+</sup> as shown in Figures 8L and 9L. It is important to *quantify voids* (Γ(**r**) = 1 − ∑ *K*+1 *i*=1 *viCi*(**r**)) at highly charged locations in channel proteins and many more biological, chemical, and nano systems. The charge space competition has been a central topic in the study of ion channels since at least [202–206]. The literature is too large to describe in detail here. Recent reviews can help [207–210]. (iv) PNPB preserves *mass conservation* due to void and size effects in contrast to PNP as shown in Figure 9R, where the total number of H2O and K<sup>+</sup> in the channel pore is 8 [211].

**Figure 8. Left (L)**: Water concentration *<sup>C</sup>*H2O(**r**) profiles. **Right (R)**: Dielectric function <sup>e</sup>*e*(**r**) profiles.

**Figure 9. Left (L)**: K<sup>+</sup> concentration *<sup>C</sup>*<sup>K</sup> <sup>+</sup> (**r**) profiles. **Right (R)**: Occupancy of H2O and K<sup>+</sup> in the GA channel pore by PNPB and PNP as [KCl] increases from 0 to 2 M. The total number of H2O and K<sup>+</sup> in the pore is 8 [211], which is conserved by PNPB but not by PNP (without steric and correlation effects).

8.3.2. L-Type Calcium Channel

L-type calcium channels operate very delicately in physiological and experimental conditions. They exquisitely tune their conductance from Na+-flow, to Na+-blockage, and to Ca2+-flow when bath Ca2<sup>+</sup> varies from trace to high concentrations as shown by the single-channel currents in femto ampere in Figure 10L (circle symbols) recorded by Almers and McCleskey [212], where the range of extracellular concentrations [Ca2+]<sup>o</sup> is 10<sup>8</sup> -fold from 10−<sup>10</sup> to 10−<sup>2</sup> M.

**Figure 10. Left (L)**: Single-channel currents in femto ampere (fA) plotted as a function of log10[Ca2+]o. Experimental data of [212] are marked by small circles and PNPB data are denoted by the plus sign and lines. **Right (R)**: The Lipkind–Fozzard pore model of L-type calcium channel, where 3 Ca2<sup>+</sup> are shown in violet, 8 O1/2<sup>−</sup> in red, 2 H2O in white and red. Reprinted with permission from (G. M. Lipkind and H. A. Fozzard, Biochem. 40, 6786 (2001)). Copyright (2001) American Chemical Society.

We used the Lipkind-Fozzard molecular model [213] shown in Figure 10R to perform PNPB simulations with both atomic and continuum methods (Algorithm 2 in [68]) for this model channel, where the EEEE locus (four glutamate side chains modeled by 8 O1/2<sup>−</sup> ions shown by red spheres) forms a high-affinity Ca2<sup>+</sup> binding site (center violet sphere) that is essential to Ca2<sup>+</sup> selectivity, blockage, and permeation. Water molecules are shown in white and red. More realistic structures would be appropriate if the work were done now, but the analysis here shows the ability of PNPB to deal with experimental data using even a quite primitive model of the structure.

PNPB results (plus symbols) in Figure 10L agree with the experimental data at [Na+]<sup>i</sup> = [Na+]<sup>o</sup> = <sup>32</sup> mM, [Ca2+]<sup>i</sup> <sup>=</sup> 0, *<sup>V</sup>*<sup>o</sup> <sup>=</sup> 0, and *<sup>V</sup>*<sup>i</sup> <sup>=</sup> <sup>−</sup><sup>20</sup> mV (the intracellular membrane potential), where the partial Ca2<sup>+</sup> and Na<sup>+</sup> currents are denoted by the solid and dotted line, respectively. These two ionic currents show the anomalous mole fraction effect of the channel at nonequilibrium, i.e., trace concentrations of Ca2<sup>+</sup> ions effectively block the flow of abundant monovalent cations [212].

#### 8.3.3. Potassium Channel

Potassium channels conduct K<sup>+</sup> ions very rapidly (nearly at the diffusion rate limit (10<sup>8</sup> per second) in bulk water) and selectively (excluding, most notably, Na<sup>+</sup> despite their difference in radius is only *a*<sup>K</sup> <sup>+</sup> − *a*Na<sup>+</sup> = 1.33 − 0.95 = 0.38 Å in sub-Angstrom range) [9]. Figure 11 shows the structure of KcsA (PDB ID 3F5W) crystallized by Cuello et al. [214], where the spheres denote five specific cation binding sites (S0 to S4) [215] in the solvent domain Ω*<sup>s</sup>* and the channel protein in Ω*<sup>m</sup>* consists of *N* = 31,268 charged atoms. The exquisite selectivity of K<sup>+</sup> over Na<sup>+</sup> by K channels can be quantified by the free energy (*G*) differences of K<sup>+</sup> and Na<sup>+</sup> in the pore and in the bulk solution, i.e., by ∆*G*(K+) = - *G*pore(K <sup>+</sup>) <sup>−</sup> *<sup>G</sup>*bulk(<sup>K</sup> +) and <sup>∆</sup>*G*(Na+) = *<sup>G</sup>*pore(Na+) <sup>−</sup> *<sup>G</sup>*bulk(Na+) [215]. Experimental measurements [216–218] showed that the relative free energy

$$
\Delta\Delta\mathcal{G}(\text{K}^+ \to \text{Na}^+) = \Delta\mathcal{G}(\text{Na}^+) - \Delta\mathcal{G}(\text{K}^+) = 5 \sim 6 \text{ kcal/mol} \tag{59}
$$

unfavorable for Na+.

**Figure 11.** The crystal structure of the K channel KcsA (PDB ID 3F5W) [214] with five cation binding sites S0, S1, S2, S3, and S4 [215] marked by spheres.

Free energies can be calculated by the electric and steric potentials [72]

$$\phi\_{\rm S2} = \frac{1}{4\pi\epsilon\_0} \left( \frac{1}{6} \sum\_{k=1}^{6} \sum\_{j=1}^{N} \frac{q\_j}{\epsilon\_p(r)|c\_j - A\_k|} + \frac{q\_{\rm S2}}{\epsilon\_b a\_{\rm S2}} \right), \\ S\_{\rm S2}^{\rm trc} = \ln \frac{1 - \frac{\overline{v}\_{\rm S2}}{\overline{V}\_{\rm S2}}}{\Gamma^B} \tag{60}$$

at the binding site S2 [215] on the atomic scale, where S2 also denotes Na<sup>+</sup> or K<sup>+</sup> (the site is occupied by a Na<sup>+</sup> or K+), *q<sup>j</sup>* is the charge on the atom *j* in the protein given by PDB2PQR [219], *ep*(*r*) = 1 + 77*r*/(27.7 + *r*) [119], *r* = |*cj*− *c*S2|, *c<sup>j</sup>* is the center of atom *j*, *A<sup>k</sup>* is one of six symmetric surface points on the spherical S2, *e<sup>b</sup>* = 3.6, and *V*S2 = 1.5*v*<sup>K</sup> <sup>+</sup> is a volume containing the ion at S2. We obtained ∆∆*G* = 5.26 kcal/mol [72] in accord with the MD result 5.3 kcal/mol [215], where *G*pore(Na+) = 4.4, *<sup>G</sup>*bulk(Na+) = <sup>−</sup>0.26, *<sup>G</sup>*pore(<sup>K</sup> <sup>+</sup>) = <sup>−</sup>0.87, *<sup>G</sup>*bulk(<sup>K</sup> <sup>+</sup>) = <sup>−</sup>0.27 kcal/mol, *<sup>φ</sup>*Na<sup>+</sup> <sup>=</sup> 7.5 *<sup>k</sup>BT*/*e*, *v*Na<sup>+</sup> *v*0 *S trc* Na<sup>+</sup> = 0.23, *φ*<sup>K</sup> <sup>+</sup> = −1.93 *kBT*/*e*, *v*K<sup>+</sup> *v*0 *S trc* K <sup>+</sup> = −0.59, and *C B* Na<sup>+</sup> = *C B* K <sup>+</sup> = 0.4 M.

The crucial parameter in (60) is the ionic radius *a*S2 = 0.95 or 1.33 Å (also in |*c<sup>j</sup>* − *A<sup>k</sup>* |) that affects *φ*S2 very strongly but *S trc* S2 weakly. Another important parameter in (60) is the bulk void fraction Γ *B* that depends on the bulk concentrations of all ions and water and links the total energy of the ion at S2 to these bulk conditions measured very far away (∼10<sup>6</sup> Å) in the baths on the atomic scale.

#### 8.3.4. Sodium Calcium Exchanger

The Na+/Ca2<sup>+</sup> exchanger (NCX) is the major cardiac mechanism that extrudes intracellular Ca2<sup>+</sup> across the cell membrane against its chemical gradient by using the downhill gradient of Na<sup>+</sup> [28]. The molecular basis of Na+/Ca2<sup>+</sup> interactions in NCX so striking to Lüttgau and Niedegerke [220] have been revealed by the cloning of NCX gene [221] and the structure of the ancient archaebacterial version NCX\_Mj determined by Liao et al. [222]. Figure 12L illustrates NCX\_Mj that consists of 10 transmembrane (TM) helices in which eight helices (TMs 2 to 5 and 7 to 10 labeled numerically in the figure) form a binding pocket of three putative Na<sup>+</sup> (green spheres) and one Ca2<sup>+</sup> (blue sphere) binding sites [222].

**Figure 12. Left (L)**: Structure of NCX\_Mj consisting of ten transmembrane helices that form a binding pocket of three Na<sup>+</sup> (green spheres) and one Ca2<sup>+</sup> (blue sphere) binding sites [222]. **Right (R)**: Schematic diagram of a cycle of Na+/Ca2<sup>+</sup> exchange in NCX consisting of five total potential states (TPS). Two Na<sup>+</sup> and one Ca2<sup>+</sup> ions enter the binding pocket in the outward- (TPS2 <sup>→</sup> TPS3 <sup>→</sup> TPS4) and inward-facing (TPS5 → TPS1) conformations, respectively. They exit in opposite conformations [70].

We developed a cyclic model of Na+/Ca2<sup>+</sup> exchange mechanism in NCX [70] using (60) to calculate five total (electric and steric) potential states (TPS) of various Na<sup>+</sup> and Ca2<sup>+</sup> ions shown in Figure 12R, where TPS1 and TPS4 are stable (with negative values) and TPS2, TPS3, and TPS5 are unstable (positive). Four extra sites in Figure 12R are determined empirically and close to entrance or exit locations of the binding pocket. The green and blue dots in the diagram represent Na<sup>+</sup> and Ca2<sup>+</sup> ions occupying the respective sites. Two Na<sup>+</sup> and one Ca2<sup>+</sup> ions enter the binding pocket in the outward- (TPS2 → TPS3 → TPS4) and inward-facing (TPS5 → TPS1) conformations, respectively. They exit in opposite conformations. The cycle consists of five steps.

*Step 1:* A conformational change is hypothetically activated [70] by a binding Ca2<sup>+</sup> at the blue site (S1) in TPS1 from inward-facing to outward-facing in TPS2.

*Step 2:* One Na<sup>+</sup> enters the binding pocket from the access site in TPS2 to the top Na<sup>+</sup> binding site (S2) in TPS3 followed by another Na<sup>+</sup> to the access site. These two coming Na<sup>+</sup> ions move the existing Na<sup>+</sup> ion from the middle Na<sup>+</sup> site (S3) to the bottom site (S4) by their Coulomb forces. TPS2 and TPS3 are unstable meaning that the two coming Na<sup>+</sup> ions have positive energies and are thus mobile. The selectivity ratio of Na<sup>+</sup> to Ca2<sup>+</sup> by NCX from the extracellular bath to the binding site S2 is *C*Na<sup>+</sup> (S2)/*C*Ca2<sup>+</sup> (*S*2) = 55.4 under the experimental conditions of the extracellular bath - Na<sup>+</sup> <sup>o</sup> <sup>=</sup> 120 mM and <sup>h</sup> Ca2<sup>+</sup> i = 1 µM [70].

o *Step 3:* The vacant site S3 in TPS3 is a deep potential well with TP = −8.89 *kBT*/*e* that pulls the two unstable Na<sup>+</sup> ions to their sites in TPS4. Meanwhile, these two moving Na<sup>+</sup> and the stable Na<sup>+</sup> at S4 extrude the Ca2<sup>+</sup> (with unstable TP = 1.65) at S1 out of the pocket to become TPS4.

*Step 4:* Now, all three Na<sup>+</sup> ions in TPS4 are stable with negative TP and the vacant site S1 has an even deeper TP = −16.02. We conjecture that this TP value may trigger a conformational change from outward-facing in TPS4 to inward-facing in TPS5. The mechanism of conformational changes in NCX is yet to be studied.

*Step 5:* Furthermore, this large negative TP in TPS5 yields a remarkably large *selectivity* ratio of Ca2<sup>+</sup> to Na<sup>+</sup> by NCX from the intracellular bath to S1, i.e., *<sup>C</sup>*Ca2<sup>+</sup> (S1)/*C*Na<sup>+</sup> (S1) = 4986.1 at h Ca2<sup>+</sup> i i = 33 µM and - Na<sup>+</sup> <sup>i</sup> <sup>=</sup> <sup>60</sup> mM. A coming Ca2<sup>+</sup> in TPS5 then extrudes two Na<sup>+</sup> ions out of the packet when it settles at S1 in stable TPS1.

Assuming that the total time T of an exchange cycle is equally shared by the 5 TPS, this model also infers that the stoichiometry of NCX is <sup>3</sup> 5 <sup>T</sup>·<sup>2</sup> Na<sup>+</sup> : 2 5 <sup>T</sup>·<sup>1</sup> Ca2<sup>+</sup> <sup>=</sup> <sup>3</sup> Na<sup>+</sup> : <sup>1</sup> Ca2<sup>+</sup> in transporting Na<sup>+</sup> and Ca2<sup>+</sup> ions [70], which is the generally accepted stoichiometry (see reviews of Blaustein and Lederer [223] and Dipolo and Beaugé [224]) since the pivotal work of Reeves and Hale [225] and other subsequent experimental results. Please note that our model does not consider the electrogenic property of NCX [223], i.e., the driving force of the electric potential gradient.

#### **9. Conclusions**

We have covered a range of aspects of the fourth-order Poisson-Nernst-Planck-Bikerman theory from physical modeling, mathematical analysis, numerical implementation, to applications and verifications for aqueous electrolyte systems in chemistry and biology. The theory can describe many properties of ions and water in the system that classical theories fail to describe such as steric, correlation, polarization, variable permittivity, dehydration, mass conservation, charge/space competition, overscreening, selectivity, saturation, and more. All these properties are accounted for in a single framework with only two fundamental parameters, namely the dielectric constant of pure water and the correlation length of empirical choice. Ions and water have their physical volumes as those in molecular dynamic simulations. The theory applies to a system at both continuum and atomic scales due to the exact definition of the total volume of all ions, water molecules, and interstitial voids.

The most important features of PNPB are that (i) ions and water have unequal volumes with interstitial voids, (ii) their distributions are saturating of the Fermi type, (iii) these Fermi distributions approach Boltzmann distributions as the volumes tend to zero, and (iv) all the above physical properties appear self-consistently in a single model not separately by various models. Most existing modified Poisson-Boltzmann models consider ions of equal size and fail to yield Boltzmann distributions in limiting cases, i.e., the limit is divergent indicating that steric energies are poorly estimated. Numerous models for different properties such as steric, correlation, polarization, permittivity are proposed separately in the past.

We have shown how to solve 4PBik analytically and PNPB numerically. The generalized Debye-Hückel theory derived from the 4PBik model gives valuable insights into physical properties and leads to an electrolyte (analytical) equation of state that is useful to study thermodynamic activities of ion and water under wide ranges of composition, concentration, temperature, and pressure.

Numerically solving the fourth-order PNPB model in 3D for realistic problems is a challenging task. There are many pitfalls that one must carefully avoid in coding. For that reason, we have particularly mentioned some methods for handling the convergence issues of the highly nonlinear PNPB system of partial differential equations and the discretization problems concerning the complicate interface between molecular and solvent domains and the Scharfetter-Gummel stability condition to ensure positivity of numerical concentrations and current preservation.

Finally, we have shown novel results obtained by PNPB for chemical and biological systems on ion activities, electric double layers, gramicidin A channel, L-type calcium channel, potassium channel, and sodium calcium exchanger. These results agree with experiments or molecular dynamics data and show not only continuum but also atomic properties of the system under far-field conditions. The fourth-order PNPB model is consistent and applicable to a great variety of systems on a vast scale from *meter* to *Angstrom*.

**Author Contributions:** J.-L.L. developed the theory, performed the computations, and wrote the manuscript. B.E. conceived the topics, verified the results, and helped shape the research, analysis, and manuscript. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was funded by the Ministry of Science and Technology, Taiwan under Grant MOST107- 2115-M-007-017-MY2 to J.-L.L.

**Conflicts of Interest:** The authors declare no conflict of interest.

### **Abbreviations**

The following abbreviations are used in this manuscript:



#### **References**


© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Modeling the Device Behavior of Biological and Synthetic Nanopores with Reduced Models**

#### **Dezs˝o Boda 1,\* ,†, Mónika Valiskó 1,† and Dirk Gillespie 2,†**


Received: 12 October 2020; Accepted: 2 November 2020; Published: 5 November 2020

**Abstract:** Biological ion channels and synthetic nanopores are responsible for passive transport of ions through a membrane between two compartments. Modeling these ionic currents is especially amenable to reduced models because the device functions of these pores, the relation of input parameters (e.g., applied voltage, bath concentrations) and output parameters (e.g., current, rectification, selectivity), are well defined. Reduced models focus on the physics that produces the device functions (i.e., the physics of how inputs become outputs) rather than the atomic/molecular-scale physics inside the pore. Here, we propose four rules of thumb for constructing good reduced models of ion channels and nanopores. They are about (1) the importance of the axial concentration profiles, (2) the importance of the pore charges, (3) choosing the right explicit degrees of freedom, and (4) creating the proper response functions. We provide examples for how each rule of thumb helps in creating a reduced model of device behavior.

**Keywords:** nanopores; ion channels; reduced models; Monte Carlo; classical density functional theory; Poisson-Nernst-Planck

*We dedicate this paper to our distinguished colleague and dear friend, Douglas Henderson (1934–2020).*

#### **1. Introduction**

When modeling anything, some approximations must be made, usually to make the calculations feasible. For example, molecular dynamics (MD) simulations use Lennard-Jones (LJ) interactions between atoms in lieu of quantum mechanical interactions. This still keeps the all-atomic nature of the simulations, but can now include more than a small number of atoms. Other models coarse-grain the system much more, reducing the physics to simply calculated properties. Electrical circuits are an example; the electrons are never explicitly modeled, nor are the quantum mechanical interactions that produce electrical resistance. Instead, the concept of resistance is reduced to a proportionality factor between the current and voltage, a kind of response function that (phenomenologically) encapsulates complicated physics in a number. A reduced model can thus be very powerful.

In many nanoscale systems, however, it is not a priori clear how (or even if) one can reduce the physics and still get sensible results. In this paper, we would like to better understand and define when and why reduced models work for certain problems, but not for others? (Ion channels can be considered as natural nanopores, a nomenclature often used in the literature. In this work, when we use the term "nanopore", we mean synthetic ones.) Why do reduced models work well for some biological ion channels and synthetic nanopores What are reduced models and what defines a "good" reduced model? Our attempt to answer these questions is based on the knowledge accumulated over 20 years [1–52] of modeling and computing permeation and selectivity in ion channels and nanopores.

#### *1.1. The Device Approach*

Reduced models are especially useful if we focus on a system as a simple device [53,54]. A device is a black box that responds to some incoming stimuli (input signals) by producing output signals. Our goal is to open the black box a little bit and peak into it to understand the inner mechanisms of the device that make the output. In the case of nanodevices, understanding necessarily means understanding molecular mechanisms due to the microscopic dimensions of the underlying processes. This is generally accomplished by modeling. In our model, we do not want to include everything; we focus on those components that are absolutely necessary to reproduce device behavior. By device behavior, we mean the relation of the input signal and output signal, also called device function.

By focusing on device function we reduce the problem at hand. We look at a complex system from an engineering point of view. While complex systems are called complex because the engineering approach tends to fail, there are systems where focusing on the important degrees of freedom allows us to reproduce and explain device behavior, which is an experimentally measurable quantity. The system gives the same response to a given signal in a reproducible manner no matter how complicated the underlying molecular processes are.

Let us take the example of a toy model of an airplane. If we want to reproduce the primary device function—the plane flies as a result of lift produced by a horizontal driving force—we do not need to model unimportant degrees of freedom like seats inside the plane and screens on the backs of the seats. We just need to model the proper shape of the plane, the wings especially. Those are the important degrees of freedom.

Similarly, in modeling ion channels, the knowledge of which amino acid residues are charged (and thus interact with the ions) is an important degree of freedom. The residues that are uncharged and are far from the pore are unimportant. For example, in our model of the 2.2 megadalton ryanodine receptor (RyR) channel (one of the largest ion channels known), we only include five charged amino acids. Moreover, as we describe later in Section 3.1.2, not having the surface charge pattern correct (because all the charged amino acids had not been identified yet) produces qualitatively incorrect results.

#### *1.2. Ion Channels and Nanopores as Devices*

A basic function of nanopores and open ion channels is to connect the bulk aqueous phases separated by a membrane and let ions through in a controlled manner [55–57]. The basic input signals of the baths+pore system are the concentrations and electrical potentials on the two sides of the membrane. A difference in any of these properties (concentration and/or electrical potential difference, for example, voltage) acts as a driving force for the diffusing ions and results in ionic currents that are the output signals of the system.

We can also consider the structural features of the nanopore as an input signal as soon as they can be changed easily. The most important feature is the surface charge pattern on the wall of the nanopore [58]. This can be modified very easily with pH [59–61] or an electrical potential [62,63] applied on the wall of a nanopore (a gate potential, to borrow a term from semiconductors) when it is made of a conducting material (typically, gold). Surface charge pattern can also be changed with chemical methods in the case of nanopores [64,65] and with point mutation techniques in the case of ion channels. Here, we restrict ourselves to bath concentrations and voltage (the boundary conditions of the problem of steady state transport) as the main input parameters also controlled by experiments.

The pore's structural features are important because they determine the current response of the device given to the driving force. That relation determines the various useful device functions that are commonly attributed to ion channels and nanopores. An especially important feature of ion channels is selectivity. Various ion channels with well-defined functions in the cell are often distinguished by the specific ion that they favor over other kinds of ions. Regardless of their strict selectivity properties, ions channels are often named on the basis of their physiological roles in the cell. This way, for example, we distinguish calcium channels, potassium channels, sodium channels, and so on.

Nanopores can be manipulated more easily, so they can exhibit a wider variety of functions [56,65–72]. They can also be selective if they distinguish cations from anions. They can behave as diodes if they let ions through at one sign of the voltage, but not at the opposite sign of voltage, a phenomenon called rectification. If we can modify the pore's properties by a third signal (gate voltage or pH, for example), we can use the pore as a transistor [45,48,73]. We can also decorate the nanopore's wall with molecules that bind certain ion species selectively. In this case, if that ion is present in the electrolyte, it can change the pore wall's properties by binding to these molecules and thus changing the current of the background electrolyte. In this way, the nanopore can be used as a sensor [43,47,49,51]. The range of applications of nanopores for specific tasks is much wider, well beyond the short list above, for example, DNA sequencing.

#### **2. Reduced Models**

The main idea of reduced models is in their name: the number of degrees of freedom that we treat in detail in the model is reduced. We build only those components into the reduced model that are necessary to reproduce and understand the device function. We call these degrees of freedom the *important* ones. The unimportant or implicit degrees of freedom are treated with less precision and are averaged into "response functions".

*A good reduced model is defined by choosing the important degrees of freedom carefully and constructing sufficiently accurate response functions for the others.*

Our aim with this paper is to illustrate how to accomplish this, with ion channels and nanopores as worked examples.

The first question is how reduced our model should be? How much detail can we ignore? In this respect, the models shown in this paper belong to a "no man's land" between the really detailed all-atom models studied by MD simulations popular in ion channel studies and mean-field continuum models (the Poisson-Nernst-Planck (PNP) theory, for example) popular in nanopore studies. We believe that our position between these two limiting cases is especially suitable to shed light on the nature of good reduced models that are appropriate for a well-specified purpose, namely, studying a device behavior.

First, we explain in a few words, why reduced models can be more suitable for ion channel devices than all-atom models, at least, in certain cases. All-atom, in this context, means that we model all water molecules and every single atom of the protein and the membrane explicitly. There are various problems with these all-atom models. They cannot always cover the physiological parameter range, small voltages or small concentrations, for example. They have sampling issues, specifically regarding the simulation of ionic currents, because this means collecting samples of rare events, for example, ions passing through the pore. The applied force fields might be problematic; they tend to overestimate interactions between multiply charged entities due to missing electronic polarization in the models [74]. Also, the models are based on X-ray structures of the protein that are not always available, and even if they are, the crystal structures often do not represent native functional states. For example, the fact that they have been obtained for a frozen structure calls into question their applicability at room temperature. Such uncertainties might be overcome with reduced models with properly adjusted parameters if the goal is to reproduce the conductance properties of the pore.

Reduced models, as soon as they contain the necessary physics, do not suffer from these shortcomings as much (they have other kinds of shortcomings, naturally). They can be simulated faster, sampled better, and the model contains only the basic physics necessary to reproduce the device behavior. One can spare oneself from computing the unimportant details. What is the important physics and degrees of freedom is always determined by the problem at hand, the intention of the investigator (to what deepness are you interested in the details, for example), and the computational resources. Computation, namely, the simulation method with which we investigate the model is a crucial point of the research, but, from the point of view of the train of thoughts of this discussion, they have secondary importance.

#### *2.1. Ionic Distribution in the Pore as a Determining Factor*

One aspect of our methodologies, however, is important and should be discussed here. In our work, we usually use the Nernst-Planck (NP) transport equation [75,76] to compute the ionic flux:

$$\mathbf{j}\_{l}(\mathbf{r}) = -\frac{1}{kT} D\_{l}(\mathbf{r}) c\_{l}(\mathbf{r}) \nabla \mu\_{l}(\mathbf{r}) \,\tag{1}$$

where **j***i*(**r**), *Di*(**r**), *ci*(**r**), and *µi*(**r**) are the flux density, the diffusion coefficent profile, the concentration profile, and the electrochemical potential profile of ionic species *i*, respectively. One important principle (rule of thumb) of this paper follows from this equation:

*1. The current carried by an ionic species as a result of a given driving force (conductance) is mainly determined by the axial concentration profile of that species inside the pore.*

One interpretation of this statement is the obvious one that if there are more ions in the pore, they will carry more current. The mechanism can, however, be more subtle than that. Pores working on the basis of excluding certain ions from the pore (sodium channels exclude K+, while nanopores with overlapping double layers exclude the coion) are controlled by depletion zones of these excluded ionic species inside the pore somewhere. These depletion zones of low concentration act as high resistance elements in a equivalent circuit if we imagine the consecutive zones of the pore as resistors connected in series. These ideas will be fleshed out below, in our worked examples.

#### *2.2. What Determines Local Concentration Inside the Pore?*

The probability that a particle is found at a given position **r** in the system depends on the potential energy, *U*(**r**), and the electrochemical potential, *µi*(**r**), of ionic species *i* at **r** (see the acceptance probability of the particle insertion/deletion step in a Grand Canonical Monte Carlo (GCMC) simulation [29]). The distribution of ions inside the pore, therefore, is influenced by (1) local interactions of the ions with pore charges, other ions, solvent molecules, and confining surfaces, and (2) external parameters such as concentration and electrical potential in the baths (the boundary conditions).

If local interactions dominate (*U*(**r**) dominates over *µi*(**r**)), such as in the crowded selectivity filters of calcium channels (see Section 3.1), the concentration profiles are not so sensitive to boundary conditions. In wide nanopores (Sections 3.2 and 3.3), on the other hand, changes in voltage or bath concentration can significantly influence the concentration profile. In bipolar nanopores, for example, changing the sign of the voltage reduces the depletion zones of ions even further, reducing current and resulting in a diode behavior.

Of these two factors, however, it is the local interactions that are more important for our discussion. These local effects determine the shape of the concentration profile, where it has peaks and where it has depletion zones. They determine the basic device characteristics of the pore and they determine how the pore responds to changes in the external conditions.

We can narrow what is important more specifically. Because free particles (ions and water) just respond to changes in *U*(**r**) and *µi*(**r**), it is the features (structure) of the pore that determines device function. Moreover, because the ions are charged, their Coulomb interactions with pore charges are dominant; dipolar and higher-order terms in the multipole expansion are secondary both in strength and range. Concentration profiles, therefore, depend sensitively on the distribution of the pore charges. From all our work on channels and pores [5–31,33–52] up to this day, we can conclude the following principle:

*2. We need to build the pore charges into the model properly if we want to reproduce local concentration, and, consequently, device function.*

In summary, pore charges are important degrees of freedom, as is the geometry of the pore (length, radius, shape). But what can we say about important and unimportant degrees of freedom?

#### *2.3. Important vs. Unimportant Degrees of Freedom*

Charges (monopoles) are the first, and strongest term of the multipole expansion. The second, and weaker term is the dipolar one that appears in the interaction of an ion with water molecules. The big question arises whether we need to take the water molecules into account explicitly (as in all-atom MD models), or can we replace them with response functions such as a dielectric constant or a diffusion coefficient?

The answer to this question also depends on the system at hand. In the case of ion channels, it is obvious that explicit water molecules are crucial in potassium channels; the selectivity of that channel is the result of a subtle balance between the interactions of the permeating ions with the atoms of the selectivity filter and with water molecules [77]. Calcium and sodium channels, however, as our model calculations imply, work on the basis of interactions with charged side chains inside the selectivity filter and volume exclusion (discussed below).

Using implicit water is not even a question in the nanopore world, where they abundantly use transport equations and the PB theory. In this world, there is no argument about the necessity of the implicit water model. Instead, we need to argue about the necessity of sophisticated statistical mechanical methods such as classical density functional theory (DFT) or MC.

Why can water be smeared into an implicit background in one case, but not in the other case? In other words, what decides whether explicit water is an important degree of freedom or not? Or, in general, what decides whether any degree of freedom is important or not? We give an explicit answer to this question that, we hope, will be a general recipe for building reduced models:

*3. Those degrees of freedom are the important ones that depend on the input parameters of the device (voltage and concentration), while those that do not can be replaced by response functions.*

If a component of the system does not change considerably upon, for example, changing the voltage, then this component does not influence the mechanisms by which the model generates an output signal as a response to the input signal.

Let us use implicit water as an example to explain this, as this choice is sometimes controversial. Ions are screened by the surrounding water molecules no matter whether external conditions change or not. Certainly, an applied field or the presence of other ions distort the hydration shell around the ions, so screening is changed by changing voltage or concentration.

The effect of external conditions is small if they are small *relative* to primary effects, for example, to interactions with pore charges. If two degrees of freedom have a large relative difference in how they change with external conditions, then we can make the one with the small response implicit. This is a decision for the modeller, and, eventually, a matter of comparison of the model results with reference data. Reference data are primarily experimental data, but they can also be MD results for all-atom models (results will be shown for both cases).

Implicit water, although the most characteristic, is not the only way of reducing the number of explicit degrees of freedom. We can, for example, model the membrane with a slab between two hard walls. We can model the pore with a cylinder of hard wall. We can model the ion channel only by taking its selectivity filter into account, because that is the region that discriminates between ions. We can model protein side chains in a simplified way by taking only the oxygens of the carboxyl groups into account. There are a plenty of ways to simplify the model, but we need to ask ourselves at every step whether the details we just ignored are important or not.

As in the case of the mean-field PNP theory, it can happen that we ignore too much detail. It is well known that PNP cannot reproduce the selectivity behavior of calcium channels, because ionic correlations and volume exclusion that are so important in the highly charged and crowded selectivity filter of Ca channels are absent in PNP. We cannot use the approximations of PNP even in the case

of the relatively wide nanopores if multivalent ions are present. Charge inversion, a feature that is common in charged confined systems with multivalent ions cannot be reproduced with PNP [78].

The bottom line is that we need to balance between too many and too few details when we create a model for a specific purpose. If one is curious about the detailed physics of the coordination of ions at binding sites, the reduced model is too crude. If one is studying a wide nanopore with a 1:1 electrolyte in it, PNP theory is probably all right. There is, however, a wide area in between, where ionic correlations (including finite size) matter, but explicit water does not matter.

#### *2.4. What Are Good Response Functions?*

If we managed to distinguish between important and less important degrees of freedom, the next step is to decide how to smear the less important ones into response functions. There are various possibilities and it is not always obvious which one we should choose. In this respect, we suggest the following principle.

*4. When we create a response function, we should choose one whose parameters do not depend on external conditions, or, at least, we should minimize that dependence. In other words, those parameters should be transferable as much as possible.*

This rule might sound obvious because it seems quite ridiculous to refit the parameters for every state point (different values of input device parameters). A model is a model together with its parameters. If those parameters are not stable, meaning transferable between various state points, the model is probably missing some basic physics.

That is exactly the deeper meaning of the above rule. If the physics of the model is right, then it should describe the properties of the nanopore's wall or the ion channel's selectivity filter in a robust way. The model should be the same at another voltage or concentration. If the parameters depend on external conditions, they should do that in a physically well-based and explainable way. Otherwise, it is just an unsystematic fitting on the basis of a useless model. The model is useless in this case because it is unusable for prediction. Transferable parameters are the basis of predictions.

In the following, we present our results for three different case studies that illustrate the rules introduced above.

#### **3. Case Studies**

In the case studies presented in the following sections the system consists of two baths separated by a membrane that contains a pore connecting the two baths. Two electrodes in the two baths produce electrical potential difference (voltage) that is a part of the driving force of the transport of ions. Also, ionic concentrations can be different on the two sides of the membrane. Concentration difference and voltage add up to create an electrochemical potential difference that is the full driving force in the NP equation (Equation (1)).

In the model of this system we include the two baths, the membrane and the pore. The simulation cell is finite surrounded by a boundary at which different boundary conditions are prescribed for the ionic concentrations and the electrical potential on the two sides of the membrane. The electrolyte is modeled in the implicit water framework with the "Primitive Model" that, given the success of our models, is not so primitive after all.

The ions are modeled as charged hard spheres immersed in a dielectric continuum represented by the dielectric constant *e*, one of the response functions. The interaction potential is

$$u\_{i\bar{j}}(r) = \begin{cases} \infty & \text{if } \quad r < R\_i + R\_{\bar{j}}\\ \frac{1}{4\pi\epsilon\_0\varepsilon} \frac{z\_i z\_{\bar{j}}e^2}{r} & \text{if } \quad r \ge R\_{\bar{i}} + R\_{\bar{j}\prime} \end{cases} \tag{2}$$

where *R<sup>i</sup>* and *R<sup>j</sup>* are the radii of ionic species *i* and *j*, respectively, *z<sup>i</sup>* and *z<sup>j</sup>* are the valences of ionic species *i* and *j*, respectively, *e*<sup>0</sup> is the permittivity of vacuum, *e* is the elementary charge, and *r* is the distance between the two ions. The solvent also exerts its effect on the ions by hindering their diffusion via friction. This is taken into account by another response function, the diffusion coefficient *Di*(**r**) (see Equation (1)), which may include effects beyond interactions with waters, such as interactions with other ions and the confining geometry.

The membrane and the pore are defined by hard walls for simplicity. The most important difference between the test cases is that the pore is modeled differently in different cases. Basically, the shape of the pore and the representation of pore charges are different. By shape of the pore we mean an *R*(*z*) function that defines the hard wall obtained by rotating this function around the *z* axis. The models of pore charges will be described in the different cases.

The models are studied with a hybrid simulation method in which the NP equation is coupled to the Local Equilibrium Monte Carlo (LEMC) method (NP+LEMC). The LEMC method is basically a generalization of the GCMC method [79,80] for the case of non-equilibrium systems where the chemical potential is not necessarily constant, so the system is not in global equilibrium. Instead, the input of the LEMC method is the *µi*(**r**) profile, while the output is the *ci*(**r**) profile. In practice, the system is divided into small subvolumes, *V α* , in which the *µ α i* is constant (local equilibrium is assumed). The result of the simulation is the concentration in each subvolume, *c α i* . The resulting *µ α i* and *c α i* profiles are substituted into the NP equation providing a flux, **j** *α i* . An iteration process results in a self consistent *µ α i* and *c α i* pair that produces a flux density satisfying the continuity equation, ∇·**j***i*(**r**) = 0. It is an expression for local conservation of mass, while in our calculations we use the integrated form that states that the sum of inward and outward currents in and out of a volume element is zero. Details are found in previous papers [29,37,39,41].

The results of other models and computation methods will also be presented. Specifically, we will show results of DFT coupled to the NP equation and MD simulations for explicit water models. These models and methods will be described at the specific system, where they are used.

#### *3.1. The Ryanodine Receptor Calcium Channel*

The RyR is a biological ion channel that, in muscle, releases Ca2<sup>+</sup> ions from the sarcoplasmic reticulum in response to an influx of Ca2<sup>+</sup> through L-type calcium channels. In both cardiac and skeletal muscle cells, that RyR-mediated Ca2<sup>+</sup> initiates muscle contraction. While its physiological importance is obvious, RyR is also interesting from a single-channel biophysics point of view. Experimentally, its large current allows for relatively easy single-channel current/voltage (IV) recordings. Theoretically, it is a Ca2+-selective channel, but whose preference for Ca2<sup>+</sup> is much lower than the L-type calcium channel, even though they share the same selectivity filter in amino acids.

What makes an ion channel a calcium channel is the abundance of negative carboxyl groups (COO−) in and around the selectivity filter. Generally, four glutamate (E) and/or aspartate (D) amino acids line the selectivity filter, which is a short and narrow region of the pore. An important turning point in the understanding of the physics of Ca2<sup>+</sup> versus monovalent cations selectivity was a reduced model by Nonner et al. [4] They imagined the selectivity filter of a calcium channel as a high-density fluid where the two oxygens of each of the four COO− groups were modeled as independent hard-sphere O1/2<sup>−</sup> ions (with radius 0.14 nm). When both Na<sup>+</sup> and Ca2<sup>+</sup> ions compete for space in this "electric stew" [81], the competition is won by Ca2<sup>+</sup> ions because they provide twice the charge of Na<sup>+</sup> ions while occupying the same volume (as they have similar Pauling radii).

This mechanism was later called "Charge-Space Competition" [5] because, while the four negative charges of the selectivity filter attract cations, the crowding of those COO− groups and the permeating ions inside the very small selectivity filter imposes entropic and energetic penalties for permeating ions (Figure 1). In this scheme, there is a competition between entropic and enthalpic components, creating an advantage for small and/or high-valence cations over large and/or low-valence cations. This effect is amplified when the dielectric constant of the protein surrounding the pore is lower than the dielectric constant of the selectivity filter lumen [13].

**Figure 1.** Model of the RyR channel [37]. (**A**) The 3D model is obtained by rotating the shaded gray area about the *z*-axis (the models have rotational symmetry). The arrows indicate the regions into which the 8 O1/2<sup>−</sup> ions representing the respective amino acids are confined. The charges of the E4902 residues of the RyR channel are modeled by eight point charges on a ring. The dielectric constants is *e*w = 78.5 in the whole system. The entire simulation cell is enclosed in a large cylinder. The geometry for the NP+LEMC calculations can be found in Figure 1 of Reference [39]. The brown line indicates the countour of the 1D model of Gillespie [16]. (**B**) A snapshot of the simulation. The blue, green, light blue, and red spheres represent Na+, Ca2+, Cl−, and O1/2<sup>−</sup> ions, respectively. This figure was prepared with vmd [82].

When this model of the L-type selectivity filter was incorporated into a pore and studied with GCMC simulations, the model was successful in reproducing the micromolar Ca2<sup>+</sup> selectivity of the L-type calcium channel (EEEE locus). Specifically, it reproduced the seminal experiment of Almers and McCleskey [83] where, in 32 mM NaCl, 1 µM Ca2<sup>+</sup> in the bath blocks Na<sup>+</sup> current, reducing it to half that in the absence of Ca2+. The block works because Ca2<sup>+</sup> ions displace Na<sup>+</sup> ions in the selectivity filter even though they are present in the bath at much smaller concentrations than the Na<sup>+</sup> ions. The model also reproduced [17,21,24] other mole fraction experiments (e.g., Ca2<sup>+</sup> vs. Ba2<sup>+</sup> [84–86], Li<sup>+</sup> vs. Na<sup>+</sup> [87]) and Gd3+-block of ionic current [88]. Lastly, we were able to interpret [14] the experiments of Heinemann et al. [89] where a DEKA→DEEA mutation converted a sodium channel without a Ca2<sup>+</sup> blockade into a calcium channel with 10−<sup>4</sup> M affinity.

Concurrent to this work on the physics of L-type calcium channel selectivity, one of us (DG) created a 1D reduced model of RyR using DFT based on the Nonner et al. independent-O1/2<sup>−</sup> model of the COO− groups [11]. Here, we focus on a second, improved version of this 1D DFT model [16], as it included more charged amino acids that are outside of the selectivity filter yet play an important role in cation permeation [90] (following the second principle of reduced models). The D4945, D4938, D4899, and E4900 amino acids (four copies of each of them due to the homotetrameric RyR structure) were modeled by confining eight half charged oxygen ions, O1/2<sup>−</sup> (with radius 0.14 nm), in the regions indicated by arrows in Figure 1. The E4902 amino acids were placed in a ring at the luminal entrance of the pore.

The purpose of this RyR model was to determine whether a reduced model of this channel could reproduce and predict experimental data. (RyR is more useful for this than L-type calcium channel because of the vast amounts of IV data available for RyR.) Both the model [16] and its subsequent applications [19,20,32,35] showed that this is indeed the case, reproducing all the available IV data from the labs of Gerhard Meissner (University of North Carolina, Chapel Hill) and Michael Fill (Rush University Medical Center, Chicago). Moreover, in these papers the model predicted (before confirming experiments were done) a number of counterintuitive and nonlinear selectivity phenomena in RyR.

Later, a 3D reduced model of RyR was created by Boda et al. [37,41]. The purpose of this model was partly to understand the success of the 1D model, trying to define the effects of radial ion distributions that are ignored in the 1D model (which assumed homogeneity in the radial direction). The profile of the pore radius is indicated by the gray shaded area in Figure 1. Here, we focus on the 3D model because it has been less well analyzed in detail and because it uses the same NP+LEMC simulation technique that is also used for the nanopores, described later, that serve as different case studies of reduced models.

Both the 1D and 3D models reproduce dozens of IV curves, some shown in the Supplementary Information for the 3D model. This indicates that both models seem to capture the basics of the RyR device physics in the axial direction. Therefore, we will discuss how each of the principles of reduced models for nanopores works in these RyR models.

#### 3.1.1. Ionic Concentrations and Current

How the ionic profiles determine the species current has several interesting subtleties in RyR. First, given that the 3D model performs equally well as the 1D model, it seems that any radial ion packing effects do not contribute significantly to the current. Figure 2 shows examples for Na<sup>+</sup> and Ca2+. The profiles are monotonic in the radial dimension, so the cross-section averaged axial concentration profiles are the main determinants of current. This explains the success of the 1D model.

**Figure 2.** Concentration profiles, *c<sup>i</sup>* (*z*,*r*), of Na<sup>+</sup> and Ca2<sup>+</sup> over the (*z*,*r*) plane for 100 mM NaCl and 1 mM CaCl2.

Second, high concentration of a species inside the pore does not always translate into high current for that species. This is exemplified in mole fraction experiments, where two cation species compete for the pore (Figures 3 and 4). We distinguish two basic kinds of mole fraction experiments: (1) In one kind, we add one type of cation (e.g., divalents) to a fixed background of the other type of cation (e.g., monovalents), for example, adding CaCl<sup>2</sup> to a fixed 100 mM NaCl (or CsCl) solution; (2) In the other kind, we keep the total salt concentration (or ionic strength) fixed while changing the mole fraction of the two salt, for example, a NaCl/CsCl mixture at 250 mM total concentration.

Total current, *I*, or chord conductance, *G* = *I*/*U* (*U* is the applied voltage), can be considered a primary device function in the case of ion channels. But, currents carried by the ionic species are also interesting, and we show those as well. Figure 3 shows the currents as functions of composition expressed either as lg[CaCl2] for the added-salt experiment or the mole fraction of Na<sup>+</sup> ([NaCl] + [CsCl] = 250 mM) for the mole fraction experiment.

In the added-Ca2<sup>+</sup> experiment with Na+, it is seen that 10−<sup>3</sup> M Ca2<sup>+</sup> affects the current; against Na+, RyR has millimolar Ca2<sup>+</sup> selectivity. This [Ca2+] is smaller for Cs<sup>+</sup> because Ca2<sup>+</sup> can compete more easily with the larger Cs+. In both cases, the total current has a minimum, called the anomalous mole fraction effect (AMFE), for experiments (gray spheres), the 1D DFT RyR model (magenta lines), and the 3D NP+LEMC model (green triangles). There is also an AMFE for mixtures of Na<sup>+</sup> and Cs+.

**Figure 3.** Added-salt experiments for Ca2<sup>+</sup> vs. Na<sup>+</sup> and Ca2<sup>+</sup> vs. Cs<sup>+</sup> competition (**top**), and mole-fraction experiments for Na<sup>+</sup> vs. Cs<sup>+</sup> competition (**bottom**). Details are in the main text.

**Figure 4.** Axial concentration profiles for Ca2<sup>+</sup> vs. Na+, Ca2<sup>+</sup> vs. Cs+, and Na<sup>+</sup> vs. Cs<sup>+</sup> competition. In the top row, Ca2<sup>+</sup> is added to either Na<sup>+</sup> or Cs+. Profiles are for 10−<sup>5</sup> M and 10−<sup>3</sup> M added Ca2+. In the bottom row, the Na+/Cs<sup>+</sup> mole fraction profiles are shown for 0, 0.5, and 1 Na<sup>+</sup> mole fractions.

To understand the origin of the minimum in current, we first note that at the extremes all the single-species currents are very similar: the all-Na<sup>+</sup> current (and all Cs<sup>+</sup> current) at 10−<sup>6</sup> M Ca2<sup>+</sup> or 0 Cs<sup>+</sup> (0 Na+) mole fraction is similar to the all-Ca2<sup>+</sup> current at 10−<sup>2</sup> M Ca2+. Why then does current decrease with added Ca2<sup>+</sup> (added-salt experiment) or added Na<sup>+</sup> (mole fraction experiment) and then increase?

Part of the answer lies in the axial concentration profiles for select Ca2<sup>+</sup> concentrations and Na<sup>+</sup> mole fractions shown in Figure 4. In the added-salt experiment (top row of figure panels), Ca2<sup>+</sup> displaces Na<sup>+</sup> and Cs<sup>+</sup> throughout the pore. Ca2<sup>+</sup> has a much stronger effect on Cs+, indicating that RyR has a higher preference for Ca2<sup>+</sup> than Cs<sup>+</sup> (compared to Ca2<sup>+</sup> versus Na+). Specifically, 10−<sup>3</sup> M Ca2<sup>+</sup> displaces almost half the Na<sup>+</sup> in the pore and two-thirds of the Cs+. Interestingly, however, the Ca2<sup>+</sup> current in Figure 3 (top row) is below the Na<sup>+</sup> current and equal to the Cs<sup>+</sup> current at [Ca2+] = 1 mM. Recall that all single-species currents are nearly identical. Therefore, just because Ca2<sup>+</sup> has a large (or even the largest) concentration in the pore, it does not produce as much current as would be predicted from those intra-pore concentrations.

In previous work [17,20], we traced this anomaly to the fact that Ca2<sup>+</sup> is at low concentration in the baths, even though it is extremely high (relatively) in the selectivity filter. This produces the counterintuitive result that the bath has a high resistance to Ca2<sup>+</sup> flowing, while the selectivity filter has a low resistance. Usually it is the opposite. Only when the bath Ca2<sup>+</sup> concentration is relatively high is there an appreciable amount of Ca2<sup>+</sup> current. This is physiologically relevant, as resting luminal SR Ca2<sup>+</sup> concentration is between 0.5 to <sup>1</sup> mM, and during contractions this is Ca2<sup>+</sup> depleted to <sup>∼</sup> <sup>50</sup> % levels in cardiac myocytes and even lower in skeletal myocytes. The physiological cardiac ion species currents are described in Reference [32].

This is an extreme example of the depletion zones we will discuss for the nanopores later. A depletion zone (a place where ions are absent for the most part) can have as large an effect on current as the regions of high concentration. This is because the axial direction for current flow is made of several regions, the bath, the access region (at the mouth of the channel or pore), the pore, another access region, and another bath. Each of these has a resistance to current flow and the highest resistance element can dominate. In a channel this is usually the selectivity filter because it is commonly physically narrow. However, if it is highly charged, then it will always have ions in it at high concentration and so the bath resistance may dominate the current. In general, the absence of ions in a region can be as consequential as high concentrations.

#### 3.1.2. Accurate Representation of Pore Charges is Important for Reproducing Device Function

As stated above, the first 1D RyR model [11] did not include all the charged groups that the second one [16] does. In fact, it originally included only the two then-known charged groups (Asp-4899 and Glu-4900). But, no parameters could be found to make the computed IV curves resemble, even qualitatively, the experimental curves. Only by hypothesizing the existence of a region of negative charge on the cytosolic side of the selectivity filter did the curves begin to match up. Later, it was determined that two other aspartate groups (Asp-4945 and Asp-4938) also significantly affect ion permeation and selectivity [90]. Only with the explicit addition of these and another charged group (Glu-4902) did the model reproduce all the experimental data and predict even more (which were later confirmed by experiments [16,19,20,32,35]).

#### 3.1.3. Important versus Unimportant Degrees of Freedom

The results of both the 1D and 3D models indicate that the essential important degrees of freedom were captured. One that was left out was ion dehydration. This is crucial for the physiological function of potassium channels [77,91] and excludes Mg2<sup>+</sup> from many other calcium channels [92]. However, in RyR it does not seem to play a role, as indicated by both experiments and the models. In experiments, Mg2<sup>+</sup> (which has a very large ion dehydration energy compared to the otherwise similar Ca2+) permeates RyR equally as well as Ca2+, indicating no large energetic barrier for Mg2<sup>+</sup> entry by stripping off waters. In the two models, missing an important piece of physics ought to result in (large) deviations from the experimental data, especially in Mg2<sup>+</sup> versus monovalent cation

competition experiments. That this was not seen implies (but does not prove) that ion dehydration is not significant for RyR.

One degree of freedom we have in both the 1D and 3D models that may be superfluous is the flexibility of the O1/2<sup>−</sup> to move within their regions of confinement. Our previous work on the L-type calcium channel [27] indicates that their movement in response to other ions being nearby is unimportant for selectivity. Specifically, for that model pore the selectivity behavior of the channel does not change much if we fix the positions of the O1/2<sup>−</sup> ions. Seemingly, the important characteristics is the density of the O1/2<sup>−</sup> ions inside the pore, while their exact position is secondary. We continue to include the flexibility because it is easy to include and extensive studies would be needed to verify that it is indeed superfluous.

#### 3.1.4. Transferability of Parameters

The main parameters we had that were not based on known RyR structure and that had to be fitted to data were the ionic diffusion coefficients. For both the 1D and 3D models, after these were fit, they were never changed. Therefore, they were used at low and high ionic bath concentrations, low/high and negative/positive applied voltages, and in ionic mixtures. This indicates that they truly are transferable and independent of external conditions.

The one caveat to that statement relates to one of the differences in constructing the 1D and 3D models. In the 3D model, we used only one adjustable *D* pore *i* value in the selectivity filter and interpolated in the vestibules to the bulk. (Values are shown in Table 1.) In the 1D model, on the other hand, there were fitted diffusion coefficients not only in the selectivity filter, but also in the vestibules on either side, in the D4938 and E4900 regions (Figure 1). These were fit for K<sup>+</sup> based on data of RyR in symmetric 0.25 M KCl for native RyR (i.e., fully charged) and two charge-neutralizing mutations (D4938N and E4900Q). With these, the 1D model reproduces the nonlinear IV curve of another charge-neutralizing mutation (D4899N) that was not used in fitting the diffusion coefficients. This further shows the transferability of the diffusion coefficients. (All non-K<sup>+</sup> cation species were fitted with one experimental data point for the selectivity filter diffusion coefficient and the vestibule values were determined from ratios of the K<sup>+</sup> diffusion coefficients in different areas of the pore.)

**Table 1.** Parameters of ions as used in the NP+Local Equilibrium Monte Carlo (LEMC) simulations. The last column shows the density functional theory (DFT) value *D* pore *i* , the diffusion coefficient in the selectivity filter, for comparison; the values for the vestibules are found in Reference [16]. *<sup>a</sup>* This value was not fitted due to the fact that the channel does not let Cl− through.


The 3D model, on the other hand, does not reproduce these charge-neutralizing experiments (data not shown). Therefore, its diffusion coefficients are not as robust against changes to external conditions (although such mutations are large perturbations). This indicates that caution is always in order when interpreting a reduced model outside its established (i.e., tested against experiments) range of external conditions.

#### *3.2. Nanopores of Different Device Functions from Different Charge Patterns*

In a recent work [46], we considered synthetic nanopores with varying charge patterns on their walls along the *z*-axis (Figure 5). Although our rules of thumb were not formulated explicitly back then, we practically organized that study along the lines of the four rules of thumb:


**Figure 5.** Schematics of the cylindrical nanopores with different charge patterns. There are two regions of lengths *H*<sup>L</sup> and *H*<sup>R</sup> carrying *σ*<sup>L</sup> and *σ*<sup>R</sup> surface charges. We consider either bipolar (top row) or unipolar (bottom row) nanopores. In the bipolar cases, the left-hand region is always negative (*σ*<sup>L</sup> <sup>=</sup> <sup>−</sup>*σ*<sup>0</sup> with *<sup>σ</sup>*<sup>0</sup> <sup>=</sup> 0.4835 *<sup>e</sup>*/nm<sup>2</sup> ), while the right-hand region is positive (*σ*<sup>R</sup> = *σ*0). In the unipolar cases, the same is true, but the other side is neutral. The dimensionless net charge, *Q*, increases from left to right, while the fraction of the left region, *x*<sup>L</sup> (Equation (3)), changes as indicated by the arrows.

A cylindrical nanopore was considered with radius *R*pore = 0.97 nm and length *H* = 6.4 nm. The wall of the pore was divided into two regions along the *z*-axis: a left (L) region of length *H*<sup>L</sup> carrying *σ*<sup>L</sup> surface charge, and a right (R) region of length *H*<sup>R</sup> = *H*−*H*<sup>L</sup> carrying *σ*<sup>R</sup> surface charge. The geometry can be characterized by the dimensionless parameter *x*<sup>L</sup> = *H*L/*H*. We gradually increased *H*L, while keeping the total length, *H*, fixed, so we increased *x*<sup>L</sup> from 0 to 1. We performed two series of calculations.


In order to characterize charge pattern, we introduced a dimensionless net charge, *Q*, ranging from −1 to 1, defined as

$$Q = \mathfrak{x}\_{\rm L} \frac{\sigma\_{\rm L}}{\sigma\_0} + (1 - \mathfrak{x}\_{\rm L}) \frac{\sigma\_{\rm R}}{\sigma\_0}. \tag{3}$$

This value is uniquely related to *x*<sup>L</sup> in the cases depicted in Figure 5. Its value is −1 for the 'nn' pore, 1 for the 'pp' pore, 0 for the 'np' pore, −0.5 for the 'n0' pore, 0.5 for the '0p' pore, and 0 for the 'np' and '00' pores. We found that the pore's basic behavior is correlated with this parameter (Figure 6).

**Figure 6.** (**A**) Ionic currents as functions of *Q* for the bipolar (left panel) and unipolar (right panel) nanopores. Blue and red colors correspond to Na<sup>+</sup> and Cl−, respectively. Symbols and lines correspond to MD and NP+LEMC results, respectively. Filled symbold and solid lines refer to the ON state (200 mV), while open symbols and dashed lines refer to the OFF state (−200 mV). (**B**) Diffusion coefficients in the pore, *D* pore *i* , normalized by the bulk values, *D*bath *i* , fitted to MD currents in the ON states of the bipolar pore. The fit was done for every *Q* separately.

In order to relate our implicit-water NP+LEMC simulations to explicit-water MD simulations, we constructed an all-atom version of the model. While we did our best in building the all-atom model that is, apart from the treatment of water, is as similar to the reduced model as possible, there are differences:


The most serious difference between the two systems is the treatment of water, so we consider this study as a test of the implicit-water approximation for this nanopore system.

A continuous surface charge was mimicked by placing partial point charges at the carbon atoms of the CNT. The CNT consisted of hexagons of side width 0.142 nm. There were 1682 partial charges of strength 0.0112 *e* on the grid for the 'pp' pore. These same partial charges were used in the NP+LEMC calculations. This fine resolution of the pore charges was necessary, because we also compared to the PNP theory in Reference [46] (PNP results are not shown here).

The electrolyte was NaCl (for the ionic parameters see Reference [46]) at bulk concentrations 1 M. The asymmetric pores were rectifying when we applied voltages 200 and −200 mV (ON and OFF states, respectively).

#### 3.2.1. Concentration Profiles and Device Functions

The MD simulation results are our gold standard, so we fit the diffusion coefficients inside the pore, *D* pore *i* , to MD current data for the bipolar pore in the ON state (Figure 6B). Because we decided to use only one adjustable parameter (*D* pore *i* ), it was necessary to make its value *Q*-dependent, because the pore's behavior is severely different at different *Q* parameters as also shown by the concentration profiles (Figure 7).

**Figure 7.** Cross-section averaged axial concentration profiles of Na<sup>+</sup> (blue) and Cl<sup>−</sup> (red) ions for (**A**) the bipolar and (**B**) unipolar cases. In each case, top row and bottom row show the ON and OFF states, respectively. Symbols and solid lines refer to molecular dynamics (MD) and NP+LEMC results, respectively.

As the pore charge, *Q*, increases, Na<sup>+</sup> current decreases and Cl<sup>−</sup> current increases (Figure 6A). One of the device functions, selectivity, changes with *Q*, with the pore being non-selective at *Q* = 0. When the charge pattern is asymmetrical, the pore rectifies, namely, the ON current is larger than the OFF current (Figure 6A). Rectification (the other device function) has a maximum at *Q*≈0 in the bipolar case, while it has maxima between *Q* = −1 and 0 as well as between *Q* = 0 and 1 in the unipolar case. The selectivity and rectification curves as functions of *Q* are shown in Reference [46] (their Figure 7).

The axial concentration profiles (Figure 7) determine the current, as in the case of the RyR ion channel. The major difference compared to the RyR channel is that the depletion zones have decisive roles inside the pore here, not only in the access regions as in the case of the RyR. Briefly, if an ionic species has a depletion zone somewhere inside the pore along the *z*-axis, its current is suppressed. This statement is intuitive if we imagine the pore as a collection of layers along the *z*-axis that, in turn, are imagined as resistors connected in series. If any of the resistors has a large resistance due to a depletion zone in that layer, the whole circuit has a large resistance.

We can also support our statement with a quantitative analysis. In Appendix A, we outline our slope-conductance approach that shows that the resistance of the pore is related to the integral of *c* −1 *i* (*z*) (Equation (A6)). Depletion zones give large contributions to that integral, and, therefore, to resistance.

#### 3.2.2. Charge Pattern Determines Device Behavior

The decisive effect of pore charge pattern does not require special verification here; the studies of Reference [46] shown in Figure 5 were devised for the purpose of studying that effect. Figure 6A for the current and Figure 7 for the concentration profiles clearly show that the charge pattern characterized by the *Q* parameter squarely determines device behavior.

When electrostatic attractions and repulsions play the primary role in forming the shape of the ionic concentration profiles—namely, defining which are the coions and which are the counterions to define where depletion zones and peaks are formed—it is not a surprise that charge pattern dominates over other factors.

#### 3.2.3. Water Molecules as Unimportant Degrees of Freedom

The decisive roles of Coulomb interactions and charge patterns also explain why water molecules can be smeared into a continuum background. Both the axial concentration profiles (Figure 7) and currents (Figure 6A) show that the device works qualitatively the same way in the case of the explicit-water (MD) and implicit-water (NP+LEMC) models.

We devoted a whole paper to this question [42], so we summarize the results of that paper. We showed that the implicit-water and explicit-water models produced qualitatively similar behavior of the current for different voltages and model parameters. Looking at the details of concentration and potential profiles, we found profound differences between the two models. However, these differences did not influence the basic behavior of the model as a device because they do not influence the *z*-dependence of the concentration profiles, which we found are the main determinants of current. Therefore, our simulations showed that reduced models can still capture the overall device physics correctly because they included the physics that is necessary from the point of view of device function. This is despite the fact that they get some important aspects of the molecular-scale physics quite wrong (e.g., radial ion packing produced by the structure of the water molecules).

#### 3.2.4. Transferability of the Fitted Diffusion Coefficient

We emphasized that it is the *qualitative* behavior that is the same on the two modeling levels. If we want *quantitative* agreement, we need to fit the parameter(s) of the reduced model to MD or to experimental data. In general, we can say that if we observe an overall *qualitative* agreement, the reduced model does its job and there is a good chance that our response function that replaces the smeared degrees of freedom is transferable. The question is what transferability means. What are the external conditions that influence the response function and what are those that do not?

This question has been already touched on with the RyR ion channel, where we stated that our choice of a single adjustable parameter (the diffusion coefficient in the selectivity filter, *D* pore *i* ) does not make it possible to create a response function that is transferable over charge patterns, namely, over mutations. It was, however, transferable over voltages, concentrations, and electrolyte compositions. The situation here is the same. We attempted to create a diffusion coefficient profile that is independent of *Q*, but due to uncertainties in MD simulations and computational demand of NP+LEMC simulations, we abandoned these efforts. Instead, we realized that the difference between the MD and NP+LEMC concentration profiles (Figure 7) depends on *Q* systematically. For example, as *Q* increases, the Na<sup>+</sup> profiles as obtained from MD and NP+LEMC become increasingly different. (At the same time, Cl− profiles become increasingly similar.) Exactly this difference is what must be balanced by the diffusion coefficient in the pore.

Therefore, we decided to use a single *D* pore *i* (*Q*) value all along the pore that is allowed to vary with *Q*. We fitted *D* pore *i* to one case (bipolar/ON), and investigated transferability for the remaining

three cases (bipolar/OFF, unipolar/ON, unipolar/OFF). So, we fixed the *D* pore *i* (*Q*) values fitted to the bipolar/ON case and used them at other cases for the same *Q*. These values are shown in Figure 6B as functions of *Q*.

As an example, let us consider *Q* values close to −1. This is close to the 'nn' geometry, namely, a cation selective pore. Cl− ions have depletion zones in this case in both models, but they are deeper in MD than in NP+LEMC. We need, therefore, a very small *D* pore Cl<sup>−</sup> value to bring the NP+LEMC profiles (and, therefore, currents) down to the values yielded by MD. As *Q* increases, the difference between the Cl− profiles decreases and they are pretty similar for *Q* = 1, namely, for the 'pp' pore. In general, we can state that the implicit-water approximation works better (compared to MD) in the case of peaks than in the case of depletion zones.

To summarize, one job of the *D* pore *i* (*Q*) function is to take the differences in the explicit and implicit water models into account. The diffusion coefficient in the pore, therefore, is more than a transport coefficient that, in principle, could be calculated from autocorrelations functions or mean square displacements. It carries more information that stems from differences between the reduced model and the more realistic experimental data or MD simulations. Eventually, it is an adjustable parameter of the reduced model as a whole.

#### *3.3. Selectivity Inversion Due to Charge Inversion*

In the two case studies so far radial profiles were relatively unimportant. The narrow RyR pore was a crowded high density region (Figure 2) but without layering (oscillatory concentration profiles) in the radial dimension. The case of the wider nanopore in Section 3.2, however, is much more complex. The radial distribution of the ions is important because it determines the behavior of the axial profiles. This was discussed in detailed in our recent studies [50,52].

In the first study [50], we showed that bipolar nanopores exhibit a scaling behavior for a fixed *<sup>σ</sup>* <sup>=</sup> <sup>±</sup><sup>1</sup> *<sup>e</sup>*/nm<sup>2</sup> . Specifically, we constructed a scaling parameter, *ξ* = *R*pore/*λ* p *z*+|*z*−|, where *λ* is the characteristic screening length of the electrolyte computer either as the Debye length (for a point-ion model) or the Mean Spherical Approximation screening length (for Primitive Model ion). (Note that screening works differently near surfaces of different curvatures (flat, concave, convex). Different equations for the capacitance can be given with an unchanged value of the Debye length. [93]) We found that for different pore sizes and different electrolyte concentrations that had the same *ξ* the device function (this time rectification) was the same; that is, for a given *z*+:*z*<sup>−</sup> electrolyte, the relationship of *R*pore and *λ* determines device behavior. If *R*pore*λ*, the double layers formed at the nanopore's wall in the radial dimension overlap. In that case, the counterions will be at high concentration in the middle of the pore, while coions will be at relatively low concentration. This forms depletion zones for the excluded coions. If *R*pore*λ*, the double layers do not overlap, a bulk electrolyte is present in the pore's center line, and depletion zones are not formed. Depletion zones are necessary for selectivity and rectification. In Section 3.2, this was not discussed because *R*pore and *λ* (concentration) were fixed. Double layer overlap was present.

In the second study, [52] we considered the dependence of bipolar nanopores on *σ* for different electrolytes (1:1, 2:2, 2:1, 3:1). If multivalent ions are present, a deviation from the above scaling behavior (basically a mean-field phenomenon) appears because strong ionic correlations cause peculiar phenomena such as overcharging (overcharging means that more counterions are attracted to the surface than necessary to compensate the surface charge) and charge inversion [78] (charge inversion is the appearance of a layer of excess coions that produces a change in the sign of the electrical potential in this layer). Specifically, these correlations cause an increase in coion concentration in the second layer of ions behind the dense counterion first layer near the charged wall. Consequently, the electrostatic potential can change sign (relative to the potential at the charged wall). We showed that this accumulation of coions (anions) produces an anion leakage current, and this causes non-monotonic behavior in the device function (rectification) as *σ* increases. Charge inversion always manifests itself in the dimension perpendicular to the charged wall, which for pores is the radial dimension.

In this section, we present new results for the phenomenon of selectivity inversion in a negatively charged nanopore (*<sup>σ</sup>* <sup>=</sup> <sup>−</sup><sup>1</sup> *<sup>e</sup>*/nm<sup>2</sup> ) as the electrolyte is changed from 1:1 through 2:1 to 3:1. This phenomenon was shown experimentally in our paper with the group of Zuzanna Siwy [22] and interpreted with the help of GCMC simulations. It was observed that while the pore is cation selective for a 1:1 electrolyte (KCl), it becomes anion selective for a 3:1 electrolyte (CoSepCl3). The GCMC simulations supported the idea that the basic reason of this selectivity inversion is charge inversion. The trivalent cations stick to the negatively charged surface, overcharge it, and remain paralyzed; they do not contribute to the current significantly because their mobility near the pore is reduced by being trapped in an energy well.

Here, we show that this behavior can be reproduced without explicitly changing the mobilities of the ions (i.e., decreasing *Di*(*r*) near the wall) by using localized charges instead of a surface charge that is smeared over the surface relatively uniformly as it was in Section 3.2. In fact, this model is much closer to the experimental reality, because the negative charges are localized in chemical groups on the surface of an insulator, specifically, in COO− groups for the PET nanopores used by Siwy et al.

Here, we show that adopting this idea can produce strong charge inversion around the binding sites now both in the *z* and *r* dimensions. The nanopore is practically the same as the one in Section 3.2: it is a cylindrical pore with *R*pore = 1 nm and *H* = 6 nm with *c* = 0.1 M electrolytes on both sides (ionic radii are *R*<sup>+</sup> = *R*<sup>−</sup> = 0.15 nm). We place fractional point charges on a rectangular grid on the pore's surface of width ∆*z* in a way that the surface charge density is kept constant at *<sup>σ</sup>* <sup>=</sup> <sup>−</sup><sup>1</sup> *<sup>e</sup>*/nm<sup>2</sup> . Having ∆*z* = 1 nm, where −*e* point charges are sitting on the grid, corresponds to the experimental situation.

#### 3.3.1. Axial Concentration Profiles Determine Selectivity

Cation selectivity defined as *I*+/(*I*++*I*−) is shown in the bottom panel of Figure 8A as a function of ∆*z* for different electrolytes (1:1, 2:1, and 3:1) for a constant *R*pore. The top panel shows the ionic currents from which selectivity was computed. While cation selectivity is insensitive to the fineness of the grid (the degree of localization of surface charge) in the 1:1 case, cation current (and cation selectivity with it) quickly drops as ∆*z* increases above 0.8 nm in the 3:1 case (thick red lines).

The explanation follows from the axial cationic concentration profiles (i.e., cross-sectionally averaged concentrations) in Figure 8B. For a fine grid similar to that used in Section 3.2 and in our earlier studies (∆*z* = 0.2 nm), [42–52] the cation profiles are practically constant inside the pore for all the electrolytes from 1:1 to 3:1. For the case of localized charges (∆*z* = 1 nm), depletion zones appear along the *z*-dimension that are much deeper in the case of 2:1 and, especially, 3:1 electrolytes. As the axial depletion zones get deeper, cation currents decrease as ∆*z* increases.

Anion currents, on the other hand, do not change significantly as ∆*z* changes because the anion profiles do not change (Figure 8B). This statement is valid for the anion profiles too (see Figure 8B). This is not a surprise because the anions are far from the charged surface on average, so their distribution is less influenced by the localization of the pore charges. This indicates that it is the behavior of the cations that is responsible for selectivity inversion.

#### 3.3.2. Charge Localization Is an Important Degree of Freedom

The appearance of those depletion zones, however, can be fully understood only if we take into account both the *z*- and *r*-dependence of the ionic distributions. Although the statement that current primarily depends on the axial profiles remains true (Equation (A6)), understanding why the axis profiles look the way they look requires the complete picture.

Figure 9A shows the *c*3+(*z*,*r*) concentration profiles for trivalent cations and ∆*z* = 1 nm. The figure shows the large peaks near the localized pore charges and deep depletion zones between the peaks (note the logarithmic scale). Also, the cationic concentration profiles decline as *r*→0, namely, approaching the centerline of the pore.

**Figure 8.** (**A**) The top panel shows ionic currents as functions of ∆*z* for various electrolytes (green, blue, and red color refer to 1:1, 2:1, and 3:1, electrolytes). Solid and dashed lines refer to cations and anions, respectively. The bottom panel shows the cation selectivities computed as *I*+/(*I*<sup>+</sup> + *I*−). Values above and below 0.5 correspond to cation and anion selectivities, respectively. (**B**) Axial concentration profiles of cations (solid lines) and anions (dashed lines) in the three elecrolytes. Different panels refer to different values of ∆*z* (0.2, 0.8, and 1 nm from top to bottom). Colors have the same meaning as in Figure 8A.

These phenomena can be observed better if we plot the radial profiles for fixed *z* values that correspond to either a peak (red) or a depletion region (blue). (For the actual values of *z*, see the caption of Figure 9B.) The left panel of Figure 9B shows radial profiles for ∆*z* = 0.8 nm; the corresponding axial profiles were shown in the middle panel of Figure 8B. The important thing to note is that the radial profiles do not differ much for different values of *z*. Depletion zones, therefore, are not formed in this case (see solid blue line with filled squared). It is important to point out that charge inversion is present in this case in the radial profiles; the anion profiles are larger in and around the center line of the pore. It is, however, only present in the radial direction.

(**A**)

**Figure 9.** (**A**) The *c*3+(*z*,*r*) concentration profiles of the trivalent cations for the ∆*z* = 1 nm case. (**B**) Radial concentration profiles of trivalent cations (C3+, filled symbols with solid lines) and monovalent anions (A−, open symbols with dashed lines) for selected values of *z*. In the left panel (∆*z* = 0.8 nm), the values *z* = 0.3 nm and *z* = 0.7 nm correspond to a peak and a depletion region, respectively. In the right panel (∆*z* = 1 nm), the values *z* = 0.5 nm and *z* = 0.9 nm correspond to a peak and a depletion region, respectively.

The right panel of Figure 9B shows the radial profiles for ∆*z* = 1 nm. This small difference in ∆*z* results in a significant change in the behavior of the ions. The cation profiles show the large peaks for *z* = 0.5 nm (solid red line with filled circles), while they exhibit depletion zone for *z* = 0.9 nm (solid blue line with filled squares). This different behavior at *z* = 0.5 nm and *z* = 0.9 nm produces the oscillating axial concentration profiles with the axial depletion zones of Figure 8B. In this case, therefore, we have charge inversion in both the radial and axial directions.

Taken together, these results show that the way we place the pore charges on the wall matters from the point of view of reproducing device function (pore selectivity and specifically its change due to charge inversion). Specifically, modelers probably need to step beyond the continuous surface charge distribution and to build localized pore charges into the reduced model. Counterion interactions with pore charges depend on the distance from a local binding site in all directions. In the case here, charge inversion around a local binding site produced important axial depletion zones. However, even when not considering cases with charge inversion, different ion correlations around localized pore charges can potentially produce similar important axial effects that are missed with a uniformly charged wall.

#### 3.3.3. Future Work

While it is clear that the location and discreteness of pore charges are an important degree of freedom, whether we need to use explicit particles to model the atoms of the COO− groups is a subject of ongoing research. We suspect it is not vital since the charge inversion at the core of the device behavior is a product of charge itself, not the shape or mobility of the atoms producing the charge.

Also, we do not know whether we need to change the diffusion coefficient, *Di*(*z*,*r*), in the radial dimension in order to fit to experiments or to dynamic simulations. Work is currently underway with all-atom MD simulations to determine this.

#### **4. Conclusions**

In reduced models, some degrees of freedom (the important ones) are modeled explicitly, while the rest (the unimportant ones) are taken into account implicitly in some way, via response functions, for example. Before the age of computers, all models were reduced. When MD simulations became an everyday computational tool, atomic models became the new standard in certain areas of chemistry, physics, and biology. While understanding nanoscale physics is vital, we believe that the ease of use of MD has sometimes caused the baby to be thrown out with bath water. Rather, we think that what is needed are clever models that are necessarily reduced to some degree to be computationally feasible.

Modeling of ion channels and synthetic nanopores is a case in point. This modeling is inherently difficult as nanoscale interactions and physics directly translate into measurable phenomena (what we call device functions). By simplifying the physics to be modeled, reduced models have a number of advantages over all-atom simulations. However, building such models is in many ways more art than science. Here, we have taken both old and new data from our simulations of ion channels and nanopores and distilled from them four rules of thumb (principles) for constructing reduced models for nanopores. These are


Our goal is to offer insights into how to think about reduced model, but also to point out the subtleties and consequences of the choices a modeler might make. Specifically, for each rule of thumb we showed that its interpretation is not as straightforward as it might seem. For example, while large ion concentrations are important, so are areas with small concentrations which act as large resistors that can dominate the current. Also, charged groups seemingly far from the key locations (e.g., the selectivity filter of an ion channel) can grossly change current/voltage curves. Overall, testing and probing to find the important degrees of freedom that capture the axial-direction physics is the key to reproducing device function and understanding the physics behind the device function; for example, using uniform versus discrete pore charges can have measurable consequences. Once these have been identified, approximating other physics as response functions is a lot easier.

Lastly, we note that while reduced models are important to understand these devices, they are only one part of the continuum of modeling levels that are possible. All-atom and even quantum mechanical simulations play key roles as well in defining the physics of nanopores at the atomic and molecular levels. The role of reduced models is on a larger scale, namely to identify the physics of the device as a whole using the nanoscale physics defined at more detailed levels of modeling. They are the last step to couple atoms to experimental measurements.

**Supplementary Materials:** The following are available online at http://www.mdpi.com/1099-4300/22/11/1259/s1 .

**Author Contributions:** Conceptualization, D.B. and D.G.; Methodology, D.B. and D.G.; Software, D.B. and D.G.; Validation, D.B. and M.V. and D.G.; Investigation, D.B. and M.V. and D.G.; Writing—Original Draft Preparation, D.B.; Writing—Review & Editing, D.B. and D.G.; Visualization, D.B. and M.V.; All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the National Research, Development and Innovation Office (NKFIH) grant number K124353.

**Acknowledgments:** Present article was published in the frame of the project GINOP-2.3.2-15-2016-00053. The authors acknowledge the contributions of many talented colleagues with whom they worked together during the last 20 years: Douglas Henderson, Wolfgang Nonner, Bob Eisenberg, Tamás Kristóf, Zuzanna Siwy, Zoltán Ható, Claudio Berti, Simone Furini, Attila Malasics, Eszter Mádai, Dávid Fertig, Róbert Kovács, Janhavi Giri, Michael Fill, Gerhard Meissner.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A. Slope Conductance Theory**

Let us assume rotational symmetry, so quantities depend on the variables *z*,*r*. Let assume, furthermore, that *µ* does not depend on *r*; results (not shown) support this assumption. Let us integrate the NP equation (Equation (1)) over the cross section:

$$\mathbf{H}\_{i} = -z\_{i}e \int\_{A(z)} j\_{i}(z,r) da = \frac{z\_{i}e}{kT} \left[ \int\_{A(z)} \mathbf{D}\_{i}(z,r) c\_{i}(z,r) da \right] \frac{d\mu\_{i}(z)}{dz} = \frac{z\_{i}e}{kT} \mathbf{N}\_{i}(z) \frac{d\mu\_{i}(z)}{dz} \tag{A1}$$

for any *z* inside the pore with

$$N\_i(z) = \int\_{A(z)} D\_i(z, r) c\_i(z, r) da.$$

Let us rearrange and integrate over the pore

$$I\_i \int\_{H\_1}^{H\_2} \frac{dz}{N\_i(z)} = \frac{z\_i e}{kT} \int\_{H\_1}^{H\_2} d\mu\_i(z) = \frac{z\_i e}{kT} \Delta \mu\_i. \tag{A2}$$

If we assume that bulk concentrations are the same on the two sides of the membrane, the electrochemical difference is

$$
\Delta \mu\_i = z\_i e \mathcal{U}\_\prime \tag{A3}
$$

where *U* is the voltage across the pore. Substituting into Equation (A2), we obtain that

$$\mathrm{I}\_{i} \int\_{H\_{1}}^{H\_{2}} \frac{dz}{N\_{i}(z)} = \frac{z\_{i}^{2}e^{2}}{kT} \mathrm{II} \tag{A4}$$

from which the resistance (the reciprocal of conductance) is obtained as

$$\frac{1}{G\_i} = \frac{\mathcal{U}}{I\_i} = \frac{kT}{z\_i^2 e^2} \int\_{H\_1}^{H\_2} \frac{dz}{N\_i(z)}. \tag{A5}$$

If we assume that *D* pore *i* (*z*) does not depend on *r* inside the pore, we can write that

$$\frac{1}{G\_i} = \frac{kT}{z\_i^2 e^2} \int\_{H\_1}^{H\_2} \frac{dz}{D\_i^{\text{Pore}}(z) A(z) c\_i(z)}\,\text{}\tag{A6}$$

where *ci*(*z*) = <sup>1</sup> *A*(*z*) R *A*(*z*) *ci*(*z*,*r*)*da* is the radially-averaged concentration. If *ci*(*z*) is very small somewhere in the pore along the *z*-axis, the integral, the resistance, becomes large. This analysis was used in several works [17,18,21,23,24,27,46,48,52] to relate concentration profiles to currents.

#### **Reference**


**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Electrophysiological Properties from Computations at a Single Voltage: Testing Theory with Stochastic Simulations**

**Michael A. Wilson 1,2 and Andrew Pohorille 1,3,\***


**Abstract:** We use stochastic simulations to investigate the performance of two recently developed methods for calculating the free energy profiles of ion channels and their electrophysiological properties, such as current–voltage dependence and reversal potential, from molecular dynamics simulations at a single applied voltage. These methods require neither knowledge of the diffusivity nor simulations at multiple voltages, which greatly reduces the computational effort required to probe the electrophysiological properties of ion channels. They can be used to determine the free energy profiles from either forward or backward one-sided properties of ions in the channel, such as ion fluxes, density profiles, committor probabilities, or from their two-sided combination. By generating large sets of stochastic trajectories, which are individually designed to mimic the molecular dynamics crossing statistics of models of channels of trichotoxin, p7 from hepatitis C and a bacterial homolog of the pentameric ligand-gated ion channel, GLIC, we find that the free energy profiles obtained from stochastic simulations corresponding to molecular dynamics simulations of even a modest length are burdened with statistical errors of only 0.3 kcal/mol. Even with many crossing events, applying two-sided formulas substantially reduces statistical errors compared to one-sided formulas. With a properly chosen reference voltage, the current–voltage curves can be reproduced with good accuracy from simulations at a single voltage in a range extending for over 200 mV. If possible, the reference voltages should be chosen not simply to drive a large current in one direction, but to observe crossing events in both directions.

**Keywords:** computational electrophysiology; electrodiffusion model; stochastic simulations; current– voltage dependence; reversal potential; committor probabilities

#### **1. Introduction**

Ion channels are ubiquitous in living systems in which they mediate ion transport across cell walls [1–3]. Although all confirmed structures of ion channels are either bundles of *α*-helices or *β*-barrels organized around a transmembrane, water-filled pore lined largely with hydrophilic side chains, they markedly differ in their properties. Their activity is regulated by a variety of signals, such as voltage, ligands, pH or mechanical tension. Some channels are made of peptides that barely span the membrane, while others are among the largest protein assemblies in a cell. In terms of ionic conductance, defined as the ratio of ionic current to voltage, channels differ by more than two orders of magnitude and conductance is not correlated with size. For example, the single-channel conductance of a bacterial homolog of pentameric ligand gated ion channels (pLGICs), GLIC, which consists of 317 residues per subunit is 8 pS [4], similar to the lowest conductance level of a channel made of antimicrobial peptide, alamethicin, which is built of 20 amino acids [5]. Another channel-forming peptide trichotoxin (TTX), consisting of 7 helices, each containing 18 residues conducts ions at 850–900 pS [6], which is close to the conductance of mechanosensitve channels MscS containing 250–1100 residues [7], approximately equal to

**Citation:** Wilson, M.A.; Pohorille, A. Electrophysiological Properties from Computations at a Single Voltage: Testing Theory with Stochastic Simulations. *Entropy* **2021**, *23*, 571. https://doi.org/10.3390/e23050571

Academic Editor: Peter V E McClintock

Received: 1 December 2020 Accepted: 28 April 2021 Published: 6 May 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

1 nS [8]. Some channels exhibit exquisite selectivity whereas others are non-selective. Single point mutations can not only markedly affect conductance and selectivity but even render a channel inactive or constitutively open [9–11]. How variations in a common, general architecture translate to a variety of electrophysiological behaviors is of great interest not only for understanding regular biological systems but also for explaining a number of diseases associated with improper function of ion channels [12–14].

The availability of high-resolution structural models of ion channels has created opportunities to connect structure and function. Molecular dynamics (MD) computer simulations can contribute to this goal by providing mechanistic and thermodynamic descriptions of ion transport that is not readily accessible from experimental studies [15–26]. For a recent, comprehensive review, see Flood, et al. [27]. Furthermore, MD simulations can be used to validate experimentally derived structural models, which do not always correspond to the native structures of channels [28], select the native structure among several candidates [29], and predict functional effects of mutations. These simulations, however, have to be validated by demonstrating that they can be used to reproduce measured electrophysiological properties with satisfactory reliability.

Calculating electrophysiological properties from MD simulations with applied voltage can be done simply by way of computing the current across the simulation cell [15,30,31] or counting the number of ions that cross the channel [16,24,25,32]. However, this direct method, especially when applied to obtain I-V curves and reversal potentials, requires significant computational effort, as it involves MD simulations at a number of applied voltages. To obtain the same accuracy, channels with low conductance generally require longer simulations than channels with high conductance. For example, in simulations of TTX, we counted almost 200 K<sup>+</sup> crossing events in 900 ns at 50 mV [33], whereas only 23 Na<sup>+</sup> crossing events were observed in a 7.7 µs simulation of GLIC at 100 mV (Wilson and Pohorille, unpublished). Since the ionic currents from both simulations appear to obey Poisson statistics, we expect the relative errors in the conductance of K<sup>+</sup> in TTX and Na<sup>+</sup> in GLIC to be approximately 7% and 20%, respectively. To achieve the same relative errors for currents in GLIC as in TTX, a MD trajectory of over 60 µs in length would be required. This means that calculating the I-V curve might present a considerable challenge. To deal with this challenge, it has been common to improve statistics for ion crossing events by applying high voltages, sometimes substantially above their physiological values, or increasing ionic concentration in bulk solution [15–17,34–38]. This approach, however, is fraught with dangers, as it might lead to the disruption of membrane structure or saturation effects for ion entry to a channel [37]. Furthermore, if high voltages are used, I-V curves in physiologically relevant ranges are obtained via interpolation or extrapolation procedures of unknown accuracy [35].

If the motion of ions through the channel can be satisfactorily described as diffusion in the applied electric field and the potential of mean force (PMF) exerted by all other components of the system, which is assumed to be independent of voltage, then the computational effort can be markedly reduced. Several approaches not based on MD take advantage of this description. Methods based on Poisson-Nernst-Planck (PNP) theory rely on solving the electrodiffusion (ED) equation for electrical current in which the mobile ions are represented as a mean-field concentration profile whose distribution and motion is determined by electrostatic forces [39–44]. In Brownian Dynamics (BD), channel conductance is calculated by way of solving the Langevin equation in which both short-ranged interactions with a static model of the channel and long-ranged, electrostatic interactions are taken into account [40,45–48]. In both PNP and BD approaches, electrostatic forces are are obtained from the Poisson equation and the medium is represented as a continuum. From this perspective, we take an approach in which atomistic, dynamic information offered by MD is combined with the efficiency of the ED equation. Instead of carrying out a series of extensive MD calculations of a channel over a range of applied voltages, a substantially reduced set of simulations is combined with the one-dimensional ED model in a steady state. In this approach, the actual electrophysiological properties, such as the

current, are calculated from the ED equation, whereas the quantities needed to solve this equation are supplied from MD simulations.

Not all channels conform to the ED model. This model cannot be applied directly if ion crossing events are not statistically independent [49,50], ion diffusion is single file rather than Fickian [51], there are strong binding sites for ions in the channel, the channel changes its structure in response to applied voltage in the range of interest [26] or ions experience saturation in the mouth of the channel. Despite these limitations, it appears that ion movement through many channels satisfies the assumptions of the ED model. A number of small, naturally occurring and synthetic channels and pLGICs belong to this category. The channels discussed in this paper were found to be well-described by the model. More generally, a number of different equations that are special cases of the ED equation, such as Goldman-Hodgkin-Katz (GHK) equation, have been extensively and successfully used as basic tools in experimental and computational electrophysiology for nearly 80 years, for example to determine ionic selectivity from the measured reversal potential [1,52]. Further, basic assumptions of the ED model, such as independence of ion crossing events or Fickian nature of diffusion, can be tested without substantial, additional effort. This was previously done for a number of channels [25].

Previously, we developed an approach to calculate electrophysiological properties from the integrated form of the ED equation [19,24,25]. Instead of MD calculations at several voltages, the system is simulated at a single voltage (or with no applied voltage) to obtain the PMF for each ion in the channel. Subsequently, markedly shorter simulations at voltages of interest are required to determine the densities of the ions near the ends of the channel. Calculating the currents from simulations at these voltages is not required. In addition, ionic diffusivity along the channel has to be determined. Both boundary density values and diffusivity obtained by any of these methods are burdened with errors, contributing to inaccuracies in the calculated currents.

Recently, we developed two formalisms for calculations of electrophysiological properties, including I-V curves and reversal potential, from a single MD simulation at one voltage [33]. From this simulation, the PMF, nonequilibrium density profiles and committor probabilities for ions in the channel are obtained and used to calculate currents at different voltages after appropriate transformations of the ED equation. Additional calculations to obtain the density boundary terms at different voltages and diffusivity are no longer needed. These formalisms were tested on a simple model of the TTX channel, comprised of 7 straight *α*-helices, each containing 18 amino acids [6], and were shown to perform very well. The improved efficiency of this novel approach derives from the fact that only one MD simulation instead of multiple ones is needed to obtain the I-V curve or reversal potential.

As is the case for any new approach, it is essential to establish the intrinsic accuracy of our formalism. This is the goal of this paper. Specifically, we focus on the question: how reliable are our approaches to calculating electrophysiological properties, independent of other sources of errors, such as inaccuracies in force fields and insufficient simulation times? Separating errors due to the proposed methods from other error is not simple. It cannot be done through a direct comparison with experiments because, for example, of inaccuracies due to force fields. In principle, it can be done via comparison with accurate MD simulations of the PMF and currents at several applied voltages, but, in practice, it is expensive to obtain sufficiently accurate free energies and currents. Although PMFs for ions in a number of channels were obtained from MD simulations, in most cases no estimates of errors were provided [21,23,26,53–61]. In a few cases in which errors are available by way of either direct estimates or comparisons of PMF obtained via different methods [20,24,25,62–68] they are of the order of 0.2–0.7 kcal/mol, which is similar to what is expected to be the intrinsic accuracy of the formalisms studied here. This means that if there were differences between electrophysiological properties obtained from MD simulations at several applied voltages and reconstructed from simulations at a single voltage it would not be possible to determine whether these differences were due to insufficient accuracy of the simulations or to inaccurate reconstruction from the new methods. For these reasons, we take a different approach.

We assume that the ED model describes ion transport with satisfactory accuracy and that the underlying PMF is known. Then, the ED equation is solved many times by way of stochastic simulations to ascertain how statistical errors depend on the number of stochastic trajectories. Even though the stochastic simulations employed here do not involve time explicitly, the number of trajectories considered in a stochastic dataset can be related to the MD simulation time. In a simple case of TTX, we demonstrate that this can be done consistently. For each set of stochastic trajectories, the PMF and the electrophysiological properties at different applied voltages are reconstructed by way of the new theoretical approaches considered here. Due to the stochastic nature of each solution to the ED equation and the limited number of trajectories in each set, which can be related to a specific simulation time, the quantities of interest obtained from each reconstruction differ between themselves and from the accurate values associated with the underlying PMF. Further, the calculated quantities also depend on the theoretical formalism used. If a sufficient number of trajectories has been generated, statistical errors on the quantities of interest can be estimated as a function of the number of trajectories or equivalently, simulation time and performance of each theoretical approach can be systematically assessed. For sets with a large number of trajectories, which corresponds to long simulation times in MD, the underlying PMFs will be reconstructed accurately. For sets with a smaller number of trajectories, the accuracy will not be as good and is expected to deteriorate as the number of trajectories is reduced. A similar systematic study cannot easily be done in practice by way of MD simulations because the computational effort to generate many MD trajectories of different length would have been prohibitive. Furthermore, no analytical method for error analysis exists for this problem.

In principle, this type of analysis can be carried out for any underlying PMF, even if it is unrelated to real ion channels. This is, however, not the direction that we pursue. Instead, we use the PMFs that we previously obtained from simulations of three actual channel models and, for the purpose of this study, assume that they are accurate. The models were selected such that they differ in size, pore structure, conductance and selectivity. The first model is TTX, which exhibits relatively high conductance, very little structure inside the pore and a weak selectivity for cations.

The second model is the high-resolution NMR structure proposed by OuYang et al. [69] for a hexameric channel p7 from the hepatitis C virus. Each subunit consists of 63 amino acids. The model has an unusual architecture not found in any other channel. The channel does not exist as a bundle of *α*-helices, which is the most common structural motif among membrane proteins, but instead forms an interlocked structure in which each subunit assumes a horseshoe conformation with each side comprised of a short, *α*-helical section. Because of these atypical features there have been concerns about the veracity of this model [28]. Recently, we calculated conductance and ionic selectivity of this model by way of MD simulations and showed that both properties differ significantly from those measured experimentally (Shannon et al., unpublished). Specifically, in contrast to the electrophysiological data, the model exhibits high conductance and strong selectivity for Cl<sup>−</sup> over K+. These results strongly suggest that the proposed model does not represent the native structure of the channel, demonstrating that computational electrophysiology can be used not only to support but also to disprove structural models of ion channels.

The third model is based on the crystal structure of a pentameric, cation-selective ion channel, GLIC, from a cyanobacterium Gloeobacter [70]. This channel is a bacterial homolog of receptors belonging to the family of pentameric ligand-gated ion channels. Its main electrophysiological characteristics are low-conductance (9.3 pS) and strong selectivity for cations [70]. Molecular models of all three channels are shown in the Supplementary Materials (SM), Section S4.

Both the p7 and GLIC models have a markedly more varied pore structure than TTX and, consequently, a more complex PMF. Although we will use the names of these three channels further in the text, we do not claim that the underlying PMFs faithfully represent the PMFs for these channels in their native open forms (this does not appear to be the case for p7) and, therefore, we do not compare the electrophysiological properties calculated from stochastic simulations with the same properties obtained experimentally. Instead, we fully concentrate on assessing the accuracy of the underlying theory.

#### **2. Theory and Method**

In this section, we briefly outline the theory behind three different approaches to calculating the PMF and electrophysiological characteristics of an ion channel. Two of them require simulations at only one applied voltage. A more detailed derivation of the basic formulas, which follows closely the earlier development [33], is provided in Supplementary Materials, Section S1. Next, we describe how the properties of interest can be obtained from stochastic simulations under the assumptions of the ED model specified in the introduction. Note that while the theory is developed in the context of MD simulations, here, we use the results of the theory to compute the PMF and I-V curves from density profiles and committor probabilities that were obtained from stochastic simulations.

#### *2.1. Calculating the Potential of Mean Force*

If the concentrations of ions on both sides of the membrane and the applied voltage remain constant in time, the system is in a steady state, which means that the flux of ions through the channel, *J*, is also constant in time. These are the conditions most often considered in both experiments and simulations aimed at extracting electrophysiological properties of channels. Then, the one-dimensional ED equation for a given type of ions can be written as

$$J = -D(z)\left(\frac{d\rho(z)}{dz} + \beta\rho(z)\frac{dE(z)}{dz}\right),\tag{1}$$

where *D*(*z*) is the diffusivity that, in general, depends on position *z* along the reaction coordinate **z**. For a transmembrane channel embedded in the membrane located in the x,y-plane, a convenient reaction coordinate is the position of an ion along the pore of the channel, which can be measured along the z-coordinate. *ρ*(*z*) is the line density of ions, which is usually recorded as a histogram in computer simulations. *β* = *kBT*, where *k<sup>B</sup>* is the Boltzmann constant and *T* is temperature. *E*(*z*) is given by

$$E(z) = A(z) + qV(z),\tag{2}$$

where *A*(*z*) is the PMF, *V*(*z*) is the applied voltage and *q* is ionic charge. In a constant electric field, E *el*, acting along **z**, which is the most frequent experimental condition,

$$V(z) = \mathcal{E}^{\varepsilon l}(z - z\_a). \tag{3}$$

Even though the electric field is applied across the whole system [15,30], it acts only between *z<sup>a</sup>* and *z<sup>b</sup>* in the non-polar phase, which has been identified as corresponding to the hydrophobic core of the membrane [25,33]. Thus, electric field is a boxcar function that is equal to E *el* in the range [*za*, *z<sup>b</sup>* ] and zero otherwise. This can be formally written as E *el*[*H*(*<sup>z</sup>* <sup>−</sup> *<sup>z</sup>a*) <sup>−</sup> *<sup>H</sup>*(*<sup>z</sup>* <sup>−</sup> *<sup>z</sup><sup>b</sup>* )], where *H* is the Heaviside function. Although we will not use this notation for simplicity, the range in which E *el* is non-zero has to be kept in mind.

Integrated with the integrating factor exp[*βE*(*z*)] and resolved for *J*, the ED equation takes the form

$$J = \frac{\rho(z\_{\min}) \exp[\beta E(z\_{\min})] - \rho(z\_{\max}) \exp[\beta E(z\_{\max})]}{\int\_{z\_{\min}}^{z\_{\max}} \frac{\exp[\beta E(z)]}{D(z)} dz}. \tag{4}$$

For a system in a steady state, *J* does not formally depend on the limits of integration *zmin* and *zmax*. This means that these limits do not have to coincide with the edges of the channel. In practice, the limited precision of MD simulations introduces some dependence on the limits of integration, as analyzed elsewhere [25].

To calculate *J* from this equation, *E*(*z*) has to be known, which in turn requires determining *A*(*z*). This can be done in equilibrium simulations in the absence of voltage. A host of methods exist for this purpose [71–73]. *A*(*z*) can be also calculated from nonequilibrium simulations at an applied voltage. If the ED equation is integrated with the integrating factor 1/*ρ*(*z*) then

$$J = \frac{\ln \frac{\rho(z\_{\text{min}})}{\rho(z\_{\text{max}})} - \beta \left[ A(z\_{\text{max}}) - A(z\_{\text{min}}) + q \mathcal{E}^{el}(z\_{\text{max}} - z\_{\text{min}}) \right]}{\int\_{z\_{\text{min}}}^{z\_{\text{max}}} \frac{1}{D(z)\rho(z)} dz}. \tag{5}$$

Since *J* is independent of the limits of integration, *zmax* can be substituted by *z*. After simple rearrangements, it yields a formula for the PMF relative to its value at *zmin*, ∆*A*(*z*, *zmin*) = *A*(*z*) − *A*(*zmin*)

$$
\Delta A(z, z\_{\rm min}) = -k\_B T \left[ \ln \frac{\rho(z)}{\rho(z\_{\rm min})} + \int\_{z\_{\rm min}}^z \frac{1}{D(z')\rho(z')} dz' \right] - q \mathcal{E}^{el}(z - z\_{\rm min}).\tag{6}
$$

We call this method for determining PMF the Integrated Electrodiffusion Equation Method (IEEM).

To solve Equations (5) and (6), *D*(*z*) has to be known in the full range of *z*. *D*(*z*) can be determined by way of calculating the mean square displacement of the ion at several points along the channel obtained from a series of short MD trajectories after subtracting the PMF [19], from the force-force autocorrelation function acting on a stationary ion at different positions in the channel [74], or by way of a Bayesian fitting method [75–77]. See Supplementary Materials, Section S5 for a discussion of how diffusivity was computed in our MD simulations.

Once ∆*A*(*zmax*, *zmin*) and *D*(*z*), which are both assumed to be independent on voltage, are known, the boundary density terms *ρ*(*zmin*) and *ρ*(*zmax*) have to be obtained from either MD or stochastic simulations at each voltage of interest. Since the full knowledge of *ρ*(*z*) is not needed, these simulations can be markedly shorter than simulations to determine the PMF. Then, ∆*A*(*zmax*, *zmin*), *D*(*z*), *ρ*(*zmin*) and *ρ*(*zmax*) are used in Equation (5) to calculate *J* at a given voltage. Previously, we demonstrated that this method performs satisfactorily for simple channels [19,24,25].

Recently, we developed two alternative approaches to calculating the PMF and electrophysiological properties that require markedly less computational effort [33]. Both rely on separating the total ionic current, *J*, to currents moving in two opposite directions – from *zmin* to *zmax* and from *zmax* to *zmin*. We abbreviate them *J <sup>f</sup>* and *J <sup>b</sup>* and call them forward and backward currents, respectively.

$$J^f = \frac{\rho^f(z\_{\min}) \exp[\beta E(z\_{\min})] - \rho^f(z) \exp[\beta E(z)]}{\int\_{z\_{\min}}^z \frac{\exp[\beta E(z')]}{D(z')} dz'},\tag{7}$$

$$J^b = \frac{\rho^b(z) \exp[\beta E(z)] - \rho^b(z\_{\rm min}) \exp[\beta E(z\_{\rm min})]}{\int\_{z\_{\rm min}}^z \frac{\exp[\beta E(z')]}{D(z')} dz'},\tag{8}$$

$$J = J^f - J^b. \tag{9}$$

Here, *ρ f* (*z*) and *ρ b* (*z*) are densities of ions that entered the range [*zmin*, *zmax*] at *zmin* and *zmax*, respectively. We assume that both forward and backward currents are in a steady state and, therefore, their values do not depend on the limits of integration. This allows for setting the upper limit to *zmin* < *z* ≤ *zmax*.

Assume that *J <sup>f</sup>* > 0 and *J <sup>b</sup>* > 0 and take the ratio of Equation (7) to Equation (8). This yields

$$\frac{J^f}{J^b} = \frac{\rho^f(z\_{\rm min}) - \rho^f(z) \exp[\beta \Delta E(z, z\_{\rm min})]}{\rho^b(z) \exp[\beta \Delta E(z, z\_{\rm min})] - \rho^b(z\_{\rm min})},\tag{10}$$

where

$$
\Delta E(z, z\_{\rm min}) = E(z) - E(z\_{\rm min}).\tag{11}
$$

Combined with Equations (2) and (3), Equation (10) can be solved for ∆*A*(*z*, *zmin*)

$$
\Delta A(z, z\_{\rm min}) = -k\_B T \ln \frac{\mathcal{J}^b \rho^f(z) + \mathcal{J}^f \rho^b(z)}{\mathcal{J}^b \rho^f(z\_{\rm min}) + \mathcal{J}^f \rho^b(z\_{\rm min})} - q \mathcal{E}^{\varepsilon l} (z - z\_{\rm min}). \tag{12}
$$

From this equation it follows that the PMF can be obtained from non-equilibrium simulations at applied electric field E *el* simply from an average of ion densities in the forward and backward directions weighed by the backward and forward currents, respectively. We call this method for calculating the PMF the Current-Weighted Density Method (CWDM). Knowledge of diffusivity is not necessary in CWDM. The denominator in the argument of the logarithmic function sets the reference value of the PMF at *zmin*.

If we abbreviate the number of crossing events in forward and backward direction as *n <sup>f</sup>* and *n b* , respectively, then, assuming that crossing events are governed by the Poisson statistics, the corresponding errors will be approximately 1/ p (*n f* ) and 1/ p (*n b* ). This means that if *n <sup>f</sup>* or *n b* is small, ∆*A*(*z*, *zmin*) calculated from Equation (12) may become inaccurate. Thus, we developed another, related theoretical approach for determining the PMF from non-equilibrium simulations that does not suffer from this disadvantage. Since it requires calculating committor probability, *P*(*z*), we will call it the Committor Probability Method (CPM). For a diffusive process considered here, *P*(*z*) referenced to the forward direction is defined as the probability that a particle (ion) in position *z* will reach *zmax* before it reaches *zmin*. *P*(*z*) can be calculated either directly during computer simulations or in post-processing, as described in Supplementary Materials, Section S3. A general discussion of committor probabilities in more than one dimension and their application to chemical kinetics can be found elsewhere [78–81].

The PMF can be calculated from ion densities in the forward or the backward direction. The corresponding formulas are

$$\exp\left[\beta\Delta E(z, z\_{\min})\right] = \frac{\rho^f(z\_{\min})[1 - P(z)]}{\rho^f(z) - \rho^f(z\_{\max})},\tag{13}$$

$$\exp\left[\beta\Delta E(z, z\_{\rm min})\right] = \exp\left[\beta\Delta E(z\_{\rm max}, z\_{\rm min})\right] \frac{\rho^b(z\_{\rm max})P(z)}{\rho^b(z) - \rho^b(z\_{\rm min})}.\tag{14}$$

Their derivation closely follows our earlier work [33] and is given in Supplementary Materials Section S1.

Both equations allow for calculating the same quantity— the PMF. Individually, each of them is not expected to be accurate in the full [*zmin*, *zmax*] range of *z*, especially away from the entry point. Specifically, as *z* becomes close to *zmax* both *ρ f* (*z*) − *ρ f* (*zmax*) and 1 − *P*(*z*) approach zero. Since numerical inaccuracies in Equations (13) and (14) affect mainly the opposite sides of the [*zmin*, *zmax*] range, these two equations can be profitably combined. Then, *ρ f* (*z*) − *ρ f* (*zmax*) and *ρ b* (*z*) − *ρ b* (*zmin*) can be considered as two biased distributions representing the same unbiased distribution *h*(*z*). The problem of merging them to reconstruct *h*(*z*) such that statistical error on ∆*A*(*z*, *zmin*) is minimized can be solved by way of the Weighted Histogram Analysis Method (WHAM) [82]. This yields the following formula for reconstructing the PMF from non-equilibrium simulations:

$$\Delta A(z, z\_{\rm min}) = \mathbb{C} - k\_B T \ln \left[ \frac{h(z)}{\rho^f(z\_{\rm min})(1 - P(z))P(z)} \right] - q \mathcal{E}^{\varepsilon l}(z - z\_{\rm min}), \tag{15}$$

where neither *C*, which is a constant that only shifts the energy scale, nor *ρ f* (*zmin*), which is independent of *z* and is needed to ensure that the PMF at *z* = *zmin* is equal to zero, influences the shape of ∆*A*(*z*, *zmin*). Similarly to Equation (12), no knowledge of diffusivity is required.

Typically, MD simulations would be carried out on the channel system of interest at some applied voltage *V*. From this, the committor probability, *P*(*z*) and the 1-sided density profiles, *ρ f* (*z*) and *ρ b* (*z*), and the number of forward and backward crossing events would be determined, and used to calculate the forward and backward fluxes, *J f* and *J b* , respectively. The PMF can be determined from either CWDM (Equation (12)) or CPM (Equation (15)). As will be discussed later, we created synthetic data sets of 106–10<sup>8</sup> stochastic trajectories. For each data set, we calculate the same quantities that would be calculated in MD, *P*(*z*), *ρ f* (*z*) and *ρ b* (*z*), and then use these to calculate the PMF. Note that the free energy depends on ratios of density profiles, so the absolute normalization of the density profiles is not important. Similarly, the CWDM requires only ratios of the forward and backward currents, so the magnitudes are not required.

#### *2.2. Calculating I-V Dependence from Simulation at a Single Voltage*

If the PMF, the current, *Jµ*, and the density, *ρµ*(*z*), or the committor probability, *Pµ*(*z*), are known from simulations at an applied voltage, ∆*Vµ*, the current, *Jν*, at a different voltage ∆*V<sup>ν</sup>* can be obtained without any calculations at this voltage. This allows for reconstructing the I-V curve from simulations at a single voltage.

If Equation (1) is integrated with the same integrating factor, exp[*βEν*(*z*)], for both voltages, ∆*V<sup>µ</sup>* and ∆*Vν*, we obtain

$$\begin{split} J\_{\mu} &= \\ \frac{\int \rho\_{\mu}(z\_{\min}) \exp\left[\beta E\_{\nu}(z\_{\min})\right] - \rho\_{\mu}(z\_{\max}) \exp\left[\beta E\_{\nu}(z\_{\max})\right] + \beta q(\mathcal{E}\_{\nu}^{\mathrm{el}} - \mathcal{E}\_{\mu}^{\mathrm{el}}) \int\_{z\_{\min}}^{z\_{\max}} \rho\_{\mu}(z) \exp\left[\beta E\_{\nu}(z)\right] dz}{\int\_{z\_{\min}}^{z\_{\max}} \frac{\exp[\beta E\_{\nu}(z)]}{D(z)} dz} \end{split} \tag{16}$$

and

$$J\_{\nu} = \frac{\rho\_{\nu}(z\_{\min}) \exp[\beta E\_{\nu}(z\_{\min})] - \rho\_{\nu}(z\_{\max}) \exp[\beta E\_{\nu}(z\_{\max})]}{\int\_{z\_{\min}}^{z\_{\max}} \frac{\exp[\beta E\_{\nu}(z)]}{D(z)} dz} \tag{17}$$

The latter but not the former equation is the standard integrated form of the ED equation, Equation (4).

If we take the ratio of currents *Jµ*/*J<sup>ν</sup>* then, after some algebra given in Supplementary Materials, Section S2 we obtain

$$\frac{J\_{\mu}}{J\_{\nu}} = 1 + \beta q (\mathcal{E}\_{\nu}^{\mathrm{el}} - \mathcal{E}\_{\mu}^{\mathrm{el}}) \int\_{z\_{\mathrm{min}}}^{z\_{\mathrm{max}}} f\_{\mu}(z) \exp\{\beta q \left[V\_{\nu}(z) - V\_{\mu}(z)\right]\} dz,\tag{18}$$

where

$$f\_{\mu}(z) = \exp\left[\beta \Delta E\_{\mu}(z, z\_{\min})\right] \frac{\rho\_{\mu}(z)}{\rho\_{\mu}(z\_{\min})}.\tag{19}$$

or

$$f\_{\mu}(z) = 1 + \exp\left[\beta \Delta E\_{\mu}(z, z\_{\rm min})\right] \frac{\rho\_{\mu}(z\_{\rm max})}{\rho\_{\mu}(z\_{\rm min})} - P\_{\mu}(z), \tag{20}$$

depending on whether it is preferred to calculate *J<sup>ν</sup>* from ion density, *ρµ*(*z*), or committor probability, *Pµ*(*z*). In both instances, neither diffusivity nor quantities at the applied voltage ∆*V<sup>ν</sup>* are needed. Equation (19) is expected to be less accurate that Equation (20) because ∆*Eµ*(*z*, *zmin*) and *Pµ*(*z*) that enter the latter equation are estimated on the basis of both forward and backward simulations, whereas *ρµ*(*z*) in the former equation is a one-sided density that looses accuracy away from the entry point.

Unlike the free energy, Equation (18) gives only ratios of forward or backward currents with respect to a reference voltage. Consequently, to calculate the I-V curves, we need currents at this reference voltage.

#### *2.3. Stochastic Simulations*

The electrodiffusion equation was solved by generating trajectories on a free energy surface *E*(*z*) that included the PMF and applied electric field with diffusivity *D*(*z*) or average diffusion coefficient, < *D* >, at temperature *T* [83,84]. This allowed us to generate the channel crossing statistics, density profiles and committor probabilities for the ions for this free energy surface. As the crossing events that we have observed in MD simulations appear to obey Poisson statistics, independently for both ions, we consider the ED equation for each ion separately. Then, we calculated statistical errors in recovering the underlying PMF and the I-V curves as functions of the number of trajectories.

As above, we define the channel boundaries as *zmin* and *zmax*, and absorbing boundary conditions were located at these points. Trajectories were initiated at a point just inside the boundary at either *zmin* for forward trajectories or *zmax* for backward trajectories, and propagated until they reached either of the absorbing boundaries. Forward and backward trajectories are considered separately as we are interested in the 1-sided density profiles and committor probabilities, as well as their 2-sided combination. A trajectory that crossed from *zmin* to *zmax* is said to be a crossing trajectory in the forward direction. Similarly, trajectories that cross from *zmax* to *zmin* are crossing trajectories in the backward direction. For simplicity, these will be referred to as forward or backward crossing events. Since the trajectories are initiated near the absorbing boundaries, the majority of trajectories in either direction do not cross, but they do contribute to the 1-sided density profiles and firstpassage statistics that are used to compute the committor probabilities (see Supplementary Materials, Section S3 for further details).

The number of trajectories initiated per data set were typically 10<sup>6</sup> , 10<sup>7</sup> or 10<sup>8</sup> , further abbreviated as N6, N7 and N8, respectively. These numbers were chosen because the average number of crossing events for PMFs corresponding to the models of TTX and p7 observed for *N* = 10<sup>6</sup> is of the same order of magnitude as the numbers of crossing events observed in our MD simulations of 0.5–2 µs. For the cation-selective GLIC channel, in which the free energy barrier to permeation of Na<sup>+</sup> is markedly higher, simulations with 10<sup>6</sup> trajectories yielded too few crossing events to be useful. In this case, the number of crossing events observed at the N7 level approximately corresponds to the number seen in MD simulations of 8 µs. See Supplementary Materials, Section S4 for details of the MD simulations.

The free energy surfaces for the stochastic simulations were obtained by adding the voltage ramp to the PMFs. We use a set of PMFs from our MD calculations. For our problem, the PMFs are the equilibrium free energy surfaces for moving an ion along the 1-dimensional reaction coordinate of the ion with respect to the center-of-mass of the protein channel, at the bulk ion densities of the MD. For this study, we used average diffusion coefficients, < *D* > obtained by averaging the diffusivities estimated from MD (Supplementary Materials, Section S4). Strict matching of diffusivity is not necessary for the primary purpose of this study, but it provides a more realistic connection between statistical errors estimated in stochastic simulations and time scales of MD simulations. Additional details are given in Supplementary Materials, Section S7. The forward and backward ion density profiles were obtained from histograms of either the forward or backward trajectories in each data set. Committor probabilities (Supplementary Materials, Section S3) were calculated from the first passage statistics of the forward and backward trajectories. The density histograms and committor probabilities were computed for each data set, and not as an average over the individual trajectories in the data set. Averages were constructed over multiple data sets.

#### **3. Results and Discussion**

#### *3.1. Connection with Molecular Dynamics*

To compute the currents for the I-V curves from stochastic simulations, some connection to MD is required. MD simulations provide both forward and backward ion trajectories as part of the simulation, unless the channel is strongly rectifying or a large voltage is applied. The net current due to a particular ion is *J* = *J <sup>f</sup>* <sup>−</sup> *<sup>J</sup> b* (Equation (9)). The total current is the sum of these net currents over all types of ions. As mentioned in the introduction, *J* can be obtained from MD simulations by way of combining the fluxes from forward and backward crossing events or calculating the ionic displacement currents. In stochastic simulations, only the former method can be used. Therefore, we tested whether both method yield the same results for MD and found that this was indeed the case. Both methods and the results of the tests are described in Supplementary Materials, Section S6.

In addition, the detailed balance condition connecting forward and backward crossing events has to be satisfied. In MD simulations this problem is implicitly solved: if there is no external voltage, simulations of transmembrane systems will exhibit no net current to within statistical errors, which means that the number of forward and backward crossing events is equal, again to within statistical errors. In stochastic simulations, detailed balance also has to be satisfied, which means that trajectories in both directions have to be combined with the correct weights. To determine these weights, we carried out sets of 10<sup>8</sup> simulations with no applied voltage to obtain the well converged, average numbers of forward and backward crossing events. From these simulations, the ratio of forward to backward trajectories that satisfies the detailed balance condition was established and subsequently used to compute the density profiles, committor probabilities and the PMFs at different voltages.

Once the ratio of forward to backward trajectories needed to satisfy the detailed balance condition is known, the average numbers of crossing events in both directions, *n f* (∆*V*) and *n b* (∆*V*), can be obtained from stochastic simulations at a given voltage ∆*V*. This, however, is still insufficient to determine currents; additional information about time scales is required. This can be obtained from a MD simulation of the system. We abbreviate the number of forward and backward crossing events observed in MD simulations at applied voltage, ∆*Vre f* , as *m<sup>f</sup>* and *m<sup>b</sup>* . Then, the length of the MD trajectory, *tMD*, can be used to estimate a stochastic time, *tS*, corresponding to the number of stochastic trajectories that produced *n f* (∆*Vre f*) and *n b* (∆*Vre f*) crossing events at the voltage ∆*Vre f* . A simple way to make such estimate is to use the number of crossing events in one direction. It is, of course, recommended to choose the direction that provides better statistics. Assuming that there are more forward than backward crossing events in the MD simulations,

$$t\_{\mathcal{S}} = t\_{MD} \frac{n^f (\Delta V\_{ref})}{m^f}$$

.

If the backward events dominate, *t<sup>S</sup>* would be estimated using *n b* (∆*Vre f*) and *m<sup>b</sup>* . Once *t<sup>S</sup>* has been determined, the stochastic currents at voltage ∆*V* can be calculated:

$$J\_{\mathbb{S}}(\Delta V) = f\_{\mathbb{S}}^{f}(\Delta V) - f\_{\mathbb{S}}^{b}(\Delta V) = [n^{f}(\Delta V) - n^{b}(\Delta V)]/t\_{\mathbb{S}}.$$

where *JS*(∆*V*), *J f S* (∆*V*) and *J b S* (∆*V*) are the total, forward and backward currents at voltage ∆*V*.

#### *3.2. Committor Probabilities*

The committor probabilities for p7 are shown in Figure 1. The committor probabilities in Figure 1a have been calculated from Supplementary Materials, Equation S30, in which ions arriving at *z* from both sides are included. The statistical errors associated with *P*(*z*) at different voltages are small, even at the N6 level. As can be seen in Figure 1b, this is not the case for one-sided *P*(*z*), obtained using Supplementary Materials, Equations

S28 and S29. This is due to the decreasing number of ions from one direction as they approach the opposite side of the channel. The inset of Figure 1b shows the numbers of first-passage events from the forward and backward calculations as well as the combined number of events. At 140 mV, the statistics are satisfactory only in the forward direction, in which most of crossing events occur, whereas no reliable probabilities are obtained for the backward direction over the full range of *z*. The opposite is true for −140 mV; *P*(*z*) in the forward direction is unreliable. Thus, combining information about *P*(*z*) in both directions is preferable whenever possible.

As voltage changes from −140 mV to 140 mV, the position of the transition state for K<sup>+</sup> permeation through p7, defined as the x,y-plane at which *P*(*z*) = 0.5, Ref. [85] shifts substantially and systematically from 7 Å to −13 Å with respect to the center of mass of the membrane. Such large shifts, however, are not universal. As we have shown in the example of TTX [33], the position of the transition state changes markedly less with voltage if the underlying PMF is strongly peaked.

**Figure 1.** (**a**) Committor probabilities for Cl<sup>−</sup> in p7 at −140 mV (red), −70 mV (green), −35 mV (black), 0 V (blue), 70 mV (cyan) and 140 mV (magenta). Error bars are shown for the N6 data sets at −35 mV and 140 mV; (**b**) Committor probabilities for p7 at 140 mV from the N6 data set for 1-sided forward (green) and backward (blue) trajectories, respectively, 2-sided data set in the backward direction (red lines), and average in the forward direction with error bars (red symbols). In the inset we show the number of first passage trajectories to reach *z* for one N6 data set in the forward (green) and backward (magenta) directions and the total (light blue).

Calculating *P*(*z*) for GLIC is more difficult. This is a slow channel and even at the N7 level, which approximately corresponds to a MD trajectory of 8 µs in length (see Supplementary Materials, Section S4, the number of crossing events is small. At 100 mV only an average number of 0.5 forward and 29 backward crossing events were observed. In particular, N7 simulations of forward trajectories frequently produce no crossing events. At the same voltage, *P*(*z*) in the backward direction is often equal to 1 over a relatively wide range of several Å near *zmax*, which means that all ions that reached this range exit the channel at *zmax*. In such circumstances, calculation of *P*(*z*) from Supplementary Materials, Equation S30 is no longer possible. A different approach is needed.

Direct calculation of the committor probability requires that some number of trajectories successfully cross the channel, *N<sup>b</sup>* (*zmin*) > 0 and *N<sup>f</sup>* (*zmax*) > 0. If one of these conditions is not met, for example, if *N<sup>f</sup>* (*zmax*) = 0, then *P f* (*z*) = *N<sup>f</sup>* (*zmax*)/*N<sup>f</sup>* (*z*) = 0. If we consider position *z* 0 (*z* <sup>0</sup> < *zmax*) at which *N<sup>f</sup>* (*z* 0 ) > 0, then we can write the committor probability *P*(*z*) = *αN<sup>f</sup>* (*z* 0 )/*N<sup>f</sup>* (*z*), where *α* is unknown, though formally would be equal to *N<sup>f</sup>* (*zmax*)/*N<sup>f</sup>* (*z* 0 ) if complete sampling of the forward direction were available. *α* can be determined in a self-consistent manner. Using Equation S30 from Supplementary Materials, Section S3, we can write the total committor probability in the region *z* < *z* 0 and the backward committor probability *z* > *z* 0 :

$$P(z) = \begin{cases} \frac{aN^f(z') + N^b(z) - N^b(z\_{\min})}{N^f(z) + N^b(z)} & \text{if } z < z'\\ 1 - \frac{N^b(z\_{\min})}{N^b(z)} & \text{if } z > z'. \end{cases}$$

If we require that *P*(*z*) is continuous at *z* = *z* 0 , then *<sup>α</sup>* = 1 <sup>−</sup> *<sup>N</sup><sup>b</sup>* (*zmin*)/*N<sup>b</sup>* (*z* 0 ). Other ways of determining *P*(*z*) for this problem are possible.

#### *3.3. The Potential of Mean Force*

Typically, the PMFs for ions in channels are calculated in simulations in the absence of electric field using enhanced sampling techniques (see the recent review by Flood, et al. [27]). In contrast, the methods outlined here allow for reconstructing PMF from steady-state simulations with an applied electric field. The underlying PMFs for p7 and GLIC used in the present study were obtained by way of this method (see Supplementary Materials, Section S1). Since TTX is a bundle of straight *α*-helices surrounding a featureless water pore, the PMFs for K<sup>+</sup> and Cl<sup>−</sup> are quite generic, which is characteristic of several very simple channels (see Figure 2a) [24,25]. For K+, the PMF is fairly flat over a wide range of approximately 18 Å inside the channel, which is reminiscent of classical models of ionic conductance in which it is assumed that the PMF is a step function constant inside the channel [1,86]. For Cl−, the PMF is peaked near the center of the bilayer, which can be attributed to the Born barrier experienced by an ion permeating a rigid, featureless non-polar lamella [87]. If an ion is transferred across a membrane through a water-filled pore, the general shape of the PMF remains the same, but the barrier is substantially reduced [87,88]. For TTX, it still remains approximately 1.5 kcal/mol higher than the barrier for K+, which is consistent with a weak selectivity of this channel toward cations.

The PMF for permeation of Cl− in the OuYang et al. model of p7 [69] is more structured than the PMF for TTX (see Figure 3a). The barriers are low, which explains high chloride current predicted by this model [37]. In contrast to TTX, the barriers to Cl− permeation in p7 are located near the mouths of the channel due to the presence of positively charged residues at these locations. Compared to permeation of Cl−, the current of K<sup>+</sup> in this model is quite low, which indicates that the channel should be anion-selective. Both predicted selectivity and total currents are at variance with electrophysiological data [89], thus contributing to arguments that the proposed high-resolution structure [69] is not native.

The PMF representing permeation of Na<sup>+</sup> through GLIC is also markedly more structured than the PMF for ions in TTX (see Figure 2b). The barrier is substantially higher than in the other two channels. As a result, the conductance of this channel is relatively low [70]. This presents a challenge because the number of crossing events in both directions is small. The PMF for Cl− in this channel is not considered because no crossing events of this ion have been observed in MD simulations.

**Figure 2.** (**a**) PMFs for K<sup>+</sup> (lower curves) and Cl<sup>−</sup> (upper curves) in TTX from stochastic simulations with an applied voltage of 50 mV. The PMFs have been reconstructed by way of CWDM at the N6 (blue) and N7 (gold) levels or by way of CPM at the N6 (green) and N7 (magenta) levels; (**b**) PMF for Na<sup>+</sup> in GLIC from stochastic simulations with applied voltage of 100 mV. The PMF has been reconstructed by way of CWDM at the N7 (blue) and N8 (gold) level or by way of CPM at the N7 (green) and N8 (magenta) level. In both panels, the underlying PMF is in red.

Taken together, the PMFs considered here are quite different from one another, but are typical of the variety seen in ion channels. In spite of these differences, all three PMFs were successfully reconstructed from non-equilibrium simulations by way of

Equations (12) and (15) associated, respectively, with the CWDM and CPM methods. The applied voltages were 50 mV for TTX, 140 and −35 mV for p7 and 100 mV for GLIC. For TTX and p7, reconstruction was carried out at the N6, N7 and N8 levels. For GLIC, the number of crossing events at the N6 level was quite small or equal to zero. Thus, only N7 and N8 levels were considered. At the N6 level, 50 and 100 data sets of trajectories were generated for TTX and p7, respectively. At the N7 level, 20 sets of trajectories were generated for TTX and p7, and 50 sets were generated for GLIC. At the N8 level, the number of generated sets was 4, 8 and 25 for TTX, p7 and GLIC, respectively. At each level, all reconstructed PMFs are found to be tightly clustered and their averages at each level are close to the underlying PMF, as shown in Figures 2 and 3a.

**Figure 3.** (**a**) PMF for Cl− in p7 from stochastic simulations with an applied voltage of 140 mV. The PMFs have been reconstructed by way of CWDM at the N6 (blue) and N7 (gold) levels or by way of CPM at the N6 (green) and N7 (magenta) levels. The input PMF (red) is shown for reference. PMFs at the N8 level are not shown, as they coincide with the underlying PMFs and statistical errors associated with this level arequite small and are poorly visible at this scale; (**b**) PMFs for P7 reconstructed by way of one-sided forward trajectories (green) using Equation (13) and backward trajectories (blue) using Equation (14) from stochastic simulations at the N6 level with applied voltage of 140 mV. Two-sided reconstruction (magenta) and the underlying PMF (red) are shown for comparison. Note that one-sided, but not two-sided reconstructions are burdened with large errors at the ends.

In these figures, statistical errors associated with dispersion of the reconstructed PMFs are marked. For ∆*A*(*zmax*, *zmin*), these errors are approximately ±0.3 kcal/mol at the N6 level for TTX and p7 and at the N7 level for GLIC. As expected, they are reduced by approximately a factor of 3 with each level in which the number of sets increases by an order of magnitude. The mean PMFs obtained by way of CWDM and CPM at different levels are quite close to the underlying PMF and the corresponding statistical errors are very similar, indicating that both methods are successful in reproducing the underlying PMFs. Only for GLIC at the N7 level, does the ∆*A*(*zmax*, *zmin*) reconstructed by way of CWDM appear to be systematically underestimated. At this level, no crossing events in one direction are observed for a considerable fraction of data sets, which makes reconstruction of the PMF from Equation (12) impossible. This systematically biases the sample in favor of sets with higher counts of crossing events and, consequently, lower ∆*A*(*zmax*, *zmin*). From the comparison between the PMFs reconstructed for p7 from trajectories at 140 and −35 mV, it appears that precision of the reconstruction depends somewhat on applied voltage. If the forward and backward densities are well balanced, precision improves.

In CPM, the PMFs can be calculated from one-sided quantities, Equations (13) and (14), or by combining them. Here, the latter has been done by way of WHAM, Equation (15). As can be seen in Figure 3b, this approach yields improved agreement with the underlying PMF. In one-sided formulas, the densities can become quite low near the exit and, as a result, precision in this range suffers.

In summary, both CWDM and CPM provide a reliable means for reconstructing PMFs from non-equilibrium simulations. However, the relation between statistical errors obtained in stochastic and MD simulations is not straightforward. Even if the assumptions of the ED model are satisfied, precision of stochastic simulations is expected to be higher than precision in MD simulations of equivalent length. Specifically, it is usually uncertain if all degrees of freedom perpendicular to the reaction coordinate have been properly equilibrated on the time scale of the simulations. Torsional angles in the side chains of residues lining the pore or motion of whole helices are examples of degrees of freedom that might undergo slow equilibration and, by doing so, influence the calculated PMF and electrophysiological properties. The same concern applies to all other methods for calculating these quantities.

#### *3.4. Current-Voltage Dependence*

Once the PMFs for the ions permeating the channel have been reconstructed and the committor probabilities for these ions have been calculated for a reference voltage, the full current-voltage (I-V) curves can be calculated from Equations (19) and (20) without the need for additional simulations. This is the principal gain in efficiency of the method: the I-V curve can be obtained from a single MD simulation instead of multiple simulations. For example, if constructing the I-V curve required MD simulations at five different voltages in the range of interest the efficiency of our methods would be approximately five-fold. Since numerical results indicate that Equation (19) yields less accurate results than Equation (20), this equation will not be further considered. The results for TTX, p7 and GLIC are shown in Figures 4 and 5. The reference applied voltages are the voltages used for reconstructing PMFs, described in the previous subsection. For comparison, currents calculated directly from stochastic trajectories at several voltages are also shown.

**Figure 4.** (**a**) I-V curves for K<sup>+</sup> (green) and Cl<sup>−</sup> (blue) in TTX reconstructed from simulations at 50 mV at the N6 level. Blue and green dots are currents obtained from direct simulations at specific voltages.; (**b**) I-V curves for Cl− in p7 reconstructed from simulations at 140 mV at the N6 (blue), N7 (green) and N8 (red) level, and for −35 mV at the N6 level (magenta). N7 and N8 curves are not shown because they are almost identical to the N6 results. Black dots are currents obtained from direct simulations at specific voltages. All reconstructions were done using the PMFs obtained by way of CPM. The results of reconstructions using the PMFs from CWDM are not displayed because they are nearly identical.

As we can see in Figure 4, the agreement between the I-V curves for both K<sup>+</sup> and Cl<sup>−</sup> in TTX calculated directly and by way of Equations (18) and (20) is excellent, even at the N6 level, for the full range of voltages studied here, which extends from −100 to 100 mV. As shown in Figure 6, the I-V curves obtained for different sets of trajectories are closely clustered and deviate from each other only at the largest absolute applied voltages by no more than a few pA.

**Figure 5.** I-V curves for Na<sup>+</sup> in GLIC reconstructed from simulations at 100 mV at the N7 level with PMF from CPM (blue), at the N7 level with PMF from CWDM (magenta), and N8 with PMF from CWDM (red). N8 with CPM (not shown) is almost identical to N7 CPM. Black dots are currents obtained from direct simulations at specific voltages.

**Figure 6.** Reconstructions of I-V curves in TTX from individual sets of trajectories for K<sup>+</sup> (**a**) and Cl− (**b**). The PMFs were obtained from CPM (upper panels) or CWDM (lower panels). The curves were calculated by way of Equation (20) (blue) or Equation (18) (green). All reconstructions were carried out from simulations at applied voltage of 50 mV at the N6 level. Note that blue curves, but not green curves, are tightly clustered together indicating that Equation (20) is more accurate than Equation (18).

For p7, the agreement is not as good if the the reference voltage of 140 mV is used for calculating the I-V curve. The Cl− currents calculated directly and from Equation (18) agree well for positive voltages, but diverge for negative voltages, away from the reference state. The corresponding statistical errors also increase and become quite large below −50 mV. The source of this disagreement can be traced to the integrand in Equation (18). As the difference between the reference and the target voltage increases, the exponential term also increases, which magnifies inaccuracies in function *f*(*z*). If the reference voltage is chosen to be −35 mV, the agreement over the full range of voltages −150 to 150 mV improves markedly, with modest deviations only at high, positive voltages (see Figure 4). A similar situation was observed for GLIC. For the reference voltage of 100 mV, the I-V curves at the N7 level satisfactorily reproduce currents calculated directly for positive voltages. For negative voltages, the performance of the method progressively deteriorates. Again, if one is interested in an I-V curve that extends to both positive and negative voltages, a different choice of reference voltage may yield significant improvements in accuracy.

As pointed out in the introduction, a number of previous studies have used unrealistically large applied voltages to increase the number of crossing events and, by doing so, improved precision of the calculated currents [15,23,35,37]. Furthermore, as discussed earlier, this may lead to electroporation of the membrane, saturation effects during the intake of ions at the mouth of the channel and involves extrapolation or interpolation to the voltages by way of *ad hoc* procedures of unknown accuracy. The approach developed here is more efficient and accurate and has a substantially stronger theoretical basis than procedures used previously, even though only calculations at the reference applied voltage are necessary. In this approach, the accuracy of the reconstructed I-V curves can be substantially improved through a judicious choice of this reference voltage. This choice

depends on the range of voltage that is of interest and on several properties of a channel, in particular its rectification, which characterizes an asymmetry of currents in response to the change in direction of applied voltage. In general, maximizing total ion current through applying high voltage is not the optimal strategy. Instead, it is often better to choose a voltage that yields good statistics in both directions.

#### *3.5. Reversal Potential*

The reversal potential, ∆*VR*, is the applied voltage at which there is no net current. If ionic concentrations on both sides of the membrane are equal, ∆*V<sup>R</sup>* = 0. Experimentally, the reversal potential is measured by maintaining different concentrations on the *cis* and *trans* side of the membrane, and then used in conjunction with the GHK equation to estimate channel selectivity [1,6]. In MD, asymmetric concentrations have to be maintained to measure directly the reversal potential [90], which markedly complicates simulations. We have only considered the situation where the concentrations of ions are the same on both sides of the membrane, as this corresponds to the conditions under which we have carried out MD. We wish to expand this to a range of concentrations.

We expect that the net number of crossing events, from which we calculate the I-V curve, depends on this concentration. If the bulk concentrations are low and the channel is not saturated, then we expect the number of crossing events, and hence the currents, to be linearly dependent on the concentrations of ions. For example, if the concentration is doubled on both sides of the membrane, the net currents will also double. Under these assumptions, we can calculate the reversal potential from our formalism. We simply need to scale the fluxes of all ion types on one side of the membrane to match the desired concentration difference.

The K+/Cl<sup>−</sup> selectivity of TTX obtained from the MD simulations is 2.2 [24,33]. Using currents scaled by 5:1 in the I-V reconstruction from Equation (20), we obtain a reversal potential of −9 mV. This corresponds to a GHK selectivity of 1.7, which is reasonably close to the selectivity found in MD. Note that experimentally, the reversal potential is −27 mV, corresponding to a K+/Cl<sup>−</sup> selectivity of 6 estimated from the GHK Equation [6]. This cannot be compared directly to our results because the actual channel structure is unknown, there are uncertainties due to force fields, and the GHK equation itself is an approximation.

#### **4. Conclusions**

Stochastic simulations were used to investigate the reliability of two new methods to calculate PMFs for ion transport across transmembrane ion channels and electrophysiological properties of these channels within the general framework of the electrodiffusion model. Both methods have the desirable features that only simulations at a single voltage are needed and information on the diffusivity is not required. In CPM, knowledge of the committor probability is required. Stochastic simulations containing 10<sup>6</sup> trajectories were shown to have similar numbers of crossing events for models of TTX and p7 in MD simulations of 1–2 µs in length. Analysis of 50 or 100 of such simulations indicate that errors in the free energy profiles are approximately ±0.3 kcal/mol. For a model of a slow channel, GLIC, 10<sup>7</sup> trajectories, which approximately corresponds to MD simulations of 10 µs in length, are needed to achieve a similar statistical error. For both TTX and for p7 at lower applied voltages, CPM and CWDM yield similar results. In CWDM, one-sided fluxes are used directly, and for cases in which few crossing events are observed in one direction, either due to large applied voltages, such as p7 at ±140 mV, or because the channel is rectifying, such as GLIC, CPM performs better because two-sided quantities are employed in this method. Similarly, even though one-sided CPM calculations are possible, the errors near the end of the channel become substantial because the density becomes quite small, yielding large relative errors.

Stochastic simulations were also used to investigate the reliability of a new expression to calculate the ionic currents at different voltages, ∆*V*, given knowledge of the PMF, committor probabilities and density profiles at a reference voltage ∆*V*ref. We found that

the I-V dependence could be reconstructed over a range of ±100 mV, with respect to the reference voltage. Judicious choice of ∆*V*ref can markedly improve the accuracy of the reconstruction. Specifically, the I-V reconstruction for p7 is much better for ∆*V*ref = −35 mV than for ∆*V*ref = 140 mV. Although much of the error can be attributed to the large voltage ramp for voltages away from ∆*V*ref (3.2 kcal/mol at 140 mV), some of the error is due to the poor statistics in the direction against the field. This is also evident in the reconstruction of the I-V curve for GLIC, for which some simulations yielded no crossing events against the field.

Common goals of simulations of ion channels are to obtain the free energy profiles of ions translocating the channel and to determine electrophysiological properties of the channel. In some instances, a reliable estimate of the numbers of crossing events, from which the ionic currents can be calculated, is difficult to obtain from MD even for long simulation times. We have shown that the new methods perform very well both to obtain reliably the free energy profile across the channel and to allow for accurate determination of the I-V curves. In the latter case, it is desirable to use a reference voltage that yields good crossing statistics in both directions rather than a voltage that maximizes the total number of crossing events. In summary, if transport of ions through a channel can be satisfactorily described by the ED model, the new methods offer substantial reductions of computational effort without sacrificing accuracy. Our approach is amenable to extensions in which the advantages of MD and stochastic simulations are further combined on reliable theoretical grounds.

**Supplementary Materials:** The following are available at https://www.mdpi.com/article/10.339 0/e23050571/s1 , Figure S1: Pictures from MD simulations of TTX, p7, and GLIC, Figure S2: Total displacement charge calculated from ion crossing statistics and displacement current.

**Author Contributions:** Conceptualization, A.P.; methodology, A.P. and M.A.W.; software, M.A.W.; validation, M.A.W. and A.P.; formal analysis, A.P. and M.A.W.; writing–original draft preparation, A.P.; writing–review and editing, A.P and M.A.W.; visualization, M.A.W.; project administration, A.P.; funding acquisition, A.P. All authors have read and agreed to the published version of the manuscript.

**Funding:** Support for this research was provided by NASA's Planetary Science DivisionResearch Program.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

#### **Abbreviations**

The following abbreviations are used in this manuscript:


#### **References**


## *Article* **Maxwell Equations without a Polarization Field, Using a Paradigm from Biophysics**

**Robert S. Eisenberg 1,2**


**Abstract:** When forces are applied to matter, the distribution of mass changes. Similarly, when an electric field is applied to matter with charge, the distribution of charge changes. The change in the distribution of charge (when a local electric field is applied) might in general be called the induced charge. When the change in charge is simply related to the applied local electric field, the polarization field **P** is widely used to describe the induced charge. This approach does not allow electrical measurements (in themselves) to determine the structure of the polarization fields. Many polarization fields will produce the same electrical forces because only the divergence of polarization enters Maxwell's first equation, relating charge and electric forces and field. The curl of any function can be added to a polarization field **P** without changing the electric field at all. The divergence of the curl is always zero. Additional information is needed to specify the curl and thus the structure of the **P** field. When the structure of charge changes substantially with the local electric field, the induced charge is a nonlinear and time dependent function of the field and **P** is not a useful framework to describe either the electrical or structural basis-induced charge. In the nonlinear, time dependent case, models must describe the charge distribution and how it varies as the field changes. One class of models has been used widely in biophysics to describe field dependent charge, i.e., the phenomenon of nonlinear time dependent induced charge, called 'gating current' in the biophysical literature. The operational definition of gating current has worked well in biophysics for fifty years, where it has been found to makes neurons respond sensitively to voltage. Theoretical estimates of polarization computed with this definition fit experimental data. I propose that the operational definition of gating current be used to define voltage and time dependent induced charge, although other definitions may be needed as well, for example if the induced charge is fundamentally current dependent. Gating currents involve substantial changes in structure and so need to be computed from a combination of electrodynamics and mechanics because everything charged interacts with everything charged as well as most things mechanical. It may be useful to separate the classical polarization field as a component of the total induced charge, as it is in biophysics. When nothing is known about polarization, it is necessary to use an approximate representation of polarization with a dielectric constant that is a single real positive number. This approximation allows important results in some cases, e.g., design of integrated circuits in silicon semiconductors, but can be seriously misleading in other cases, e.g., ionic solutions.

**Keywords:** polarization; maxwell equations; gating current; dielectric constant

#### **1. Introduction**

When forces are applied to matter, the distribution of mass changes. Similarly, when electrical forces are applied matter with charge, the distribution of charge changes.

The electric field **E** *x*, *y*, *z t*; ρ*Q*(*x*, *y*, *z*|*t*; **E**) changes the spatial distribution of charge **P**(*x*, *y*, *z*|*t*; **E**) producing polarization that has a central role in electrodynamics. In general, the change in charge distribution induced by the electric field will depend on time and electric field in a complex nonlinear way. We will discuss that situation later. But even

**Citation:** Eisenberg, R.S. Maxwell Equations without a Polarization Field, Using a Paradigm from Biophysics. *Entropy* **2021**, *23*, 172. https://doi.org/10.3390/e23020172

Received: 13 December 2020 Accepted: 26 January 2021 Published: 30 January 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

when the induced charge is that of a polarization field characterized by a single dielectric constant (a real number), the actual definition of the polarization field **P**(*x*, *y*, *z*|*t*; **E**) is problematic, as major textbooks point out. Purcell and Morin [1], p. 500–507, show how the same structure can be described by different fields **P**(*x*, *y*, *z*|*t*). They conclude "The concept of polarization density **P** is more or less arbitrary" (slight paraphrase of [1], p. 507) and leads to an auxiliary variable that "is an artifice that is not, on the whole, very helpful" [1], p. 500.

Feynman shares this view. Feynman's text says (on p. 10–17 of [2]) "One more point should be emphasized. An equation like **D** = *εrε*0**E** is an attempt to describe a property of matter. But matter is extremely complicated, and the equation is in fact not correct.", as he then explains in some detail [3]. (Zangwill [2] uses quantum electrodynamics (p. 160) to deal with **P** and avoids (p. 44) the auxiliary variable **D.** He concentrates on the fundamental variable **E**, as we do here.) Neither Purcell nor Feynman propose a general explanation for the ambiguity in **P**.

The significance of the Purcell and Morin and Feynman's statements is great. If the concept of polarization is 'more or less arbitrary' (Purcell and Morin's words); and the distinction between bound and free charge is 'ambiguous', then the formulation of the Maxwell equations in textbooks is ambiguous and arbitrary.

I hope it is not necessary to say the obvious: something as important as the Maxwell equations should not be presented in a way that two Nobel Laureates (Purcell and Feynman) think is ambiguous and arbitrary (their words, not mine). It seems that " . . . the conventional theory of electrodynamics inside matter needs to be redesigned": p. 13 of [4].

A general explanation is presented here following Griffiths, Ch. 4, [5]. The ambiguity in the definition of polarization arises from a mathematical property of vector fields and not from a particular physics or structure of charges. Only the divergence of the polarization field enters into the equations for the electric field **E** and so very different functions can be added to **P** without changing the observable electric field. Specifically, the curl of any function can be added to **P** without changing the electric field because the divergence of the curl of any function is zero. Thus, measurements of **E** cannot determine the polarization field **P** uniquely. Different structures of polarization charge can give the same electric field and so measurements of the electric field cannot determine the structures producing polarization or there the structures of charge itself.

A paradigm widely used in biophysics to define gating current allows resolution of this ambiguity in many cases beyond biophysics. This paradigm cannot be universally applied but when it can be applied it is very useful. The dependence of polarization on the electric field is as complicated as the motions of matter in an electric field. These motions are nearly as complicated as the motions of matter in general. It is unlikely that any single paradigm will be universal. Nonetheless, the gating current paradigm of biophysics may be generally useful and will surely make specific what is needed for paradigms in general.

The paradigm of biophysics was developed to resolve the nonlinear displacement (i.e., capacitive) current of nerve that Hodgkin and Huxley [6] suggested might be the voltage sensor of nerve. This 'gating current' was measured in nerve [7] using a paradigm developed by Schneider and Chandler [8,9] and significantly improved by Bezanilla and Armstrong [10,11] and has been studied in great detail [7,12–19] because of the insight it gives [17,18,20,21] into the physical mechanism of conformation change in a most important biological protein and process. The conformation change of the voltage sensor determines many properties of the action potential, which is the signal used by the nervous system, skeletal and cardiac muscle to send signals more than a few micrometers.

The ambiguity of **P** arises from the history of electrodynamics, in my view. Faraday and Maxwell thought all charge depends on the electric field ([3], p. 36; [22–24]. All charge would then be polarization.

Maxwell used the **D** and **P** fields as fundamental dependent variables. Charge only appeared as polarization, usually over-approximated [25–35] by a dielectric constant *ε<sup>r</sup>* that is a single real positive number. Charge independent of the electric field was not included,

because the electron had not been discovered: physicists at Cambridge University (UK) did not think that charge could be independent of the electric field. The electron was discovered some decades later, in Cambridge, ironically enough [36,37]. (Thomson's monograph"intended as a sequel to Professor Clerk-Maxwell's Treatise on electricity and magnetism" [38] does not mention charge, as far as I can tell. Clarendon Press: 1893. "intended as a sequel to Professor Clerk-Maxwell's Treatise on electricity and magnetism"; does not mention charge, as far as I can tell. Faraday's chemical law of electrolysis was not known and so the chemist's 'electron' postulated by Richard Laming and defined by Stoney [39] was not accepted in Cambridge as permanent charge, independent of the electric field. It is surprising that the physical unit 'the Faraday' describes a quantity of charged particles unknown to Michael Faraday. Indeed, he did not anticipate the existence or importance of permanent charge on particles or elsewhere.) It then became apparent to all that the permanent charge of an electron is a fundamental source of the electric field. The electron and permanent charge must be included in the equations defining the electric field, e.g., Equations (1) and (6) as it is in every textbook I have examined. cluded, because the electron had not been discovered: physicists at Cambridge University (UK) did not think that charge could be independent of the electric field. The electron was discovered some decades later, in Cambridge, ironically enough [36,37]. (Thomson's monograph"intended as a sequel to Professor Clerk-Maxwell's Treatise on electricity and magnetism" [38] does not mention charge, as far as I can tell. Clarendon Press: 1893. "intended as a sequel to Professor Clerk-Maxwell's Treatise on electricity and magnetism"; does not mention charge, as far as I can tell. Faraday's chemical law of electrolysis was not known and so the chemist's 'electron' postulated by Richard Laming and defined by Stoney [39] was not accepted in Cambridge as permanent charge, independent of the electric field. It is surprising that the physical unit 'the Faraday' describes a quantity of charged particles unknown to Michael Faraday. Indeed, he did not anticipate the existence or importance of permanent charge on particles or elsewhere.) It then became apparent to all that the permanent charge of an electron is a fundamental source of the electric field. The electron and permanent charge must be included in the equations defining the electric field, e.g., Equations (1) and (6) as it is in every textbook I have examined.

that is a single real positive number. Charge independent of the electric field was not in-

For physicists today, the fundamental electrical variable is the **E** field that describes the electric force on an infinitesimal test charge. **D** and **P** fields are auxiliary derived fields that many textbooks think unnecessary, at best. For physicists today, the fundamental electrical variable is the field that describes the electric force on an infinitesimal test charge. and fields are auxiliary derived fields that many textbooks think unnecessary, at best.

#### **2. Theory 2. Theory**

The setup used here is described in many fine textbooks and so detail is omitted [1–5,40,41]. The specifics of the setup used to measure gating currents is described later, see Figures 1 and 2. The setup used here is described in many fine textbooks and so detail is omitted [1– 5,40,41]. The specifics of the setup used to measure gating currents is described later, see Figures 1 and 2.

*Entropy* **2021**, *23*, x FOR PEER REVIEW 3 of 23

Maxwell's first equation for the composite variable **D** relates the 'free charge' ρ*<sup>f</sup>* (*x*, *y*, *z*|*t*), units cou/m<sup>3</sup> , to the sum of the electric field **E** and polarization **P**. It is usually written as

$$\mathbf{div}\,\mathbf{D}(\mathbf{x},y,z|t) = \varrho\_f(\mathbf{x},y,z|t) \tag{1}$$

$$\mathbf{D}(x, y, z | t; \mathbf{E}) \stackrel{\triangle}{=} \varepsilon\_0 \mathbf{E}(x, y, z | t) + \mathbf{P}(x, y, z | t; \mathbf{E}) \tag{2}$$

The physical variable **E** that describes the electric field is not visible in the classical formulation Equation (1). Maxwell embedded polarization in the very definition of the dependent variable **D** , *ε*<sup>0</sup> **E** + **P**. *ε*<sup>0</sup> is the electrical constant, sometimes called the 'permittivity of free space'. Polarization is described by a vector field **P** with units of dipole moment per volume, cou-m/m<sup>3</sup> , that can be misleadingly simplified to cou-m−<sup>2</sup> . The charge ρ*<sup>f</sup>* cannot depend on **D** or **E** in traditional formulations and so ρ*<sup>f</sup>* is a permanent charge.

*Entropy* **2021**, *23*, x FOR PEER REVIEW 4 of 23

**Figure 2.** shows the response to a step function change in potential and the charges measured that are proposed as an operational definition of polarization. **Figure 2.** shows the response to a step function change in potential and the charges measured that are proposed as an operational definition of polarization.

Maxwell's first equation for the composite variable relates the 'free charge' ρ (, , |), units cou/m<sup>3</sup> , to the sum of the electric field and polarization . It is usually written as When Maxwell's first equation is written in a style appropriate since the discovery of the electron **E** is the dependent variable, as textbooks make clear. The source terms are ρ*<sup>f</sup>* and the divergence of **P**.

$$
\varepsilon\_0 \mathbf{div} \, \mathbf{E}(\mathbf{x}, y, z | t) = \rho\_f(\mathbf{x}, y, z | t) - \mathbf{div} \, \mathbf{P}(\mathbf{x}, y, z | t; \mathbf{E}) \tag{3}
$$

(, , |; ) ≜ <sup>0</sup> (, , |)+ (, , |; ) (2) The physical variable that describes the electric field is not visible in the classical formulation Equation (1). Maxwell embedded polarization in the very definition of the **P** does not have the units of charge and should not be called the 'polarization charge'. **P** does not enter the equation by itself. Only the divergence of **P** appears on the right-hand side of Equation (3).

dependent variable ≜ <sup>0</sup> + . <sup>0</sup> is the electrical constant, sometimes called the 'permittivity of free space'. Polarization is described by a vector field with units of dipole moment per volume, cou-m/m<sup>3</sup> , that can be misleadingly simplified to cou-m−2. The **D**(*x*, *y*, *z*|*t*) and the polarization **P**(*x*, *y*, *z*|*t*) are customarily over-approximated in classical presentations of Maxwell's equations: the polarization is assumed to be proportional to the electric field, independent of time.

$$\mathbf{P}(\mathbf{x}, y, z | t) \triangleq (\varepsilon\_r - 1)\varepsilon\_0 \operatorname{E}(\mathbf{x}, y, z | t) \tag{4}$$

$$\mathbf{D}(\mathbf{x}, y, z | \mathbf{t}) \triangleq \varepsilon\_r \varepsilon\_0 \mathbf{E}(\mathbf{x}, y, z | \mathbf{t}) \tag{5}$$

ρ and the divergence of . 0 (, , |) = ρ (, , |)− (, , |; ) (3) does not have the units of charge and should not be called the 'polarization charge'. does not enter the equation by itself. Only the divergence of appears on the right-hand side of Equation (3). (, , |) and the polarization (, , |) are customarily over-approximated in The proportionality constant (*ε<sup>r</sup>* − 1)*ε*<sup>0</sup> involves the dielectric constant *ε<sup>r</sup>* which must be a single real positive number if the classical form of the Maxwell equations is taken as an exact mathematical statement of a system of partial differential equations. If *ε<sup>r</sup>* is generalized to depend on time, or frequency, or the electric field, the form of the Maxwell equations changes. If *ε<sup>r</sup>* is generalized, traditional equations cannot be taken literally as a mathematical statement of a boundary value problem. They must be changed to accommodate the generalization.

classical presentations of Maxwell's equations: the polarization is assumed to be proportional to the electric field, independent of time. Polarization and thus *εr*—however generalized—depend on time or frequency in complex ways in all matter as documented in innumerable experiments [33–35,42–44].

charge.

Many of the most interesting applications of electrodynamics arise from the dependence of polarization and *ε<sup>r</sup>* on field strength.

*ε<sup>r</sup>* should be taken as a constant only when experimental estimates, or theoretical models are not available, in my view.

It is difficult to imagine a physical system in which the electric field produces a change in charge distribution independent of time (see examples shown towards the end of Discussion). The time range in which Maxwell's equations are used in the technology of our computers, smartphones, and video displays starts around 10−<sup>10</sup> s. The time range in which Maxwell's equations are used in biology start around 10−<sup>15</sup> s in simulations of the atoms that control protein function. The time range of the X-rays that determine protein structure is ~10−<sup>19</sup> s. The time range used to design and operate the synchrotrons that generate X-rays is very much faster than that, something like 10−<sup>23</sup> s. The Maxwell equations describe experiments to many significant figures over this entire range.

It is evident that a dielectric constant *ε<sup>r</sup>* independent of time is an inadequate overapproximation in many cases of practical interest today, in biology, engineering, chemistry, and physics.

Maxwell's first equation for **E** is well described in many textbooks, although the inadequacies of the usual representation of polarization with a single dielectric constant are not emphasized, if mentioned at all. Students are then often unaware of the overapproximation, particularly if they have a stronger background in biology or mathematics than the physical sciences.

$$
\varepsilon\_r \varepsilon\_0 \mathbf{div} \, \mathbf{E}(\mathbf{x}, y, z | t) = \mathfrak{p}\_f(\mathbf{x}, y, z | t) \tag{6}
$$

Polarization is particularly well described in Griffiths [5].

It is wise, in my view to combine the fields on the right-hand side of Equation (3) with the definition

$$\rho\_Q(\mathbf{x}, y, z | \mathbf{t}; \mathbf{E}) \triangleq \rho\_f(\mathbf{x}, y, z | t) - \mathbf{div} \, \mathbf{P}(\mathbf{x}, y, z | \mathbf{t}; \mathbf{E}) \tag{7}$$

yielding the version of Maxwell's first law that does not involve a polarization field **P**(*x*, *y*, *z*|*t*) at all.

$$
\varepsilon\_0 \mathbf{div} \, \mathbf{E}(\mathbf{x}, y, z | t) = \mathfrak{p}\_{\mathbb{Q}}(\mathbf{x}, y, z | t; \mathbf{E}) \tag{8}
$$

We adopt this version of Maxwell's first equation here.

#### **3. Results**

The traditional formulation of the differential equations shown in Equations (1) and (6) is ambiguous in an important way (Integral forms of the Maxwell equations show more clearly the need for boundaries. They display the charge on the surface as an integral and explicit part of the general solution of Poisson's equation for the electrical potential, for example). They do not mention the shape or boundaries of the regions in question. In fact, if **P** varies from region to region, but is constant within each region, charge is absent within each region: when **P** is constant, **div P** = 0. Charge accumulates only at the boundaries of the regions. In many situations involving dielectrics, including most of those described in classical textbooks Only the boundary charge has effects on the Maxwell Equations (1) and (6). The **P** field in the Maxwell Equation (7), and implied in Equations (1) and (6), is zero; only the boundary values of **P** are important and they are not visible in the Maxwell Equations (1) and (6) themselves.

We turn now to applications in biology where the issue of charge at boundaries is particularly important, not to say that it is unimportant in semiconductor devices as well. Dielectric boundary charges have a particular role in biological systems involving membranes or proteins. The membrane capacitance, so important in determining the electrical properties of cells, particularly cells with action potentials like nerve and muscle, is a boundary phenomenon. Boundary charges are of great importance in channel proteins that allow (nearly catalyze) ion flow through membranes, see Appendix A on Proteins and [45].

Turning back to classical electrodynamics, we remember that most of the properties of dielectric rods studied by Faraday—and predecessors going back to Benjamin Franklin, if not earlier—arise from the dielectric boundary charges. Textbooks typically spend much effort teaching why polarization charge appears on dielectric boundaries in systems with constant **P** where **div P** = 0 (e.g., Ch. 6 of [3]). Students wonder why regions of dielectrics without polarization charge have polarization charge on boundaries.

A general principle is at work here: a field equation in itself—like Equations (1) and (6) that are partial differential equations without boundary conditions—is altogether insufficient to specify an electric field. A model is needed that has boundary conditions. Applications of electrodynamics to biology, electrochemistry, and semiconductors are not useful until they specify models and boundary conditions that realistically describe the system of interest.

The model needs to include an explicit structure. It needs to describe the spatial variation of **P**. Indeed, the spatial variation of **P** may be a main determinant of properties [46–48] in (for example) many biological systems (e.g., channels), electrochemical systems (electrodes of batteries), and semiconductor devices. Without specifying boundary conditions (defined explicitly in specific structures), using **P** in the differential Equation (7), and implied in Equations (1) and (6), is ambiguous and confusing. Indeed, using **P** without boundary conditions is so incomplete that it might be called incorrect.

The general nature of the ambiguity in **P** becomes clear once one realizes that:

$$\text{Adding } \mathbf{curl \, C}(\mathbf{x}, y, z | t) \text{ to } \mathbf{P}(\mathbf{x}, y, z | t) \text{ in Maxwell's first equation, Equation (7)} \quad (9)$$

changes nothing (Ch. 4 of [5]) because [49,50]

$$\mathbf{div}\,\mathbf{curl}\,\mathbb{C}(\mathbf{x},y,z|t) \;\equiv \mathbf{0} \;\text{;}\tag{10}$$

The ambiguity in **P** in the Maxwell differential equations means that any model **P***model*(*x*, *y*, *z*|*t*) of polarization can have **curl** Ce(*x*, *y*, *z*|*t*) added to it, without making any change in the **div P**(*x*, *y*, *z*|*t*) in Maxwell's first equation (7), and implied in Equations (1) and (6).

In other words, the polarization **div P**(*x*, *y*, *z*|*t*) in Maxwell's first Equations (7), and implied in Equations (1) and (6), does not provide a unique structural model of polarization **P***model*(*x*, *y*, *z*|*t*). In particular, a model drawn from an atomic detail structure can be modified by adding a polarization Pe(*x*, *y*, *z*|*t*) , **curl** Ce(*x*, *y*, *z*|*t*) to its representation (i.e., 'drawing') of polarization without changing electrical properties at all: **div P** <sup>≡</sup> **div P** + Pe .

Models of the polarization **P** 1 *model* and **P** 2 *model* of the same structure written by different authors may be strikingly different but they can give the same electrical results even though the models can appear to be very different. The **curl** Ce(*x*, *y*, *z*|*t*) field can be quite complex and hard to recognize in a model, particularly for structural biologists who may not be comfortable with vector calculus and its **curl** and **div** operators. The two models **P** 1 *model* and **P** 2 *model* produce the same charge distribution **div P**<sup>1</sup> *model* and div **P** 2 *model* in Maxwell's first equation Equation (11) and so they cannot be distinguished by electrical measurements.

As we have seen, the **P** field is arbitrary, as certainly has been known previously Ch. 4 of [5]. Purcell and Morin [1], see pp. 500–507, describe structural models and ways to construct different fields **P**(*x*, *y*, *z*|*t*) from the same structure as stated in the introduction to this paper. **P** fields are not unique.

Purcell and Morin are not guilty of overstatement—indeed they may be guilty of understatement—when they say "The concept of polarization density **P** is more or less arbitrary" (slight paraphrase of [1], p. 507) and the **D** field is "is an artifice that is not, on the whole, very helpful" [1], p. 500.

The classical approach criticized by Purcell and Morin [1] does not allow unique specification of a polarization field **P**(*x*, *y*, *z*|*t*) from electrical measurements.

An arbitrary artificial formulation is prone to artifact and likely to produce misunderstanding and unproductive argument: "what is the true description of a dielectric object

(e.g., protein)?" is a question likely to arise and be unanswerable if the polarization field **P** is itself not unique.

The **P**(*x*, *y*, *z*|*t*) of classical theory is not a firm foundation on which to build an understanding of the structural basis of the phenomena of polarization, or the electrodynamics of matter, with problems particularly apparent in the understanding of the polarization arising from the structure of proteins (see Appendix A).

It seems clear that most formulations of electrodynamics of dielectrics in classical textbooks are "more or less arbitrary" and depend on an "artifice" (quotations from Feynman and Purcell and Morin). Because dielectrics, polarization and a dielectric constant (as a single real number) are central to the classical treatments of electrodynamics, the conclusion (p. 13) of a modern monograph on electrodynamics, using mathematics (exterior differential forms) appropriate for relativistic theories of electrodynamics, [4] quoted previously seems worth restating "We believe that the conventional theory of electrodynamics inside matter needs to be redesigned". That redesign begins with a revised treatment of polarization that reflects the ambiguity of the curl, see [5]. Ambiguity and its problems can be avoided if Maxwell's First Equation is rewritten without a polarization field **P**(*x*, *y*, *z*|*t*) as shown previously in Equation (8). The phenomena of polarization—the response of charges to an electric field—is then included in a variable ρ*Q*(*x*, *y*, *z*|*t*; **E**), specifically as (part of) its dependence on **E** :

$$\mathbf{div}\,\varepsilon\_0\mathbf{E}(\mathbf{x},y,z|t) \,:= \mathfrak{q}\_{\mathbb{Q}}(\mathbf{x},y,z|t;\mathbf{E})\tag{11}$$

Here ρ*Q*(*x*, *y*, *z*|*t*; **E**) describes all charge whatsoever, no matter how fast, small or transient are their movements, including what is usually called dielectric charge and permanent charge, as well as charges driven by other fields, like convection, diffusion or temperature. The charge ρ*<sup>Q</sup>* can be parsed into components in many ways (see Equations (1), (3), (6) and (8) and [43,51]). Updated formulations of the Maxwell differential equations [43,51] are needed, in my opinion, to avoid the problems produced by ambiguous **P** and over-simplified *ε<sup>r</sup>* .

We turn now to a quite different property of charge matter, the flow of charges.

Most applications of electrodynamics involve flow. The most prominent application of electrodynamics is surely computational and semiconductor electronics [52–61] and that involves flow, usually described by Kirchhoff's current law. Semiconductor electronics has remade our world increasing computer power by nearly 109<sup>×</sup> in the last seventy years [62–67]. Biology and electrochemistry (batteries) scarcely exist without flow: what physical chemists call equilibrium (no flows of any kind) is hardly worth studying in biological or electrochemical systems. Unlike thermodynamics, electrodynamics nearly always involves flow.

Thus, we study the flux of charges ρ*<sup>Q</sup>* as well as their density. Maxwell's second equation describes the flow of charges, electrical current, and the magnetic field. It is understandable that Maxwell—and his Cambridge contemporaries and followers—had difficulty understanding current flow when their models did not include permanent charge, electrons or their motions.

Maxwell's extension of Ampere's law describes the special properties of current flow **J***total* (Equation (13) that make it so different from the flux of matter. Maxwell's field equations include the ethereal current *ε*0*∂***E**/*∂t* that makes the equations resemble those of a perfectly incompressible fluid: the ethereal current always exists, whether matter is present or not, unlike the dielectric current (*ε*<sup>r</sup> − 1)*ε*0*∂***E**/*∂t* that exists only when matter is present.

Maxwell's field equations describe the incompressible flow **J***total* over the dynamic range of something like 10<sup>16</sup> that is safely accessible within laboratories. The dynamic range of the Maxwell equations is much larger if one includes the interior of stars, and the core of galaxies in which light is known to follow the same equations of electrodynamics as in our laboratories.

Maxwell's field equations are different from material field equations (like the Navier– Stokes equations) because they are meaningful and valid universally [68], both in a vacuum devoid of mass and matter and within and between the atoms of matter [43].

The ethereal current *ε*0*∂***E**/*∂t* responsible for the special properties of Maxwell's equations arises from the Lorentz (un)transformation of charge. Charge does not vary with velocity, unlike mass (this is the mass that determines inertia, called the 'relativistic mass' nowadays. This was the meaning of the word 'mass' in Einstein's original papers, presumably because he wanted an operational definition of 'mass' that was based on the observable properties, inertia and momentum, and that was independent of Lorentz transformations, and theoretical considerations) [69], length, and time, all of which change dramatically as velocities approach the speed of light, strange as that seems. This topic is explained in any textbook of electrodynamics that includes special relativity. Feynman's discussion of 'The Relativity of Electric and Magnetic Fields' was an unforgettable revelation to me as a student, see Section 13-6 of reference [2]: an obervers moving at the same speed as a stream of electrons sees zero current, but the forces measured by that observer are the same as the forces measured by an observed who is not moving at all. The moving observer describes the force as an electric field **E**(*x*, *y*, *z*|*t*). The unmoving observer describes the force as a magnetic field **B**(*x*, *y*, *z*|*t*). The observable forces are the same, whatever they are called, according to the principle and theory of relativity. (The principle and theory of relativity are confirmed to many significant figures every day in the GPS (global positioning systems) software of the map apps on our smartphones, and in the advanced photon sources (synchrotrons) that produce X-rays to determine the structure of proteins).

The ethereal current reveals itself in magnetic forces which have no counterpart in material fields. The ethereal current is apparent in the daylight from the sun, that fuels life on earth, and in the night light from stars that fuels our dreams as it decorates the sky. The ethereal current is the term in the Maxwell equations that produces propagating waves in a perfect vacuum like space.

Magnetism **B** is described by Maxwell's version of Ampere Law, Maxwell's Second Equation:

$$\frac{1}{\mu\_0} \text{curl } \mathbf{B} = \mathbf{J}\_Q + \varepsilon\_0 \frac{\partial \mathbf{E}}{\partial t} \tag{12}$$

$$\mathbf{J}\_{total} \triangleq \mathbf{J}\_{\mathcal{Q}} + \varepsilon\_0 \frac{\partial \mathbf{E}}{\partial t} \tag{13}$$

$$\frac{1}{\mu\_0} \text{ curl } \mathbf{B} = \mathbf{J}\_{total} \tag{14}$$

If we are interested in flux and current, we must turn to Maxwell's second equation and deal explicitly with magnetism, even if magnetic fields themselves do not carry significant energy (as in almost all biological applications). Only by dealing with Maxwell's second equation can we derive conservation of total current and compare it with the conservation of charge. Indeed, the derivation of the continuity equation used here depends on equations involving the magnetic field.

Note that **J***<sup>Q</sup>* includes the movement of all charge ρ*<sup>Q</sup>* with mass, no matter how small, rapid or transient. It includes the movements of charge classically approximated as the properties of an ideal dielectric. It describes all movements of the charge described by ρ*Q*(*x*, *y*, *z*|*t*; **E**); ρ*<sup>f</sup>* is one of the components of ρ*Q*. Indeed, **J***<sup>Q</sup>* can be written in terms of **v***<sup>Q</sup>* the velocity of mass with charge. In simple cases, such as a plasma of ions each with charge **Q***<sup>Q</sup>*

$$\mathbf{J}\_{\mathcal{Q}} = \mathbf{v}\_{\mathcal{Q}} \mathbf{Q}\_{\mathcal{Q}} \mathbf{N}\_{\mathcal{Q}} \tag{15}$$

where Q<sup>Q</sup> is the charge per particle and N*<sup>Q</sup>* is the number density of particles. In a mixture, sets of fluxes **J** *i Q* , velocities **v** *i Q* , charges Q *i Q*, number densities N*<sup>i</sup> Q* , and charge densities ρ *i Q* are needed to keep track of each elemental species *i* of particles. Plasmas are always mixtures because they must contain both positive and negative particles to keep electrical forces within safe bounds, as determined by (approximate) global electroneutrality.

In cases other than plasmas, the relationship of **J***<sup>Q</sup>* , **J***total* and Q*<sup>Q</sup>* to material properties is complex. The relationship often involves convection and diffusion fields and extends over a range of scales from atomic to macroscopic, in both space and time. For example, the Maxwell equations do not describe charge and current driven by other fields, like convection, diffusion, or temperature. They do not describe constraints imposed by boundary conditions and mechanical structures. Those must be specified separately. If the other fields, structures, or boundary conditions involve matter with charge, they will respond to changes in the electric field. The other fields and constraints thus contribute to the phenomena of polarization and must be included in a description of it, as we shall discuss further below in the examples shown towards the end of Discussion. The theory of complex fluids has dealt with many such cases, often with the label 'micro macro', spanning scales, connecting micro (even atomic) structures with macro phenomena.

The charge density ρ*<sup>Q</sup>* and current **J***total* can be parsed into components in many ways, some helpful in one historical context, some in another. References [33,43,51,70–75] define and explore those representations in tedious detail. Simplifying those representations led to the treatment in this paper.

Maxwell's Ampere's law Equation (12) implies two equations of great importance and generality. First, it implies a continuity equation that describes the conservation of charge with mass. The continuity equation is the relation between the flux of charge with mass and density of charge with mass.

**Derivation**: Take the divergence of both sides of Equation (12), use **div curl** = 0 [49,50], and get

$$\mathbf{div}\,\mathbf{J}\_Q = \mathbf{div}\left(-\varepsilon\_0 \frac{\partial \mathbf{E}}{\partial t}\right) = -\varepsilon\_0 \frac{\partial}{\partial t} \mathbf{div}\,\mathbf{E} \tag{16}$$

when we interchange time and spatial differentiation.

However, we have a relation between **div E** and charge ρ*<sup>Q</sup>* from Maxwell's first equation, Equation (11), giving the Maxwell Continuity Equation:

$$\mathbf{div}\,\mathbf{J}\_{\mathcal{Q}} = -\,\varepsilon\_0\,\varepsilon\_0 \frac{\partial \rho\_{\mathcal{Q}}}{\partial t} \tag{17}$$

$$\mathbf{div}\left(\mathbf{v}\_{Q}\mathbf{Q}\_{Q}\mathbf{N}\_{Q}\right) = -\varepsilon\_{0}\frac{\partial\rho\_{Q}}{\partial t},\tag{18}$$

for a biophysical or astrophysical plasma of ions.

Note that sets of fluxes **J** *i Q* and sets of charge densities ρ *i Q* are needed to keep track of each elemental species *i* of particles in a mixture, along with sets of velocities **v** *i Q* , charges Q *i Q*, and number densities N*<sup>i</sup> Q* , as described near Equation (15).

Maxwell's Ampere's law Equation (12) implies a second equation of great importance. Indeed, it is this equation that allows the design of the one-dimensional branched circuits of our digital technology using the relatively simple mathematics of Kirchhoff's current law [72,74].

**Derivation**: Taking the divergence of both sides of Maxwell's Second law Equation (12) yields Conservation of Total Current

$$\mathbf{div}\,\mathbf{J}\_{total} \stackrel{\scriptstyle \Delta}{=} \mathbf{div}\left(\mathbf{J}\_{\mathcal{Q}} + \varepsilon\_0 \frac{\partial \mathbf{E}}{\partial t}\right) = \mathbf{0} \tag{19}$$

$$\mathbf{div}\,\mathbf{J}\_{total} = 0\tag{20}$$

or

$$\mathbf{div}\,\mathbf{J}\_{total} \triangleq \mathbf{div}\left(\mathbf{v}\_Q \mathbf{Q}\_Q \mathbf{N}\_Q \mathbf{J}\_Q + \varepsilon\_0 \frac{\partial \mathbf{E}}{\partial t}\right) = \mathbf{0} \tag{21}$$

It is easy to overlook the importance of one-dimensional systems. They may seem trivial, almost unworthy of analysis using the powerful beauty of vector calculus. However, one-dimensional systems are of great importance despite, or because of their simplicity.

Nearly all of our electronic technology occurs in one-dimensional systems, networks of branching one-dimensional conductors. Our electronic technology is driven by batteries that are one-dimensional systems. Our technology is at the hands of animals, humans in which all information transfer is done by one-dimensional circuits, unbranched in ion channels, and barely branched in nerve cells. Branched one-dimensional systems describe the metabolic pathways of biological cells that make life possible.

The importance of one-dimensional systems may come from their design. The design of one-dimensional systems is relatively easy for engineers or evolution. Design requires Kirchhoff's laws and little else. One-dimensional systems are widely used for another reason. They are reliable. The dimensionality of these circuits rules out spatial singularities. Systems are more robust when steep slopes near infinities are not present to create severe sensitivity.

Kirchhoff's laws are used to design semiconductor circuits that work over an enormous range of sizes and times, from say 10−<sup>10</sup> s to many minutes, from 10−<sup>19</sup> m to 10<sup>4</sup> m or longer. Current flow over these ranges of time space involves a wide range of physics, described by many constitutive equations.

Current is not just the movement of point permanent charges as assumed in the textbook derivations of Kirchhoff's current law I have consulted, both in electrical engineering and electrodynamics. The derivations of Kirchhoff's current law are usually restricted to the simplest case of the long-time translation of point permanent charges, although it is very well known that is a poor model for current flow under conditions actually found in the integrated circuits of our digital technology. It is possible to show, however, that current flow in one-dimensional systems can be described accurately by a simple generalization of Kirchhoff's current law that arises naturally from the treatment of Maxwell's equations found in this paper: all the **J***total* that flows into a node must flow out [51,72–74]. This result seems to be rather new, although of course it seems elementary and obvious. Indeed, it is so obvious that it must exist somewhere in the literature, even though I do not know where.

Kirchhoff's current law take on simplest form in unbranched one-dimensional systems. Unbranched one-dimensional systems are important despite their utter simplicity. Indeed, the ion channels of biological systems control a wide range of biological function and are unbranched one-dimensional series systems. They cannot be considered degenerate. Nor can be the diodes of electronic technology that are also series systems. However, the greatest importance of unbranched one-dimensional systems may be the insight they give to the importance of the ethereal current *ε*0*∂***E**/*∂t*, as we shall soon see.

Unbranched one-dimensional systems have components in series, each with its own current voltage relation arising from its microphysics. In a series one-dimensional system, the total current **J***total* is equal everywhere at any time in every location no matter what the microphysics of the flux **J***<sup>Q</sup>* of charge with mass. The current through a battery is an exceedingly complicated mixture of the microphysics of electrodes, ion movement and electron flow. If that battery is connected by a wire to a vacuum capacitor, the microphysics of the vacuum capacitor *icapacitor* = *Areaε*0*∂***E**/*∂t*, is as simple as the microphysics of the battery is complex, yet the total currents in the capacitor and the battery are equal at any time, in any conditions. Indeed, the microphysics of the wire linking the capacitor and the battery is totally different from the microphysics of the capacitor and battery. The microphysics of the wire actually resemble that of a waveguide at frequencies important in our digital integrated circuits. The microphysics of the wire, capacitor and battery do not change the fact that the total current through each is exactly the same, always, at every location and at every time.

How can that possibly be true? The answer is found in the Maxwell equations. They can be solved for the electric field and magnetic fields that make the total currents equal.

The solutions of Maxwell's equations ensure that the ethereal current *ε*0*∂***E**/*∂t*, and the other dependent variables, take on the values at every location and every time needed to make the total currents **J***total* equal everywhere. A practical example, not difficult to build in any laboratory, including resistor, capacitor, diode, capacitor, cylinder of salt water, and wire is described in detail near Figure 2 of [73].

There is no spatial dependence of total current in a series one-dimensional system. No spatial variable or derivative is needed to describe total current in such a system [75], although of course spatial variables are needed to describe other variables, including (1) the density of mass with charge **Q***<sup>i</sup> Q* (2) the flux **J***<sup>Q</sup>* of charge with mass (3) the electrical current **J** *i total* of individual elemental species (4) the velocities, charge, and number densities **v***Q*, Q*Q*, ρ*Q*, and N*Q*.

It is important to realize that the flux of charge with mass **J***<sup>Q</sup>* is not conserved, only the total current **J***total* is conserved. Charges carry **J***<sup>Q</sup>* can accumulate. In fact, **div J***<sup>Q</sup>* = **div v***Q*Q*Q*N*<sup>Q</sup>* supplies the flow of charge that is the current *∂*ρ*<sup>Q</sup>* /*∂t* necessary to change **div** (*ε*0*∂***E**/*∂t*) as described by the following continuity equation.

$$\mathbf{div}\,\mathbf{J}\_Q = \mathbf{div}\,\varepsilon\_0 \frac{\partial \mathbf{E}}{\partial t} = \frac{\partial}{\partial t} \mathbf{div}\,(\varepsilon\_0 \mathbf{E}) = \frac{\partial \rho\_Q}{\partial t}.\tag{22}$$

That is to say, **J***<sup>Q</sup>* can accumulate as Q*Q*. Total current **J***total* cannot accumulate, not at all, not anywhere, not at any time.

Because conservation of total current applies on every time and space scale, including those of thermal motion, the properties of **J***<sup>Q</sup>* differ a great deal from the properties of **J***total*. For example, in one-dimensional channels, the material flux **J***<sup>Q</sup>* can exhibit all the complexities of a function of infinite variation, like a trajectory of a Brownian stochastic process, that reverses direction an uncountably infinite number of times in any interval. A Brownian trajectory of a Brownian stochastic process is a continuous function that does not have a (well defined) time derivative anywhere.

In marked contrast to the infinite variation of **J***Q*, the electrical current **J***total* has no spatial variation at all. It is spatially uniform [75].

The fluctuations of *ε*0*∂***E**/*∂t* (in time and space) and other variables are exactly what are needed to completely smooth the infinite fluctuations of **J***<sup>Q</sup>* into the spatially uniform **J***total*.

Maxwell's equations serve as the perfect low pass (spatial) filter converting the infinite variation of Brownian motion into a spatial constant, as strange as that seems.

These universal and exact properties of Maxwell's equations are hidden in the usual treatment of Maxwell's equations. The usual treatment includes a grossly approximate treatment of polarization as the property of a perfect dielectric. Everyone knows how bad this approximation is, so everyone understands that Maxwell's equations as usually written are not universal or exact. They are as sloppy as is the dielectric constant as a description of the polarization of matter.

*ONLY* when Maxwell's equations are written without a dielectric constant, with a perfectly general treatment of induced charge, does it become clear that Maxwell's equations are universal and exact independent of any property of matter.

How then is polarization included in a modified version of the Maxwell equations that does not include a dielectric constant. One needs an explicit model of polarization appropriate for the system of interest.

It is obvious that one cannot describe material flow unless one knows how matter moves in response to forces. It should be obvious that one cannot describe the flux of charges unless one knows how material charge moves in response to forces.

The use of a single real dielectric constant in Maxwell's equations is no more necessary than the use of a single spring constant (i.e., elasticity) is in material equations. But Maxwell's equations describe the total electrical current—that includes the ethereal current—not the flux of charges. Because of the ethereal current, Maxwell's equations describe light in the vacuum of space between stars.

Because of the ethereal current, Maxwell's equations are universal and exact. They describe total current as exactly as they describe anything, and their description of total current flow is entirely independent of the properties of matter. Total current flow depends on no constitutive equations, except perhaps the constitutive equation of a vacuum, more or less determined by special relativity. Electrodynamics are very different in this respect from the equations of material movement. They always depend on constitutive equations in important respects. The fundamental properties of electrodynamics do not depend on constitutive equations.

#### **4. Discussion: From Electrodynamics to Biophysics and Back**

A fundamental question arises with the updated version of Maxwell's equations. How is the phenomenon of polarization included in Equation (11) and Equation (14)?

To answer this question, we first need a general paradigm to define polarization, even when dielectrics are far from ideal, when they might be time and frequency dependent, and voltage dependent as well. We need a paradigm that describes how the charge distribution varies with the electric field in as general a system as possible, including systems with charge movement driven by forces not in the Maxwell equations at all, such as convection and diffusion.

It seems obvious that a general paradigm cannot be found. After all the motions of matter in response to a change in electric field are more or less as complex as the motions of matter itself! Nonetheless, a paradigm of that may be helpful in many cases has been in use for many years, even if it is not perfectly general.

This problem has been addressed in membrane biophysics. A community of scholars has studied the nonlinear currents that control the opening of voltage sensitive protein channels for nearly fifty years, [7,12–19] inspired by [6]. They have developed protocols that may be useful in other systems, as they have been in biophysics. Schneider and Chandler followed by Bezanilla and Armstrong are responsible for this paradigm, more than anyone else [7–9].

The basic setup used in these experiments is that of an electrochemical cell modified to deal with a cylindrical cell as shown in Figure 1. Membrane potential is measured across a biological membrane, with defined concentrations on both sides of the membrane. Current is applied through electrodes to control the potential, in the classical voltage clamp set up of Cole [76] and Hodgkin, Huxley, and Katz [77,78]. It is best to apply that current in electrodes different from those that record membrane potential using a so-called four electrode setup [79–81], like those described in textbooks of electrochemistry.

I propose using the operational definition of 'gating current' used to define nonlinear, time and voltage dependent polarization by biophysicists as a useful setup and definition of many types of polarization. Obviously, this definition is not general, but the hope is that it may be generally useful.

The basic idea is to apply a set of step functions of potential across the system—in biology across the membrane—and observe the currents that flow. The currents observed are transients that decline to a steady value, often to near zero after a reasonable (biologically relevant) time. The measured currents are perfectly reproducible. If a pulse is applied, the charge moved (the integral of the current) can be measured when the voltage step is applied. The integration goes on until *t*<sup>1</sup> when the current *i leak* is nearly independent of time, often nearly zero. That integral is called the ON charge **QON**.

When the voltage is returned to its initial value (the value that was present before the ON pulse), another current is observed that often has quite different time course [7–9], much more so than in Figure 1. The integral of that current is the OFF charge **QOFF**.

If **QON** = **QOFF**, and the physical processes involved depend fundamentally on potential and not its time derivative, the biophysical paradigm is likely to be useful. In other cases, another paradigm is needed. If the current produced by the step in potential is in fact actually transient, the steady current will be what it was before the voltage step was applied. The transient will disappear with time as the word 'transient' implies. In that case it seems that the biophysical paradigm is not only useful but may even provide a unique definition of gating current and the corresponding polarization.

Gating current as measured in biophysical experiments depends on the membrane voltage before the step, as well as the voltage just after and during the step. It also depends separately on the voltage after the step, although Figure 1 does not illustrate the dependence documented in the literature [7–9]. The voltage and time dependence arises from the molecular motions underlying the gating current. The voltage and time dependence defines the mean molecular motions [7,16,17,19,21,82–86] and is called 'the gating current' in the biophysics literature.

If the ON charge is found experimentally to equal the OFF charge, for a variety of pulse sizes and range of experimental conditions, the current is said to arise in a nonlinear (i.e., voltage dependent) polarization capacitance and is interpreted as the movement of charged groups in the electric field. The charged groups move to one location after the ON pulse, and return to their original location following the OFF pulse. The charge is called 'gating charge', and the current that carries the charge is called 'gating current'.

The macroscale current observed in the set-up is equal to the sum of the micro (actually atomic scale) currents carried by the charged groups inside a channel protein, even though the recording electrodes are remote from the protein. Indeed, there might be 10<sup>18</sup> charged atoms (ions) between the electrodes and the protein.

The currents in the electrodes and the channel protein are equal because the setup is designed to be an unbranched one-dimensional circuit with everything in series. In a one-dimensional series setup the total current is equal everywhere in the series system at any one time, even though the total current varies significantly with time. The Maxwell equations guarantee spatial uniformity of total current (including the ethereal current *ε*0*∂***E**/*∂t*) independent of the microphysics of movement of charge (with mass): Figure 2 of [73], and [43,75,87]. The equality of current can be checked by measuring current in different locations in the experiment. The spatial equality of current needs also to be checked in simulations as in [18,21,88] because tiny inadvertent errors in numerical procedures or coding can produce substantial deviations from spatial equality and thus misleading artifacts. Imposing periodic boundary conditions on nonperiodic systems is another possible source of such artifacts.

If the currents reach a steady value independent of time, but not equal to zero, as in Figure 1, the signal is not transient, in the strict meaning of the word. In biophysics, the steady current *i leak* is then usually considered to flow in a resistive path that is time independent, but perhaps voltage dependent, in parallel with the path or device in which the gating charges QON and QOFF flow. If the current does not reach a steady value, or if the areas in Figure 2 are not equal, the currents are not considered 'capacitive' and are interpreted as those through a time and voltage dependent 'resistor'. This is a biological and biophysical assumption. It is not a physical or mathematical necessity. Thus, it is important to investigate the properties of the currents through the resistive path—e.g., those that are not transient and do not return to zero and those that make QON 6= QOFF by *independent* methods to see if they are time independent. In biophysics, currents can be done by blocking the resistive path with drugs, or with mutations of the channel protein. If the resistive currents are not time independent, the definition of QON and QOFF in Figure 1 needs to be changed. Indeed, experiments of another type must be designed that allow separation of polarization from conduction currents. The simplest version of the biophysics paradigm then needs to be extended.

Clearly, this approach will only work if step functions can reveal all the properties of the underlying mechanism. If the underlying mechanisms depend on the time rate of change of voltage, step functions are clearly insufficient because *∂V*/*∂t*, is zero or infinity but nothing else in a step function. In the classical language of membrane biophysics, the ionic conductances *gNa* and *g<sup>K</sup>* must not depend on the rate of change of voltage.

Much work has been conducted showing that step functions are enough to understand the voltage dependent mechanisms in the classical action potential of the squid axon [89–91], starting with [78], Figure 10 and Equation (11). Hodgkin kindly explained the significance of this issue to colleagues, including the author (around 1970). He explained the possible incompleteness of step function measurements: if sodium conductance had a significant dependence on *∂V*/*∂t*, the action potential computed from voltage clamp data

would differ from experimental measurements. He mentioned that this possibility was an important motivation for Huxley's heroic hand integration [6] of the Hodgkin Huxley differential equations. Huxley confirmed this in a separate personal communication, Huxley to Eisenberg. Those computations and many papers since [89–91] have shown that voltage clamp data (in response to steps) is enough to predict the shape and propagation of the action potential in nerve and skeletal muscle. It should be clearly understood that such a result is not available for biological systems in which the influx of Ca++ drives the action potential and its propagation [92].

The conductance of the voltage activated calcium channel has complex dependence on the current through the channel because the concentration of Ca++ in the cytoplasm is so low (~10−<sup>8</sup> M at rest) that the current almost always changes the local concentration in and near the channel on the cytoplasmic side. Those concentration changes, in turn, alter the gating and selectivity characteristics of the channel protein, as calcium ions are prone to do int many physical and biological systems, particularly at interfaces.

It seems unlikely that the resulting properties of voltage dependent calcium channels can be comfortably described by the same formalism [6] used for voltage-controlled sodium and potassium channels of nerve and skeletal muscle. That formalism uses variables that depend on membrane potential and not membrane current because Cole [93] and Hodgkin [94–96] guessed that neuronal action potentials were essentially voltage dependent, not current dependent. They found action potentials in 'space clamped' axons with wires down their middle [76,77,97,98] that ensured spatial uniformity of potential. These axons had very different patterns of current flow from normal axons, and so Cole and Hodgkin were confirmed in their view that the membrane processes generating the action potential were voltage dependent, much more than current dependent (personal communication Cole to Eisenberg 1960; Hodgkin to Eisenberg 1961, et al.).

Hodgkin, Huxley, Katz, and Cole did not know of action potentials driven by calcium channels [99–103], nor of the extraordinarily small concentration of calcium ions inside cells. There may of course be other reasons the formalism [6] is inadequate. In summary, experiments, theory, computations and perhaps simulations are needed to show that responses to steps of voltage allow computation of a calcium driven action potential.

The polarization protocol described here can be applied to simulations of polarization as well as experimental measurements of polarization. Indeed, the operational definition of polarization has been applied even when theories [18] or simulations are enormously complicated by atomic detail that includes the individual motions of thousands of atoms [21,88].

Another question of general interest is how does the polarization defined this way correspond to the polarization −**P** = (*ε<sup>r</sup>* − 1)*ε*0**E** in the classical formulation of the Maxwell Equations (7) and implied in Equations (1) and (6)? Does the estimated polarization equal **P**?

The answer is not pleasing. Polarization cannot be defined in general. The variety of possible responses of matter to a step of potential prevents a general answer. Indeed, a main point of this paper is that polarization must be defined by a protocol in a specific setting that specifies how the local electric field changes the distribution of charge.

Polarization cannot be defined in general because there are too many possible motions of mass with charge in response to a change in the electric field. Every possible motion of mass (with charge), including rotations and translations and changes of shape and density of charge, would produce a polarization. Polarization currents can be as complicated as the motions of matter.

In mechanical systems in general these issues do not attract much attention. It seems obvious that one must have a model and theory of how a system changes shape (and distribution of mass) when forces are applied. Seeking a general treatment is silly. In electrodynamics, for illogical reasons of history, tradition, and respect for our elders, scientists have sought the general treatment that would be considered silly for mechanical systems.

Scientists, certainly including me, have used the simple electromechanical model of an ideal dielectric to describe how charge moves in response to an electric field, using the name polarization to describe the phenomena. They have tried to apply it everywhere, as is seen because that model is embedded in the traditional formulation of the Maxwell equations found universally in textbooks.

It seems to me time to abandon this forlorn hope of a general description of the response of charged matter to a change in the electric field, and to move to a more reasonable approach, in which explicit models of the response of charge to the electric field are constructed, with different models for different systems.

Insight can be developed into various kinds of polarization by constructing 'toy' models of simple systems. Those models must specify the mechanical variables **v***Q*, **Q***Q*, ρ*<sup>Q</sup>* and **N***<sup>Q</sup>* (or their equivalent) and solve the field equations of mechanics, perhaps including diffusion, along with the Maxwell equations. The models are then studied using the operational definition of polarization, described previously (Figure 1) or other operational definitions more suitable for other systems. One can hope some of the models resemble some of the more elaborate models of polarization already in the literature [26–29,31,32,34].

Toy models might include:


These examples, taken together, will help form a handbook of practical examples closely related to the classical approximations of dielectrics.

These problems have time dependent solutions except in degenerate, uninteresting cases. Time dependence poses particular problems for the classical formulations of Maxwell equations. As stated in [51] on p. 13.

"It is necessary also to reiterate that *ε<sup>r</sup>* is a single, real positive constant in Maxwell's equations as he wrote them and as they have been stated in many textbooks since then, following [108–110]. If one wishes to generalize *ε<sup>r</sup>* so that it more realistically describes the properties of matter, one must actually change the differential Equation (6) and the set of Maxwell's equations as a whole. If, to cite a common (but not universal) example, *ε<sup>r</sup>* is to be generalized to a time dependent function, (because polarization current in this case is a time dependent solution of a linear, often constant coefficient, differential equation that depends only on the local electric field), the mathematical structure of Maxwell's equations changes".

Perhaps it is tempting to take a short cut by simply converting *ε<sup>r</sup>* into a function of time *εr*(*t*) in Maxwell's equations, as classically written. "Solving the equations with a constant *ε<sup>r</sup>* and then letting *ε<sup>r</sup>* become a function of time creates a mathematical chimera that is not correct. The chimera is not a solution of the equations." The full functional form, or differential equation for *εr*(*t*) must be written and solved together with the Maxwell equations. This is a formidable task in any case, but becomes an even more formidable challenge if convection or electrodiffusion modify polarization, as well as the electric field.

If one confines oneself to sinusoidal systems (as in classical impedance or dielectric spectroscopy [27,42,111,112]), one should explicitly introduce the sinusoids into the equations and not just assume that the simplified treatment of sinusoids in elementary circuit theory [113–117] is correct. It is not at all clear that Maxwell's equations joined with constitutive equations; and boundary conditions always have steady state solutions in the sinusoidal case. The Maxwell equations joined with diffusion and convection equations (like Navier–Stokes [118–135] or PNP = Poisson Nernst Planck = drift diffusion [52,53,55,57,59,61,123,136–145]) certainly do not always have solutions that are linear functions of just the electric field [146–149]."

It seems clear that the classical Maxwell equations with the over-approximated dielectric coefficient *ε<sup>r</sup>* cannot emerge in the time dependent case. Of course, the classical Maxwell equations cannot emerge when polarization has a nonlinear dependence on the electric field, or depends on the global (not local) electric field, or depends on convection or electrodiffusion.

Indeed, in my opinion, when confronted with the models of polarization listed on the previous page, the classical Maxwell equations will be useful only when knowledge of the actual properties of polarization is not available. All the models listed involve time dependence in the polarization fields that are not included in the classical Maxwell equations as usually written.

#### **5. Conclusions**

A generalization of Maxwell's **P** useful in a range of systems may emerge. The generalization would describe how the local electric field changes the distribution of charge, as one imagines that Maxwell hoped **P** and **D** would be.

Until then, one is left with:


What should be done when little is known? Sadly, the actual properties of polarization are often unknown. Then, one is left with the over-approximated Equation (6) or nothing at all. It is almost never wise to assume polarization effects are negligible. Equation (6) is certainly better than nothing: Equation (6) can be particularly helpful if it is used gingerly: toy models can successfully represent an idealized view of a part of the real world of technological or biological importance, for example, electronic circuits or several properties of ion channels.

In some cases, the toy models can be enormously helpful. They allow the design of circuits in our analog and digital electronic technology [150–153]. They allow the understanding of selectivity [107,154–156] and current voltage relations of several important biological channel proteins in a wide range of solutions [107,157–159]. In other cases—for example, the description of ionic solutions with many components—the toy models can be too unrealistic to be useful. Experiments and experience can tell how useful the toy model actually is in a particular case: pure thought usually cannot.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not available.

**Acknowledgments:** I thank the reviewers for suggestions and criticisms that substantially improved the paper. It is a particular pleasure to thank my friend and teacher Chun Liu for his continual encouragement and advice, and for patiently correcting mistakes in my mathematics as these ideas were developed over many years. Mistakes may remain, sad to say. All are my responsibility.

**Conflicts of Interest:** The author declares no conflict of interest.

## **Appendix A P**(*x*, *y*, *z*|*t*) **in Proteins**

Ambiguities in the meaning of the polarization field **P**(*x*, *y*, *z*|*t*) can cause serious difficulties in the understanding of protein function. Understanding protein function is greatly aided by knowledge of protein structure. The protein data bank contains 173,754 structures in atomic detail today (24 January 2021) and the number is growing rapidly as cryo-electron microscopy is used more and more.

Protein structures are usually analyzed with molecular dynamics programs that assume periodic boundary conditions and chemical equilibrium, i.e., no flows. Most proteins control large flows as part of their natural biological functions. Equilibrium hardly ever occurs in living biological systems. It seems obvious that equilibrium systems cannot provide general insight into flows, any more than a nonfunctional amplifier without a power supply can show how a functional amplifier works. Proteins are not periodic in their natural setting. It seems obvious that periodic systems with flow cannot conserve total current **J***total* in general—or perhaps even in particular—as required by the Maxwell equations, see Equation (19). In other words, it is likely that molecular dynamics analyses of periodic structures do not satisfy the Maxwell equations, although almost all known physics does satisfy those equations.

It is also unlikely that standard programs of molecular dynamics compute electrodynamics of nonperiodic systems correctly, despite their use of Ewald sums, with various conventions, and force fields (tailored to fit macroscopic, not quantum mechanical) data. Compare the exhaustive methods used to validate results in computational electronics [61] with those in the computation of electric fields in proteins.

The electrostatic and electrodynamic properties of proteins are of great importance. Many of the atoms in a protein are assigned permanent charge greater than 0.2**e** in the force fields used in molecular dynamics, where **e** is the elementary charge, and these charges tend to cluster in locations most important for biological function, just as they cluster at high density near the electrodes of batteries and other electrochemical systems. Enormous densities of charge (>10 M, sometimes much larger) are found in and near channels of proteins [107,160–162] and in the 'catalytic active sites' [163] of enzymes. Such densities are also found near nucleic acids, DNA and (all types of) RNA and binding sites of proteins in general.

It seems likely that a hierarchy of models of different resolutions will be needed to compute the electrodynamics of proteins accurately enough to explain how the electrical properties of side chains (polarizability [21] and others) of a protein determine biological function. Analysis of gating currents suggests such an approach is feasible [17,18,20,21].

#### **References**


## *Article* **Unraveling of a Strongly Correlated Dynamical Network of Residues Controlling the Permeation of Potassium in KcsA Ion Channel**

**Salvatore M. Cosseddu † , Eunju Julia Choe and Igor A. Khovanov \***

> School of Engineering, University of Warwick, Coventry CV4 7AL, UK; salvatore.cosseddu@gmail.com (S.M.C.); e.j.choe.00@gmail.com (E.J.C.)

**\*** Correspondence: i.khovanov@warwick.ac.uk

† Current address: Viseca Payment Services SA, Hagenholzstrasse 56, 8050 Zürich, Switzeland.

**Abstract:** The complicated patterns of the single-channel currents in potassium ion channel KcsA are governed by the structural variability of the selectivity filter. A comparative analysis of the dynamics of the wild type KcsA channel and several of its mutants showing different conducting patterns was performed. A strongly correlated dynamical network of interacting residues is found to play a key role in regulating the state of the wild type channel. The network is centered on the aspartate D80 which plays the role of a hub by strong interacting via hydrogen bonds with residues E71, R64, R89, and W67. Residue D80 also affects the selectivity filter via its backbones. This network further compromises ions and water molecules located inside the channel that results in the mutual influence: the permeation depends on the configuration of residues in the network, and the dynamics of network's residues depends on locations of ions and water molecules inside the selectivity filter. Some features of the network provide a further understanding of experimental results describing the KcsA activity. In particular, the necessity of anionic lipids to be present for functioning the channel is explained by the interaction between the lipids and the arginine residues R64 and R89 that prevents destabilizing the structure of the selectivity filter.

**Keywords:** ion channels; protein dynamics; molecular dynamics

#### **1. Introduction**

Over the last few decades, the bacterial K<sup>+</sup> ion channel KcsA [1] found in *Streptomyces lividans* has been widely studied in order to understand the structural and functional features of potassium ion channels. It continues to be of interest [2–8] in part due to its sequence similarity to eukaryotic K<sup>+</sup> channels, and in part because of its role as an archetype for ion permeation, selectivity, and the complex interplay of the different "gates" which governs a variety of current patterns observed experimentally in the K<sup>+</sup> channel superfamily [9–11]. These patterns are defined by small structural rearrangements of the pore region once the inner gate is opened [4,10–13]. The local rearrangements are mostly obscure as current experimental techniques are unable to provide the combination of spatial and temporal resolution needed to identify the underlying atomistic-level mechanisms. Structural studies showed that the current patterns depend on a number of residues, some of which are located relatively far from the pathway of K<sup>+</sup> permeation, and physiological recordings revealed the strong influence of the K<sup>+</sup> concentration in the outer bulk on the patterns [1,10,14–19]. The anionic phospholipids modulate the function of the channel [20–22] and the addition of phosphatidic acid lipid significantly affects the permeation [23].

As with most of the K<sup>+</sup> ion channels, KcsA contains a highly conserved amino acid sequence motif TXXTXGYGD known as the *signature sequence*, which corresponds to residues 75 to 79 in the reference X-ray structure 1K4C [1], where "X" in position 76 is

**Citation:** Cosseddu, S.M.; Choe, E.J.; Khovanov, I.A. Unraveling of a Strongly Correlated Dynamical Network of Residues Controlling the Permeation of Potassium in KcsA Ion Channel. *Entropy* **2021**, *23*, 72. https://doi.org/10.3390/e23010072

Received: 10 November 2020 Accepted: 2 January 2021 Published: 6 January 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/ licenses/by/4.0/).

replaced by valine. The whole quaternary structure of KcsA is divided into three functional regions: the selectivity filter, a water-filled cavity, and an inner gate associated with large movements in the transmembrane helices for opening the channel [16,24–27]. The selectivity filter (SF) is the narrowest part of the pore. The SF consists of five well-defined binding sites for K<sup>+</sup> ions by exposing the backbone carbonyl groups of the residues toward the channel axis [28]. These sites are commonly labeled as S0 (below T75), S1 (between T75 and V76), S2 (between V76 and G77), S3 (between G77 and Y78), and S4 (between Y78 and G79). The permeation is forced to occur in a single file fashion as a hopping of an ion from one site to another site.

The filter plays a role in both ion selectivity and modulating the current. The latter corresponds to random-like switching (gating) between zero and finite values of the current. Once the inner gate is opened, the current is regulated by small structural rearrangements. They are responsible for different gating processes, such as the *C-type inactivation* and the *modal-gating*, from which complex patterns of ion current arise [13,14,16,17,28–30]. The C-type inactivation corresponds to very long inactive (zero current) time intervals under steady-state conditions. The modal gating is associated with three different modes of the single-channel currents in KcsA. Two modes correspond to a high and low probability of the pore to be in the conducting (active) state, respectively. Third mode is a high-frequency flicker mode representing in bursts of fast switching back and forth between active and inactive states [13,14].

The inactivation in the KcsA channel is a common feature in functioning potassium channels, including eukaryotic ones [10]. Therefore, the C-type inactivation has been extensively studied using a variety of different experimental techniques such as crystallography, NMR, ssNMR, fluorescence measurements, and computational studies, leading to several hypotheses reflected in the recent detailed review [6]. A combination of structural (X-ray) studies and physiological measurements of the wild type (WT) of KcsA and its different mutants suggests that several residues behind the SF could be involved in filter's structural rearrangements during the inactivation [9,14,15,18,19,29–33]. These studies led to the suggestion of four channel's states with an open or closed inner gate and a conducting or non-conducting SF [4,11]. One of the hypotheses [6,11,34] suggests that the activation by opening the inner gate simultaneously alters the SF via allosteric coupling [35,36]. This coupling leads to a slow (on a time scale of seconds) collapse of the SF to a non-conducting configuration. Although structures corresponding to an inactive channel with closed and open inner gate were reported [11,28,30], a structure of an active channel with an open gate and a conducting SF is still missing. Note that the canonical structure 1K4C [28] with a conductive configuration of the SF has a closed inner gate. Another set of experiments used mutagenesis of residues in the SF and demonstrated that the ion occupancy in specific sites controls the inactivation [37–39]. This result leads to the second hypothesis that the SF alone could play the role of an "inactivating gate" without the involvement of the inner gate [6]. This hypothesis tightly links to experimental observations that the SF's conformational dynamics in the WT KcsA and its mutants govern gating properties in the KcsA channel [14,40,41]. Although these two hypotheses are sometimes considered controversial [6,42], they could coexist and reflect the complexity of the KcsA channel.

An additional complication to this gating-permeation picture is the dependence of the K <sup>+</sup> current and the filter rearrangements on the extracellular K<sup>+</sup> concentrations, common among numerous K<sup>+</sup> channels [9,10,38,43,44]. The probability of the inactivation grows with decreasing K<sup>+</sup> concentration. This effect has been suggested to link to a "footin-the-door" mechanism in which an ion resident in the filter stabilizes the conductive conformation and reduces the inactivation probability [9,16,38]. The exact location of the binding site responsible for the effect is unknown. However, it is suggested such site can be located either at the extracellular mouth or in the central region of the selectivity filter [10,38,45].

In the majority of these studies static (crystallographic) X-ray structures were used for describing the function. However, these static pictures do not provide details of the

essentially dynamical picture of the inactivation. Therefore, general mechanistic knowledge of the gating behavior, which comprises transitions between various states, remains obscure [4,10,11,13]. Recent applications of solid-state NMR [41], 2D IT spectroscopy [2], and florescences measurements [46] for analyzing channel dynamics could address the uncertainties in functional relevance of crystallographic structures. However, a mechanistic picture of the filter's rearrangements with simultaneous dynamical analysis of ions and water molecules is beyond the current experimental techniques. Molecular dynamics (MD) simulations offer valuable tools for exploring dynamical properties at the atomic level [47–49]. For example, MD helped discriminate between "knock-on" and "snug-fit" mechanisms of the permeation in the KcsA channel [50]. In turn, the structural study [51] recently resolved some controversy in MD simulations [2,3] on water involvement in knock-on mechanisms.

The inactivation hypotheses were also discussed by applying MD approaches [34,52]. These computational studies concluded that the activation via opening the inner gate affects the low site S0 in the SF by enhancing the permeation [34] and controlling the SF's stability [52]. The latter result leads to new perspectives [53] for the inactivation mechanism as a process tightly controlled by the inner gate, which could be in different partially open states [52]. The dynamics near SF becomes less important in this picture. A quick collapse of the SF in the case of a widely open inner gate was observed [52]. The collapse happens on much shorter than broader time scales of the inactivation, which can be the order of seconds. The same time scale for collapsing the SF was reported in recent unbiased simulations of a similar open structure [54]. The former result [34] partly supports these new perspectives as it shows that configurations of the inner gate affect the permeation. However, a collapsed SF has not been reported for the performed biased MD simulations for the open inner gate [34]. As mentioned above, a crystallographic structure of the KcsA channel with a conducting SF and an open inner gate is not available, so such a structure was created in silico [34,52] using combinations of the reported structures [55]. Differences in structures used for creating proteins with an open inner gate could explain some contradictions in those MD approaches [34,52].

Heer et al. [34] also reported that the permeation barrier in the canonical (a conducting SF and a closed inner gate) structure 1K4C [1] is too high to consider its SF configuration as conducting. This conclusion was derived from biased simulations using the umbrella sampling method [56]. The obtained barrier was found to be too high for observing the permeation rate according to experimental recordings [57]. This result is in line with the work reported earlier by Fowler et al. [58]. In contrast, other unbiased MD simulations [59–61] confirmed the conducting state of SF. Note that the SF of structure 1K4C was used in the majority of the simulations mentioned above. Two major factors could explain such discrepancies. The first factor is the use of either biased or unbiased MD approaches. The second factor is defined by differences between obtained in silico structures with an open inner gate. While generating a new structure in silico applied a tight control of SF backbones and ions' and water molecules' locations, other residues were not over-sighted. In biased approaches, just one or two so-called collective variables (typically ions locations) were considered assuming that the dynamics of all other variables (water molecules and residues) can be averaged out. Yet, in unbiased approaches some constraints are applied on the protein during MD simulations.

Thus, conformations and behavior of many residues, especially in the region of the SF, were kept out of the consideration despite the experimental studies that identified a number of residues strongly altering the inactivation and gating [9,14,15,18,19,29–33]. A series of papers by Cordero et al. [9,14,29] suggested that the stability of the SF depends on a hydrogen-bond (H-bond) network formed by the triad of residues E71-D80-W67. In particular, the substitution of glutamate E71 with alanine A71 suppresses the inactivation, and the conduction is observed even in low K<sup>+</sup> concentrations [14]. Therefore, there is a gap in understanding how states of this triad are linked to the permeation. In this manuscript, we aim to provide a mechanistic picture of rearrangements in the WT KcsA protein and

discuss the mechanisms by which residues behind the SF interacts with the backbones of the SF, and ions and water molecules within. This picture is an essential piece of the inactivation puzzle and in addressing issues of MD biased simulations.

A large number of residues in the SF region of the KcsA protein means that a brute force (combinatoric) consideration of all possible combinations of different residues states is unrealistic. The state-of-the-art microseconds MD simulations [52,54] show that structure 1KC4 adapts one of the multistable states and no rearrangements of residues behind the SF were reported. In the present work, therefore, we first conduct a comparative analysis between the WT protein and different mutants (E71A, Y82A, R64A, and L81A) (see Figure 1) where key residues are replaced by the short, weakly interacting alanine. The selection of the mutated structures is based on previous experiments [9,14,15,18,19,29–33] which reported different probability of the inactivation. MD simulations were combined with biased free-energy methods, well-tempered metadynamics [62], and statistical analysis. The biased simulations introduce additional perturbations into the protein and, therefore, verify the stability and thermodynamics of different states of the SF. The results of MD simulations are critically assessed against published experimental and computational investigations. The study was designed to unveil the complex dynamics that underlie the permeation path in the WT KcsA protein and has allowed us to identify a cooperative network of dynamically interacting residues located near the SF. Note that preliminary results of this study were reported in work [63].

In this paper, first, an analysis of residues' dynamics in mutated structure E71A is presented. The relationship between conformational changes at the SF and rearrangements of residue D80, located at the channel's outer entrance, is explored. Second, a network of residues, which affect the ion permeation, is identified by comparing the dynamics of proteins WT, Y82A, R64A and L81A. Third, a thorough description of the network dynamics, including energetics of transitions in the network, and its influence on the filter structure and the ion permeation is presented.

**Figure 1.** A region near the SF in the different proteins: WT, E71A, Y82A, R64A, and L81A, is shown. With the exception of the mutated residues, the other residues are in the X-ray conformation [28]. Ions are shown as purple spheres interacting with oxygen atoms (red color) of residues in the SF. The key residues are highlighted by different colors, mutated residues are shown in blue.

#### **2. Methods**

#### *2.1. Setup of the Simulations*

The simulations were performed using NAMD 2.8 and 2.9 [64] in the NPT ensemble with pressure 1.01 bar and temperature 310 K. A multiple timestep algorithm was used [65,66]. In the case of unbiased simulations the integration step size was 1 fs, nonbonded nonelectrostatic interactions were calculated every 2 fs, and electrostatic forces [67] every 4 fs. In biased simulations, the step size was 2 fs, nonbonded nonelectrostatic interactions were calculated every 2 fs, and electrostatic forces every 6 fs. The CHARMM27 force field (FF) was used for the protein, with a modification in the Lennard–Jones term to represent the interaction between K<sup>+</sup> and the carbonyl oxygens of the protein, CHARMM36 for the lipids, and TIP3P for water were applied [50,68–72]. The system was prepared by embedding the X-ray structure (pdb code 1K4C; solved at 2 Å resolution [28]) with 2 K<sup>+</sup> in the SF and 1 K<sup>+</sup> in the cavity, in a membrane patch of 222 molecules of 1-palmitoyl-2-oleoylphosphatidylcholine (POPC), and solvated by 17740 water molecules [73–75]. A potassium concentration in the aqueous phase of 0.2 M was obtained with 63 K<sup>+</sup> ions, and the system was neutralized by 75 Cl− ions. The ions were distributed over the whole simulation box. Relaxation of the system and preparation of the mutants is described in Supplementary Materials.

Coordinates, if not otherwise stated, were considered every 2 ps, ignoring an initial equilibration period of 1 ns.

#### *2.2. Collective Variables and Order Parameters*

Collective variables (order parameters) used in this work are defined as follows. (i) Variables *ψ*<sup>76</sup> and *ψ*<sup>81</sup> are the *ψ* dihedral angles measured for residues indicated in the subscripts, and they follow the standard definition. (ii) Variable *χ*1<sup>81</sup> is the *χ*1 dihedral angle of the L81 residue, which follows the standard definition as well. (iii) Variable SC<sup>80</sup> is the position of the D80 side chain considered as the distance between C*<sup>γ</sup>* atom of D80 and a reference atom, C*<sup>α</sup>* of A73. Note that the latter residue shows the lowest fluctuations in RMSD analysis. (iv) The distance D80–R89 is between C*<sup>γ</sup>* atom of D80 and the C*<sup>ζ</sup>* atom of the closest R89 residue in the quaternary structure. (v) SF length, the length of the TVGYG sequence, is measured as the distance between the C*<sup>α</sup>* atoms of residues T75 and G79. (vi) The distance R64–SF is measured between C*<sup>ζ</sup>* atom of R64 and the center of mass (COM) of the selectivity filter. (vii) The distance E71–D80 is between C*<sup>γ</sup>* atom of D80 and the H-bond donor oxygen of E71. (viii) The coordinates *zK*<sup>1</sup> and *zK*<sup>2</sup> are the *z* coordinates of the K<sup>+</sup> ions bound to the filter (ions labeled as K1 and K2 in Figure 5); the coordinate system has been centered with respect of the COM of the SF, in order to remove the components associated with the protein diffusion in the membrane.

The COM of the SF was defined by the atoms N, C*α*, and C of residues from 74 to 78 of all four subunits.

#### *2.3. Free Energy Calculations—Metadynamics*

Different approaches are used to enhance the sampling when high energetic barriers between states do not allow an appropriate sampling for the investigation of rare events and the reconstruction of the free energies. These are often based on non-Boltzmann sampling.

Well-tempered metadynamics (wt-metaD) is a non-Boltzmann sampling method based on a history-dependent bias potential, created as a sum of Gaussians centered along the trajectory of specified collective variables (CVs) [62,76,77]. In wt-metaD technique, the height of Gaussians added is history-dependent, and this dependence is associated with a parameter ∆*T* having the dimension of temperature. This parameter was adjusted for each simulation. The NAMD package [64] includes module *colvar* for performing wtmetaD. Additional details of the implementation of wt-metaD and the selection of the relevant parameters are reported in the Supplementary Materials, section "Well-tempered Metadynamics".

#### *2.4. Initialization of WT-R64D80 Simulation*

For the simulation denoted as WT-R64D80, an equilibrated conformation of WT KcsA simulated for 6 ns was used. During first 20 ps of the relaxation, residues L81 and R64 were restrained. Every residue L81 was restrained towards the flipped state by the harmonic potential with a spring constant of 24 kcal/mol degree<sup>2</sup> and centered on 185◦ . The harmonic potential (spring constant 20 kcal/mol degree<sup>2</sup> centered on −160◦ ) was applied on *χ*1 dihedral angle of each R64. Note that the latter restraints were added to speed up the calculation, but are not strictly necessary to obtain the desired configuration. A further 25 ps of relaxation were performed without any restraint.

#### *2.5. Statistical Analysis*

The statistical analyses were performed using VMD 1.9 [78] and R software environment [79]. Several packages for R were used in addition to the core functions: bio3d, ggplot2, car, and MASS [80–84].

All the free-energy surfaces (FES) presented in this work were smoothed via cubic smoothing spline (grid length 80) and thin plate spline methods (grid sizes 80 × 80) which are implemented in R packages stats v2.15.3 [79] and fields v6.7.6 [85], respectively.

#### **3. Comparative Analysis of Dynamics of WT and Mutated Proteins**

#### *3.1. Considered Proteins*

The simulations commenced from relaxed systems, prepared from the X-ray structure solved at 2 Å resolution [28], as explained in the previous section "Methods". The KcsA channel has a tetrameric structure, and the four subunits of the KcsA are referred by capital letters A, B, C, and D. The SF is described as a five-site pore [24,25] through which ions and water molecules move in a single-file fashion. The standard notation of the sites is used: S0 to S4 starting from the outer site. The configurations of the SF are described by a fivecharacter string (from S0 to S4), where a "K" represents a K<sup>+</sup> ion, "w" a water molecule, "0" a vacancy; when a K<sup>+</sup> is present in the cavity a "K" is appended, separated from the filter occupancy by sign "+". For example, the configuration wKwKw+K means the presence of K <sup>+</sup> ions in S1, S3, and the cavity separated by water molecules. In comparison, KwK0K implies the presence of K<sup>+</sup> ions in S0, S2, and S4, a water molecule in S1, a vacancy in S3 and no a K<sup>+</sup> ion in the cavity. Consistent to the previous literature [14,86], the results are described by considering the extracellular region as an outer region and "up" in the frame of reference, while the intracellular region is considered as inner and "down".

Among the numerous mutants, which differ from the WT in the gating behavior, three proteins have been considered: (i) E71A is be resistant to the inactivation, (ii) R64A shows a sharp reduction of the inactivation, and (iii) Y82A demonstrates an enhancement of the rate and extent of the inactivation [14]. It is, therefore, possible to specify a trend in the inactivation probability of these proteins: E71A < R64A < WT < Y82A. An additional mutant, L81A, was created for testing the roles of residues L81 and R64, and their coupled motions.

#### *3.2. Dynamics of Mutant E71A*

The link between residues E71 and D80 is considered to be an important one for KcsA functioning. A special patch in the force fields was introduced to tune the link for observing ions' conduction [86]. However, the mutation of glutamate (E71) to alanine (A71) does not affect the conductivity and, moreover, it suppresses the inactivation. This observation means that other residues play an essential role to keep the SF in a conducting configuration.

The mutation by replacing glutamate E in position 71 by alanine A results in the structure E71A which was studied experimentally by Cordero et al. [14]. The authors demonstrated that the permeation path undergoes large conformational rearrangements in the non-inactivating mutant E71A. The rearrangements primarily occur in the region of V76 residue. Additionally, the authors [14] reported a strong upward movement of residue D80 relative to its position in the WT structure, leading to the "flipped E71A" structure.

For understanding the influence of the mutation on the dynamics and interactions of residues, an unbiased MD simulation of length 24 ns was performed. Several rearrangements in the permeation path were observed during the simulation. The most noticeable changes happened among the residues of the TVGYG sequence in the subunit B. Rotation of the V76-G77 peptide group occurred at 9 ns, and the rotation remained stable until the end of the simulation. Similar transitions have been reported in the literature for both WT and E71A. Many hypotheses [30,58,87–90] have been proposed for explaining the transitions which are usually referred to as "flipping of V76". However, understanding the significance and origin of the transitions is still missing.

The flipping of V76 has been suggested by different authors to be able to generate nonconductive conformations associated with the C-type inactivation or flicker mode [30,88]. We performed various permeation tests on V76 flipped conformations of E71A and WT by performing unbiased simulations with two ions in the cavity (see Supplementary Materials, section "Permeation in the V76 flipped configurations of E71A and WT"). The simulations revealed that reverse transitions of V76 occurring easily in the case of K<sup>+</sup> permeation. This result supports the hypothesis of Domene et al. [89] that flipping of V76 is not responsible for the C-type inactivation. Furthermore, the observed conductivity suggests that the flipping of V76 alone is not sufficient even for short-living inactive states, which are associated to the modal-gating, and that additional conformational readjustments are necessary for generating meta-stable non-conductive states.

The simulation of E71A showed that interactions between D80 and an arginine nearby (R89) could trigger structural rearrangements of the filter. D80 side chains, which are negatively charged, demonstrated relatively large fluctuations towards the extracellular region (see Supplementary Materials, Figure S2). These fluctuations are promoted by strong inter-domain electrostatic interactions with positively charged arginines R89. The interaction between the residues D80 and R89, and the corresponding rearrangements of the SF, are illustrated in Figure 2. The conformational space in Figure 2a is defined by the three order parameters (conformational changes for subunit B only are shown): (i) the dihedral angle *ψ* of V76 (*ψ*76); (ii) the position of the D80 side chain (SC80); (iii) the distance between D80 and nearest R89 (D80–R89). Initially, the dynamics of D80 and R89 appear uncorrelated (blue clouds). Some correlations arose (light blue cloud) as the time advanced because of an intermittent creation of a H-bond (D80–R89 distance is around 3 Å) between D80 and R89. Note that similar H-bonds between D80 and R89 have been reported in the literature also occurring for the WT structure [91]. The presence of the D80–R89 H-bond in E71A protein is associated with a small drift in the position of D80, SC<sup>80</sup> is changed from 13.5 to 13.8 Å (see Supplementary Materials, Figures S3 and S4, for more details). The temporary strengthening and stabilization of the H-bond was accompanied by a distortion of the filter structure (in Figure 2a clouds blue to green, and in Figure 2b structure green to colored). Residue V76 assumed a *partially flipped* conformation (*ψ*<sup>76</sup> ≈ 50◦ ) in the distorted structure. This observation is an important result since it demonstrates that the backbone structure of the sequence GYGD is rigid enough for delivering a perturbation from D80 to the V76-G77 peptide group. It is shown below that the rigidity of the GYGD backbone strongly affects the SF flexibility.

Time series reported in Supplementary Materials (Figure S3) further demonstrated that, in turn, V76 partial flipping affected the permeating K<sup>+</sup> ions, causing an inward shift of the outermost ion K1. Thus, ions' dynamics are linked with the dynamics of residues behind the filter, D80 and R89. The partially flipped conformation of V76 appears to be unstable and evolved into a complete flipping of V76. The D80–R89 H-bond caused additional small transitions in the TVGYGD sequence until a slight movement of the D80 towards the extracellular side (see Figure S3 in Supplementary Materials and Figure 2b colored to yellow) restored the initial uncorrelated motions of the D80 and R89 (red clouds in Figure 2a) causing a breakage of the H-bond.

**Figure 2.** The influence of residues D80 and R89 on structural rearrangements in mutant E71A. (**a**) The evolution of the system (subunit B, initial 1 ns ignored as the relaxation interval) revealed that the stress induced by the D80–R89 H-bond led to rearrangements in the filter structure (the flipping of V76) and to an outward transition of D80. The conformational space is defined by a set of the order parameters (see "Methods"): (i) SC<sup>80</sup> the position of side chain D80; (ii) D80–R89 distance, where residue R89 belongs to the neighboring subunit; and (iii) *ψ*76. Time evolution of the system in the conformational space is coded by color scale shown in the colorbar. (**b**) Superposition of snapshots from the simulation of E71A: an initial configuration (green drawing); a configuration with the D80–R89 H-bond and partially flipped V76 (colored drawing); and a configuration at the end of the simulation (yellow drawing).

Although the described path is one among many available toward a V76 flipped configuration in protein E71A, these results demonstrate that the creation of H-bonds with residue D80 can trigger structural rearrangements which propagate to the filter because of the relative rigidity of the GYGD sequence backbone. The arginine R89 is able to promote the triggering transitions by creating a strong H-bond with residue D80. In the following sections, further evidence is presented for confirming that all the residues which can form H-bonds with D80 play a significant role in conformational rearrangements of the permeation path.

#### *3.3. Correlated Dynamics of L81 and R64 Residues*

The results of the previous subsection indicate that residue E71 plays an essential role in the inactivation and, therefore, in the dynamics of the WT protein. In mutant E71A, alanine in position 71 does not form bonds with D80 and residue D80 is very flexible. In contrast, in the WT protein, residue D80 is restrained by a strong link between D80 and E71. For the identification of residues that affect the permeation path, we performed a comparative analysis of three different proteins in which E71 is present. The selected proteins are the WT protein and mutants Y82A and R64A. These mutants show distinct behaviors for the inactivation: Y82A has significantly higher, and R64A has reduced the inactivation probability in comparison to WT. All three structure were simulated starting with the same initial configuration (excluding mutated residues) for different but comparable intervals: 38 ns, 28.5 ns and 23 ns for WT, Y82A and R64A respectively. Note that in the WT protein, residue R64 directly interacts with L81, which is a neighbor of residue Y82 (Figure 1).

The root mean square displacements (RMSDs) of the backbone atoms of each residue reveal residues which showed different behaviors across the three selected proteins; the X-ray structure of WT was used as the reference [28]. The results are reported in Figure S5 in the Supplementary Materials. The RMSDs analysis shows that fluctuations of the arginine R89 are wider in the proteins with a higher probability of inactivation, WT and Y82A, than in R64A. This observation additionally supports the hypothesis of a particular role of this arginine in the conformational variability of the pore. However, residue R89 in proteins WT and Y82A show similar RMSDs, and the difference in RMSDs of R89 in R64A and WT structures is relatively small. These facts imply that the dynamics of R69 by itself cannot account for the substantial diversity in the inactivation between these three proteins.

A closer inspection reveals the importance of second arginine residue, R64, which has relatively large RMSDs in WT and Y82A. The mutation of this arginine with alanine in structure R64A leads to a significant reduction of the RMSDs of the residue in position 64. In WT and Y82A proteins, arginine R64 can approach and interact with D80 and create strong H-bonds similarly to R89 in mutant E71A (Figure 3a). The possibility of a H-bond between R64 and D80 is important considering that R64 is located relatively far from D80 in the static structure provided by X-ray experiments [28] (D80-R64 distance = 9.3 Å). Residue R64 fluctuated over wide ranges and, more importantly, it can destabilize linkages between the triad of E71-D80-W67 via the interaction with residue D80 (Figure 3a). This interaction occurs more prominently in mutant Y82A, the simulation of which ended with a broken triad E71-D80-W67 in two subunits. As a result of the R64–D80 interaction, residue D80 can rotate around the dihedral angle *χ*1 and such rotations were observed a few times during simulations (see Figure S6 in the Supplementary Materials). In Y82A and WT proteins, the flexibility of D80 promoted by R64 leads to several multistable configurations, one of which includes a broken E71–D80 link. Note that this link is stable though the whole simulation of mutant R64A. Thus, residues R64 in WT and Y82A proteins play the destabilizing role.

In both WT and Y82A proteins, arginine R64 can interact with D80, but these structures demonstrate different inactivation behavior. Our simulations indicate that the difference in the inactivation has a dynamical origin. Residue R64 moves faster and creates quicker a H-bond with D80 in mutant Y82A than in WT. The rate of H-bond creation depends on the conformation of the leucine in position 81 (L81). This rate primarily controls by the rotation of L81 side chain, which can open by *flipping*, when angle *χ*1<sup>81</sup> changes from −63◦ to 185◦ , or obstruct, when residue L81 is in that conformation found in the crystallographic structure, the path toward forming the D80–R64 H-bond (Figure 3a). Conformational changes of L81 have, therefore, a critical regulatory role in the dynamics of residue R64.

**Figure 3.** (**a**) The disruption of E71–D80–W67 linkages in Y82A and WT, promoted by the D80–R64 interaction. Snapshots and the final configurations for simulations of Y82A and WT are reported. Residues in the snapshots are superimposed with respect of the heavy atoms of the SF. The color sequence in the snapshots is (i) blue, an initial state; (ii) green, a transition state (distinguished only for the WT and characterized by the E71 *χ*1 angle of 120 degrees); and (iii) red, E71-D80-W71 linkages disrupted. Distances in the figures are reported in Å. (**b**) Comparison of the probability density of the D80–R64 distance in WT, Y82A, and L81A calculated for 10 ns of simulation. The initial D80–R64 distance is 9.3 Å and shown with the dashed magenta line. Distance D80–R64 which is less than 5.2 Å indicates the formation of the D80–R64 H-bond. Data from all the four subunits were used.

In turn, the dynamics of L81 are associated with additional readjustments in the amino acid sequence L81-X82-P83-V84, roughly definable as pivoting around the residue in position 82 (*X82-pivoting* (Figure 3a). Collective motions of this sequence can promote the flipping of L81 and a small drift of its backbone. The X82-pivoting is different in three considered proteins. The main differences are reflected in the RMSDs of residues surrounding residue 82 (X82) (Figure S5 in the Supplementary Materials). In all the three proteins, the RMSDs are similar for X82 (where X is tyrosine Y in the WT and R64A and alanine A in Y82A). In contrast, the RMSDs of the surrounding residues (L81, P83, and V84) correspond to the inactivation probabilities R64A < WT < Y82A, that is, the RMSDs are larger for Y82A and smaller for R64A than for WT. The lowest RMSDs for mutant R64A are due to the absence of a residue in position 64, which is capable of interacting with L81 via X82-pivoting. In WT and Y82A proteins, the R64–L81 interaction is controlled by bulky tyrosine Y82 and non-bulky alanine A82, respectively. In the WT protein, therefore, the motion of L81 is slower and more limited than in mutant Y82A, while in Y82A, the dynamics of L81 are faster and accompanied by a noticeable backbone drift (Figure 3a). Note that the described X82-pivoting can furthermore explain the conformational rearrangements of Y82 suggested in the experimental investigation of the C-type inactivation [17].

Thus, the mutation in position 82 changes the dynamics of residues close to the filter region, mainly affecting the conformation of L81. The enhancement of L81 transitions in the deep inactivating mutant Y82A causes the promotion of D80–R64 interactions because the dynamics of L81 and R64 are strongly coupled. Note that the X82-pivoting also alters the dynamics of residues V84 which can access D80 in a similar manner as R64. A comparable influence of V84 on D80, therefore, can be hypothesized. However, if such influence exists, it was masked by a stronger R64–D80 interaction.

For verifying the regulatory role of L81, mutant L81A (Figure 1) was additionally considered. The probability of the creation of the D80–R64 H-bond was compared for three proteins: WT, Y82A, and L81A. All the proteins were simulated with the same

initial configuration, and the probability was calculated for the same time interval 10 ns. Consistently with the presented above results, the simulations confirm that the rate of the H-bond creation depends on residues L81, with a trend in the probability L81A > Y82A > WT (Figure 3b). The probability is larger in Y82A with respect to the WT by the enhanced fluctuations of L81. The probability becomes even larger in L81A when L81 is directly substituted by the small alanine which interferes less with R64 motion.

The results of this subsection show that proteins WT, R64A, and Y82A differ from each other in the dynamics of a few residues: primarily arginines R64 and R89, and leucine L81 that regulates the D80–R64 interaction. The cooperative dynamics of these three residues have a destabilizing effect on the triad E71-D80-W67 and, therefore, affect the pore region. Note that glutamate in position 71 (E71) has a non-trivial influence on the inactivation. In the absence of E71, mutant E71A is very flexible, but the inactivation is suppressed entirely. The presence of E71 is, therefore, essential to observe the inactivation, as E71 limits the flexibility of the pore region and strongly affects the motion of D80.

#### **4. The Interactions of Residues and Ions in the WT Protein**

#### *4.1. Influence of Arginines R64 and R89 on D80, the SF and Ions*

For understanding the action of arginines R64 and R89 in the WT protein, conformations with residue R64 close to residue D80 were investigated by a simulation started with a particular initial configuration. In order to enhance the probability of the R64-D80 interaction, an unbiased simulation (denoted *WT-R64D80*, duration of 45 ns) of the KcsA WT protein was commenced from a conformation with residues R64 were near D80 in all the four subunits. Details of how the initial configuration was obtained are given in the section "Methods". Figure 4b shows the initial configuration characterized by the filter occupancy wKwKw+K; the flipped state of L81 and R64 is close to D80 in all subunits. Note that in subunit C, residue R64 forms a H-bond with D80 during a short equilibration in a preparation stage (see Figure S7 in the Supplementary Materials, starting point).

Two positively charged arginines R64 and R69 can exert a sufficiently strong combined upward force on negatively charged residue D80 to overcome the strong downward attraction toward E71. From the beginning of simulation WT-R64D80, this force resulted in a large mobility of the pore region. A long breakage of E71-D80-W67 linkages (for 17 ns) occurred in subunit C as well as brief disruptions of the linkages in other subunits were observed. Conformational rearrangements in residues and content (K<sup>+</sup> ions and water molecules) of the SF accompanied these disruption events. Representative snapshots of changes in the SF are shown in the Supplementary Materials, Figures S7 and S8. The rearrangements observed in the subunit C were analyzed using three order parameters: angle *ψ*76, distance SC<sup>80</sup> and the length of the TVGYG sequence of the subunit, SF length. Figure 4 shows the trajectory, which reflects the time evolution of the system, in the conformational space defined by these three order parameters for the first 22.5 ns of the simulation. Initially, several transitions of residues V76 (angle *ψ*<sup>76</sup> switches back and forth between −50◦ and 145◦ ) were observed. These transitions demonstrate the inherent flexibility of the V76/G77 peptide group, which is sensitive to changes in the SF. The trajectory also shows that after 5 ns the E71-D80-W67 triad broke and residues D80 moved outward (distance SC<sup>80</sup> changes from 13.5 Å to 15.5 Å). All these changes were promoted by residue R64. Residue D80 accommodated an upward state with H-bonds formed between either D80 and R64, or D80 and R89, or D80 and both arginines (see Figure S8 in Supplementary Materials). This upward state of D80 caused stretching of the TVGYGD sequence of the SF (SF length increases), and residue V76 switched to a meta-stable flipped state. The described changes correspond to the transition from the state **A** to the state **B** in the conformational space (Figure 2). The two-dimensional density for distance SC<sup>80</sup> and angle *ψ*<sup>76</sup> shown in Figure 2b emphasizes a meta-stable character of the distorted state **B** and its dependence on the position of D80 side chain.

**Figure 4.** Panels (**a**,**b**) compare of the X-ray structure of the WT protein (a) and the initial configuration for simulation WT-R64D80 (b). Panel (**c**) shows the correlation between the states of D80 and the SF by means of the two-dimensional probability density for position SC<sup>80</sup> and angle *ψ*<sup>76</sup> (see "Methods") during first 22.5 ns of simulation WT-R64D80, i.e., before ion configuration KwK0K was reached. Two meta-stable states are denoted by letters **A** and **B**. Snapshots corresponding to each of the two states are shown on the right side panels. State **A** is the initial state in which E71-D80-W67 linkages were present. State **B** is characterized by an outer movement of D80 which followed by the break of E71-D80-W67 linkages and TVGYGD rearrangements. Panel (**d**) reports time evolution of subunit C in the conformational space defined by (i) *ψ*76, (ii) SC80, and (iii) the length of the TVGYG sequence, SF length. The trajectory in the conformational space is coded by color scale shown in the colorbar. Letters **A** and **B** indicate the same states as in panel (c).

Figure 5 and Figure S9 in Supplementary Materials show significant consequences of the distortions in subunit C on the elements bound to the filter, and in particular, on the permeating K<sup>+</sup> ions, which facilitate in spreading the distortions among the other three subunits. The changes in the permeation path can be characterized by the correlation between the positions of K<sup>+</sup> ions in the SF. Let us stress that strongly correlated motion of ions was considered as being the fundamental feature of the knock-on mechanism of the permeation in previous works [92,93]. Simulations started from the X-ray configuration demonstrate the presence of such correlated dynamics of ions in the SF (see Figure 5a): Pearson's coefficient is large (around 0.75) and the positions of ions K1 and K2 are linearly correlated. In the distorted state observed in simulation WT-R64D80 (state **B** in Figure 2b), the correlation between K<sup>+</sup> ions is lost; Pearson's coefficient is close to zero (Figure 5a). Ions in the SF become more flexible in the binding sites, that leads to weakening in the spatial definition of the K<sup>+</sup> sites (Supplementary Materials, Figure S9). An unexpected transition of the innermost ion (K2) toward the intracellular side was observed (wKwKw+K wKw0K+K, Figure 5b). This transition occurred in the reverse direction with respect to the permeation path. Note that such a transition was not observed in simulations started from the X-ray conformation of the WT protein. The observation of the inverse transition is particularly important because it reveals the influence of protein distortions on single K<sup>+</sup> permeation events. This influence additionally can explain the different free-energy barriers obtained for the permeation path using biased approaches which induce distortions of some parts of the KcsA protein [58,86,90,94,95].

**Figure 5.** Panel (**a**) illustrates the correlation of the position of permeating K<sup>+</sup> ions (K1 and K2) for two different conformations of WT: (i) the X-ray conformation in which R64 was far from D80 in all subunits; and (ii) the confirmation used in simulation WT-R64D80 with R64 near D80 in all the subunits. Only a part (durations of 7 ns for (i) and 15 ns for (ii), respectively) of the simulations with identical filter occupancy wKwKw+K were considered. For simulation WT-R64D80 the part corresponds to broken E71-D80-W67 H-bonds. Representative snapshots and *z*-positions of two ions in the SF on the state plane of *z*K1 and *z*K2 for each configuration are shown. Panel (**b**) depicts a configuration of the channel after the inward transition of the innermost ion K2 (wKwKw+K wKw0K+K) occurred. All four subunits denoted by letters A–D are shown.

The described distorted state of the SF is observed during the initial part of simulation WT-R64D80. In the later stage, ions underwent several further rearrangements. One of the rearrangements is the ion (K3) from the cavity enters the SF (wKwKw+K −→ wKwKK) that leads to the re-establishment of E71-D80-W67 linkages. Then, the transition of the outermost ion (K1) to the site S0 (wKwKK −→ KwK0K) led to a configuration close to those observed in the conductive state of the X-ray structure. The latter result suggests that the conformation in which a K<sup>+</sup> ion is bound to site S0 stabilizes the filter structure.

Thus, simulation WT-R64D80 demonstrates that conformational changes of the SF are dependent on a strongly correlated network of residues, in which aspartate D80 plays the central role. States of D80 with the broken E71-D80-W67 triad are promoted by the combined action of arginines R64 and R89. Furthermore, these states of D80 can destabilize the SF and cause filter's distortions. The latter affects the dynamics of ions and can prevent the permeation of ions. In turn, ions permeation through the SF can either enhance or suppress the destabilization effect.

#### *4.2. Energetics of the Arginine Motions*

Simulation WT-R64D80 was started from a particular initial configuration, and the observed changes in the SF are transient. In this subsection, therefore, the energetics of the changes are studied using a technique called well-tempered metadynamics (wtmetaD). Wt-metaD (see Supplementary Materials for further details) is a theoretical method which belongs to the family of the biased methods and has been successfully applied for both to accelerate the observation of rare events and to reconstruct free energy surfaces (FES) [62,76].

The analysis of the dynamics of mutants and simulation WT-R64D80 demonstrate that the interactions of two arginines R64 and R89 with D80 can trigger rearrangements which change the shape of the channel pore, hence alter the ion permeation in KcsA. The dynamics of R64 is strongly coupled with leucine L81 which regulates the D80–R64 interaction. Therefore, the motion of R64 needs to be analyzed together with the motion of L81. Arginines R89 is not directly controlled by neighboring residues and can be studied alone. Energetics of the motion of R69 are described in the Supplementary Materials (section "Energetics of the arginine motions").

For characterizing the dynamics of R64 and L81 the two-dimensional FES was calculated for the following order parameters: (i) the distance between R64 side chain and the center of mass of the SF (*R64–SF*), and (ii) angle *χ*1<sup>81</sup> (more details in section "Methods" and Supplementary Materials). The total sampling length of the wt-metaD simulation was 122 ns. The computed FES, shown in Figure 6, confirms the interplay between R64 and L81, and the regulatory role of the latter. There are several multi-stable states on the two-dimensional FES. State *S<sup>a</sup>* is with non-flipped residue L81 (*χ*1<sup>81</sup> ≈ 297◦ ) and residue R64 which is far from the SF (R64–SF > 19 Å). This state is close to the X-ray structure of KcsA. It demonstrates that when L81 is in the non-flipped conformation, R64 tends to be away from the filter and D80. On the other hand, when L81 is in the flipped conformation (*χ*1<sup>81</sup> ≈ 185◦ ) residue R64 can approach closer to the SF (states *S<sup>b</sup>* and *Sc*, R64–SF < 18 Å). Residues D80 and R64 form a H-bond in the state *Sc*. Two minimal-energy paths *S<sup>a</sup>* → *S<sup>c</sup>* are shown by dotted lines in the FES plane. The first path, highlighted by the magenta line, consists of an initial flipping of the L81 side chain (*χ*1<sup>81</sup> from ~297◦ to ~185◦ ) followed by the subsequent movement of R64 towards the SF along a downward gradient. The second path, highlighted by the black line, involves the creation of an initially relatively unstable D80–R64 H-bond which is lately stabilized by the flipping of the L81 side chain. Both paths have a similar energy barrier (5 kcal/mol).

Note that the energy barrier for the inverse transition *S<sup>c</sup>* → *S<sup>a</sup>* is significantly higher (13–15 kcal/mol) than for *S<sup>a</sup>* → *Sc*. It means that state *S<sup>c</sup>* corresponds to the global minimum of the FES and the configuration with a H-bound between R64 and D80 and with L81 in the flipped conformation should be observed in X-ray structural studies [28]. An analogous result, with the R89–D80 H-bond in the most probable state, was also obtained by free-energy calculations of the R89 motion for two out of three configurations of ions (see Supplementary Materials, section "Energetics of the arginine motions").

**Figure 6.** In the middle, two-dimensional FES computed via the wt-metaD approach is shown with respect to the distance between R64 and the SF (R64–SF) and angle *χ*181. In order to aid the visualization, angle *χ*1<sup>81</sup> is reported in the range (0, 360), instead of the standard (−180, 180). The FES is shown in kcal/mol, lines in the contour plot are drawn every 1 kcal/mol. Configurations of residues for three different multi-stable states: *Sa*, *S<sup>b</sup>* and *Sc*, are shown on the sides of the FES plot. These three states are denoted on the FES. State *S<sup>a</sup>* is with non-flipped residue L81 and close to the X-ray structure of KcsA. L81 is in the flipped conformation for states *S<sup>b</sup>* and *Sc*. Residue R64 forms a H-bound to D80 in the state *Sc*. The global minimum of the FES is state *Sc*.

Thus, the most probable positions of arginines R64 and R89 observed in the wt-metaD calculations are different from those in the X-ray structure [28]. These positions difference can be explained by interactions between the protein and surrounding lipids. In fact, numerous experiments indicate that in common with other K<sup>+</sup> channels, KcsA channel is stabilized in the conductive state by the presence of the anionic lipids. In contrast, the channel is primarily non-conductive for the non-anionic lipids [20,33]. Deol et al. [96] revealed, by means of molecular dynamics simulations, that R64 and R89 can form strong, long-lived H-bonds with the head groups of the anionic lipids. Later, this result was experimentally confirmed [33]. This arginine–lipid interaction could bring the arginines in positions close to those determined by the X-ray experiment [28].

Our wt-metaD simulations were performed in the absence of anionic lipids, using neutral POPC lipids which as shown experimentally have no specific interaction with KcsA [33]. However, the radial distribution function that characterizes the interaction of Cl− ions in the bulk with residues R64 and R89, confirms the strong affinity between the arginines and negatively charged species (see Figure S11 in the Supplementary Materials). Because of this affinity, the computed FESs (Figure 6 and Figure S10 in Supplementary Materials) show that the most stable position of R64 and R89 are located in proximity to the negatively charged D80. The presence of the anionic lipids would make this position less probable by additional interactions between the arginines and these lipids. Another factor affecting the arginines is locations of ions in the SF. For example, when ions occupy sites S0, S2 and S4, the probabilities of finding R89, respectively, in proximity to D80 and far from D80 are equal (see Figure S10 in Supplementary Materials). The influence of ions' configuration on the dynamics of R64 is considered in the next subsection.

#### *4.3. Opposite Influence of R64 and a K*<sup>+</sup> *Ion Bound to S0 on the E71–D80 H-bond*

For characterizing the simultaneous action of arginine R64 and ions in the SF on the strong H-bond between E71 and D80, we calculated the FES for the interaction of E71 and D80 in different configurations of the SF. The distance between E71 and D80 residues is selected as the order parameter for the FES. The calculated FESs for arginine R89 (see Supplementary Materials, Figure S10) suggest that a K<sup>+</sup> ion in site S0 stabilizes the E71–D80 H-bond by reducing the probability of the R89–D80 interaction. Free energy calculations were, therefore, performed for two ions conformations: one is "KwK0K" with

an ion bound to S0, and the other is "wwK0K" without an ion in S0. Two different positions of R64 with respect to D80, near and far away respectively, were additionally considered. Thus, the FES were calculated for four different configurations of R64 and ions in the SF (see Figure 7). Further details of the FES calculations are reported in Supplementary Materials (Section 7). Note that configurations with a water molecule occupied site S3 ("KwKwK" and "wwKwK") were also considered and the corresponding results are reported in Supplementary Materials (Figure S12). These results are consistent with those presented below.

The FESs in Figure 7 demonstrate the strong mutual influence between the filter occupancy and the position of R64 on the E71–D80 H-bond. The interaction between E71 and D80, therefore, does not merely depend on the nature of the residues and the nearby solvent molecules (water). Still, it originates from many different elements which constitute a strongly interacted (correlated) system.

If an ion is absent in site S0 and simultaneously R64 is far from D80 (Figure 7b), the E71–D80 H-bond is the only stable state in the FES. However, the proximity of R64 to D80 makes breaking the E71–D80 H-bond possible and leads to new meta-stable states without the bond (Figure 7d). The energetic barrier for the breaking the H-bond is relatively small (around 2.5 kcal/mol) and slightly higher (by 0.2 kcal/mol) than the barrier for reestablishing the H-bond. These new states without the interaction between E71 and D80 are close to those that led to distorted configurations in the SF observed during simulation WT-R64D80.

**Figure 7.** Graphs in the middle show the FESs for the distance between E71 and D80 in different cases: (**a**) R64 is far from D80 and ions configuration "KwK0K", (**b**) R64 is far from D80 and ions configuration "wwK0K", (**c**) R64 is close to D80 and ions configuration "KwK0K", and (**d**) R64 is close to D80 and ions configuration "wwK0K". A starting configuration for each wt-metaD simulation is shown on the left side of the figure. Examples of a configuration with a broken H-bond between E71 and D80 are shown on the right side of the figure for each corresponding initial configuration.

The presence of a K<sup>+</sup> ion in site S0 changes the observed picture. In the case of R64 located far from D80, the presence of an ion leads to new states with the broken E71–D80 H-bond (compare Figure 7b and Figure 7a). However, the new states are less stable than the state with the H-bond, and the energetic barrier for the bond breaking is high (around 6.5 kcal/mol). When R64 is close to D80, the occupation of site S0 increases the energetic barrier for the breaking the H-bond and makes states without the H-bond significantly less stable (compare Figure 7d and Figure 7c). In this case, the barrier for the bond breaking is around 4 kcal/mol, and for the re-establishing, the barrier is four times less (Figure 7c). Thus, a K<sup>+</sup> ion occupied site S0 opposes the destabilizing influence of arginine R64, favoring the presence of the E71-D80 H-bond. The absence of an ion in site S0 induces a widening of the site that facilitates the approach of R64 to D80 and destabilizing the E71–D80 H-bond.

These results demonstrate broad cooperation between residues and ions in controlling the dynamics of the pore region. Note that the described role of the occupation of site S0 by an ion provides a mechanistic and energetic insight to the hypothesis of a 'foot-in-thedoor' mechanism, widely discussed in the literature for interpreting some experimental results [9,16,38]. In particular, the strong dependence of the current on the extracellular K<sup>+</sup> concentration was observed experimentally [9,10,38,43,44]. For explaining this strong dependence, different authors have hypothesized that ion's occupancy in the SF rises for the high concentration of ions and an ion resident in the filter stabilizes the conductive conformation. This hypothesis was supported further by the evidence that ions with a longer occupancy (Rb+, Cs+, and NH<sup>+</sup> 4 ) slow down the switching of the ion channel into the inactivated state [9,38]. According to our results, an ion in site S0 appears as the most valuable candidate for playing the role of the "foot-in-the-door".

As previously mentioned, Cordero et al. [14] reported a flipped structure in mutant E71A, where the replacement of glutamate E71 by alanine A71 effectively remove the E71– D80 H-bond that leads to broad outward movement of D80 and large rearrangements in the V76 region. In all our simulations, the WT protein never adopted a similar configuration, even for states with considerable free energies. It implies that residue E71 consistently plays a dual role in shaping the WT ion channel through the strong electrostatic interaction between E71 and D80 and through a steric hindrance of large rearrangements in the region of V76.

#### **5. Conclusions**

In this work, a comparative analysis of the dynamics of the WT KcsA ion channel and mutants E71A, Y82A, R64A, and L81A was conducted using molecular dynamics simulations. This analysis helped us to identify a set of residues which control the state of the SF. The interactions between the identified residues and the interdependence between the residues and ions in the SF were characterized by free-energy calculations using welltempered metadynamics [62]. A detailed description was provided for the residues which most prominently outlined the state of the SF and the influence of the ion permeation path. Our investigations revealed that the permeation path is regulated by a strongly interconnected dynamical system. The system is centered on aspartate D80, which is linked to neighboring H-bond donors, includes ions in the SF and residues located far from the SF. Key features of this interconnected system were described, that provides a consistent and unifying picture for some experimental results on the regulation of KcsA activities. These features are highlighted below.

First, the highly conserved aspartate D80 plays the critical role in changing the structure of the SF by translating broader dynamics of the protein to the filter structure because of the relatively rigid backbone of the conserved sequence GYGD of the SF. Thus, movements of residue D80 can trigger significant rearrangements of the whole pore.

Second, two arginines (R64 and R89) can strongly interact with D80 via H-bonding. This interaction facilitates movements of D80 that triggers the changes in the protein pore. While the D80–R89 interaction was previously described in the literature [91], the possibility of the D80–R64 H-bond and the destabilizing consequences of the combined action of these two arginines on D80 were described for the first time in this work. Between

the two arginines, R64 was found to exert the strongest influence on D80, and thus on the ion flow.

Third, the local dynamics of the region behind the filter is regulated by conformational changes of leucine residue L81. These changes, in turn, are linked to collective motions of the amino acid sequence L81-Y82-P83-V84, in particular to a pivoting action on residue Y82. Additionally, the simulations provided the unambiguous evidence for the regulatory role of L81: the flipping of the L81 side chain facilitates the establishment of the D80–R64 H-bond.

Fourth, the destabilization effect of arginines R64 and R89 on states D80 is reduced by the presence of a K<sup>+</sup> ion in the outermost binding site (S0) of the filter since the resulting electrostatic interactions stabilize the conductive structure.

We showed that the interactions between the two arginines (R64 and R89) and D80 induces the breaking the E71–D80 H-bond that could lead to a non-conducting state of the pore. This result provides an explanation of the necessity of the anionic lipids for observing the current in KcsA channel as the lipids can interact with both arginines [96], and this interaction reduces the probability of breaking the E71–D80 H-bond. Additionally, we showed that the occupancy of site S0 by an ion also stabilize the E71–D80 H-bond. The stabilizing influence of the ion bound to S0 offers an important insight into the "foot-inthe-door" mechanism proposed by various authors for explaining the influence of the extracellular K<sup>+</sup> concentration in stabilizing the conductive state [9,38,43].

Our comparison between the dynamics of the WT protein and mutant E71A revealed a vital role of glutamate residue E71 in response to perturbations of the pore region. In the WT protein, the residue E71 participates in E71-D80-W67 linkages, which are considered as being an essential factor driving the filter toward non-conductive conformations [9,29]. Our results demonstrated that these linkages represent just a part of the more extensive strongly correlated network which dynamically and collectively participates in determining the state of the SF. The mutation of E71 with alanine in mutant E71A generated a noninactivating pore with freely moving D80 [14]. We showed that in mutant E71A, residue D80 interacts with arginines R64 and R89. This interaction induces the strain on the SF, which adapts and relieves the perturbation through a flipping of V76 and a transition of D80 toward the extracellular (outer) region. As a result of this adaptation, the filter remains in a conducting state. The presence of E71 in the WT protein prevents such adaptation when D80 interacts with the two arginines. This interaction, therefore, leads to distorted configurations with complicated dynamics. The resulting complex picture is defined by ions and water molecules in the filter as well as by residues interacting or controlling the interaction between the arginines and D80.

The summarized complex picture provided by this research can be represented as a network of weighted nodes which affect the permeation path (Figure 8). The sizes of the nodes are weighted according to the number of edges connecting each node. This figure reveals the primary importance of the residue D80, being the main hub. It forms the core of the network with the neighbouring H-bond donors E71, W67, and arginines R64 and R89 which mutual dynamical influence defines states of D80. The collaborative dynamics of the residues result either in the stabilization of the conductive conformation or in distorted states of the TVGYGD sequence of the SF. Note that the sequence belongs to the highly conserved signature sequence TXXTXGYGD observed in many potassium channels [16,29,31,97]. In these channels, the aspartate residue D, similar to D80 in KcsA, is surrounded by different H-bond donors. Thus, the existence of similar complex network might be a general feature in the regulation of the current in the K<sup>+</sup> ion channels.

The significant mutual influence between the residues behind the SF and the ion occupancy in specific sites means that perturbations imposed on either residues or ions affect the KcsA channel's state. It is reasonable to expect that numerous networks' states have distinct permeation properties. Recent experiments [7,23] with modified phospholipids showed that the interaction of arginines R64 and R89 with added phosphatidic acid lipid enhances the conduction in the KcsA channel. These experiments confirm the results of

Section 4, which describes the particular role of these arginines in regulating the network (Figure 8). In turn, the change of ion locations by an additional artificial force in biased MD simulations [34,58] could alter the residues behind the SF resulting in a non-conducting state with a high permeation barrier. It means that all the network components (Figure 8) should be included as collective variables in a biased MD simulation. Alternatively, a set of biased MD calculations for different states of the networks should be considered. The self-organized dynamics of the whole network define conducting or non-conducting states of the KcsA channel and considering the SF and ions only is not sufficient.

This dynamical network (Figure 8) is identified for the canonical structure 1KC4 with a closed inner gate. It is shown [35,36] that opening the inner gate leads to the perturbations on the backbones of the SF. Therefore, the inner gate should be included in this dynamical network as well. However, the possibility of an opposite influence of ions and residues near the SF on the inner gate is an open question. Several structures [34,52,54,59,60] with a conducting SF and an open inner gate were generated in silico by combining different crystallographic structures. Creating such structures should include a slow adaptation of the whole network to changes in the gate. The applied constraints on the SF backbones and ions only does not guarantee a realistic configuration of the SF. In this context, MD simulations of a transition of the inner gate from closed to open state are an essential missing link for clarifying the influence of the inner gate on the whole channel.

Current physical models (see, for example, the recent work in [98] and references therein) of the ion permeation in the KcsA channel consider a part of this network: ions and their interaction with the residues in the SF. Incorporating the whole network in physical models would lead to a more complex model, for example, the Markov state type, but a more realistic one. The representation of the protein's complexity via this network would lead to a comprehensive description of complicated patterns of currents observed experimentally.

Results of Section 4.1 show that one of network's states is non-conducting, and the channel in that state is inactive. This observation means that the inactivation can result from the dynamics of this network alone without the involvement of the gate residues. Future work will address the role of the network in the C-type inactivation.

**Figure 8.** The network of residues that are determinant for the permeation path is drawn following certain rules: (i) blue-dashed lines represent non-bonded electrostatic interactions that can eventually lead to strong H-bonds; (ii) black lines represent connections through the backbones of the WT protein; and (iii) green dotted lines represent all the remaining non-bonded interactions, such as steric interactions or repulsions between positive or partially-positive charged groups. The sizes of the nodes are weighted according to the number of edges connecting each node. The label "SF" indicates the selectivity filter. The network was created using software package Gephi [99].

**Supplementary Materials:** The Supplementary Materials containing details of the methods, additional results and figures are available online at https://www.mdpi.com/1099-4300/23/1/72/s1.

**Author Contributions:** Conceptualization, S.M.C., E.J.C., and I.A.K.; methodology, S.M.C. and I.A.K.; formal analysis, investigation, and writing—original draft preparation, S.M.C.; writing—review and editing, S.M.C., E.J.C., and I.A.K. All authors have read and agreed to the published version of the manuscript.

**Funding:** The work has been supported by the Engineering and Physical Sciences Research Council (UK) under grants No. EP/G070660/1 and EP/M016889/1. Computational facilities were provided by the MidPlus Regional Centre of Excellence for Computational Science, Engineering and Mathemtatics, under EPSRC grant No. EP/K000128/1.

**Data Availability Statement:** Data related to this research are openly available from the University of Warwick archive at https://wrap.warwick.ac.uk/143572.

**Acknowledgments:** The authors thank M. Sansom, S. Takahama, D. Quigley, and M. Allen for useful discussions and critical comments. We dedicate the paper to the memory of Mark P Rodger. We are greatly indebted to him for contributions to our research.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


#### **Subin Sahu 1,2,3 and Michael Zwolak 1,\***


Received: 19 October 2020; Accepted: 6 November 2020; Published: 20 November 2020

**Abstract:** Ionic transport in nano- to sub-nano-scale pores is highly dependent on translocation barriers and potential wells. These features in the free-energy landscape are primarily the result of ion dehydration and electrostatic interactions. For pores in atomically thin membranes, such as graphene, other factors come into play. Ion dynamics both inside and outside the geometric volume of the pore can be critical in determining the transport properties of the channel due to several commensurate length scales, such as the effective membrane thickness, radii of the first and the second hydration layers, pore radius, and Debye length. In particular, for biomimetic pores, such as the graphene crown ether we examine here, there are regimes where transport is highly sensitive to the pore size due to the interplay of dehydration and interaction with pore charge. Picometer changes in the size, e.g., due to a minute strain, can lead to a large change in conductance. Outside of these regimes, the small pore size itself gives a large resistance, even when electrostatic factors and dehydration compensate each other to give a relatively flat—e.g., near barrierless—free energy landscape. The permeability, though, can still be large and ions will translocate rapidly after they arrive within the capture radius of the pore. This, in turn, leads to diffusion and drift effects dominating the conductance. The current thus plateaus and becomes effectively independent of pore-free energy characteristics. Measurement of this effect will give an estimate of the magnitude of kinetically limiting features, and experimentally constrain the local electromechanical conditions.

**Keywords:** ion transport; nanopore; graphene; crown ether

#### **1. Introduction**

Ionic transport through nano- and sub-nano-scale pores elicits a tremendous amount of interest due to its relevance in cellular processes including neurotransmission, muscle contraction, and other biological processes [1,2], as well as its application in technologies such as desalination [3–5], osmotic power generation [6–8], and bio-chemical sensing [9–13]. Ref. [14] provides a recent review of the use of pores in 2D materials for these applications. Understanding transport mechanisms, particularly in biological settings, has remained challenging due to their complexity and dependence on atomic details [15]. Furthermore, even for uncharged membranes, the region outside the pore can play a significant role in determining ionic transport, either via access resistance [16–18] or via diffusion limitations [19].

Synthetic nanopores offer the ability to study the factors that underpin transport mechanisms, such as the role of dehydration [20–24] and functional groups [25–28], and give rise to functional behavior, such as ion selectivity [27,29]. Pores in 2D membranes, in particular, have a larger access resistance, compared to their pore resistance, due to their small aspect ratio, *ap*/*hp*, where *a<sup>p</sup>* is the effective pore radius and *h<sup>p</sup>* is the effective pore length. Therefore, 2D materials provide a unique opportunity to study geometric effects in transport [30], atomic changes in area via precise control in pore fabrication and height via layering [24], and the interplay of various length scales relevant to the problem [14], which will help in the design of separation membranes [14,31,32] and delineating factors relevant to biological channels [14]. In particular, the effective thickness is not the geometric thickness of the membrane (e.g., for graphene, twice the van der Waals radius of the carbon atoms), since ion size, hydration layer radii, and even membrane charge and the build up of charge layers give rise to an effective thickness [16]. When both the bulk and pore are within the continuum drift-diffusion regime, measuring or calculating the dependence of the conductance on pore radius quantifies the effective pore length through the equation

$$R = \gamma \left(\frac{1}{2a\_p} + \frac{h\_p}{\pi a\_p^2}\right),\tag{1}$$

which assumes a homogeneous resistivity *γ* and a cylindrical pore geometry. The former entails that there are no concentration gradients and that the medium in the pore has the same resistivity as the bulk (otherwise, it requires an independent determination of the pore resistivity). For graphene, this approach yields an effective pore thickness of about 1 nm, both experimentally [33] (for unknown rim functionalization) and computationally [14,16] (for unfunctionalized rims), provided that simulations properly include the influence of the bulk via the golden aspect ratio or associated scaling analysis [16,34], as well as properly determine the pore radius [16]. The effective pore length is mostly due to the van der Waals radii and first hydration layer of the ions, which are both reflected in a build up of charge layers about 0.5 nm from the membrane.

More recently, it was demonstrated that applying strain to 2D pores can elucidate the conditions under which optimal transport and ion selectivity arise by modulating the balance of dehydration and electrostatic interactions [27]. It is unclear, however, whether 2D pores, and synthetic pores more generally, offer a means to investigate diffusion limitations. These arise due to fine details of pore structure typically thought to be out of our control. We will demonstrate here that the control provided by strain can tune atomically thin biomimetic pores into a diffusion-limited regime. Finding transitions into these regimes will help delineate and probe the electromechanical environment of nanopores, and elucidate diffusion-limited phenomena in more complex, biological settings.

Ionic transport through a pore becomes diffusion-limited when the permeability of ions in the pore is large and the current is only restricted by the rate of diffusion of the ions from the bulk to the pore mouth [35,36]. In this diffusion-limited regime, the current does not increase with voltage as expected from Ohm's law. This is similar to the diffusion-limited processes in chemical reactions [37] and other transport processes [38,39]. Diffusion-limited ionic currents are a regular occurrence in biological pores [40,41], as these can have the necessary conditions: narrow channels with high permeability for specific ions [1] and the presence of "inert" ions [42]. Pores in atomically thin membranes, such as graphene, MoS2, and hBN, also provide a very high permeability for ions due to their sub-nanoscale channel length [14]. Thus, in these membranes, the diffusion of the ions from the bulk to the mouth of the pore may become the limiting factor in the ion transport. It is the objective of this work to determine under what conditions bulk diffusion becomes the limiting factor, particularly when drift is also present.

For the transport to be diffusion-limited, however, the drift contribution to the current in the bulk has to be small. This condition is hard to realize in pores in 2D materials under electrically driven transport,

since a large portion of the applied voltage drops in the bulk solution, which in general has a higher resistance—in the form of access resistance—than the pore itself [14,16,17,33]. This is also true of small aspect ratio—short and wide—biological channels, where access resistance becomes dominant at low ion concentration [18]. There are regimes, though, where diffusion limitations may appear. Sub-nanoscale pores in graphene, for instance, have a high pore resistance [23]. As a result, most of the applied potential will drop across the pore, thus diminishing the drift current in the bulk and giving an opportunity to observe the diffusion-limited transport. At this sub-nanometer length scale, the translocation barriers and potential wells due to ion dehydration and electrostatic interactions play a major role in determining transport through such pores [27]. When one or the other interaction dominates, translocation through the pore is barrier-limited.

Under the right conditions—determined by pore size and charge, dehydration energy, etc. —the permeability of the pore will be large [43,44] (e.g., ion channels with binding sites, in particular, can have an inverse relationship between permeability and conductance [45]). To understand the conditions for having a small pore conductance and high pore permeability, one can look at the continuum-limit expressions

$$G\_p = q \, c\_p \, \mu\_p \, A\_p / h\_p \tag{2}$$

and

$$P\_p = D\_p / h\_{p\_1} \tag{3}$$

where *c<sup>p</sup>* is the ion concentration in the pore, *q* is the charge of the ion, *µ<sup>p</sup>* is the pore mobility, *D<sup>p</sup>* (=*µpkBT*/*q*) is the diffusion coefficient, *A<sup>p</sup>* is the pore area, *h<sup>p</sup>* is the pore length, *k<sup>B</sup>* is Boltzmann's constant, and *T* is the temperature. We briefly note that all pore quantities (which have a subscript *p*) are effective quantities, as will be abundantly clear throughout this work. An overall barrier in the pore can limit the concentration by exponentially reducing the partition coefficient into the pore. When such a barrier is present, without features internal to the pore, the mobility and diffusion can be unaffected. Similarly, the internal features, and hence mobility, can be altered with very small relative changes to pore size (and vice versa, reducing the cross-sectional area of the pore can reduce the conductance but with little effect on mobility when the pore size is relatively large). Essentially, strain and voltage will give the right knobs to tune some pores into a diffusion-limited regime by modulation of *c<sup>p</sup>* and *µp*, while retaining a small *A<sup>p</sup>* (i.e., a large pore resistance).

Here, we will examine the 18-crown-6 pore in graphene (see Figure 1) under the influence of a homogeneous strain in the plane of the membrane and cross-membrane voltage for different local electronic pore environments. Crown ether pores were seen by scanning transmission electron microscopy of graphene membranes that were made by exfoliation of graphite [46]. Even though strain only changes the pore area by a minuscule amount, the change in the free-energy barrier and hence the ionic current is substantial [27]. We find both barrier-limited and diffusion-limited regimes depending on the strain, voltage, and local environment. In the barrier-limited regime, the conductance increases with applied voltage, as it helps ions overcome the barrier. In the diffusion-limited regime, the pore conductance decreases with voltage as the bias depletes the charge carriers in the pore and at its entrance. The transition between these regimes depends on the charge separation in the pore (e.g., the local dipole moment at pore rim), which is not experimentally known, nor is there sufficient thermodynamic or kinetic data from experiment to constrain it. Measurement of the pore conductance versus strain and voltage, therefore, gives a possible route to determining the electromechanical environment, and thus constraining the magnitude of charge separation at the pore rim.

**Figure 1. Ion trajectories in a graphene crown ether pore.** The trajectories of K<sup>+</sup> ions around the pore with *<sup>q</sup>*<sup>O</sup> <sup>=</sup> <sup>−</sup>0.54 *<sup>e</sup>* at 4% strain, with an applied bias of 0.25 V along the *<sup>z</sup>*-axis (making K<sup>+</sup> move in the positive *z*-direction). We plot the trajectories in *z*-*r* 2 space to keep the representation of volume constant. The red trajectories are for ions that translocate through the pore and the cyan trajectories are for ions that reflect back. The green dashed line shows the geometric boundary of the membrane determined by the van der Waals radii of the pore atoms. The effective boundary extends to |*z*| . 0.5 nm due to the size of the hydrated ions. The inset shows a portion of the graphene crown ether (red and grey spheres are oxygen and carbon atoms, respectively) and a three-dimensional trajectory of a potassium ion (purple spheres, separated in time by 10 ps, connected with purple lines) crossing the pore. Connecting lines are a guide to the eye only.

#### **2. Methods**

We performed all-atom molecular dynamics (MD) simulations using the NAMD2 simulations package [47]. The details of the simulations were the same as in Ref. [27]. We applied a voltage between 0.1 V to 1.0 V and calculated the ionic current for a 1 mol/L KCl solution by counting the ions that crossed the pore. Since the pore rim is negatively charged and sub-nanoscale in size (i.e., the electrostatic interactions are not significantly screened), only cation currents were present in all cases. We calculated the free-energy barrier using the adaptive biasing force (ABF) method [48] in a cylinder of radius 0.28 nm and height 3 nm, centered at the pore. A portion of our simulation cell and a set of ion trajectories are shown in Figure 1. We employed the golden aspect ratio method, as it is the only method that can properly capture bulk access effects [16,17], and without it, one cannot explore bulk diffusion limitations with all-atom simulations.

We calculated the ionic current and free-energy barrier at various homogeneous strains from 0% to 10% on the graphene membrane (we note that most of the features we observe occur at strains of 4% to 6%, and graphene can survive strains above 20% [49,50]). The strain was within the membrane and thus tended to enlarge the pore (albeit by small amounts) and expand in-plane distances between atoms. In the unstrained pore, the nominal pore radius (measured from the pore center to the center of the oxygen atoms) was approximately 0.29 nm and increased by about 7.5 pm for each 1% strain (this reflects a small—a factor of ≈2—geometric amplification [27]). This yielded nominal pore sizes from 0.29 nm (at 0% strain) to about 0.37 nm (at 10% strain), with a roughly linear relationship with strain. Though the change in pore size was minuscule, the energy landscape changes substantially. The landscape also depended on the pore charge, which is, however, not known. For the crown ether pore in graphene, the charge per oxygen atom (*q*O) could be between −0.2 *e* and −0.7 *e* [27,51–53]. We thus used two representative test charges, −0.24 *e* and

−0.54 *e*. In the former, there was an energy barrier, and in the latter, there was a potential well at the center of the unstrained pore. Each of the 12 carbon atoms—the ones adjacent to the six oxygen atoms in the pore—had charge −*q*O/2, and the rest of the carbon atoms were neutral. When we refer to the pore charge, we are referencing the local polarization of charge from the carbon atoms of the graphene near the pore and the oxygen atoms on the pore rim. For each data point (i.e., for a particular value of *q*O, strain, and voltage), we performed five parallel production runs for a total simulation between 250 ns to 500 ns. This allowed for an error estimation using the standard error, SE = √ var/*n<sup>r</sup>* , where var is the variance between the *n<sup>r</sup>* = 5 parallel runs.

#### **3. Results**

**Ionic current through a graphene crown ether pore:** Figure 2 shows the potassium current, *I*K, through the graphene crown ether pore versus strain at various voltages. Only potassium ions contribute to the total current, as the negatively charged pore edge does not allow any chloride ions to translocate (on the timescale of the simulations). We also plot the conductance of potassium ions, *G*K, in order to demonstrate the non-Ohmic behavior of ionic current. At low voltage, the current increases by several fold for a minute strain—a couple percent strain changes the conductance by a couple hundred percent. This dramatic amplification is an example of colossal ionic mechano-conductance [27]. The current eventually maximizes around 3% strain and either decreases (for *q*<sup>O</sup> = −0.24 *e* at small voltage) or plateaus (for all other cases). Since the pore size does not change substantially with a small strain, the colossal change in the ionic conductance is the result of a modification of the translocation barriers. The translocation landscape veers toward barrierless transport as strain tunes transport to its optimum [27]. Furthermore, the change in conductance with voltage displays non-Ohmic behavior. In some regimes, such as the colossal mechano-conductance, the conductance increases with voltage, indicating an activated process. In other regimes, the conductance decreases with voltage, indicating the diffusion-limited process. The depletion of charge carriers in the pore, see Figure 3, is also consistent with the decrease in conductance and points to a diffusion-limited regime.

**Translocation barriers in sub-nanoscale pores:** The single-ion energetics of transport through functionalized sub-nanoscale pores may be approximately expressed as

$$
\Delta F\_{\rm V} \approx \sum\_{i} \eta\_{i} f\_{i\nu} E\_{i\nu} + \sum\_{\nu'} \frac{q\_{\nu} q\_{\nu'} \eta\_{\nu'}}{4\pi \epsilon\_0 \varepsilon \, r\_{\nu\nu'}},\tag{4}
$$

where *fi<sup>ν</sup>* and *Ei<sup>ν</sup>* are the fractional dehydration and energy of *i* th hydration layer for ion *ν*, *q<sup>ν</sup>* <sup>0</sup>'s and *n<sup>ν</sup>* <sup>0</sup> are the charge and number of atom species *ν* 0 in the pore, and *e*<sup>0</sup> is the vacuum permittivity. The parameter *ηi* is an O(1) factor to account for the increased binding of water molecules with the ion as dehydration increases [23], essentially giving the non-linear response of the hydration energy to the removal of water molecules. The relative permittivity of water, *e*, under nanoscale confinement is significantly smaller than the bulk value and depends on atomic details [27,29,54]. Specifically, in the case here, when there are not intervening water molecules between the ion and charged groups in the pore, the dielectric constant is around 4 [27] and the electrostatic interaction is very large. The small dielectric constant and short distances involved give rise to the large electromechanical susceptibility of ions within the pore. The fractional dehydration *fν*<sup>1</sup> also changes with the position of ion [23,24,27] and can be estimated with geometric arguments [20,22–24]. As an ion approaches the pore, the free-energy change will remain small (<*kBT*) even at 1 nm distance from the pore, because the ion is still fully hydrated. The electrostatic interaction between the fully hydrated ion and the pore charge is weak due to the dielectric screening of the solution. However, when the ion is about 0.5 nm from the pore, it starts to dehydrate (initially in the second hydration shell and then in the first), and consequently the dehydration energy increases sharply. Simultaneously, the electrostatic energy also rises rapidly since the ion will be significantly closer to the negatively charged oxygen atoms compared to the positively charged carbons, and the effective dielectric constant of water at this distance will be strongly diminished due to the removal of intervening molecules.

**Figure 2. Colossal rise, non-monotonicity, and saturation of the ionic current.** (**top panel**) Potassium current (*I*K) versus strain at various voltages across the graphene crown ether pore within 1 mol/L KCl. At small voltages and minute strains, *I*<sup>K</sup> increases rapidly with strain due to the large electromechanical susceptibility of the pore [27]. A further increase of the strain causes *I*<sup>K</sup> to either decrease (for *q*<sup>O</sup> = −0.24 *e*) or saturate (for *q*<sup>O</sup> = −0.54 *e*), albeit the latter will also decrease when the electrostatic well disappears and dehydration begins to control the current. At large voltage, the current becomes less sensitive to strain because the applied bias dominates over the energy landscape of the pore, self-consistently washing out relevant features—ones that are contributing to resistance—of the landscape. As voltage increases further still, the current saturates at a smaller strain where the relevant free energy features are commensurate with the voltage drop (**bottom panel**). The conductance versus strain shows that for *q*<sup>O</sup> = −0.24 *e* there is an intricate interplay of voltage and strain, indicating that the variation of free energy features with these two parameters is playing a defining role. At larger strain (greater than about 6%), the conductance tends to increase with the voltage (i.e., superlinear behavior). This is a telltale sign of an activated process, where the voltage helps overcome an overall barrier, but does not yet wash it out. In this particular case, this is due to a reduction in electrostatic compensation of dehydration as strain pulls away the counteracting negatively charged oxygen atoms. For *q*<sup>O</sup> = −0.54 *e*, the conductance increases with the voltage at smaller strain (superlinear behavior) and decreases with voltage at larger strain (sublinear behavior). The superlinear behavior indicates barrier-limited transport and sublinear behavior diffusion-limited. The error bars are plus/minus one SE from five parallel runs. Connecting lines are a guide to the eye only. Purple shaded regions in the upper panels approximately delineate the region where bulk limitations control the current.

**Figure 3. Voltage dependence of the ion concentration.** Concentration of potassium ions near a graphene crown ether pore with *q*<sup>O</sup> = −0.54 *e* at (**a**) 0% and (**b**) 4% strain for various voltages. For 4% strain, where we see diffusion limitations of the ionic current, we also see the depletion of ions in the pore as the voltage increases. See the Supplemental Material (SM) for additional plots with different parameter values.

Equation (4) gives these qualitative features of the energy landscape and helps to understand why electrostatics can play such a strong role even in high salt solutions. Still, we use all-atom molecular dynamics (MD) simulation for the calculation of quantitative landscape—see the Methods and later discussion about an additional entropic barrier to move into the ABF constriction. The free-energy profiles from MD are shown in Figure 4. Since we are driving the ionic current through a nanopore by an external voltage, we also calculate the energy landscape of ion transport with an applied bias. The equilibrium free-energy barrier alone does not fully represent the energy landscape of ion transport, especially when the applied bias is large compared to the features in the free energy. We note, of course, that even the energy landscape with the bias does not fully capture the current due to kinetic prefactors and averaging effects.

The equilibrium free-energy profiles (blue lines in Figure 4) exhibit a potential barrier for *q*<sup>O</sup> = −0.24 *e* and potential well for *q*<sup>O</sup> = −0.54 *e* at the center of the pore in the unstrained membrane. In the former, the electrostatic energy (between the ion and the pore charges) is less than the dehydration barrier, while the opposite is true for the latter. Additionally, there can be small potential wells just outside the pore where the ion maintains a larger hydration yet stays close to the negatively charged oxygen atoms of the pore. The energy landscape changes markedly with strain, which is primarily due to the change in the electrostatic interactions within the pore and dehydration outside of the pore. An increase in the pore size—by picometers—due to the strain causes the attractive electrostatic energy to decrease rapidly. The dehydration energy penalty in the pore also decreases with strain but, for small strain, it does not change as rapidly as the electrostatic energy. Consequently, there is a net increase in the energy of the ion at the center of the pore. As a result, the barrier in *q*<sup>O</sup> = −0.24 *e* increases, and the potential well in *q*<sup>O</sup> = −0.54 *e* flattens and then disappears at large strain (a dehydration-based barrier does appear in the middle of the *q*<sup>O</sup> = −0.54 *e* pore, a feature which is already present in the *q*<sup>O</sup> = −0.24 *e* pore at 0% strain due to the lower electrostatic compensation). In contrast, the effect of the strain on the free-energy outside the pore is in the opposite direction. The barrier outside the pore decreases with strain as ion can hydrate better with reduced hindrance from the pore oxygen atoms. The electrostatic energy, however, does not change as rapidly as in the center of the pore. The basic mechanism behind these large changes in free energies is that at the 0.1 nm to 0.5 nm scale; picometer changes in atomic configuration result in large changes in electrostatic and dehydration energies [27]. Dielectric screening (from the solution), in particular, is not that effective at this length scale.

**Figure 4. Equilibrium and voltage-dependent landscape for ion transport.** The free-energy profile of K<sup>+</sup> going through a graphene crown ether pore at 0% and 4% strain for equilibrium and non-equilibrium (*V*ext = 0.25 V) cases. The charge of the oxygen atoms of the crown ether is either −0.24 *e* or −0.54 *e* (and adjacent carbon atoms have half this charge). The potential wells and barriers are mainly the result of competition between the electrostatic attraction and the dehydration. The applied voltage reduces the features (barriers and wells) in the energy landscape but some sharp features still remain, either due to the barriers' size or due to their *irrelevance*, a term we use operationally, see the main text and Figure 5. Irrelevance of barriers occurs since the influence of a barrier on the current is both a kinetic and thermodynamic effect, and other bottlenecks (e.g., diffusion limitations) can exist, i.e., these are the relevant processes at a given voltage and strain. In particular, for 4% strain and *q*<sup>O</sup> = −0.54 *e*, the supply of ions from bulk has a much larger influence on the ionic transport than the dissociation from the well at 0.2 nm. Hence, that barrier remains roughly unchanged. Error bars are plus/minus one SE from five parallel runs.

The free-energy landscape explains many of the features seen in the ionic current versus strain, Figure 2. At small voltages, the current changes significantly with strain because of the change in the energy landscape of the pore. For *q*<sup>O</sup> = −0.24 *e*, the entrance barrier, just outside the pore, initially decreases with strain and the current increases rapidly. Eventually, the increase in the energy at the center of the pore will negate the decrease in the outer barrier, and the current subsequently decreases, thus giving a turnover behavior with the optimal current around 3% strain. At very large strains, ion hydration will increase in the pore, and thus the energy barrier at the center will start to disappear. For *q*<sup>O</sup> = −0.54 *e*, the potential well at the center of the pore becomes shallow with strain, making it easier for ions to dissociate from the pore and contribute to the increase in the current. A common principle for the colossal mechano-conductance change is that the free energy veers toward a barrierless landscape for both these example pore environments. At intermediate strains (4% to 10%), while strain does influence barriers, bulk limitations have kicked in and the barrier change will not be manifest in the current versus strain. Even at small strain, the current will become flat if the applied voltage is large enough, since the larger voltage can wash out larger free energy features. There are, however, irrelevant free energy features—ones that are not rate limiting—that remain even as voltage increases, for which we introduce discrete-barrier and one-way rate analyses below that help identify relevant and irrelevant features. The evolution of features under strain can also suggest their relevance (i.e., if the current is constant versus strain, yet a large feature disappears, that feature is likely—but not guaranteed, since other factors can conspire together—to be irrelevant).

To elucidate the effect of the applied bias on the energy landscape, and hence the current, we calculated the free-energy of the potassium ion in the presence of an external voltage (*V*ext = 0.25 V). Figure 4 shows that the applied voltage raises the potential on one side of the membrane and decreases on the opposite side and the overall potential roughly drops over |*z*| < 0.5 nm (we also see this drop in the calculation of the electrostatic potential). Note that although the graphene is only 0.3 nm thick, the double layer of cations and anions on the opposite side of the membrane will be separated by a distance of about 1 nm due to their hydrated radii. Nonetheless, even a small voltage will result in a large electric field in the pore which can suppress the energetic features. Yet, some sharp features have a spatial variation larger than the applied field and are still prominent in the free energy landscape with an applied voltage.

To capture how the features in the equilibrium free-energy profile change with applied voltage, we plot in Figure 5 the discrete gradients from each energy minimum, *i*, to the next maximum in positive *z*-direction, i.e., (∆*F i* max − ∆*F i* min)/(*z i* max − *z i* min). The gradient of applied voltage, which is in the opposite direction to these gradients, reduces the barrier to transport (we do not plot the gradients in negative *z*-direction, which assist rather than hinder the ion translocation). Figure 5 shows that some of the gradients are larger than the electric field from the applied bias (*V*ext = 0.25 V), and thus these barriers are still present in the energy landscape with applied bias. More importantly, though, the examination of how these discrete gradients change with voltage enables one to identify rather large features that remain unchanged at finite voltages, such as the well at 0.2 nm for the *q<sup>O</sup>* = −0.54 *e* and 4% strain case. This well (and associated barrier) is not a limiting factor in transport at this strain and thus the applied bias does not self-consistently remove it. This type of plot (and a related plot we will examine later) give a clear depiction of what features are influencing transport, including indirectly the influence of kinetic prefactors. We further note that the largest gradient for the unstrained pore at *q<sup>O</sup>* = −0.54 *e* is about 40 *kBT*/nm and thus will require *V*ext ≈ 1 V to effectively wash it out, which is reflected in Figure 2. Once the applied voltage produces local fields larger than the relevant discrete gradients, the ionic current will have little dependence on the equilibrium landscape of the pore, which explains the saturation of the ionic current across all values of strain for large voltages, as we see in Figure 2. Saturation at smaller voltage is a combination of this same washing out plus the presence of irrelevant features due to high kinetic rates (compared to other rates, such as diffusion and entrance-side feeding; see the Supplemental Material (SM) for additional plots of the equilibrium and non-equilibrium free energy barriers).

Some features in the energy-landscape, though, are beyond 0.5 nm from the pore, albeit they are small. These features can survive to large applied voltages. Thus, while they matter little for smaller voltages, they eventually can become important when their energy- and kinetic-scales are commensurate with the other renormalized features. Thus, ions will eventually have to overcome additional entrance barriers. These barriers will directly affect the rate at which ions can enter and exit the pore and thus influence the saturation current through the pore. Conversely, the barrier on the exit side, though significant, has a smaller influence due to the larger dissociation rate, which we will discuss later when examining the interpretation of the rate constants within the model.

We note that since the K<sup>+</sup> ion is confined to a cylindrical region during the ABF calculations, the free energy we present does not include the entropic, 'constriction' barrier to move an ion from bulk to the ABF cylindrical constriction of radius *r*ABF = 0.28 nm, and vice versa (on the exit side). The ABF constriction allows other ions (both coions and counterions) to be in the volume. The ratio of accessible states is thus approximately ΩABF = *πr* 2 ABF*l*/*l* 3 , where *l* is the typical distance between co-ions in bulk (≈1.2 nm at 1 mol/L KCl). Thus, the contribution to the free energy of this constriction penalty is −*kBT* ln ΩABF ≈ 1.7 *kBT*. From within the ABF constriction, the entropic penalty to then go into the pore is included within the ABF calculation. For comparison, this contribution can also be estimated as follows: The geometric pore radius is *r<sup>p</sup>* = 0.137 nm, taken as the pore center to oxygen center, 0.29 nm, minus oxygen's van der Waals radius, 0.152 nm. A typical approach to estimate the entropic penalty

is the formula, <sup>−</sup>*kB<sup>T</sup>* ln 1 − (*r*K<sup>+</sup> /*ap*) 2 . However, *r*K<sup>+</sup> ≈ *a<sup>p</sup>* (see, e.g., Ref. [55] for ionic sizes) and this approach will lead to large errors and, in fact, does not include important physical processes, such as the movement of oxygen atoms at the pore rim. A better approach is to estimate the entropy from the actual trajectories of ions going through the pore. Potassium ions cross the pore within a radius of about *r<sup>c</sup>* ≈ 0.02 nm from the origin. Assuming that the ions are not localized in a well, but still are locally in equilibrium, the entropic penalty is approximately <sup>−</sup>*kB<sup>T</sup>* ln *r* 2 *<sup>c</sup>*/*r* 2 ABF ≈ 5 *kBT*. The presence of a well of size *l<sup>W</sup>* ≈ 0.3 nm in some cases gives an additional contribution −*kBT* ln (*lW*/*l*) ≈ 1 *kBT* to 2 *kBT*.

**Figure 5. Discrete gradients and the renormalization of energy barriers.** The discrete energy gradients encountered by a K<sup>+</sup> ion in graphene crown ether pores at 0% and 4% strain for 0 V (blue) and 0.25 V (red) applied bias. The gradients are between each local minimum and the next maximum in the positive *z*-direction in Figure 4 and we plot them against the mean position of the barrier. The gradient from the applied bias (*V*ext = 0.25 V), ideally about 10 *kBT*/nm between *z* = ±0.5 nm (shown with black, dashed line), reduces the translocation barrier, completely eliminating it in some cases (shown with a green arrow). To the first approximation (in particular, ignoring the self-consistent development of the potential drop), the sharp features that are larger than the ideal electric field will have a significant influence on the ionic current, as they are still present (though reduced) when the voltage is applied. This ideal behavior is approximately occurring in the *q<sup>O</sup>* = −0.24 *e* pore, as for both values of strain shown the gradients within the membrane region are being collectively diminished. For *q<sup>O</sup>* = −0.54 *e*, more complex behavior is occurring, with some features changing more than others. Examining the change in discrete energy gradients upon application of a voltage gives a clear indication of the presence of irrelevant features—these barriers do not change, as they are not rate limiting and do not create a self-consistent potential drop around themselves. The errors are due to the uncertainties in both the position and the magnitude of minima and maxima.

**Radius of the pore:** Before moving forward, we address an issue that permeates the whole field of transport in sub-nanoscale pores and is apparent in the proceeding paragraph—that of the pore radius. For smooth, uncharged pores, the radius or open area (when not circular) can be rigorously defined with all-atom simulation: One samples the trajectories of ion crossings and takes a weighted average of discrete area elements (see Ref. [16], where the current density was roughly uniform, enabling a direct and intuitive treatment). However, when pore charge is present or the pore has structure, whether steric or energetic, along its length, there is clearly no simple answer for pore radius. The effective radius that defines access resistance, for instance, will not be the same as the geometric radius of the pore mouth. This is easy to see when charge is present at the pore mouth, since the effective opening within a continuum approximation will increase by about a Debye length due to electrostatic attraction of counterions [14]. The fact that effective sites are present will change this picture further. For instance, there are association-side sites that form a staging area from around *z* = −0.4 nm to −0.3 nm with a spread *r<sup>s</sup>* of about 0.1 nm, which is related to *a<sup>p</sup>* but can be influenced by other factors (their numerical values here are the same). It is this region that has to be "accessed."

Moreover, if the pore has a conical shape (e.g., even for this graphene crown ether pore, ions seem to follow a coarse canonical shape, see Figure 3), what radius is relevant to defining the "open area" of the pore, especially when energetic features are present? When variation in size or energy is large on the scale of inter-ion separation and the ion mean-free path, this issue can be handled simply by assuming local equilibrium and appropriately averaging. The graphene pore examined here, as well as other pores in 2D membranes and biological channels, do not have such a simple separation of scales. Fortunately, here, the important length scales that define size fall within the range *r<sup>c</sup>* ≈ 0.02 nm (spread of trajectories of ion crossings) to *r<sup>s</sup>* ≈ 0.1 nm (spread of association-side sites) to *r<sup>p</sup>* = 0.137 nm (geometric radius) to *λ<sup>D</sup>* ≈ 0.3 nm (Debye length). We will take the effective pore radius as *a<sup>p</sup>* ≈ 0.1 nm. This value is in the middle of this range and thus, except for a few particular quantities such as the entropic barrier, it gives a reasonable starting point for estimating values of different pore characteristics.

**Incoming rates:** Before discussing the modeling of these pores, we first introduce a simple tool to further assess the influence of different energetic features. Figure 1 shows the trajectories of K<sup>+</sup> ions moving toward the graphene crown ether pore. The trajectories of ions that eventually translocate through the pore are shown with a red line, and others are shown in cyan lines. Only a few non-translocating trajectories go into the range of *z* = −0.3 nm. This becomes more apparent by plotting the trajectories near the pore (within the radial distance of 0.6 nm from the center of the pore) versus *z* and time, as seen in Figure 6 upper panel. Information regarding the rejection of ions would thus be helpful. In Figure 6, we thus also plot the incoming rate *J*in of ions crossing a *z*-plane versus the *z*-distance at various applied voltage. Initially, *J*in drops rapidly with *z*, as ions have to go through a diffusion constriction and also get reflected by the entrance barrier. At a certain location, *J*in becomes flat, indicating all ions that made it to that distance will complete the translocation. For *q*<sup>O</sup> = −0.24 *e*, for example, the rate drop sharply going from *z* = −0.5 nm to *z* = −0.2 nm in the unstrained pore due to the presence of an occupation barrier. The rate then becomes flat, as ions cannot go back (we note that we do see some ion crossing events that go backward, up the potential gradient. These are few and far between, but the small gap in Figure 6 for some cases quantifies this magnitude of these events). For 4% strain, the rate continues to drop until *z* = 0, as there is a large barrier at the center of the pore.

Similar observations can be made for *q*<sup>O</sup> = −0.54 *e*. For the unstrained pore at small voltage, we see a large drop in the incoming rate between *z* = −0.5 nm and *z* = −0.2 nm due to the repulsion from the ion already in the pore. There is a smaller drop due to dissociation of the ion from the pore. Importantly, both of these drops are due to dissociation, with the former due to a blockade (many-body) effect and the latter being actual ion dissociation. We note that many-body and single-ion effects can be unraveled by comparing the free energies at finite concentration to the free energy of a single ion pair in solution [27], which shows that the satellite barriers for the unstrained, *q*<sup>O</sup> = −0.54 *e* pore are due to the presence of an ion in the pore. For the 4% strain (and for the unstrained pore at larger voltage), the drop in the rate is small and it essentially saturates at *z* = −0.3 nm. This means that ions do not feel a significant barrier going through the pore and the total current is only limited by the rate at which ions arrive at the mouth of the pore. As with the discrete barrier gradients, the plot of the one way ion rate allows for the identification of what features matter. For *q*<sup>O</sup> = −0.54 *e* at strain at about 4% and above, the reduction in ion flux at the entrance side is due to the diffusion constriction and entrance barriers. These incoming ion rate plots thus provide both qualitative and quantitative information. We will use this to motivate the modeling choices below (specifically, the use of a staging site and the assumption of one-way current flow in the pore).

**Reaction rate model:** The 18-crown-6 pore in graphene can only fit a single ion at a time. It is thus intuitive to analyze the ionic transport process using rate theory [42]: Ions arrive at the pore at a certain rate and depart at a certain rate, which together provide the ionic current.

The simplest case would be to assume a single site and that ions only move in one direction. The latter takes into account that the bias is sufficiently large that ions cannot move backward, up

the potential gradient (this is a reasonable assumption for the voltages in this work, as we saw above, but cannot correctly reproduce equilibrium conditions). In that case, the ionic current through the pore is given as *I*/*q* = (1/*k<sup>a</sup>* + 1/*kd*) −1 , where *k<sup>a</sup>* (*k<sup>d</sup>* ) is the rate constant for association (dissociation) of ions into (from) the pore. In terms of the site occupancy (equivalently, probability of being occupied), *I*/*q* = *k<sup>a</sup>* (1 − *P*) = *k<sup>d</sup> P* and *P* = *ka*/(*k<sup>a</sup>* + *k<sup>d</sup>* ). These latter equations make it clear that, with a strain independent association rate *ka*, the current will linearly depend on occupancy and thus cannot plateau, as seen in Figure 2, until *P* is effectively zero. This limit, *k<sup>a</sup> k<sup>d</sup>* , gives *I* = *q ka*, in which case the ionic current is fully determined by the incoming rate.

**Figure 6. Translocation events and one-way rates.** (**Top panel**) Time trace of the *z*-position of potassium ions that translocate (red) through the graphene crown ether pore and reflect (cyan) after coming within 0.6 nm of the center of the pore. For *q*<sup>O</sup> = −0.24 *e*, ions cross the pore very quickly and the association rate is the primary determinant of the current. In the unstrained pore with *q*<sup>O</sup> = −0.54 *e*, the ions spend a significant time in the pore, and thus the dissociation rate determines the current. (**Bottom panel**) The inward rate of K<sup>+</sup> ions versus *z*-distance at different applied voltages. The dashed horizontal lines gives the net rate. For small voltage, *J*in near the pore is much smaller than the bulk diffusion rate, and thus the current is limited by the barriers to transport. Error bars are plus/minus one SE. Connecting lines are a guide to the eye only.

We will see that *P* is still substantial on some of the plateau. Thus, while the fit to a single-site model is reasonable when allowing *k<sup>a</sup>* to have some voltage dependence (i.e., *k<sup>a</sup>* = *ka*<sup>0</sup> + *κaV*), the model is not qualitatively consistent with the data, as the model current still increases when the actual current has leveled off. This assessment of the single-site model is the same regardless of whether only one way motion is assumed or not: Allowing fluctuations in and out of the pore on both sides of the membrane still gives a linear dependence on *P* with a similar coefficient.

Instead, we examine a three-site model, despite the fact that the channel is atomically thin. The data in Figure 3 show that there are multiple localized regions of enhanced K<sup>+</sup> density. On the association side (left side of the figure), there are candidate sites—staging sites—at about −0.3 nm and about −0.4 nm (offset from each other also in the radial direction), and similarly on the dissociation side. That is, there are 4 or 5 candidate sites in the parameter regimes of that figure (ambiguity results from the fact that the candidate sites on the dissociation side are not fully disconnected—there is a non-negligible probability to find an ion in between some locations). These sites are due to ripples in the free energy, which extend outside the pore, as discussed above and seen in Figure 4. Due to the proximity of the association-side staging sites, we will assume they are the same and employ a three site model. Moreover, the one-way rate data in Figure 6 supports this view of the pore, as well as the assumption that current (mostly) flows in one direction at the pore binding site.

The kinetic equations for the three site system are

$$
\dot{P}\_1 \quad = \
& \left(1 - P\_1\right) - k\_b' P\_1 - k\_a P\_1 (1 - P\_2) \tag{5}
$$

$$
\dot{P}\_2 \quad = \
&k\_d P\_1 (1 - P\_2) - k\_d P\_2 (1 - P\_3) \tag{6}
$$

$$
\dot{P}\_3 \quad = \; k\_d P\_2 (1 - P\_3) + k\_{b'} P\_3 - k\_{b'}' (1 - P\_3) \, , \tag{7}
$$

where *P<sup>i</sup>* is the occupancy of the site *i* =1, 2, and 3, *k<sup>b</sup>* (*k* 0 *b* ) is the incoming (outgoing) rate from bulk on the association side, and *k<sup>b</sup>* <sup>0</sup> (*k* 0 *b* <sup>0</sup>) the dissociation side. Again, the set of equations assume only one way motion into (the association side), and out of (the dissociation side), the internal pore site *i* = 2. Backward fluctuations can easily be included, but this adds extra parameters to be fitted and will only influence the fit in a minor way. We will apply this model only to the behavior of the *q<sup>O</sup>* = −0.54 *e* pore, since the *q<sup>O</sup>* = −0.24 *e* pore has more intricate behavior that would ultimately require association rates that are strain-dependent, i.e., that depend on the variation of the free energy landscape. We have discussed the *q<sup>O</sup>* = −0.24 *e* pore extensively already in Ref. [27], including the origin and scale of the free energy variation. We only note here that, as seen in Figure 2, the 0.5 V and 1 V biases for *q<sup>O</sup>* = −0.24 *e* also give an entrance-limited region. The magnitude of the currents in this region are lower than *q<sup>O</sup>* = −0.54 *e* by only an order one factor for the same voltages and strain. The similarities in current are expected for bulk-limited behavior. The fact that they are lower by a small amount is likely due to the increased capture effectiveness of the higher charge pore. The specific estimates for parameters will thus apply in this case, albeit with some small modifications of effective radii and rates.

Even with the assumptions regarding one-way rates at the *i* = 2 site and the symmetry of bulk rates, there are a number of parameters. Instead of direct fitting, we can employ main pore site (*P*2) occupancy data from MD to reduce the number of parameters and see if a consistent model results. Considering only Equation (5) and setting *P*˙ <sup>1</sup> = 0 yields the occupancy of the first site

$$P\_1 = \frac{k\_b}{k\_b + k\_b' + k\_a(1 - P\_2)}.\tag{8}$$

This site and the third site are the least well-defined, and thus eliminating them from the expressions is key to reducing mathematical and computational acrobatics in defining and fitting the quantities in the model. The particle current is given by the last term in Equation (5) (or, equivalently in the steady state, the sum of the first two terms),

$$I/q \quad = \; k\_d P\_1 (1 - P\_2) = \frac{k\_b k\_d (1 - P\_2)}{k\_b + k\_b' + k\_d (1 - P\_2)} \tag{9}$$

$$\mathcal{L} = \left(\frac{1}{k\_b} + \frac{1}{\tilde{k}\_d (1 - P\_2)}\right)^{-1},\tag{10}$$

where ˜*k<sup>a</sup>* = *kaP* eq 1 is the effective association rate and *P* eq <sup>1</sup> = *kb*/(*k<sup>b</sup>* + *k* 0 *b* ) is the equilibrium density of site 1 in the absence of its connection to the main pore site (in this absence, we can examine equilibrium of *P*1). We do not have to separately determine or fit *P* eq 1 , since we can examine solely ˜*k<sup>a</sup>* for association and only *k<sup>b</sup>* to give the influence of bulk. Note that Equation (10) has made no assumptions regarding the relative magnitude of the dissociation rate, or, for that matter, the influence of any of the factors that appear in Equations (6) and (7), other than the *kaP*1(1 − *P*2) term common with Equation (5). Thus, the model can be thought of as just Equation (5), which has only the assumption that there is a negligible backward rate from the site 2 to site 1, which as we have seen is justified for much of the parameter ranges examined for *q<sup>O</sup>* = −0.54 *e*. The form of the bulk rates on the dissociation side and the lack of backward processes on that side is thus inconsequential. Moreover, whether the model is two or three sites is also irrelevant due to our approach. The inclusion of (1 − *P*2) in the model, which will be directly extracted from MD, captures the influence of all potential processes on the dissociation side, whether included in Equations (6) and (7) or not.

Figure 7 shows the occupancy of the pore, *P*2, and the model results overlaid with the current data. Note that we only fit the model for select points (the one for which occupancy data is shown). Since the conductance depends on voltage, we let *k<sup>b</sup>* = *kb*<sup>0</sup> + *κbV*, which together with ˜*k<sup>a</sup>* gives a three parameter fit. The resulting fit parameters are *<sup>k</sup>b*<sup>0</sup> = (0.50 <sup>±</sup> 0.03) <sup>×</sup> <sup>10</sup><sup>9</sup> ion/s, *<sup>κ</sup><sup>b</sup>* = (2.2 <sup>±</sup> 0.2) <sup>×</sup> <sup>10</sup><sup>9</sup> ions/(V·s), and ˜*k<sup>a</sup>* = (1.2 <sup>±</sup> 0.3) <sup>×</sup> <sup>10</sup><sup>10</sup> ion/s, with uncertainties given by the standard error of the fit. We will discuss these parameters shortly, including their agreement with back-of-the-envelope estimates, as well as providing a quantification of diffusion limitations versus drift-supplied ions.

**Figure 7. Pore occupancy versus strain and model fit to the current for the** *q<sup>O</sup>* = −0.54 *e* **pore.** (**a**). Pore occupancy versus strain for the four voltages indicated. The occupancy is decreasing exponentially with strain and voltage, with some additional, minor features and an apparent threshold behavior with voltage at zero strain. These data are reproduced in the SM along with data for the *q<sup>O</sup>* = −0.24 *e* pore. (**b**). Current versus strain at the four voltages labeled, along with the model. The latter was fitted using current-voltage and *P*<sup>2</sup> data at 0% to 10% strain at 2% increments (i.e., the *P*<sup>2</sup> data shown in **a**). The continuous model plot is found by linearly interpolating the *P*<sup>2</sup> data. The model is very good when accounting for diffusion and access limitations. When using the interpolated *P*<sup>2</sup> data from 2% increments (solid line), there is some deviation at 0.5% and 1% strain. However, using the interpolated *P*<sup>2</sup> data including those two additional points (dashed line) shows that the issue is that *P*<sup>2</sup> has features not captured by interpolation at 2% increments (see the SM for the additional *P*<sup>2</sup> data). The *R* <sup>2</sup> and adjusted *R* 2 for the fit are 0.998 and 0.997, respectively (for data from 2% increments). The step-like features are solely due to employing (linear) interpolation to create a continuous curve. Error bars are plus/minus one SE. Connecting lines are a guide to the eye only.

When the model and data are viewed in tandem, the physical behavior is apparent. When the current plateaus versus strain, it is due to combined diffusion and entrance/access limitations, for which without some component of the latter, the current would not increase substantially with voltage (the voltage could only decrease local ion density, increasing and eventually saturating the diffusive contribution in the process). For smaller strains and voltages, the current is dominated by the ˜*ka*(<sup>1</sup> <sup>−</sup> *<sup>P</sup>*2) component. That is, the current is dictated by a many-body effect: localization of a K<sup>+</sup> ion prevents current flow until that ion dissociates, in which case an effective particle current of ˜*k<sup>a</sup>* flows while the pore is empty (i.e., in more concrete terms, this regime can be thought of as a current of zero flowing, while the *i* = 2 site is occupied and ˜*k<sup>a</sup>* otherwise, giving *<sup>I</sup>*/*<sup>q</sup>* <sup>=</sup> <sup>0</sup> · *<sup>P</sup>*<sup>2</sup> <sup>+</sup> ˜*ka*(<sup>1</sup> <sup>−</sup> *<sup>P</sup>*2) and considering *<sup>P</sup>*2, which is between 0 and 1, to be a probability). The many-body nature of transport in this regime is further supported by a decreasing ionic current versus concentration, which shows the saturating nature of the process; see the concentration figures in the SM.

The expression in Equation (10) quantitatively captures the current versus strain and voltage behavior for most of the data. Where it gives the least fidelity to the full simulation result (small strain and low voltage), it still qualitatively captures the trend in the current. For small strain and/or low voltages, this is precisely where backward motion that was neglected in the model is most important, as well as the fact that it is where the sites (except the main binding site) are the least well-defined, see Figure 3. We have seen from molecular dynamics simulations, as well, that there is a small, backward moving current, even at quite large voltage drops. Despite neglecting these effects, we still conclude here that Equation (10) is sufficient to understand and capture ionic transport through the graphene crown ether pore at *q<sup>O</sup>* = −0.54 *e*, as well as *q<sup>O</sup>* = −0.24 *e* at 0.5 V and higher (with a slight modification of rates).

**Rate constants:** The rate constants *k<sup>b</sup>* , *ka*, and *k<sup>d</sup>* depend on attempt frequencies and free-energy barriers that ions encounter during the translocation from one side of the membrane to the other [42,56]. We will consider *k<sup>b</sup>* to have separate diffusion and drift components and for the other two rates to have explicit barriers. When *U<sup>a</sup>* and *U<sup>d</sup>* are the barriers to enter the pore and exit the pore, respectively, then *k<sup>a</sup>* = *k* 0 *a e* <sup>−</sup>*Ua*/*kB<sup>T</sup>* and *k<sup>d</sup>* = *k* 0 *d e* −*Ud*/*kBT* , where *k* 0 *<sup>a</sup>* and *k* 0 *d* are the rate constants for barrierless transport.

Bulk rate constant—We first consider the rate constant from bulk, *k<sup>b</sup>* = *kb*<sup>0</sup> + *κbV*, to the sites on the association side. For pure diffusion, the standard result is to take a capture radius equal to the pore radius *a<sup>p</sup>* and solid angle Θ [1], which would give

$$k\_{b0} \approx \Theta \, D \, \text{c} \, a\_{p\prime} \tag{11}$$

where *D* and *c* are the diffusion coefficient and the bulk ion concentration. We note that here one could argue that we should take *r<sup>s</sup>* (the spread of staging sites) or some modification depending on the Debye length. However, *r<sup>s</sup>* and *a<sup>p</sup>* are related and, indeed, they are equal in this work (as discussed above). The influence of electrostatic interactions is even less clear, as the pore rim is charge neutral on the scale of the Debye length. Thus, we consider *a<sup>p</sup>* only, but there could be further refinement of the estimates of the model and parameters. The solid angle, Θ, is generally taken to be 4*π* in chemical reactions [37,57] and 2*π* for transport through pores [38,39]. Instead, considering the pore to be a circular disc rather than a sphere, one obtains Θ = 4 [58,59]. These estimates of Θ implicitly assume that the particle size is negligible compared to the capture radius. A similar estimation for sub-nanoscale pores is difficult due to the commensurate length scales involved: pore size, hydrated ion size, and Debye length are all similar. For example, Läuger pointed out that the effective capture radius of the pore can be as small as the difference between the geometric radius and the ion radius [35], which, of course, would give rise to similar issues that we discussed in the context of the pore radius.

Assuming that a region of radius *<sup>a</sup><sup>p</sup>* mimics the capture of a circular disc gives *<sup>k</sup>b*<sup>0</sup> <sup>≈</sup> 0.5 ns−<sup>1</sup> for one molar concentration. Different assumptions about the capture geometry yield only an order one deviation

in this estimate. Thus, this is in excellent agreement with that found by fitting the current data to the model, employing the pore occupancy directly from MD, which gives 0.5 ns−<sup>1</sup> also. We note that the same scale of *k<sup>b</sup>* is used for biological ion channels [42].

The driven component of the incoming rate from bulk is also inline with heuristic expectations: Ignoring diffusion and when the current is determined predominantly by the pore itself, see Equation (1), the current will be

$$I \approx \frac{V}{\gamma\_p h\_p / (\pi a\_p^2)}\tag{12}$$

in the continuum limit with pore resistivity *γp*. Again ignoring diffusion, the bulk drift has to supply this same amount of current. Converting to a rate, this gives

$$
\kappa\_b V \approx \pi a\_p^2 V / (q \gamma\_p h\_p). \tag{13}
$$

Alternatively, one can think of this scenario as one where the voltage drop in bulk on one side of the membrane is *V<sup>b</sup>* ≈ *πapγbV*/(4*γphp*), which comes from taking the exact—assuming a continuum with bulk resistivity *γb*—voltage drop on one side of the bulk *I* · *R<sup>a</sup>* and approximating the current as in Equation (12). This partial voltage drop then supplies ions at a rate determined by its bulk, access resistance, *Vb*/*R<sup>a</sup>* (note that here *R<sup>a</sup>* = *γb*/(4*ap*) as we are dealing with one side of the membrane). This yields a bulk rate identical to Equation (13). The bulk resistivity is *γ<sup>b</sup>* = 0.071 Ω·m for 1 mol/L KCl in rigid TIP3P water [14]. Putting in approximate values *h<sup>p</sup>* ≈ 1 nm (see, e.g., Ref. [16]) and *a<sup>p</sup>* ≈ 0.1 nm yields either 1.4 <sup>×</sup> <sup>10</sup><sup>9</sup> ions/(V·s) when using just the K<sup>+</sup> resistivity (*γ<sup>p</sup>* <sup>≈</sup> <sup>2</sup>*γ<sup>b</sup>* ) in Equation (12) or 2.8 <sup>×</sup> <sup>10</sup><sup>9</sup> ions/(V·s) when using the KCl resistivity as the pore resistivity (in the bulk, we use *γ<sup>b</sup>* ), which is in reasonable agreement with the extracted value of (2.2 <sup>±</sup> 0.2) <sup>×</sup> <sup>10</sup><sup>9</sup> ions/(V·s). Again, some parameters, such as *a<sup>p</sup>* and *hp*, may be different, including when one is looking at different characteristics (access versus pore resistance), but at most this will give an order one change—for instance, employing *a<sup>p</sup>* ≈ 0.13 nm and *<sup>γ</sup><sup>p</sup>* <sup>≈</sup> <sup>2</sup>*γ<sup>b</sup>* would give 2.3 <sup>×</sup> <sup>10</sup><sup>9</sup> ions/(V·s). In these estimates, we do allow *γ<sup>p</sup>* 6= *γ<sup>b</sup>* , but this is imposed from above rather than a consequence of free energy barriers or concentration gradients, both of which have more complex repercussions. Computing these even within a continuum picture would require a self-consistent solution, including without local electroneutrality. Barriers in the pore, though, are easy to incorporate, they lower the current and thus lower the drift-induced feeding *κ<sup>b</sup>* (equivalently, they reduce *Vb* ). The proximity of the estimates, though, suggests that the pore in the plateau regime is similar to that of a small open pore—"open" meaning no free-energy features. While the pore does have energetic features for smaller strain (i.e., 2% to 6%, see the SM), this entails that those features are irrelevant in the sense developed above. For larger strain (8% and 10%), the pore is basically barrierless, even in a more strict sense (see the SM). The agreement between treating the drift rate as that in response to a small but otherwise open pore (*h<sup>p</sup>* ≈ 1 nm and *a<sup>p</sup>* ≈ 0.1 nm) may be coincidental, however, as the ion crossings happen at a smaller scale in the middle of the pore (≈0.02 nm). We will discuss this further below.

We note also that the values for *kb*<sup>0</sup> and *κ<sup>b</sup>* are in rough agreement with the one way rates shown in Figure 6. Those rates seemingly would suggest a *kb*<sup>0</sup> about 4 times higher. However, these are one-way rates to cross a whole *z*-plane. Therefore, they will be larger than the rate to go into the staging sites. There are simply more fluctuations in both directions across a *z*-plane far from the pore when one is in the charge layers that maintain the potential drop. If one instead looks at the rates crossing a hemispherical surface (see the SM), the magnitudes are about a factor of two different than *kb*<sup>0</sup> . This agreement is thus still only approximate, but it does suggest consistency of the model and MD data. The agreement with *κb* is also reasonable – the increase of the one-way rates when voltage goes from 0.1 V to 1 V is about 10<sup>9</sup> ion/s to 2 <sup>×</sup> <sup>10</sup><sup>9</sup> ion/s, which agrees with the extracted *<sup>κ</sup><sup>b</sup>* <sup>≈</sup> 2.2 <sup>×</sup> <sup>10</sup><sup>9</sup> ion/s.

Finally, we discuss an alternative potential interpretation (pun intended): Above, we considered rates given separately by bulk diffusion and bulk drift. However, it could be that small entrance barriers, specifically into the association-side staging sites, are giving a weakly-activated process, and hence the voltage enters through the exponent, i.e., *k<sup>b</sup>* = *kb*<sup>0</sup> *e βV*/*kBT* . There are small features in the free energy around which ions would associate into the staging sites, as well as depleted ion density there, see Figures 3 and 4 (and similar figures in the SM). Using this as a fitting form also results in a reasonable fit, albeit slightly worse than the form we use above, especially at low voltage. The resulting fit parameters are *<sup>k</sup>b*<sup>0</sup> = (0.79 <sup>±</sup> 0.06) <sup>×</sup> <sup>10</sup><sup>9</sup> ion/s, *β* = (1.3 ± 0.1)*kBT*/V (at room temperature), and ˜*k<sup>a</sup>* = (1.0 <sup>±</sup> 0.4) <sup>×</sup> <sup>10</sup><sup>10</sup> ion/s, with uncertainties given by the standard error of the fit. All these numbers are inline with the heuristic estimates.

The major difference between these two interpretations is the behavior at small voltage and that the drift-based interpretation better captures the data at the smallest voltage we examine (0.1 V). Otherwise, it will be difficult to discern the exact diffusion/entrance mechanism: The expected one-sided access-induced potential drop is 1.5 *kBT* at room temperature when the total applied voltage is 1 V (i.e., about 40 *kBT*). From a homogeneous drift theory [58], about half of this is expected to drop within a distance *a<sup>p</sup>* from the pore (i.e., within the "Hille" hemisphere [19])—that is, one would have to dissect small changes in the free energy and potential with voltage in the same spatial vicinity. The precision to which the calculations would have to be performed is astounding—not just statistical precision, which can be made smaller than this, but non-scaling finite-size effects would have to be nearly completely removed [16,17]. Clearly, studying the temperature dependence can help further delineate these two interpretations by revealing activation energies (provided that the temperature dependence of other factors, such as the resistivity, can be accounted for), as can an even more comprehensive study including smaller voltages (where activation will be more clearly visible) and larger simulation cells (to completely—to within more than *kBT*—remove non-scaling finite-size effects [16,17]). While the data here favor the drift-based interpretation, it is not conclusive but it does not affect the main findings of a diffusion-limited regime around 0.1 V. Yet another alternative model is to just retain *κ<sup>b</sup>* in *k<sup>b</sup>* (i.e., *kb*<sup>0</sup> = 0). This assumes that just drift is feeding ions to the pore. However, this model gives a poorer fit and is not consistent with the data. Thus, diffusive contributions from the bulk are present.

Association rate constant—We next consider *ka*. The model fit did not directly give us this parameter, but instead the effective association rate ˜*k<sup>a</sup>* = *kaP* eq <sup>1</sup> = (1.2 <sup>±</sup> 0.3) <sup>×</sup> <sup>10</sup><sup>10</sup> ion/s. This rate is three times larger than *k<sup>b</sup>* at 1 V, and 30 times larger than *k<sup>b</sup>* at 0.1 V. Thus, the only time this component of the resistance matters is when the factor, 1 − *P*2, multiplying it in Equation (10), is small, which occurs only when both strain and voltage are small. It is difficult to estimate this parameter a priori without sufficient gymnastics as to obscure the truth of the matter. However, there are two qualitative features that support its magnitude. The first is that *P* eq 1 is relatively small, which can be seen from Figure 3, probably around 1/10 or smaller. This means that *k<sup>a</sup>* is an order of magnitude or more larger. From Figure 4, a large value of *k<sup>a</sup>* is expected. There is only a small barrier around −0.4 to −0.3 nm for the *q<sup>O</sup>* = −0.54 *e* pore, and then the ion will be driven downhill into the pore binding site. That is, we do expect a large *k<sup>a</sup>* for an ion already in the staging area.

Dissociation rate constant—We next consider *k<sup>d</sup>* . This parameter does not participate in the model fitting at all, since we instead used the computationally determined *P*2, which enabled us to only employ the first of the three equations, Equation (5), in the model. However, we can employ the outgoing current from the second site, i.e., the second term in Equation (6), to estimate *k<sup>d</sup>* . We can do this by noticing that *P*<sup>3</sup> is also small (just as *P*<sup>1</sup> is). Since here, 1 − *P*<sup>3</sup> is present, the estimate assuming *P*<sup>3</sup> is small will be less sensitive to this assumption compared to *P* eq 1 and *ka*. On the plateau, this entails that *kdP*<sup>2</sup> = *I*/*q* = constant. Examining this relation, data point by data point, gives *k<sup>d</sup>* estimates between 10<sup>9</sup> ion/s to 10<sup>11</sup> ion/s, with well defined trends versus strain and voltage. For instance, at 1 V, one obtains *<sup>k</sup><sup>d</sup>* <sup>=</sup> 2.0 <sup>×</sup> <sup>10</sup>10*<sup>e</sup>* (0.20±0.01)*s* ion/s, where *s* gives the strain in percent and the confidence interval of the prefactor is [1.9 <sup>×</sup> <sup>10</sup>10, 2.2 <sup>×</sup> <sup>10</sup>10] ion/s. Thus, the *<sup>k</sup><sup>d</sup>* varies from 2.0 <sup>×</sup> <sup>10</sup><sup>10</sup> ion/s at 0% strain to 1.5 <sup>×</sup> <sup>10</sup><sup>11</sup> ion/s at 10% strain. For completeness, the remaining voltages give fits for *<sup>k</sup><sup>d</sup>* of 2.7 <sup>×</sup> <sup>10</sup><sup>9</sup> *e* (0.28±0.03)*s* ion/s, 8.3 <sup>×</sup> <sup>10</sup><sup>8</sup> *e* (0.29±0.02)*s* ion/s, and 4.3 <sup>×</sup> <sup>10</sup><sup>8</sup> *e* (0.24±0.02)*s* ion/s for 0.5 V, 0.25 V, and 0.1 V, respectively, with corresponding prefactor confidence intervals [2.2 <sup>×</sup> <sup>10</sup><sup>9</sup> , 3.3 <sup>×</sup> <sup>10</sup><sup>9</sup> ] ion/s, [7.1 <sup>×</sup> <sup>10</sup><sup>8</sup> , 9.7 <sup>×</sup> <sup>10</sup><sup>8</sup> ] ion/s, and [3.7 <sup>×</sup> <sup>10</sup><sup>8</sup> , 5.0 <sup>×</sup> <sup>10</sup><sup>8</sup> ] ion/s. A reasonable fit to all the plateau data (versus voltage and strain) is 3.2 <sup>×</sup> <sup>10</sup>9*Ve*(0.043±0.003)*qV*/*kBT*+(0.22±0.01)*<sup>s</sup>* ions/(V · s) with confidence interval [2.8 <sup>×</sup> <sup>10</sup><sup>9</sup> , 3.6 <sup>×</sup> <sup>10</sup><sup>9</sup> ] ions/(V·s) (note that *V* is a prefactor out front, as well as in the exponent, since ions are driven across the pore). One takeaway from this is not only the order of magnitude of *k<sup>d</sup>* , but that strain and voltage in this regime change barriers in the pore region by O(*kBT*). This is in agreement with estimates of how barriers change due to strain, see Equation (4) and Ref. [27]. However, it is somewhat surprising that voltages, that are 10s of *kBT*, do not change the barriers by more. The reason that this is not the case is that the pore barriers simultaneously are not playing a strong role in the resistance (i.e., they are irrelevant in the language above and thus do not self-consistently get removed by the voltage) and they occur on a scale of 0.1 nm. This means that what is relevant is the ∆*V* on this scale. For 1 V that evenly drops over 1 nm, this is only 4 *kBT*, i.e., only about a factor of 2 to 4 (assuming 0.2 nm over which the barrier occurs) above the actual change found in the fitted form above. We do not expect back-of-the-envelope estimates to do much better.

To give an independent estimate of the dissociation constant for comparison, we assume that only one ion occupies the pore at a time and the translocation is driven by a constant electric field, *Ep*, across the internal pore site of length ∆*p*. Thus, the rate constant for exit from the pore may be estimated from the drift velocity-like picture (with an effective diffusion coefficient as an attempt frequency times an Arrhenius factor) as

$$k\_d = \frac{v\_d}{\Delta\_p} = \frac{q \, D \, e^{-lI\_d/k\_B T} E\_p}{k\_B T \, \Delta\_p} = \frac{q \, P\_p E\_p}{k\_B T} \, \tag{14}$$

where *P<sup>p</sup>* = *D e*−*Ud*/*kBT*/∆*<sup>p</sup>* = *Dp*/∆*<sup>p</sup>* is the permeability of the ion. Assuming that *E<sup>p</sup>* = *V*/*h<sup>p</sup>* with *h<sup>p</sup>* = 1 nm (i.e., a potential drop over the effective membrane thickness that is larger than the internal pore site), we obtain *k* 0 *<sup>d</sup>* <sup>≈</sup> <sup>76</sup> ns−<sup>1</sup> (for ∆*<sup>p</sup>* = *hp*) to 190 ns−<sup>1</sup> (for ∆*<sup>p</sup>* = 0.4 nm, which is more represented of the *<sup>P</sup>*<sup>2</sup> site) for 1 V applied voltage and the potassium mobility *<sup>µ</sup><sup>K</sup>* <sup>=</sup> *qD*/*kB<sup>T</sup>* <sup>=</sup> 7.62 <sup>×</sup> <sup>10</sup>−<sup>8</sup> <sup>m</sup>2/(V·s). This is in reasonable agreement with the *<sup>k</sup><sup>d</sup>* <sup>≈</sup> <sup>150</sup> ns−<sup>1</sup> found above for 1 V and 10% strain, where the latter has the smallest influence of barriers and is thus most similar to the barrier-free estimate here. Notably, however, this estimate decreases linearly with voltage. For 0.1 V, the estimate is 10 times too high compared with the one found from the MD data. However, it is clear from the *P*<sup>2</sup> data that occupancy is dropping faster than exponentially with voltage, meaning that *k<sup>d</sup>* increases faster than exponentially (note that the pore conductance, proportional to *kdP*<sup>2</sup> decreases with voltage, in line with diffusion-limited expectations). The form fitted above for all plateau data assumed a form *VevV*, with *v* as a positive constant. This form performs well and indicates that Equation (14) is only reasonable where the voltage is not modifying the energetic landscape at all.

Finally, we comment on the magnitude of *k<sup>d</sup>* compared to the bulk rate constant *k<sup>b</sup>* . Even taking into account the effect of the potential well in the pore, the dissociation rate constant is still larger than the association rate constant (∼0.5 ns−<sup>1</sup> ) for most cases. For plateau data, the smallest the dissociation constant becomes a factor of two larger, but for almost all data, it is an order of magnitude larger or more. Only in the unstrained pore with *q*<sup>O</sup> = −0.54 *e*, where there is a large exit barrier, are the two rate constants comparable at a small voltage. Thus, ionic transport in the crown ether pore is—outside of the colossal mechano-conductance regime—generally controlled by the rate at which ions arrive in the pore, i.e., the diffusion and drift rates, possibly with some reflection at the pore mouth due to a small

entrance barrier. The latter includes an overall barrier for pore occupation when extended to *q*<sup>O</sup> = −0.24 *e*, see Figure 4.

**Diffusion-limited currents:** The observation of diffusion-limited currents requires both that current-carrying ions spend little time in the pore and that drift component of feeding ions to the pore is small. In terms of the model, Equation (10), we need *<sup>k</sup><sup>b</sup>* ˜*ka*(<sup>1</sup> <sup>−</sup> *<sup>P</sup>*2) and that *<sup>κ</sup>bV<sup>b</sup> <sup>k</sup>b*<sup>0</sup> . To meet the former condition requires that *<sup>k</sup><sup>b</sup>* ˜*k<sup>a</sup>* (i.e., entrance barriers should not be large, or otherwise ˜*k<sup>a</sup>* will be small) and ˜*k<sup>a</sup> <sup>k</sup><sup>d</sup>* (to ensure that *P*<sup>2</sup> is not close to one), which combines to the chain of inequalities

$$k\_b \ll \tilde{k}\_d \ll k\_d.\tag{15}$$

In other words, both entrance and exit barriers should be small (i.e., transport in a near barrierless regime, where "near" is defined in terms of how fast ions arrive at the pore from bulk and thus, under realistic conditions, even barriers in the range of 5 *kBT* can be "near" barrierless for this pore, but what quantifies "near" depends on pore characteristics).

When *<sup>k</sup><sup>b</sup>* ˜*ka*(<sup>1</sup> <sup>−</sup> *<sup>P</sup>*2) in Equation (10), we get *<sup>I</sup>* <sup>=</sup> *q k<sup>b</sup>* , in which case the ionic current is fully determined by the incoming rate from bulk and is independent of pore conditions, as seen in Figure 2. Albeit, one has to compare *k<sup>b</sup>* to the association rate, which not only can have a free energy barrier associated with it, but also the equilibrium occupancy of the stating site *P* eq 1 , and thus ˜*k<sup>a</sup>* can be quite small itself.

The conditions above can in turn be employed to put conditions on the voltage. First consider a lower bound: Equation (14) gives the rate at which ions cross the pore, including both the drift (due to the local electric field) and the dissociation from a pore well (if present). This rate is proportional to the voltage. Considering the chain inequality above and considering *k<sup>d</sup>* at *U<sup>d</sup>* = 0 (i.e., the time spent in the pore without a well needs to be much greater than the bulk feeding—the presence of a well will only push this inequality toward not being satisfied):

$$k\_d \gg k\_b \approx k\_{b0} \implies E\_p \approx V/h\_p \gg \Theta \, k\_B T \, c \, a\_p \, h\_p / q,\tag{16}$$

where we can take *k<sup>b</sup>* ≈ *kb*<sup>0</sup> , since we are interested in the regime where diffusion dominates over drift. We also take ∆*<sup>p</sup>* ≈ *h<sup>p</sup>* (this only drops an order one factor). This relation indicates that the diffusion-limited current is likely to be observed in short and narrow pores, provided that entrance and exit barriers are small. Note that, although many biological ion channels are not necessarily short compared to their width, ions can move in a single-file concerted motion via "knock-on" mechanisms [27,60], which diminishes the effective length of the channel.

An approximate upper bound on the voltage to observe diffusion limitations is, as already noted, for there to be little voltage drop in the bulk. For instance, Läuger points out that the presence of excess impermeable, or "inert", electrolyte increases the impact of diffusion limitations [35], a fact that occurs in our pore (i.e., Cl− is inert). This is due to the fact that an impermeable electrolyte shifts the balance of pore and bulk resistance, making the former much larger relative to the latter. Assuming a cylindrical pore, the pore resistance is dominant if *hp*/*πa<sup>p</sup> γb*/2*γp*, where *γ<sup>b</sup>* (*γp*) is the resistivity in bulk (pore). For larger graphene pores, and even some nanoscale ones, this condition is unlikely to be true. In fact, access resistance is larger than the pore resistance for most sizes of graphene pores, and thus the majority of the voltage will drop in the bulk. In such a case, the current will be limited, not by diffusion but mostly by drift.

In the graphene crown ether pore, however, the effective pore radius is around 0.1 nm and, when under strain, near barrierless in the sense used above (there may be barriers and wells, but the prefactors—the transition rates or attempt frequencies—are still determining the hierarchy of rate scales).

Using the same estimate to find *κ<sup>b</sup>* as above, where we assume a homogeneous, continuum medium both inside and outside the pore, with the pore resistance the dominant factor, we obtain

$$
\hbar \times\_b V \approx \pi a\_p^2 V / (q \gamma\_p h\_p) = \pi a\_p^2 V c\_p \mu\_p / h\_p \ll k\_{b0} \approx \Theta D \, c \, a\_p. \tag{17}
$$

This upper bound can be derived directly from the steady state Nernst–Planck equation, assuming hemispherical symmetry (i.e., with only a radial component) and a homogeneous medium. There, one wants

$$\frac{\partial c}{\partial r} \gg \frac{qc}{k\_B T} \frac{\partial \Phi}{\partial r},\tag{18}$$

to have the diffusion contribution much larger than the drift, where Φ is the electric potential. Taking the pore mouth to be a hemisphere with radius *ap*. The RHS is *qcapVb*/(*r* 2 *kBT*). The LHS is ∆*cap*/*r* 2 , with ∆*c* the concentration bias between the bulk (infinitely far from the pore) and the hemispherical pore mouth. This presumes that the diffusive and drift components are decoupled.

Assuming further that the staging site has zero occupancy (and thus zero concentration), this gives the maximum diffusion contribution and Equation (18) results in

$$qV\_b \ll k\_B T.\tag{19}$$

This relation is interesting in itself. Its simplicity is due to Einstein's relation of mobility and diffusion coefficients, which results in additional factors dropping out, and due to comparing a maximum diffusive current occurring at the largest concentration gradient with the maximum drift current occurring at zero concentration gradient. Equation (19) indicates that for drift to be negligible, the voltage drop in bulk has to be less than the thermal energy. The latter "drives" the diffusion. It should be larger than the drive of the drift current from bulk to the pore. Plugging in the form of *V<sup>b</sup>* assuming a bulk potential drop in the presence of a dominant pore resistance (see just below Equation (13)) gives the same inequality as Equation (17) up to order one factors.

Rewriting Equation (17) together with Equation (16), assuming *c<sup>p</sup>* = *c* and *µ<sup>p</sup>* = *µ* (i.e., that these two quantities are equal to their bulk), to obtain a two-sided inequality for *V* yields

$$
\Theta \, k\_B T \, \text{c} \, a\_p \, h\_p^2 \ll qV \ll \Theta \, k\_B T \, h\_p / (\pi a\_p). \tag{20}
$$

This foundational relation gives one of the main predictions of this paper: Diffusion-limited currents appear within a sweet spot when free energy features are irrelevant. For the graphene crown ether pore, the voltage should be between about 6 mV and 300 mV. At voltages higher than this range, drift will be important and, below this range, ions will not be removed from the pore region fast enough to create a concentration gradient (and free energy features will also become relevant). The simulations and modeling validate the upper bound (at 0.25 V, the bulk drift and diffusion contributions are roughly equal), but they do not address the lower bound (in any case, free energy barriers will likely be relevant at 6 mV for the *q<sup>O</sup>* = −0.54 *e* pore, as they are with the *q<sup>O</sup>* = −0.24 *e* pore still at 100 mV and 250 mV, the relevance of which is inconsistent with the assumptions leading to Equation (20)). This range includes the voltage, 0.1 V, that we see the strongest diffusion limitations, whereas at higher voltages, drift starts to determine the current. The pore charge is important, as it determines when free energy features are irrelevant (e.g., at 0.1 V but 0% strain, the free energy landscape is dominant). Around 0.1 V is a typical value for graphene pore experiments, small enough to not degrade the membrane (i.e., 0.5 V and higher will start to see membrane degradation), but large enough that typical currents are in the 10 s of picoampere or more (while the time-resolution is irrelevant to measuring the dc conductance, we do note that pin hole leaks or other factors can set a baseline resolution of the current, around 0.5 pA at 0.1 V (see Ref. [33] where such

currents in "as-grown" membranes could vary by an order of magnitude from membrane to membrane). We note that the rate model that we are developing cannot be used at very small voltages, as it includes only one way currents in the pore which cannot capture the approach to equilibrium as *V* → 0.

In any case, there should be a very small drift current in the bulk when the applied voltage is around 0.1 V and it should start to become comparable to the diffusive component at about 0.25 V and dominant for higher voltage. We can make this quantitative using the fit to the model in Equation (10). For instance, at 0.1 V, the unstrained membrane has an effective pore associate rate, ˜*ka*(<sup>1</sup> <sup>−</sup> *<sup>P</sup>*2) of 12/µs due to the presence of a localized ion that creates a many-body blocking effect (i.e., *P*<sup>2</sup> ≈ 1). This effective rate increases to 11/ns for 10% strain. Meanwhile, the diffusive rate is 0.5/ns and the drift rate is 0.2/ns. Thus, diffusion supplies ions over drift by more than a factor of 2 over the whole range of strains, and already at 2%, the strain is smaller (though comparable) to the pore association rate (about 1.9/ns). At 0.25 V, the diffusive and drift components are comparable at 0.5/ns and 0.55/ns, respectively. These values are slightly more than the 0.25/ns effective pore association rate at 0% strain, but are the controlling factors for essentially all strains at 2% and above.

Since the smallest voltage we consider, 0.1 V, has smallest bulk drift contribution, we can postulate that the plateau resistance is the closest to *γphp*/(*πa* 2 *p* ) (i.e., without any access component). Employing *γp*/2 = *γ<sup>b</sup>* = 0.071 Ω·m (*γ<sup>b</sup>* is the resistivity of 1 mol/L KCl in TIP3P water, see Ref. [14]) and *h<sup>p</sup>* ≈ 1 nm, this gives *a<sup>p</sup>* ≈ 0.11 nm, in agreement with the effective pore radius. This is unexpected, since the potassium ions cannot make use of the full pore area for transport and there are diffusion limitations. There may be several factors that conspire to give this agreement. One is that the pore rim is not fixed but can instead move, so that *a<sup>p</sup>* can be bigger than a priori expectations. This does not, however, seem to be the case, since the density plots show that ions are translocating closer to the origin than 0.1 nm. Another factor is the role of *hp*. The effective thickness may be smaller than 1 nm (its value for unfunctionalized graphene pores [16]). Moreover, while transport veers toward barrierless transport, the pores are not becoming barrierless in the strict sense for either *q<sup>O</sup>* = −0.54 *e* (until high strain, see the SM) or *q<sup>O</sup>* = −0.24 *e* (see Ref. [27] for the discussion of the latter case). However, localized binding sites can give a rate that is similar in magnitude to free diffusion through the pore constriction, or even a higher rate, because, while ions have to jump out of the well and the barrier height thus suppresses the rate, there is still a large prefactor, since the ion is fluctuating rapidly. The enhanced density can push the currents higher than expected based on just an open area. Whether we should think about the pore as an open pore of radius 0.1 nm or whether it is a pore of radius 0.02 nm with an enhanced density due to binding, is an interesting question. Evidence—specifically the higher concentration in a smaller spatial region—suggests the latter. However, we only point out that there is still broad agreement between these two perspectives and they only inform us how we should dissect the pore resistance *R<sup>p</sup>* into component pieces (meaning, the utility of the perspectives is limited).

It is to be noted that the access resistance in an MD simulation (or any other method) depends on the simulation cell size: one can make it arbitrarily small (using a wide and short cell) or large (using tall and narrow cell) [16]. Therefore, in order to match experimental conditions, which effectively has an infinite bulk, one has to exert great care. In our simulation, we chose the simulation cell aspect ratio to be the golden aspect ratio [17], which ensures that the access resistance represents the infinite, balanced bulk resistance. Without taking this approach, one could not examine the bulk diffusion and access limitations. In the SM, we show results of simulations for several different voltages and strains, showing that the golden aspect ratio gives converged currents, ones where the bulk is properly included.

#### **4. Conclusions**

Diffusion-limited ionic currents are commonly observed in biological channels because they can provide the necessary conditions: a large pore resistance (compared to access resistance) due to the small pore radius, but also a high permeability of ions due to the functional groups that facilitate the transport of ions [61]. However, diffusion limitations have not been studied systematically in synthetic nanopores, since it is difficult to replicate the permeability of biological ion channels. In this regard, strained synthetic pores may provide a platform, not only to investigate the competition of dehydration and electrostatic interactions within precision atomic constructions that lead to optimal transport characteristics [27], but to investigate diffusion and entrance effects in ionic transport.

We have shown that there is broad agreement between a simple, many-body model developed here for the *q<sup>O</sup>* = −0.54 *e* pore and the all-atom simulations, encompassing not only the residuals and current fit, but also with the fit parameters themselves and independent estimates. This agreement suggests that, with strain, this pore transitions from a barrier-limited pore with current dictated by many-body mechanisms (i.e., a well with a localized ion that blocks the pore), to one equivalent to an open tiny pore. The pore still has a free energy structure, but this structure is irrelevant in the plateau regime: At small voltages, current-limiting regions of the landscape—dictated both by the barrier scale and the kinetics—will appear. Larger voltages will start to self-consistently remove those limiting regions by the counteracting local voltage drop. Other regions of the landscape will start to be limiting, and those regions will subsequently be washed out. For a given strain and voltage, though, the type of behavior observed depends on the pore charge and ion dehydration energy. In the particular pore here, the extent of the bulk-limited region reflects whether the unstrained pore has an internal (dehydration-dominated) barrier or (electrostatically stabilized) well.

Therefore, the transition from the barrier-limited to the diffusion-limited regime gives the opportunity to experimentally delineate and constrain the electromechanical environment of the pore, thus pushing further the limits of employing synthetic pores to understand complex mechanisms in sub-nanoscale ion transport. Functionalized pores in two dimensional membranes are thus simultaneously complicated enough to display a wide-range of ionic phenomena seen in biological pores and simple enough to be amenable to direct modeling. Moreover, if other information can be experimentally determined, such as the (equilibrium) pore occupancy (*P*<sup>2</sup> here), then measurement will enable the extraction of kinetic rates and barriers via modeling. In other words, the graphene crown ether pore is about as simple a sub-nanoscale pore as possible. Yet, it displays a wide variety of behavior: single versus many-body ion competition, optimality, diffusion limitations, relevant versus irrelevant features, etc. Its behavior, for instance, will enable quantifying aspects of transport, such as the role of precision atomic placement and charge in biological systems, and a theoretical understanding of what "near barrierless" entails in particular pores. This area is vast, and pores in 2D membranes will provide the landscape for a systematic experimental exploration and validation of theoretical models of sub-nanoscale pores and biological channels.

#### **Supplementary Materials:** The supplemental material (MS) is available online at http://www.mdpi.com/1099-4300/ 22/11/1326/s1.

**Author Contributions:** S.S. performed the numerical calculations. Both authors modeled and analyzed data, wrote the manuscript, and clarified the ideas. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Acknowledgments:** The authors thank David P. Hoogerheide, James Alexander Liddle, Jacob Majikes, and Justin Elenewski for comments on the manuscript. S.S. acknowledges support under the Cooperative Research Agreement between the University of Maryland and the National Institute of Standards and Technology Physical Measurement Laboratory, Award 70NANB14H209, through the University of Maryland.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Abbreviations**

The following abbreviations are used in this manuscript:


#### **References**




**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Review and Modification of Entropy Modeling for Steric E**ff**ects in the Poisson-Boltzmann Equation**

#### **Tzyy-Leng Horng**

Department of Applied Mathematics, Feng Chia University, Taichung 40724, Taiwan; tlhorng@fcu.edu.tw Received: 11 May 2020; Accepted: 4 June 2020; Published: 8 June 2020

**Abstract:** The classical Poisson-Boltzmann model can only work when ion concentrations are very dilute, which often does not match the experimental conditions. Researchers have been working on the modification of the model to include the steric effect of ions, which is non-negligible when the ion concentrations are not dilute. Generally the steric effect was modeled to correct the Helmholtz free energy either through its internal energy or entropy, and an overview is given here. The Bikerman model, based on adding solvent entropy to the free energy through the concept of volume exclusion, is a rather popular steric-effect model nowadays. However, ion sizes are treated as identical in the Bikerman model, making an extension of the Bikerman model to include specific ion sizes desirable. Directly replacing the ions of non-specific size by specific ones in the model seems natural and has been accepted by many researchers in this field. However, this straightforward modification does not have a free energy formula to support it. Here modifications of the Bikerman model to include specific ion sizes have been developed iteratively, and such a model is achieved with a guarantee that: (1) it can approach Boltzmann distribution at diluteness; (2) it can reach saturation limit as the reciprocal of specific ion size under extreme electrostatic conditions; (3) its entropy can be derived by mean-field lattice gas model.

**Keywords:** steric effect; Poisson-Boltzmann model; Bikerman model; entropy; specific ion size

#### **1. Introduction**

One of the major limitations of the Poisson-Boltzmann (PB) and Poisson-Nernst-Planck (PNP) models is the assumption of point-like ions without considering their sizes. These models based on mean field theories work well for dilute electrolytes, but break down when the concentration is high and ions are crowded in it. A high concentration would generally cause steric repulsions and additional electrostatic correlations among ions, that cannot be described by classical PB/PNP models [1]. For example, the concentration of counter-ions, predicted by PB, can be unrealistically high near the electrode surface, when the electrode voltage is large. Another example occurs at the selectivity filter in a potassium channel, where potassium ions are strongly attracted into this extremely narrow filter by the strong negative charges of oxygens on the backbone of the filter. Employing classical PB/PNP models would overestimate the density of potassium inside the filter and give incorrect channel current predictions. Therefore, many researchers have worked on the modification of PB/PNP to include the steric effect of ions.

Steric effect has long been approached in modeling by modifying either the internal energy or entropy in the Helmholtz free energy. Through internal energy, the steric effect has been featured as excess hard-sphere energy either by density functional theory (DFT) [2,3] or Lennard-Jones potential [4]. These energies were all formulated using non-local potentials and cause the resultant modified PB/PNP to produce a series of complicated integro-differential equations, which are hard to compute in higher dimensions. For practical implementations, localization of hard-sphere potential and simplifying integro-differential equations into pure differential equations has been conducted in [5–8] for DFT and [9] for Lennard-Jones potential.

Through an entropy approach, Bikerman modified the classical Boltzmann distribution by adjusting bulk and local ion concentrations via the excluded volume concept [10]. Borukhov et al. [11] rigorously derived the same formula independently by adding solvent entropy through excluded volume into the Helmholtz free energy. Although the localized hard-sphere model-based DFT [5–8] also captures this solvent entropy as one of the terms accounting for excess hard-sphere chemical potential, the Bikerman model [10,11] has been a more popular steric model due to its easiness of application and qualitatively good agreement with experiments [12–15].

In order to obtain the potential and further derive a neat modified PB equation from free energy, Borukhov et al. [11] treated all ions as having identical size, which has been long criticized for neglecting specific ion sizes. Researchers have tried to address this shortcoming with specific ion sizes, and many of them simply extended original Bikerman model by replacing the identical ion size with specific ones without any rigorous justification. Although the resultant model has a better agreement with experiments than the original Bikerman model [16–18], it does not have a Helmholtz free energy to support it. Here modifications of the Bikerman model to include specific ion sizes have been developed iteratively in Sections 4–6, preceded by derivations of classical PB in Section 2, and the original Bikerman model in Section 3. Finally, in the Discussion and Conclusions section a specific-ion-size Bikerman model is presented with a guarantee that: (1) it can approach Boltzmann distribution at diluteness; (2) it can reach the saturation limit as the reciprocal of specific ion size under extreme electrostatic conditions; (3) its entropy can be derived by a mean-field lattice gas model.

#### **2. Classical Poisson-Boltzmann Model**

Though the classical PB model is well known, we still derive the model here for review and comparison with its modified versions discussed later. Starting by stating the Helmholtz free energy, internal energy and entropy, we have:

$$F = \mathcal{U} - TS \tag{1}$$

$$
\mathcal{U} = \int \left[ -\frac{\varepsilon}{2} \left| \nabla \phi \right|^2 + z\_{\text{ $\mathcal{P}$ }} \text{ep}\phi + z\_{\text{ $n$ }} \text{en}\phi + q\phi + p\mathcal{W}\_{\text{sol},\text{p}} + n\mathcal{W}\_{\text{sol},\text{n}} \right] dV\_{\text{ $\mathcal{I}$ }} \tag{2}
$$

$$-TS = \int k\_B T \left[ p \, \log \frac{p}{c\_0} - p + n \, \log \frac{n}{c\_0} - n \right] dV\_\prime \tag{3}$$

where *F* is Helmholtz free energy; *U* is internal energy; *T* is temperature; *S* is entropy; φ is electric potential. *p*, *n* denote cation/anion concentrations, and *zp*, *z<sup>n</sup>* denote their valence, respectively. *e* denotes elementary charge. *q* denotes permanent charge. Permittivity ε = ε0ε*<sup>r</sup>* with ε<sup>0</sup> being the permittivity for vacuum and ε*<sup>r</sup>* being the relative permittivity or dielectric constant. *c*<sup>0</sup> is some reference concentration such as bulk concentration of electrolyte. *Wsol*,*<sup>p</sup>* and *Wsol*,*<sup>n</sup>* denote the solvation energies for cations and anions, respectively. Although the traditional PB model generally does not include solvation energy in the expression, it is important when modeling some electrolyte systems involving hydration/dehydration of ions and is therefore it is explicitly included in the energy here. Based on the Born model, the solvation energies for cations and anions are:

$$\mathcal{W}\_{\text{sol},i} = \frac{z\_i^2 e^2}{8\pi \varepsilon\_0 r\_i} \left(\frac{1}{\varepsilon\_r(\mathbf{x})} - 1\right), i = p\_\prime n. \tag{4}$$

Differentiation of *F* with respect to φ gives the Poisson equation:

$$-\nabla \cdot \left( \varepsilon \left( \mathbf{x} \right) \nabla \phi \right) = z\_p ep + z\_n en + q \tag{5}$$

By doing the differentiation of *F* with respect to *p* and *n*, we obtain the chemical potentials for *p* and *n*, respectively:

$$\frac{\partial F}{\partial p} = \mu\_p = z\_p e \phi + k\_B T \, \log \frac{p}{c\_0} + W\_{\text{sol}, p\text{\textquotedblleft}p\text{\textquotedblright}} \tag{6}$$

$$\frac{\partial F}{\partial n} = \mu\_n = z\_n e \phi + k\_B T \, \log \frac{n}{c\_0} + \mathcal{W}\_{\text{sol},n} \tag{7}$$

At equilibrium, the chemical potential is uniform everywhere and therefore the local chemical potential must be equal to its bulk value, which is usually known:

$$
\mu\_p = \mu\_{p,b\prime}\,\,\mu\_n = \mu\_{n,b\prime} \tag{8}
$$

with the bulk chemical potential for cations and anions:

$$
\mu\_{p,b} = k\_B T \log \left(\frac{p\_b}{c\_0}\right) + \mathcal{W}\_{\text{sol},p,b\prime} \quad \mu\_{n,b} = k\_B T \log \left(\frac{n\_b}{c\_0}\right) + \mathcal{W}\_{\text{sol},n,b\prime} \tag{9}
$$

where the subscript *b* denotes the bulk situation. Equations (8) and (9) can be solved solve for *p* and *n*:

$$p = p\_b e^{-\beta z\_{p} c \phi} e^{-\beta \Delta W\_{sd,p}} = p\_b e^{-\beta E\_p}, \ n = n\_b e^{-\beta z\_{b} c \phi} e^{-\beta \Delta W\_{sd,p}} = n\_b e^{-\beta E\_n} \tag{10}$$

where β = 1/*kBT*, *E<sup>p</sup>* = *zpe*φ + ∆*Wsol*,*<sup>p</sup>* , *E<sup>n</sup>* = *zne*φ + ∆*Wsol*,*n*, and:

$$
\Delta W\_{\rm sol,i} = W\_{\rm sol,i} - W\_{\rm sol,i,b} = \frac{z\_i^2 e^2}{8\pi \varepsilon\_0 r\_i} \left( \frac{1}{\varepsilon\_r(\mathbf{x})} - \frac{1}{\varepsilon\_{r,b}} \right) \; i = p\_\prime n. \tag{11}
$$

From (10), as φ → −∞, we obtain *p* → ∞, *n* → 0. Likewise, as φ → ∞, we obtain *p* → 0, *n* → ∞. These unrealistic infinite concentrations for *p* and *n* are mainly because ions are treated as particles without size in the classical PB model. This pitfall has motivated modifications of the classical PB/PNP model to account for the finite-size effect, or so-called steric effect, of ions. In reality, the limit of *p* should be at most 1/*vp*, where *v<sup>p</sup>* is the particle volume of *p*. This can be derived by considering a volume *V* fully occupied by cation *p* only, with the number of cation particles being *Np*, and then:

$$p\_{\text{max}} = \frac{N\_p}{V} = \frac{N\_p}{N\_p v\_p} = \frac{1}{v\_p}.\tag{12}$$

Likewise, the limit of *n* is at most 1/*vn*, where *v<sup>n</sup>* is the particle volume of *n*. Substituting (10) into (4), and we obtain the classical PB equation:

$$-\nabla \cdot \left( \varepsilon(\mathbf{x}) \nabla \phi \right) = z\_p e p\_b e^{-\beta E\_p} + z\_n e n\_b e^{-\beta E\_n} + q. \tag{13}$$

For *z*:*z* electrolyte without considering solvation energy, the equation above reduces to:

$$\nabla \cdot (\varepsilon(\mathfrak{x}) \nabla \phi) = 2\varepsilon \varepsilon c\_{\mathfrak{b}} \sinh(\mathfrak{\beta} \varepsilon \phi) - q\_{\prime} \tag{14}$$

with *c<sup>b</sup>* = *p<sup>b</sup>* = *n<sup>b</sup>* .

#### **3. Bikerman Model**

As stated earlier, the Bikerman model [10] has been a popular steric-effect model due to its easiness of application and qualitatively good agreement with experimental data. It modifies the free energy of the classical PB (1)–(3) by adding a solvent entropy term. This term also partially represents the excessive energy accounting for overcrowding of ions and solvent molecules in localized hard-sphere models based on DFT [5–8]. The free energy in the Bikerman model treats all species of ions and solvent molecules with an identical size, and is stated as follows:

$$F = \mathcal{U} - TS\_{\prime} \tag{15}$$

$$\mathcal{U} = \int \left[ -\frac{\varepsilon}{2} |\nabla \phi|^2 + z\_p ep \phi + z\_n en \phi + q\phi + p\mathcal{W}\_{\text{sol},p} + n\mathcal{W}\_{\text{sol},n} \right] dV\_\prime \tag{16}$$

$$-TS = \int k\_B T[p\,\log(pv) - p + n\,\log(uv) - n + w\,\log(uv) - w]dV,\tag{17}$$

where *w* is concentration of solvent (such as water); *v* is the universal particle volume. Why are all solute and solvent particles treated as having the same size? Why are specific sizes of ions and solvent molecules not used here? This was not explained in the original model [10,11], and the justification of using identical size for all species particles will be addressed later in Section 5.

If we assume that, besides occupation of ions, the rest of space is occupied by solvent molecules (which can be taken as water here). Then:

$$w = \frac{N\_{\overline{w}}}{V} = \frac{N\_{\overline{w}}v}{Vv} = \frac{V - N\_{\overline{p}}v - N\_{\overline{n}}v}{Vv} = \frac{1}{v}(1 - pv - nv),\tag{18}$$

where *V* is the whole volume of electrolyte; *Np*, *Nn*, *N<sup>w</sup>* are number of cation, anion and solvent particles in an electrolyte with volume *V*, respectively. Equation (18) can then be rewritten as:

$$
\hbar wv + pv + nv = 1,\tag{19}
$$

which simply means the sum of volume fractions of water, cation and anion is one. Note that here we assume that, besides occupation of ions, the rest of space is occupied by water.

Substituting (19) into (17) we can obtain:

$$-TS = \int k\_B T \left[ p \, \log(pv) - p + n \, \log(nv) - n + \frac{1}{v} (1 - pv - nv) \, \log(1 - pv - nv) - \frac{1}{v} (1 - pv - nv) \right] dV. \tag{20}$$

Differentiation of *F* with respect to φ again gives the Poisson equation:

$$-\nabla \cdot \left( \varepsilon \left( \mathbf{x} \right) \nabla \phi \right) = z\_p \mathbf{e} p + z\_n \mathbf{e} n + q. \tag{21}$$

By doing the derivation of *F* with respect to *p* and *n*, we obtain the chemical potentials for *p* and *n*, respectively:

$$\frac{\partial \mathcal{F}}{\partial p} = \mu\_p = z\_p e \phi + k\_B T [\log(pv) - \log(1 - pv - nv)] + \mathcal{W}\_{\text{sol}, p\text{\textquotedblleft}p\text{\textquotedblright}} \tag{22}$$

$$\frac{\partial F}{\partial n} = \mu\_n = z\_n e \phi + k\_B T [\log(nv) - \log(1 - pv - nv)] + \mathcal{W}\_{sol, p}.\tag{23}$$

At equilibrium, the chemical potential is uniform everywhere and therefore the local chemical potential must be equal to its bulk value, which is usually known:

$$
\mu\_p = \mu\_{p,b\prime} \quad \mu\_n = \mu\_{n\not\ni} \tag{24}
$$

with:

$$
\mu\_{p,b} = k\_B T [\log(p\_b \upsilon) - \log(1 - p\_b \upsilon - n\_b \upsilon)] + \mathcal{W}\_{\text{sol}, p, b\prime} \tag{25}
$$

$$
\mu\_{n,b} = k\_B T \left[ \log(n\_b \upsilon) - \log(1 - p\_b \upsilon - n\_b \upsilon) \right] + \mathcal{W}\_{\text{sol},n,b}.\tag{26}
$$

By substituting (22), (23), (25) and (26) into (24), we can relate the local ion-to-solvent volume fraction ratios (denoted as γ*<sup>i</sup>* , *i* = *p*, *n*.) to their counterparts in bulk solution in a Boltzmann manner for *p* and *n*, respectively:

$$\gamma\_p = \frac{pv}{1 - pv - nv} = \frac{p\_b v}{1 - p\_b v - n\_b v} e^{-\beta E\_p} \, \tag{27}$$

$$\gamma\_n = \frac{nv}{1 - pv - nv} = \frac{n\_b v}{1 - p\_b v - n\_b v} e^{-\beta E\_u}.\tag{28}$$

Summation of (27) and (28) gives the solute-to-solvent volume fraction ratio as:

$$\frac{pv+nv}{1-pv-nv} = \gamma\_p + \gamma\_{\nu\nu}$$

and we can further obtain the solute volume fraction:

$$pv + nv = \frac{\gamma\_p + \gamma\_n}{1 + \gamma\_p + \gamma\_n}.\tag{29}$$

From (29) and (19), we know then volume fraction for *p*, *n* and *w*, respectively.

$$pv = \frac{\gamma\_p}{1 + \gamma\_p + \gamma\_n}, \; nv = \frac{\gamma\_n}{1 + \gamma\_p + \gamma\_n}, \; uv = \frac{1}{1 + \gamma\_p + \gamma\_n} \tag{30}$$

which further gives *p* and *n* in terms of their bulk values *p<sup>b</sup>* , *n<sup>b</sup>* , identical particle size *v*, and local energy *Ep*, *En*:

$$p = \frac{p\_b e^{-\beta \mathbb{E}\_p}}{(1 - p\_b v - n\_b v) + p\_b v e^{-\beta \mathbb{E}\_p} + n\_b v e^{-\beta \mathbb{E}\_n}}, \quad n = \frac{n\_b e^{-\beta \mathbb{E}\_n}}{(1 - p\_b v - n\_b v) + p\_b v e^{-\beta \mathbb{E}\_p} + n\_b v e^{-\beta \mathbb{E}\_n}}\tag{31}$$

Equations (27) and (28) can be re-arranged to obtain:

$$p = p\_b e^{-\beta(E\_p + S^{tr})}, \ n = n\_b e^{-\beta(E\_n + S^{tr})},\tag{32}$$

with ionic steric potential *S trc* expressed as:

$$S^{\rm trc} = k\_B T \log \left( \frac{1 - p\_b v - n\_b v}{1 - p v - n v} \right). \tag{33}$$

The steric potential *S trc*, first described in [16–19], characterizes the crowding of ions and their finite-size effect by a bulk-to-local water fraction ratio. Larger local ion concentrations would have a larger steric potential.

Also, by letting 1 − *pv* − *nv* = *wv*, and 1 − *pbv* − *nbv* = *wbv*, Equations (27) and (28) can be simplified as:

$$\frac{p\upsilon}{uv} = \frac{p\_b\upsilon}{w\_b\upsilon}e^{-\beta E\_p} = \gamma\_{p\prime} \tag{34}$$

$$\frac{\partial \mathcal{w}}{\partial \mathbf{w}} = \frac{n\_b \boldsymbol{\upsilon}}{w\_b \boldsymbol{\upsilon}} e^{-\beta E\_{\boldsymbol{\eta}}} = \boldsymbol{\gamma}\_{\boldsymbol{\eta}}.\tag{35}$$

Therefore, by (30), we obtain:

$$z\_{\overline{\rho}}pv + z\_{n}nv = \frac{z\_{\overline{\rho}}\frac{p\_{\overline{\nu}}\overline{\nu}}{w\_{\overline{\nu}}\overline{\nu}}e^{-\beta E\_{p}} + z\_{n}\frac{n\_{b}\overline{\nu}}{w\_{b\overline{\nu}}\overline{\nu}}e^{-\beta E\_{n}}}{1 + \frac{p\_{b}\overline{\nu}}{w\_{b\overline{\nu}}}e^{-\beta E\_{p}} + \frac{n\_{b}\overline{\nu}}{w\_{b\overline{\nu}}}e^{-\beta E\_{n}}} = \frac{z\_{\overline{\rho}}p\_{b}v\nu e^{-\beta E\_{p}} + z\_{n}n\_{b}v\nu e^{-\beta E\_{n}}}{w\_{b}\overline{\nu} + p\_{b}v\nu e^{-\beta E\_{p}} + n\_{b}v\nu e^{-\beta E\_{n}}}.\tag{36}$$

Since *<sup>p</sup><sup>b</sup> nb* = −*z<sup>n</sup> zp* due to electric neutrality in bulk conditions, therefore:

$$p\_b v = \frac{-z\_n \mu}{z\_p - z\_n} v \,\, n\_b v = \frac{z\_p \mu}{z\_p - z\_n} v \,\, \tag{37}$$

where µ = *p<sup>b</sup>* + *n<sup>b</sup>* . Also:

$$
\Delta w\_b \upsilon = 1 - p\_b \upsilon - n\_b \upsilon = 1 - \mu \upsilon. \tag{38}
$$

then (36) becomes:

$$z\_p p + z\_{n} n = \frac{z\_p z\_n \mu \left( -e^{-\beta E\_p} + e^{-\beta E\_n} \right)}{(1 - \mu v) \left( z\_p - z\_n \right) + \mu v \left( z\_p e^{-\beta E\_n} - z\_n e^{-\beta E\_p} \right)}.\tag{39}$$

Substituting (39) into (21), we obtain the Bikerman-PB equation:

$$-\nabla \cdot \left( \varepsilon(\mathbf{x}) \nabla \phi \right) = \frac{z\_p z\_n e \mu \Big( -e^{-\beta E\_p} + e^{-\beta E\_n} \Big)}{(1 - \mu v) \left( z\_p - z\_n \right) + \mu v \Big( z\_p e^{-\beta E\_n} - z\_n e^{-\beta E\_p} \Big)} + q. \tag{40}$$

For *z*:*z* electrolyte without considering the solvation energy, the equation above becomes:

$$\nabla \cdot (\varepsilon(\mathbf{x}) \nabla \phi) = \frac{2zec\_0 \sinh(\beta z e \phi)}{1 + 2r \sinh^2(\frac{\beta z e \phi}{2})} - q\_\prime \tag{41}$$

as shown in [11] with *c*<sup>0</sup> = *p<sup>b</sup>* = *n<sup>b</sup>* , *r* = µ*v* = 2*c*0*v*.

Two important criteria need to be checked for all modified PB/PNP models accounting for steric effects:

CRITERION I: When ion concentrations *p* and *n* are dilute, will they follow the classical Boltzmann distribution?

CRITERION II: As <sup>φ</sup> → ∓∞, will *<sup>p</sup>* and *<sup>n</sup>* approach their saturation limits <sup>1</sup> *vp* and <sup>1</sup> *vn* , respectively? For CRITERION I when *p* and *n* are dilute here, it means their volume fractions are negligible, and therefore 1 − *pv* − *nv* ≈ 1, and 1 − *pbv* − *nbv* ≈ 1. Steric potential term *S trc* then vanishes, and by (32) *p* = *p<sup>b</sup> e* −β*E<sup>p</sup>* , and *n* = *n<sup>b</sup> e* <sup>−</sup>β*E<sup>n</sup>* , which follows the Boltzmann distribution.

For CRITERION II, let us consider φ → −∞ first, and φ → ∞ can be derived similarly. As φ → −∞, γ*<sup>p</sup>* → ∞, and γ*<sup>n</sup>* → 0. Therefore, *pv* → 1, and *nv* → 0 by (30), which further means *<sup>p</sup>* <sup>→</sup> <sup>1</sup> *vp* = <sup>1</sup> *v* , and *<sup>n</sup>* <sup>→</sup> 0. Likewise, as <sup>φ</sup> → ∞, we can get *<sup>n</sup>* <sup>→</sup> <sup>1</sup> *vn* = <sup>1</sup> *v* , and *p* → 0.

#### **4. The Bikerman Model with Specific Ion Sizes**

The shortcoming of the Bikerman model is the usage of a universal particle size, denoted by *v*, for cations, anions and solvents. Using specific ion and solvent sizes would be closer to reality. Taking NaCl solution as an example, the spherical diameters for Cl−, Na<sup>+</sup> and water are *DCl*<sup>−</sup> = 3.62 Å, *DNa*<sup>+</sup> = 2.04 Å, and *D<sup>w</sup>* = 2.08 Å, and then the particle volume ratio is *vNa*<sup>+</sup> : *vCl*<sup>−</sup> : *v<sup>w</sup>* = 1 : 5.59 : 1.06, in which using universal particle volume would be far from reality in the case of high ion concentrations. In appearance, it seems, and many researchers did, we can just simply modify the model to include specific ion sizes by changing *pv* and *pbv* to *pv<sup>p</sup>* and *pbvp*; similarly, *nv* and *nbv* to *nv<sup>n</sup>* and *nbv<sup>n</sup>* for (22) to (41). With this straightforward extension, we obtain *p*, *n* as:

$$p = \frac{p\_b e^{-\beta E\_p}}{\left(1 - p\_b v\_p - n\_b v\_n\right) + p\_b v\_p e^{-\beta E\_p} + n\_b v\_n e^{-\beta E\_n}} '$$

$$n = \frac{n\_b e^{-\beta E\_n}}{\left(1 - p\_b v\_p - n\_b v\_n\right) + p\_b v\_p e^{-\beta E\_p} + n\_b v\_n e^{-\beta E\_n}}\tag{42}$$

and the specific-ion-size Bikerman-PB equation:

$$-\nabla \cdot \left( \varepsilon \left( \mathbf{x} \right) \nabla \phi \right) = \frac{e \Big( z\_p p\_b e^{-\beta E\_p} + z\_n n\_b e^{-\beta E\_n} \Big)}{1 - \left( p\_b v\_p + n\_b v\_n \right) + \left( p\_b + n\_b \right) \left( z\_p v\_n e^{-\beta E\_n} - z\_n v\_p e^{-\beta E\_p} \right) / \left( z\_p - z\_n \right)} + q \tag{43}$$

For *z*:*z* electrolyte without considering solvation energy, the equation above becomes:

$$\nabla \cdot (\varepsilon(\mathbf{x}) \nabla \phi) = \frac{2z e c\_0 \sinh(\beta z e \phi)}{1 - c\_0 (v\_p + v\_n) + c\_0 (v\_n e^{\beta z e \phi} + v\_p e^{-\beta z e \phi})} - q. \tag{44}$$

Let us denote (42) as the specific-ion-size Bikerman model 1 (SISBM1) for convenience of notation. However, we can not find an energy functional like (15)–(17) to support this naive extension, which means chemical potentials (22) and (23) with universal particle size replaced by specific ion sizes cannot be derived. The correct specific-ion-size energy functional and chemical potentials should be derived as follows:

$$F = \mathcal{U} - TS\_{\prime} \tag{45}$$

$$
\delta M = \int \left[ -\frac{\varepsilon}{2} |\nabla \phi|^2 + z\_p ep \phi + z\_n en \phi + q \phi + p W\_{\text{sol},p} + n \mathcal{W}\_{\text{sol},n} \right] dV\_\prime \tag{46}
$$

$$-TS = \int k\_{\overline{B}} T \left[ p \, \log(pv\_p) - p + n \, \log(nv\_n) - n + w \, \log(wv\_w) - w \right] dV. \tag{47}$$

By *wv<sup>w</sup>* = 1 − *pv<sup>p</sup>* − *nvn*, (47) can be rewritten as:

$$-TS = \int k\_B T \left[ p \, \log(p v\_\mathcal{V}) - p + n \, \log(n v\_\mathcal{n}) - n + \frac{1}{v\_\mathcal{v}} (1 - p v\_\mathcal{p} - n v\_\mathcal{v}) \, \log \left( 1 - p v\_\mathcal{p} - n v\_\mathcal{v} \right) - \frac{1}{v\_\mathcal{v}} (1 - p v\_\mathcal{p} - n v\_\mathcal{v}) \right] dV. \tag{48}$$

Differentiation of *F* with respect to φ again gives the Poisson equation:

$$-\nabla \cdot \left( \varepsilon(\mathbf{x}) \nabla \phi \right) = z\_p ep + z\_n en + q. \tag{49}$$

By doing the differentiation of *F* with respect to *p* and *n*, we can obtain the chemical potentials for cations and anions, respectively:

$$\mu\_p = z\_p e \phi + k\_B T \left[ \log \left( p v\_p \right) - \log \left( 1 - p v\_p - n v\_n \right)^{k\_p} \right] + \mathcal{W}\_{\text{sol},p} \tag{50}$$

µ*<sup>n</sup>* = *zne*φ + *kBT* log(*nvn*) <sup>−</sup> log 1 − *pv<sup>p</sup>* − *nv<sup>n</sup> kn* + *Wsol*,*<sup>n</sup>* (51)

where *k<sup>p</sup>* = *vp vw* , *k<sup>n</sup>* = *vn vw* .

At equilibrium, the chemical potential is uniform everywhere and therefore the local chemical potential must be equal to its bulk value, which is usually known:

$$
\mu\_p = \mu\_{p,b\prime} \quad \mu\_n = \mu\_{n,b\prime} \tag{52}
$$

with:

$$\mu\_{p,b} = k\_B T \left[ \log \left( p\_b v\_p \right) - \log \left( 1 - p\_b v\_p - n\_b v\_n \right)^{k\_p} \right] + \mathcal{W}\_{\text{sol}, p, b} \tag{53}$$

$$k\_{n,b} = k\_B T \left[ \log(n\_b v\_n) - \log \left( 1 - p\_b v\_p - n\_b v\_n \right)^{k\_n} \right] + \mathcal{W}\_{\text{sol}, n, b}.\tag{54}$$

To solve *p* and *n* from (52), there is no closed form solution like (31) for *p* and *n* due to the nonlinearity, unless some simplified case such as *k<sup>p</sup>* = *k<sup>n</sup>* = 1 is considered, which is actually reduced to the original Bikerman model with *v<sup>p</sup>* = *v<sup>n</sup>* = *v<sup>w</sup>* = *v*. Like (32), *p* and *n* at most can be expressed as:

$$p = p\_b e^{-\beta(E\_p + k\_p S^{tr})}, \ \ n = n\_b e^{-\beta(E\_n + k\_n S^{tr})},\tag{55}$$

with the steric potential *S trc* being modified from (33) to include specific ion sizes:

$$S^{\text{trc}} = k\_B T \log \left( \frac{1 - p\_b v\_p - n\_b v\_n}{1 - p v\_p - n v\_n} \right) \tag{56}$$

Let us denote (55) as the specific-ion-size Bikerman model 2 (SISBM2) for convenience. Note that a similar model was also obtained in [18,19] without a rigorous derivation.

Again, we need to check criteria I and II for this specific-ion-size model. For CRITERION I, when *p* and *n* are dilute, it again means 1 − *pv<sup>p</sup>* − *nv<sup>n</sup>* ≈ 1, and 1 − *pbv<sup>p</sup>* − *nbv<sup>n</sup>* ≈ 1. Therefore *S trc* <sup>≈</sup> <sup>0</sup> by (56), and then *p* = *p<sup>b</sup> e* −β*E<sup>p</sup>* , and *n* = *n<sup>b</sup> e* <sup>−</sup>β*E<sup>n</sup>* by (55), which follows a classical Boltzmann distribution. For CRITERION II, as φ → −∞ in (50), *kBT* log *pv<sup>p</sup>* <sup>−</sup> log 1 − *pv<sup>p</sup>* − *nv<sup>n</sup> kp* should approach +∞ for µ*<sup>p</sup>* to be finite. This can only be achieved by *n* → 0, and *p* → 1 *vp* − (saturation). Applying the same reasoning for (51), as φ → ∞, *kBT* log(*nvn*) <sup>−</sup> log 1 − *pv<sup>p</sup>* − *nv<sup>n</sup> kn* should approach +∞ for µ*<sup>n</sup>* to be finite. Then *p* → 0, and *n* → 1 *vn* − (saturation). This specific-ion-size model seems correct and reasonable so far, but actually there is a pitfall. That is its entropy formula (48) cannot be derived by the traditional mean-field lattice gas model. This will be explained in the next section.

#### **5. Mixing Entropy Derivation Based on the Mean-Field Lattice Gas Model**

In this section, we would like to derive the entropy in (20) by the traditional mean-field lattice gas model. Consider the entropy for an aqueous electrolyte system:

$$TS = k\_B T \log \mathcal{W}\_\prime \tag{57}$$

where *W* is the number of microstates at equilibrium which possess a maximum number of microstates. Mixing entropy in electrolyte studies macrostates through spherical particles' (solute and solvent) occupation of identical cubic sites is based on the mean-field lattice gas model. The necessity of using identical cubic sites provides a combinatorial basis when computing the maximum number of microstates. The most probable distribution of all solute (ions) and solvent particles, reaching maximum number of microstates for each species, is that each identical cubic site generally would be at most occupied by one solute/solvent particle as depicted in Figure 1a. This is based on the concept that the size of each species' particle is infinitesimal or finite but dilute. When the actual size for each species' particle is considered and an aqueous electrolyte is extremely concentrated as depicted in Figure 1b, the most probable distribution above may not be available. The situation in Figure 1b will be addressed in the next section.

The entropy based on the most probable distribution of *K*-species solute (ions) and solvent (treated as *K* + 1-th species) particles, under dilute situation, over a total of *N* = P*K*+<sup>1</sup> *<sup>j</sup>*=<sup>1</sup> *N<sup>j</sup>* available identical sites in a system is:

$$\begin{aligned} &= \prod\_{j=1}^{N} W\_j \\ &= \binom{N}{N\_1} \binom{N-N\_1}{N\_2} \cdots \binom{N-N\_1-N\_2-\cdots-N\_{K-1}}{N\_K} \binom{N-N\_1-N\_2-\cdots-N\_K}{N\_K} \\ &= \frac{N!}{N\_1!(N-N\_1)!} \frac{(N-N\_1)!}{N\_2!(N-N\_1-N\_2)!} \cdots \frac{\binom{N-\sum\_{j=1}^{K-1} N\_j)!}{N\_K!\binom{N-\sum\_{j=1}^{K} N\_j}{N\_K+1}} \frac{\binom{N-\sum\_{j=1}^{K} N\_j}{N\_{K+1}!}}{N\_{K+1}!} \\ &= \frac{N!}{(\prod\_{j=1}^{K} N\_j)! s\_{K+1}!} \end{aligned} \tag{58}$$

where *N<sup>j</sup>* , *j* = 1, · · · , *K*, is the particle number of *j*-species ion. *NK*+<sup>1</sup> is the particle number of solvent, so the entropy becomes:

$$TS = k\_B T \log \frac{N!}{\left(\prod\_{j=1}^K N\_j!\right) \mathbb{N}\_{K+1}!}.\tag{59}$$

Using the Stirling formula log *M*! ≈ *M* log *M* − *M* with *M* 1, we can rewrite the entropy as: a similar model was also obtained in [18,19,18] without a rigorous derivation. Again, we need to check criteria I and II for this specific-ion-size model. For CRITERION I, when **Formatted:** Not Highlight **Formatted:** Not Highlight

Let us denote (55) as the specific-ion-size Bikerman model 2 (SISBM2) for convenience. Note that

*Entropy* **2020**, *22*, x FOR PEER REVIEW 8 of 15

௧ = log ቆ1− −

$$\begin{aligned} \text{SST} &= k\_B T \left[ N \log N - N - \sum\_{j=1}^{K} N\_j \log N\_j + \sum\_{j=1}^{K} N\_j - N\_{K+1} \log N\_{K+1} + N\_{K+1} \right] \\ &= k\_B T \left[ N \log N - \sum\_{j=1}^{K} N\_j \log N\_j - N\_{K+1} \log N\_{K+1} \right] \\ &= k\_B T \left[ N \log N - \sum\_{j=1}^{K} N\_j \log N\_j - \left( N - \sum\_{j=1}^{K} N\_j \right) \log \left( N - \sum\_{j=1}^{K} N\_j \right) \right] \\ &= k\_B T \left[ N \log \frac{N}{N - \sum\_{j=1}^{K} N\_j} - \sum\_{j=1}^{K} N\_j \log \frac{N\_j}{N - \sum\_{j=1}^{K} N\_j} \right] \end{aligned} \tag{60}$$

1 − −

using the following relations: In this section, we would like to derive the entropy in (20) by the traditional mean-field lattice

or unaç 
$$V = N v\_{s'} \text{ or } \frac{N}{V} = \frac{1}{v\_s} \,\tag{61}$$

$$\frac{N\_j}{V} = c\_{j\nu} \tag{62}$$

ቇ (56)

$$\frac{N\_j}{N} = \frac{N\_j v\_s}{N v\_s} = \frac{N\_j v\_s}{V} = c\_j v\_{s\prime} \tag{63}$$

where *c<sup>j</sup>* is the concentration of *j*-species particle; *V* is the volume of system; *v<sup>s</sup>* is the volume of an identical cubic site that composes the volume of system. It is naturally requested that *v<sup>s</sup>* ≥ *max*1≤*j*≤*K*+1*v<sup>j</sup>* , where *v<sup>j</sup>* is the particle volume of *j*-species particle. Usually *v<sup>s</sup>* = *max*1≤*j*≤*K*+1*v<sup>j</sup>* in aqueous electrolyte system, where solute and solvent particles are generally crowded. would be at most occupied by one solute/solvent particle as depicted in Figure 1a. This is based on the concept that the size of each species' particle is infinitesimal or finite but dilute. When the actual size for each species' particle is considered and an aqueous electrolyte is extremely concentrated as depicted in Figure 1b, the most probable distribution above may not be available. The situation in Figure 1b will be addressed in the next section.

**Figure 1.** (**a**) Moderately concentrate situation with each solute/solvent particle only occupy one identical site. (**b**) Extremely concentrate situation with each identical site can be allowed to be **Figure 1.** (**a**) Moderately concentrate situation with each solute/solvent particle only occupy one identical site. (**b**) Extremely concentrate situation with each identical site can be allowed to be occupied by multiple solute/solvent particles of the same species in order to increase packing efficiency in space.

Applying (61)–(63) to (60), the entropy density can be expressed as:

$$\frac{TS}{V} = k\_B T \left[ \frac{1}{v\_s} \log \frac{1}{1 - \sum\_{j=1}^{K} c\_j v\_s} - \sum\_{j=1}^{K} c\_j \log \frac{c\_j v\_s}{1 - \sum\_{j=1}^{K} c\_j v\_s} \right]. \tag{64}$$

For a binary electrolyte, (64) can be expressed as:

$$TS = \int k\_B T \left[ \frac{1}{v\_s} \log \frac{1}{1 - p v\_s - n v\_s} - p \log \frac{p v\_s}{1 - p v\_s - n v\_s} - n \log \frac{n v\_s}{1 - p v\_s - n v\_s} \right] dV,\tag{65}$$

$$-TS = \int k\_{\overline{B}} T \left[ p \, \log(pv\_s) + n \, \log(nv\_s) + \frac{1}{v\_s} (1 - pv\_s - nv\_s) \log(1 - pv\_s - nv\_s) \right] dV\_\prime \tag{66}$$

or:

which, without loss of generality, can be augmented as:

$$-TS = \int k\_B T \left[ p \left\{ p \eta\_3 (p v\_3) - p + n \log (u v\_3) - n + \frac{1}{v\_3} (1 - p v\_3 - n v\_3) \log (1 - p v\_3 - n v\_3) - \frac{1}{v\_3} (1 - p v\_3 - n v\_3) \right\} dV. \tag{67} \right]$$

Equation (67) is exactly the same as (20) with:

$$v = v\_s = \max\{v\_{p\prime}, v\_{\text{n}\prime}v\_{\text{w}}\}.\tag{68}$$

This means the universal particle volume *v* in the Bikerman model is actually the volume of an identical occupation site *v<sup>s</sup>* , which is limited from below by the largest particle size among all solute and solvent particles. The original Bikerman model has long suffered criticism for assuming all ions have the same size instead of using specific ion sizes in the model. The above reasoning explains why specific ion size information is left out mainly due to the need for all cubic sites to be identical in order to support the combinatorial basis demanded by the mean-field lattice gas model. Actually, information of specific ion sizes is still carried but only implicitly as shown in (68). Researchers may prefer to use SISBM2 as illustrated in Section 4, but actually its entropy formula (48) cannot be derived by the mean-field lattice gas model described above. Note that usually solute and solvent particles are treated as spheres in modeling. If *ap*, *an*, *a<sup>w</sup>* are diameters for *p*, *n*, and *w*, respectively and their maximum is *a<sup>n</sup>* for example, then *v* = *v<sup>s</sup>* = *a* 3 *<sup>n</sup>* not <sup>4</sup><sup>π</sup> 3 *an* 2 3 since the identical occupation is cubic. This is why *a* 3 , instead of <sup>4</sup><sup>π</sup> 3 *a* 2 3 , used in [10,11].

In CRITERION II described above, as φ → ∓∞, *p* and *n* should approach their saturation limits 1 *vp* and <sup>1</sup> *vn* , respectively. Here, this would be changed to approach <sup>1</sup> *vs* instead of <sup>1</sup> *vp* and <sup>1</sup> *vn* respectively, although approaching <sup>1</sup> *vp* and <sup>1</sup> *vn* sounds more physically correct. This paradoxical conclusion is from entropy rigorously derived by the traditional mean-field lattice gas model based on combinatorics requiring identical occupation sites. Can this be fixed to resume the limit approach to <sup>1</sup> *vp* and <sup>1</sup> *vn* and still holding the ground of combinatorics at the same time? An attempt at this is discussed in the next section.

#### **6. Entropy Fixing for Electrolytes under Extreme Concentration Conditions**

Here we hope to construct a steric PB model with entropy able to be derived by the mean-field lattice gas model, and at the same time showing physically correct saturation limits for ions as φ → ∓∞. The mean-field lattice gas model is fixed here such that each identical cubic site is allowed to be occupied by more than one solute particle of the same species as illustrated in Figure 1b. Although this kind of distribution is no more a most probable distribution as stated earlier, it allows more efficient packing when space is extremely limited and size among species varies largely. Again, we consider the entropy for an aqueous electrolyte system:

$$TS = k\_B T \log \mathcal{W}\_{\prime} \tag{69}$$

Let *N*e*<sup>j</sup>* , *j* = 1, · · · , *K* + 1, be the particle number of species *j* and *N<sup>j</sup>* , *j* = 1, · · · , *K* + 1, the number of identical sites occupied by *j*-species particles with *N*e*<sup>j</sup>* ≥ *N<sup>j</sup>* . This means that in an extremely concentrated situation an identical site can be occupied by more than one particle of the same species. If an identical cubic site, on average, can allow *r<sup>j</sup> j*-species particles to occupy it, we can then relate *N*e*<sup>j</sup>* and *N<sup>j</sup>* by *N*e*<sup>j</sup>* = *Njr<sup>j</sup>* , or equivalently *v<sup>s</sup>* = *rjv<sup>j</sup>* . Again, *v<sup>j</sup>* is the particle volume of species *j*. *v<sup>s</sup>* is the volume of an identical cubic site with *v<sup>s</sup>* = *max*1≤*j*≤*K*+1*v<sup>j</sup>* . The entropy based on the most probable distribution of all 'grouped' species particles over a total of *N* = P*K*+<sup>1</sup> *<sup>j</sup>*=<sup>1</sup> *N<sup>j</sup>* available identical sites in a system is:

$$\mathcal{W} = \prod\_{j=1}^{K+1} \mathcal{W}\_{j} = \frac{\mathcal{N}!}{\mathcal{N}\_{1}!(\mathcal{N}-\mathcal{N}\_{1})!} \frac{(\mathcal{N}-\mathcal{N}\_{1})!}{\mathcal{N}\_{2}!(\mathcal{N}-\mathcal{N}\_{1}-\mathcal{N}\_{2})!} \cdots \frac{\left\{\mathcal{N}-\sum\_{j=1}^{K-1}\mathcal{N}\_{j}\right\}!}{\mathcal{N}\_{k}!\left\{\mathcal{N}-\sum\_{j=1}^{K}\mathcal{N}\_{j}\right\}!} \frac{\left\{\mathcal{N}-\sum\_{j=1}^{K}\mathcal{N}\_{j}\right\}!}{\mathcal{N}\_{k+1}!} = \frac{\mathcal{N}!}{\left\{\prod\_{j=1}^{K}\mathcal{N}\_{j}!\right\}!\mathcal{N}\_{k+1}!}.\tag{70}$$

Note that, after all the ions (in group) are distributed, there are *N* − P*K <sup>j</sup>*=<sup>1</sup> *N<sup>j</sup>* = *NK*+<sup>1</sup> sites that will be filled by solvent molecules in group, so the entropy becomes:

$$TS = k\_B T \log \frac{N!}{\left(\prod\_{j=1}^K N\_j!\right) \mathbb{N}\_{K+1}!}.\tag{71}$$

Using the Stirling formula log *M*! ≈ *M* log *M* − *M* with *M* 1, we can rewrite the entropy as:

$$\begin{aligned} TS &= k\_B T \left[ N \log N - N - \sum\_{j=1}^{K} N\_j \log N\_j + \sum\_{j=1}^{K} N\_j - \left( N - \sum\_{j=1}^{K} N\_j \right) \log \left( N - \sum\_{j=1}^{K} N\_j \right) \\ &+ \left( N - \sum\_{j=1}^{K} N\_j \right) \left] = k\_B T \left[ N \log \frac{N}{N - \sum\_{j=1}^{K} N\_j} - \sum\_{j=1}^{K} N\_j \log \frac{N\_j}{N - \sum\_{j=1}^{K} N\_j} \right] \end{aligned} \tag{72}$$

Using the following relations:

$$N = Nv\_{\rm s} \text{ or } \frac{N}{V} = \frac{1}{v\_{\rm s}}.\tag{73}$$

$$\frac{N\_j}{V} = \frac{\frac{N\_j}{r\_j}}{V} = \frac{c\_j}{r\_j} \,\tag{74}$$

$$\frac{N\_j}{N} = \frac{N\_j v\_s}{N v\_s} = \frac{N\_j v\_s}{V} = \frac{c\_j}{r\_j} r\_j v\_j = c\_j v\_{j\prime} \tag{75}$$

where *c<sup>j</sup>* is the concentration of species *j*; *V* is the volume of the system.

The entropy per unit volume can be expressed as:

$$\frac{TS}{V} = k\_B T \left[ \frac{1}{v\_s} \log \frac{1}{1 - \sum\_{j=1}^K c\_j v\_j} - \sum\_{j=1}^K \frac{c\_j}{r\_j} \log \frac{c\_j v\_j}{1 - \sum\_{j=1}^K c\_j v\_j} \right]. \tag{76}$$

Compared with (64), specific ion sizes can now appear explicitly in the entropy formula (76). For a binary electrolyte:

$$TS = \int k\_B T \left[ \frac{1}{v\_s} \log \frac{1}{1 - p v\_p - n v\_n} - \frac{p}{r\_p} \log \frac{p v\_p}{1 - p v\_p - n v\_n} - \frac{n}{r\_n} \log \frac{n v\_n}{1 - p v\_p - n v\_n} \right] dV,\tag{77}$$

or:

$$-TS = \int k\_B T \left[ \frac{p}{r\_p} \log(pv\_p) + \frac{n}{r\_n} \log(nv\_n) + \frac{1}{v\_s} (1 - pv\_p - nv\_n) \log \left( 1 - pv\_p - nv\_n \right) \right] dV\_\prime \tag{78}$$

which, without loss of generality, can be augmented as:

$$-TS = \int k\_B T \left[ \frac{p}{r\_p} \log(p v\_\mathcal{P}) - \frac{p}{r\_p} + \frac{n}{r\_n} \log(n v\_\mathcal{n}) - \frac{n}{r\_n} + \frac{1}{v\_\mathcal{v}} \left( 1 - p v\_\mathcal{P} - n v\_\mathcal{v} \right) \log \left( 1 - p v\_\mathcal{P} - n v\_\mathcal{v} \right) - \frac{1}{v\_\mathcal{v}} \left( 1 - p v\_\mathcal{P} - n v\_\mathcal{v} \right) \right] dV. \tag{79}$$

by:

$$
\mu\_p = \frac{\delta f}{\delta p} = z\_p e \phi + \mathcal{W}\_{\text{sol},p} + \frac{k\_B T}{r\_p} \log \frac{pv\_p}{1 - pv\_p - nv\_n} \tag{80}
$$

$$
\mu\_{\rm n} = \frac{\delta f}{\delta n} = z\_{\rm n} e \phi + \mathcal{W}\_{\rm sol, n} + \frac{k\_B T}{r\_n} \log \frac{n v\_n}{1 - p v\_p - n v\_n} \tag{81}
$$

Again, at equilibrium, the chemical potential is uniform everywhere and therefore the local chemical potential must be equal to its bulk value, which is usually known:

$$
\mu\_p = \mu\_{p, \flat \prime} \quad \mu\_n = \mu\_{n, \flat}.\tag{82}
$$

Usually bulk solutions are dilute and chemical potentials under that condition can be formulated following (80) and (81) with *r<sup>p</sup>* = *r<sup>n</sup>* = 1:

> −*zne*φ−*Wsol*,*n kB*

*<sup>T</sup>*/*rn* ,

$$
\mu\_{p,b} = k\_B T \log \frac{p\_b v\_p}{1 - p\_b v\_p - n\_b v\_n} + \mathcal{W}\_{sol, p, b\prime} \tag{83}
$$

$$
\mu\_{n,b} = k\_B T \log \frac{n\_b v\_n}{1 - p\_b v\_p - n\_b v\_n} + \mathcal{W}\_{\text{sol}, n, b}.\tag{84}
$$

By denoting γ*<sup>p</sup>* = *e* (82) forms

µ*p*,*b*

−*zpe*φ−*Wsol*,*p kB T*/*rp*

, γ*<sup>n</sup>* = *e*

µ*n*,*b*

$$\frac{pv\_p}{1 - pv\_p - nv\_n} = \gamma\_{p\prime} \cdot \frac{nv\_n}{1 - pv\_p - nv\_n} = \gamma\_{n\prime} \tag{85}$$

and can solve for *p* and *n*:

$$pv\_p = \frac{\gamma\_p}{1 + \gamma\_p + \gamma\_n}, \ n v\_n = \frac{\gamma\_n}{1 + \gamma\_p + \gamma\_n},\tag{86}$$

or:

$$pv\_p = \frac{\left(\frac{p\_b v\_p}{1 - p\_b v\_p - n\_b v\_n}\right)^{r\_p} e^{-\beta\_p E\_p}}{1 + \left(\frac{p\_b v\_p}{1 - p\_b v\_p - n\_b v\_n}\right)^{r\_p} e^{-\beta\_p E\_p} + \left(\frac{n\_b v\_n}{1 - p\_b v\_p - n\_b v\_n}\right)^{r\_n} e^{-\beta\_n E\_n}}\tag{87}$$

$$mv\_n = \frac{\left(\frac{n\_b v\_n}{1 - p\_b v\_p - n\_b v\_n}\right)^{r\_n} e^{-\beta\_n E\_n}}{1 + \left(\frac{p\_b v\_p}{1 - p\_b v\_p - n\_b v\_n}\right)^{r\_p} e^{-\beta\_p E\_p} + \left(\frac{n\_b v\_n}{1 - p\_b v\_p - n\_b v\_n}\right)^{r\_n} e^{-\beta\_n E\_n}}\tag{88}$$

where *E<sup>p</sup>* = *zpe*φ + ∆*Wsol*,*<sup>p</sup>* , *E<sup>n</sup>* = *zne*φ + ∆*Wsol*,*n*, β*<sup>p</sup>* = *kBT*/*r<sup>p</sup>* −<sup>1</sup> , β*<sup>n</sup>* = (*kBT*/*rn*) −1 . Let us denote (87), (88) as the specific-ion-size Bikerman model 3 (SISBM3) for convenience.

Again, we need to check this new model with criteria I and II. For CRITERION II, we can easily deduce from (87) <sup>φ</sup> → −∞, *<sup>n</sup>* <sup>→</sup> 0, *pv<sup>p</sup>* <sup>→</sup> 1, *<sup>p</sup>* <sup>→</sup> <sup>1</sup> *vp* (saturation). Similarly, from (88), φ → ∞, *<sup>p</sup>* <sup>→</sup> 0, *nv<sup>n</sup>* <sup>→</sup> 1, *<sup>n</sup>* <sup>→</sup> <sup>1</sup> *vn* (saturation). There is no constraint like *<sup>p</sup>*, *<sup>n</sup>* <sup>→</sup> <sup>1</sup> *vs* as φ → ∓∞ any more, and entropy here can be derived by mean-field lattice gas model.

For CRITERION I, *p* and *n* will not approach a Boltzmann distribution *p<sup>b</sup> e* <sup>−</sup>β*E<sup>p</sup>* and *n<sup>b</sup> e* <sup>−</sup>β*E<sup>n</sup>* at diluteness unless *r<sup>p</sup>* = *r<sup>n</sup>* = 1. This violation of the Boltzmann distribution at the dilution limit is because we allow multiple ions of the same species to occupy an identical cubic site. This failure and a possible cure will be discussed in next section.

#### **7. Discussion and Conclusions**

If we wish to obtain a model for electrolytes such that: (1) it can approach a Boltzmann distribution at diluteness; (2) it can reach the saturation limit as the reciprocal of specific ion size under extreme electrostatic conditions; (3) its entropy can be derived by a mean-field lattice gas model. The only options here is SISBM3 with *r<sup>p</sup>* = *r<sup>n</sup>* = 1, since SISBM2 satisfies (1) and (2) but not (3). How can we justify *r<sup>p</sup>* = *r<sup>n</sup>* = 1 for SISBM3 here? Interpreting all ion sizes as being about the same is certainly not acceptable. Remember SISBM3 is designed for extremely high ion concentrations motivated by the more efficient packing shown in Figure 1b. Actually for situations that would give rise to extremely high ion concentrations and make the steric effect not negligible, such as the Stern layer in the electric double layer (EDL) of a charged wall (discussed next) and the selectivity filter of a K channel [20], there would be 'locally' one species only, which is the counter-ion of the local electrostatic environment, since co-ions (and even water) would be totally expelled. Taking the K channel selectivity filter as an example, its extreme narrowness and the strong negative oxygen charges inside it would definitely justify only one species being inside the selectivity filter, which is definitely potassium. This implies *r<sup>p</sup>*

to be 1 locally in the filter, and we can justify *r<sup>n</sup>* = 1 inside the filter as well since anions would be extremely dilute there due to strong electrostatic repulsion. For the rest of the K channel where ions are at most moderately concentrated, the steric effect is much less significant, and basically the original Bikerman model would be appropriate for it. Since SISBM3 with *r<sup>p</sup>* = *r<sup>n</sup>* = 1 would be a very good approximation of the original Bikerman model under mild ion concentrations, we can use SISBM3 with *r<sup>p</sup>* = *r<sup>n</sup>* = 1 globally for the whole K channel then. Notice, under *r<sup>p</sup>* = *r<sup>n</sup>* = 1, SISBM3 is actually same as SISBM1, but with a rigorous derivation now. This model has been useful and proven to fit the experimental data quite well [16–18]. Although here we just discussed steric-effect modifications for PB, modifications for PNP can be likewise derived.

Here we compare SISBM1 and PB by computing ion distributions in a 1D charged wall problem. Many researchers have used this physical model to investigate the surface differential capacitance of electrodes adjacent to electrolyte solutions [12–15]. Here (14) for PB and (44) for SISBM1 were used to calculate the ion distributions of a binary KCl electrolyte solution without considering the solvation energy and permanent charges. The associated boundary conditions are φ(0) = *Vwall*, and φ(∞) = 0. The bulk concentration of KCl as *x* → ∞ is set to be *c*<sup>0</sup> = 100 mM, and dielectric constant is set to 80 for the whole domain (0, ∞). The Debye length, featuring the order of thickness of EDL, is λ*<sup>D</sup>* = q ε0ε*rkBT c*0*e* <sup>2</sup> = 13.78 Å. The simulation result is shown in Figure 2 with Figure 2a being the distributions of [*K* <sup>+</sup>] for SISBM1 and PB under *Vwall* = 0.1 V and 2 V. Figure 2b is the counterpart plot of Figure 2a for [*Cl*−]. In Figure 2a, the [*K* <sup>+</sup>] distributions for SISBM1 and PB are very close to each other and almost indistinguishable in the graph at a weak wall voltage *Vwall* = 0.1 V. When the wall voltage increases to *Vwall* = 2 V, [*K* <sup>+</sup>] calculated by SISBM1 reaches its saturation limit 1/*v<sup>K</sup>* = 7.90 <sup>×</sup> <sup>10</sup><sup>4</sup> mM right adjacent to the wall, but [*K* <sup>+</sup>] unrealistically increases beyond the saturation limit when computed by the PB model. The main effect of SISBM1 is to offer a saturation limit for counter-ions (K here) when electrostatic attraction from electrode is strong enough, while it is very close to the result of PB when the electrostatic attraction is weak. In Figure 2b, [*Cl*−] distributions calculated by SISBM1 and PB are very close to each other for both strong and weak wall voltages due to the diluteness caused by electrostatic repulsion to the co-ion (Cl here) of the electrode. Note that, corresponding to a saturation layer of [*K* <sup>+</sup>] adjacent to wall at *Vwall* = 2 V (see Figure 2a), [*Cl*−] almost vanishes at that layer as well (see Figure 2b). This implies a total exclusion of Cl over there due to the saturation of K, and justifies the locally one-species argument above. If we use the original Bikerman model (41), in which ion sizes are universal, a similar saturating phenomenon for counter-ion concentration can still be obtained. However, specific ion sizes are particularly desired when electrolyte solutions are ternary, like a mixture of KCl and NaCl solutions since K and Na have different sizes, which would saturate at different limiting concentrations. These would be otherwise indistinguishable if using the original Bikerman model. *Entropy* **2020**, *22*, x FOR PEER REVIEW 14 of 15 different sizes, which would saturate at different limiting concentrations. These would be otherwise indistinguishable if using the original Bikerman model.

**Figure 2.** (**a**) K distributions in charged wall problem computed by SISBM1 and PB models under different wall voltages. (**b**) Cl distributions with conditions same as (**a**). **Figure 2.** (**a**) K distributions in charged wall problem computed by SISBM1 and PB models under different wall voltages. (**b**) Cl distributions with conditions same as (**a**).

Above we assume the rest of space after the occupation of ions is exclusively occupied by solvent particles such as water. [9-–12] have suggested that the rest of space should be occupied by solvent or void, so the +1 species in (58) and (70) should be interpreted as solvent or void. This may make more sense. Taking the selectivity filter of a K channel as an example, more and more evidences have Above we assume the rest of space after the occupation of ions is exclusively occupied by solvent particles such as water. [9–12] have suggested that the rest of space should be occupied by solvent or

its electrostatic behavior since water is a dipole.

suggestions and discussions at Hsinchu Taiwan.

*Phys. Rev. Lett.* **2012**, *109*, 149903.

theory of freezing. *Phys. Rev. Lett.* **1989**, *63*, 980–983.

general interactions, and plasmas. *J. Chem. Phys.* **1993**, *98*, 8126–8148.

**References** 

108-2115-M-035 -002 -MY2 and MOST 108-2218-E-035-005.

**Conflicts of Interest:** The author declares no conflict of interest.

shown the selectivity filter of a K channel is exclusively occupied by potassium and voids, and water

model. This water equation is generally hard to model due to its physical complexity, especially for

**Funding:** This research was funded by Ministry of Science and Technology of Taiwan under Grant Nos. MOST

**Acknowledgments:** The author would like to thank Robert Eisenberg and Jinn-Liang Liu for inspiring

7.1. Bazant, M.Z.; Storey, B.D.; Kornyshev, A. Double layer in ionic liquids: Overscreening versus crowding.

8.2. Rosenfeld, Y. Free-energy model for the inhomogeneous hard-sphere fluid mixture and density-functional

9.3. Rosenfeld, Y. Free energy model for the inhomogeneous fluid mixtures: Yukawa-charged hard spheres,

(http://creativecommons.org/licenses/by/4.0/).

© 2020 by the authors. Submitted for possible open access publication under the terms and conditions of the Creative Commons Attribution (CC BY) license

**Formatted:** Font: (Asian) 宋体

void, so the *K* + 1 species in (58) and (70) should be interpreted as solvent or void. This may make more sense. Taking the selectivity filter of a K channel as an example, more and more evidences have shown the selectivity filter of a K channel is exclusively occupied by potassium and voids, and water is not allowed there due to the strong solvation energy barrier. [9–12] even explicitly separate water and voids as two species in their modeling. However, that means the species transport equation (Nernst-Planck equation) of water needs to be modeled explicitly when constructing a PNP type model. This water equation is generally hard to model due to its physical complexity, especially for its electrostatic behavior since water is a dipole.

**Funding:** This research was funded by Ministry of Science and Technology of Taiwan under Grant Nos. MOST 108-2115-M-035 -002 -MY2 and MOST 108-2218-E-035-005.

**Acknowledgments:** The author would like to thank Robert Eisenberg and Jinn-Liang Liu for inspiring suggestions and discussions at Hsinchu Taiwan.

**Conflicts of Interest:** The author declares no conflict of interest.

### **References**


© 2020 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

*Article*

## **Electric Double Layer and Orientational Ordering of Water Dipoles in Narrow Channels within a Modified Langevin Poisson-Boltzmann Model**

## **Mitja Drab 1,†, Ekaterina Gongadze 1,†, Veronika Kralj-Igliˇc <sup>2</sup> and Aleš Igliˇc 1,\***


Received: 28 July 2020; Accepted: 18 September 2020; Published: 21 September 2020

**Abstract:** The electric double layer (EDL) is an important phenomenon that arises in systems where a charged surface comes into contact with an electrolyte solution. In this work we describe the generalization of classic Poisson-Boltzmann (PB) theory for point-like ions by taking into account orientational ordering of water molecules. The modified Langevin Poisson-Boltzmann (LPB) model of EDL is derived by minimizing the corresponding Helmholtz free energy functional, which includes also orientational entropy contribution of water dipoles. The formation of EDL is important in many artificial and biological systems bound by a cylindrical geometry. We therefore numerically solve the modified LPB equation in cylindrical coordinates, determining the spatial dependencies of electric potential, relative permittivity and average orientations of water dipoles within charged tubes of different radii. Results show that for tubes of a large radius, macroscopic (net) volume charge density of coions and counterions is zero at the geometrical axis. This is attributed to effective electrolyte charge screening in the vicinity of the inner charged surface of the tube. For tubes of small radii, the screening region extends into the whole inner space of the tube, leading to non-zero net volume charge density and non-zero orientational ordering of water dipoles near the axis.

**Keywords:** electric double layer; orientational ordering of water dipoles; Helmholtz free energy; modified Langevin Poisson-Boltzmann model

#### **1. Introduction**

The electric double layer (EDL) is a central phenomenon found at the boundary between a charged surface and an electrolyte solution [1–8]. The counterions are accumulated close to the charged surface and the coions are depleted from this region, resulting in a non-homogeneous distribution of ions. The physical properties of the EDL are crucial in understanding colloidal systems, transport of charged molecules across biological membrane channels or binding of charged proteins to biological surfaces.

Recently, much attention is being devoted to inorganic and organic hollow cylindrical structures in the nanometer range due to their potential benefit in technology, biology and medicine [9]. Potential applications range from microelectronics to microfluidics [10]. Ion channels or pores in biological membranes and blood capillaries are also examples for cylindrical nanotubes.

In some biological systems, the walls of organic nanotubes are charged and in contact with electrolyte solution, where the primary agents of interaction are electrostatic forces, both between charged particles and polar water molecules. Due to the surface charge of the walls, counterions and coions of the electrolyte are, respectively, accumulated and depleted near the walls. At the internal surfaces concave electrical double layers of cylindrical geometry are formed [11].

Furthermore, when bound to a cylindrical geometry, the effect of curvature on EDL properties is significant on small enough scales. Such biological cylindrical channels, where EDL interactions are important, encompass axons or tunneling nanotubes [12]. When artificially made channels, for example, those found in nanoporous materials, are used in the manufacture of electrochemical nanocapacitors, their power and energy densities are dependent on EDL characteristics such as capacitance [13–15].

EDL was first modeled by Helmholtz who assumed that the charged surface attracts the surrounding point-like counterions and a single layer is formed to screen the charge [16,17]. Later, these ions have been described by a Boltzmann distribution, forming a diffuse layer extending into the bulk [18,19]. The finite size has been incorporated by Stern with the so-called distance of closest approach [20] and later developed further by numerous authors [3,21–26]. In recent decades, EDL has been the subject of numerous analytical and numerical studies from Monte-Carlo methods, DFT theories and lattice models [3,7,27–44]. Additionally, interest in nanostructured materials [45–48] requires that theoretical models of EDL are revisited [49–51], also by taking into account the possible quantum effects [52,53].

It has been shown that close to the charged surface, orientational ordering and depletion of water molecules may result in a strong decrease in the local permittivity of the electrolyte solution [54–61]. Considering the orientational ordering of water and finite size of molecules, Outhwaite and collaborators developed a modified Poisson-Boltzmann's (PB) theory of EDL composed of a mixture of hard spheres with point-like dipoles and finite-sized ions [54,62]. Later, Szalai et al. [63] published a mean spherical approximation-based theory [64] that can reproduce simulation results for the electric field dependence of the dielectric permittivity of a dipolar fluid in a saturation regime. The problem was also considered within a discrete lattice statistics model taking into account the asymmetric size of ions and orientational ordering of water dipoles [44]. Recently, ion-ion and ion-water correlations were also considered in a mean-field approach [65,66].

In the present paper, we first discuss the relative permittivity of water molecules within a cavity field model. We then go on to the derivation of a modified Langevin Poisson-Boltzmann (LPB) equation for point-like ions and water dipoles for planar geometry and then generalize the equations for arbitrary geometry. In derivation of modified LPB equation we construct a Helmholtz free energy functional and minimize it to derive the analytical expressions for ion distributions and spatial dependence of statistical averages orientations of water dipoles. The free energy expression also includes contributions from configurational entropy of ions and rotational entropy of water dipoles. In the second part of the paper the modified LPB equation and the corresponding boundary conditions, generalized for an arbitrary geometry, are utilized to present the numerical solution for a cylindrical geometry with special emphasis given to very narrow cylindrical channels (Figure 1).

**Figure 1.** A schematic of a tubular structure with labeled independent coordinate *r* that can be at most *R*.

#### **2. Relative Permittivity of Water**

The dipole moment of an isolated water molecule is around 1.85 D (Debye is 3.336 <sup>×</sup> <sup>10</sup>−<sup>30</sup> Cm). In a solution, the dipole moment of a single water molecule differs from an isolated one since each molecule is also polarized by the electric field of the neighboring water molecules, creating an effective value of the dipole moment around 2.4 D–2.6 D [67,68]. The body of literature dealing with the dielectric permittivity of water is voluminous and comprehensive, from analytic models detailing the state of bound water molecules and water in charged crevices [69,70] to molecular dynamics simulations with nonlinear response to external electric fields [71,72].

The effect of a polarizing environment can be reproduced in the most simple way by introduction of the cavity field [61,73–75]. Cavity field is derived by solving the Poisson's equation of a model water molecule placed in an outside homogeneous electric field (for a detailed derivation, see Reference [76]). The present section deals with polarization of water dipoles that follows directly from the cavity field.

The water molecules are described within the modified Kirkwood approach [75] as point-like dipoles **p** with magnitude |**p**| = *p* at the centres of finite sized spheres, embedded in a medium with electric permittivity representing the ion-water solution *ε<sup>r</sup>* (Figure 2) [7,61]. Within this medium, a spatially homogeneous electric field, **E**, is present. Due to the built up charge at the interface between the inside and outside of the sphere, the dipole experiences the so called cavity field **E***c*. The relative permittivity of water is given by *ε<sup>r</sup>* = 1 + *P*tot/(*ε*0*E*), where *P*tot is the total polarization of water dipoles, *E* is the magnitude of the spatially homogeneous electric field and *ε*<sup>0</sup> is the permittivity of vacuum. The total polarization is the sum of electronic polarization, *P<sup>e</sup>* , and orientational polarization due to the permanent water dipoles *P*, so that *P*tot = *P<sup>e</sup>* + *P*. The electronic polarization determines the refractive index of water [51,61] *n* <sup>2</sup> <sup>=</sup> <sup>1</sup> <sup>+</sup> *<sup>P</sup>e*/(*ε*0*E*) <sup>≈</sup> 1.8 and *<sup>ε</sup><sup>r</sup>* can be expressed as

$$
\varepsilon\_r = n^2 + \frac{P}{\varepsilon\_0 E}.\tag{1}
$$

To find the expression for *P* we must take into account the constant number density of water *n<sup>w</sup>* and the statistical-average orientation of water molecules in the solution [7]:

$$P = n\_w p \langle \cos \theta \rangle. \tag{2}$$

Here, *θ* is the angle between **p** and the cavity field **E***<sup>c</sup>* acting on it (see Figure 3). Statistical averaging is labeled by h...i. To estimate hcos *θ*i, we must first find the expression for **E***c*. This involves solving the Poisson equation for a sphere with electric permittivity *n* 2 embedded in a medium with a relative permittivity *ε<sup>r</sup>* described in detail in Reference [76]. Neglecting the short range interactions between dipoles, the local electric field strength at the centre of the sphere at the location of the permanent point-like dipole (Figure 2) can be expressed as [7,76]

$$\mathbf{E}\_{\mathbf{f}} = \frac{\mathbf{3}\varepsilon\_{r}}{n^{2} + \mathbf{2}\varepsilon\_{r}} \mathbf{E}.\tag{3}$$

When the surrounding medium has a relative permittivity much larger than the refractive index of water *ε<sup>r</sup> n* 2 , it follows that

$$E\_c \approx \frac{3}{2} E \quad \rightarrow \quad \mathbf{E}\_c \approx \frac{3}{2} \mathbf{E}.\tag{4}$$

So far we have neglected the reaction field, which is the field of the point-like dipole at the center of the cavity itself. This reaction field is directly proportional to the strength of dipole **E**react ∝ **p**. In vacuum, in the case of a single isolated water molecule, the external dipole moment is also the experimentally measured dipole moment of a single water molecule **p***<sup>e</sup>* given by [7,76]:

$$\mathbf{p}\_{\varepsilon} = \frac{3}{n^2 + 2} \mathbf{p}.\tag{5}$$

Here, **p** is the permanent point-like internal water dipole at the center of the sphere. The energy of an internal point-like dipole **p** in a local field **E***<sup>c</sup>* is [61]

$$\mathcal{W}\_{\mathfrak{k}} = -\mathbf{p} \cdot \mathbf{E}\_{\mathfrak{c}}.\tag{6}$$

Substituting from Equation (4) and Equation (5), we can express the dipole energy as [61]

$$\mathcal{W}\_{\ell} = -\frac{3}{2} \left( \frac{2 + n^2}{3} \right) p\_0 E \cos \theta\_{\prime} \tag{7}$$

$$\mathcal{W}\_{\ell} \quad = \quad \gamma p\_0 \mathcal{E} \cos \omega. \tag{8}$$

Here, *p*<sup>0</sup> is the magnitude of **p***<sup>e</sup>* and *ω* is supplementary to *θ*, as shown in Figure 3. The constant *γ* equals [7,61] (see Equations (7) and (8))

$$
\gamma = \frac{3}{2} \left( \frac{2 + n^2}{3} \right). \tag{9}
$$

With this in mind, the ensemble average in Equation (2) can be calculated as:

$$
\langle \cos \omega \rangle = \frac{\int \cos \omega e^{-(\beta \gamma p\_0 \to \cos \omega)} \, d\Omega}{\int e^{-(\beta \gamma p\_0 \to \cos \omega)} \, d\Omega} = -\mathcal{L}(\beta \gamma p\_0 \to). \tag{10}
$$

Here, *β* is the Boltzmann's factor equal to *β* = 1/*kT*, where *kT* is the thermal energy. The element of solid angle is *d*Ω = 2*π* sin *ωdω*, meaning that the integral runs from 0 to *π* with assumed azimuthal symmetry. The Langevin function is defined as L(*u*) = coth *u* − 1/*u*. By taking into account Equations (1), (2), (5) and (10), we can express the relative water permittivity as [7]:

$$
\varepsilon\_r = n^2 + \frac{n\_w p\_0}{\varepsilon\_0} \left(\frac{2 + n^2}{3}\right) \frac{\mathcal{L}(\beta \gamma p\_0 E)}{E}. \tag{11}
$$

In the limit of vanishing electric field strength (*E* → 0), the above expression for the relative permittivity of water yields to the Onsager limit [7]

$$
\varepsilon\_r = n^2 + \frac{n\_w p\_0^2 \beta}{2\varepsilon\_0} \left(\frac{2 + n^2}{3}\right)^2. \tag{12}
$$

For *p*<sup>0</sup> = 3.1 D and *nw*/*N<sup>A</sup>* = 55 mol/L, [7,44], where *N<sup>A</sup>* is the Avogadro number, Equation (12) yields the value *ε<sup>r</sup>* = 78.5 at room temperature. Returning to Equation (2), we can write the final result for the orientational polarization of water dipoles *P*, which will be needed for our Helmholtz free energy minimization in the following section:

$$P = -n\_{\overline{w}} p\_0 \left(\frac{2 + n^2}{3}\right) \mathcal{L}(\beta \gamma p\_0 E). \tag{13}$$

**Figure 2.** A single water molecule is modelled by a sphere with relative permittivity *n* 2 , where *n* = 1.33 is the refractive index of water. A permanent point-like rigid dipole with magnitude, *p*, is located at the center of the sphere [61]. Due to the built up charge, the point dipole experiences the so-called cavity field **E***c*.

**Figure 3.** Relation between angles *θ* and *ω*. The water internal dipole moment is marked by **p**, the local cavity field, **Ec**, points in the opposite direction of ∇*φc*.

#### **3. Derivation of the Modified LPB Equation by Minimization of Helmholtz Free Energy**

Our model assumes the electrolyte solution is a mixture of point-like monovalent co- and counterions and permanent water dipoles, representing the water molecules. The expression for the spatial dependence of the solution permittivity *εr*(*x*), arising as a direct consequence of the spontaneous ordering of water dipoles, can be obtained by using the minimization of the Helmholtz free energy in a one-dimensional setting with the charged planar surface located at *x* = 0. In the minimization procedure, the local electric field at the positions of the hydrated point-like ions in the electrolyte solution is denoted by *E*(*x*), while the local cavity field at the positions of the water internal point-like dipoles is denoted by *Ec*(*x*). We can write the Helmholtz free energy of the system *F* as (see also Reference [58]):

$$\begin{split} F &= \underbrace{\frac{\varepsilon\_{0}\eta^{2}}{2} \int\_{\mathbb{F}\_{1}} E\_{c}^{2}(\mathbf{x}) \, dV}\_{\mathbb{F}\_{1}} + \underbrace{\frac{\varepsilon\_{0}\eta^{2}}{2} \int\_{\mathbb{F}\_{2}} E^{2}(\mathbf{x}) \, dV}\_{\mathbb{F}\_{2}} + kT \Big[ \underbrace{\int\_{\mathbb{F}} \left( n\_{+}(\mathbf{x}) \ln \frac{n\_{+}(\mathbf{x})}{n\_{0}} - (n\_{+}(\mathbf{x}) - n\_{0}) \right) dV}\_{\mathbb{F}\_{3}} + \\ &+ \underbrace{\int\_{\mathbb{F}\_{4}} \left( n\_{-}(\mathbf{x}) \ln \frac{n\_{-}(\mathbf{x})}{n\_{0}} - (n\_{-}(\mathbf{x}) - n\_{0}) \right) dV}\_{\mathbb{F}\_{4}} + \underbrace{\int\_{\mathbb{F}\_{5}} (\lambda\_{+} n\_{+}(\mathbf{x}) + \lambda\_{-} n\_{-}(\mathbf{x})) \, dV}\_{\mathbb{F}\_{5}} + \\ &+ \underbrace{\int n\_{\text{w}} \langle \mathcal{P}(\mathbf{x}, \omega) \ln \mathcal{P}(\mathbf{x}, \omega) \rangle\_{\mathbb{W}^{1}} \, dV}\_{\mathbb{F}\_{6}} + \underbrace{\int n\_{\text{w}} \eta(\mathbf{x}) \left( \langle \mathcal{P}(\mathbf{x}, \omega) \rangle\_{\omega} - 1 \right) \, dV}\_{\mathbb{F}\_{7}} \end{split} \tag{14}$$

The thermal energy is given by *kT*, while *n* is the refractive index. For greater clarity, we split the particular contributions to the free energy as marked by the underbraces in Equation (14). The mean field created by coions and counterions and the water dipoles polarization contribution are given by terms *F*<sup>1</sup> and *F*2, respectively. Mixing entropy free energy contributions of point-like counterions and coions are accounted for in terms *F*<sup>3</sup> and *F*4. The constraint of a constant number of ions in the system is given in *F*5, where *λ*<sup>+</sup> and *λ*<sup>−</sup> are the global Lagrange's multipliers for counterions and coions. The free energy that corresponds to orientational entropy of permanent water dipoles is given by *F*6, while the last term, *F*7, gives the local constraint for orientation of dipoles. P(*x*, *ω*) is the probability that a permanent water dipole located at *x* is oriented at angle *ω* with respect to the normal to the charged surface (Figure 3). The brackets h...i*<sup>ω</sup>* denote the average:

$$
\langle \mathcal{F}(\mathbf{x}, \omega) \rangle\_{\omega} = \frac{1}{4\pi} \int\_0^{\pi} \mathcal{F}(\mathbf{x}, \omega) \, 2\pi \sin \omega d\omega. \tag{15}
$$

Here, *ω* is the angle between the internal dipole moment vector, **p**, and **n***<sup>φ</sup>* = ∇*φc*/|∇*φc*| (see Figure 3). We perform variation on the Helmholtz free energy, *F*, in Equation (14), so that *δF* = 0. Let us deal with the variational approach of every contribution in Equation (14) particularly, beginning with *F*<sup>1</sup> and *F*2. For clarity of notation, direct spatial dependence will sometimes be omitted, so that for example, *n*+(*x*) ≡ *n*+.

#### *3.1. Variation Procedure*

#### 3.1.1. Electric Fields (*F*<sup>1</sup> and *F*2)

Since there are no time dependent magnetic fields, we can express the electric fields as potentials *E*(*x*) = −*φ* 0 (*x*), *Ec*(*x*) = −*φ* 0 *c* (*x*), where the prime labels the spatial derivative, and perform a variation on the electrostatic term pertaining to water dipoles.

$$
\delta \left( \frac{\varepsilon\_0 n^2}{2} \int \phi\_c'^2 \,dV \right) = \frac{\varepsilon\_0 n^2}{2} \int \mathfrak{D}\phi\_c' \delta(\phi\_c') \,dV. \tag{16}
$$

We can rearrange this term, if we consider the rules of differentiating a function product

$$\begin{aligned} (\phi\_c \delta \phi\_c')' &= \phi\_c' \delta \phi\_c' + \phi\_c \delta \phi\_c'', \\ \phi\_c' \delta \phi\_c' &= (\phi\_c \delta \phi\_c')' - \phi\_c \delta \phi\_c''. \end{aligned} \tag{17}$$

The integral in Equation (16) can be rewritten as,

$$\begin{split} \varepsilon\_{0}n^{2} \int \phi\_{c}^{\prime} \delta(\phi\_{c}^{\prime}) \,dV &= \varepsilon\_{0}n^{2} \Big( \underbrace{\phi\_{c} \delta \phi\_{c}^{\prime} |\_{0}}\_{=0} - \\ &- \int \phi\_{c} \delta(\phi\_{c}^{\prime\prime}) \,dV \Big). \end{split} \tag{18}$$

where the first term on the right-hand side equals 0 at infinity, since we impose the electric potential there to be constant and equal to 0. Taking into account the Poisson's equation for the water dipoles, namely *φ* 00 *c* (*x*) = ∇ · **P**/*ε*0*n* 2 , where **P** represents the net polarization of the permanent water dipoles, we get

$$-\varepsilon\_0 n^2 \int \phi\_c \delta(\phi\_c'') \, dV = \int \phi\_c \delta \rho\_c \, dV. \tag{19}$$

Here, *ρ<sup>c</sup>* corresponds to the bound charge density due to the dipoles' polarizations, which is related to net polarization *ρ<sup>c</sup>* = −∇ · **P**. We observe that *δ*(∇ · **P**) = ∇ · *δ***P**. The integral in (Equation (19)) can now be rewritten:

$$
\int \oint\_{\mathcal{C}} \delta \rho\_{\mathcal{C}} \, dV = - \int \phi\_{\mathcal{C}} (\nabla \cdot \delta \mathbf{P}) \, dV. \tag{20}
$$

The product rule for divergence can be used ∇ · (*φcδ***P**) = (∇*φc*) · *δ***P** + *φc*(∇ · *δ***P**), so that the integral of Equation (20) can now be written differently again:

$$\int \phi\_{\varepsilon} (\nabla \cdot \delta \mathbf{P}) \, dV = \underbrace{\int \nabla \cdot (\phi\_{\varepsilon} \delta \mathbf{P}) \, dV}\_{=0} - \int (\nabla \phi\_{\varepsilon}) \cdot \delta \mathbf{P} \, dV. \tag{21}$$

Here, the first integral on the right hand side vanishes by virtue of the divergence theorem; since the potential far away from the plates is constant and equal to zero. We therefore arrive at the final result

$$
\delta F\_1 = \int (\nabla \phi\_c) \cdot \delta \mathbf{P} \, dV. \tag{22}
$$

The polarization, **P**, is related to the average orientation of all water dipoles (Equation (2)):

$$\mathbf{P}(\mathbf{x}) = n\_{\mathcal{W}} \langle \mathcal{P}(\mathbf{x}, \omega) \rangle\_{\omega} p \mathbf{n}\_{\Phi}. \tag{23}$$

Here, *n<sup>w</sup>* is the number density of water molecules in the solution, *p* = |**p**| is the internal point-like dipole magnitude, **n***<sup>φ</sup>* = ∇*φc*/|∇*φc*| is the unit vector away from the charged plate and hP(*x*, *ω*)i*<sup>ω</sup>* is defined by Equation (15) (see Figure 3). Since our case deals with a negatively charged surface (*σ* < 0), **P** points in the direction opposite to the direction of the *x*-axis and is thus negative (for details see Reference [76]). Since the variation of **P** can be written *δ***P**(*x*) = h*nw***p***δ*P(*x*, *ω*)i*ω*, we arrive at the variation of *F*1:

$$
\delta F\_1 = n\_w \int \langle \delta \mathcal{P}(\mathbf{x}, \omega) (\nabla \phi\_c) \cdot \mathbf{p} \rangle\_{\omega} \, dV. \tag{24}
$$

Similarly, for *F*<sup>2</sup> by taking into account Equation (17), we get

$$
\delta \left( \frac{\varepsilon\_0 n^2}{2} \int \phi'^2 dV \right) = \int \phi \delta \rho\_{\text{free}} \, dV. \tag{25}
$$

The Poisson equation is different for free charges (ions): *φ* <sup>00</sup>(*x*) = −*ρ*free/*ε*0*n* <sup>2</sup> <sup>=</sup> *<sup>e</sup>*0(*n*+(*x*) <sup>−</sup> *<sup>n</sup>*−(*x*)). The variation by *φ* 00(*x*) in Equation (25) can be written with macroscopic net volume charge density *ρ*free(*x*), which in turn is the sum of the contributions of the local net ion charges. Performing the variation on ion charge distribution *ρ*free(*x*) gives

$$
\delta\rho\_{\text{free}} = e\_0(\delta n\_+ - \delta n\_-),
\tag{26}
$$

finishing the variation of the term *F*2:

$$
\delta F\_2 = \int e\_0 \phi (\delta n\_+ - \delta n\_-) \, dV. \tag{27}
$$

#### 3.1.2. Ion Mixing (*F*3, *F*<sup>4</sup> and *F*5)

It makes sense to perform the variation of the ion mixing terms (*F*<sup>3</sup> and *F*4), together with their Lagrange multipliers (*F*5), since the variation *δn*<sup>+</sup> and *δn*<sup>−</sup> will be a common term for positive and negative ions, respectively. It is easily shown from Equation (14) that

$$
\delta \mathcal{F}\_3 + \delta \mathcal{F}\_4 + \delta \mathcal{F}\_5 = kT \int \delta n\_+ (\lambda\_+ + \ln \frac{n\_+}{n\_0}) \, dV + kT \int \delta n\_- (\lambda\_- + \ln \frac{n\_-}{n\_0}) \, dV. \tag{28}
$$

#### 3.1.3. Dipole Mixing (*F*<sup>6</sup> and *F*7)

Variation of the terms *F*<sup>6</sup> and *F*<sup>7</sup> is straightforward. Since the bulk water number density, *nw*, is taken to be constant, the variation of *F*<sup>6</sup> is

$$\delta \mathcal{F}\_6 = kT n\_w \int \left( \langle \delta \mathcal{P}(\mathbf{x}, \omega) \ln \mathcal{P}(\mathbf{x}, \omega) + \delta \mathcal{P}(\mathbf{x}, \omega) \rangle\_{\omega} \right) dV. \tag{29}$$

The last variation of *F*<sup>7</sup> is performed over the probability, P(*x*, *ω*), and the Lagrange multiplier, *η*(*x*). Expanding and applying the product rule, we find that

$$\delta \mathbf{F}\_7 = kT n\_{\overline{w}} \int (\delta \eta(\mathbf{x}) \langle \mathcal{P}(\mathbf{x}, \omega) \rangle\_{\omega} + \eta(\mathbf{x}) \langle \delta \mathcal{P}(\mathbf{x}, \omega) \rangle\_{\omega} - \delta \eta(\mathbf{x})) \, dV. \tag{30}$$

#### *3.2. Euler-Lagrange Equations*

Combining the variations of all the integrals given in Equation (14), their sum *δF* gives us the variation of Helmholtz free energy. Factoring all the variation terms with respect to *n*+(*x*), *n*−(*x*),P(*x*, *ω*) and *η*(*x*) gives

$$\begin{split} \delta F &= \int dV \delta n\_{+}(\mathbf{x}) \Big[ kT \Big( \ln \frac{n\_{+}(\mathbf{x})}{n\_{0}} + \lambda\_{+} \Big) + \phi \mathbf{e}\_{0} \Big] + \int dV \delta n\_{-}(\mathbf{x}) \Big[ kT \Big( \ln \frac{n\_{-}(\mathbf{x})}{n\_{0}} + \lambda\_{-} \Big) - \phi \mathbf{e}\_{0} \Big] + \\ &+ \int dV n\_{w} \langle \delta \mathcal{P}(\mathbf{x}, \omega) \Big( \nabla \phi\_{\mathbf{c}} \cdot \mathbf{p} + \frac{\ln \mathcal{P}(\mathbf{x}, \omega) + \eta(\mathbf{x}) + 1}{\beta} \rangle\_{\omega} + kT \int dV n\_{w} \delta \eta(\mathbf{x}) \Big( \langle \mathcal{P}(\mathbf{x}, \omega) \rangle\_{\omega} - 1 \Big). \end{split} \tag{31}$$

The volume differentials in a planar geometry are *dV* = *S dx*. Since the minimization condition demands *δF* = 0, the expressions multiplied by *δn*+(*x*), *δn*−(*x*), *δ*P(*x*, *ω*) and *δη*(*x*) in the last equation must equal zero, resulting in a system

$$kT\Big(\ln\frac{n\_+(\mathbf{x})}{n\_0} + \lambda\_+\Big) + \phi \mathbf{e}\_0 = \mathbf{0},\tag{32}$$

$$kT\left(\ln\frac{n\_{-}(\chi)}{n\_{0}}+\lambda\_{-}\right)-\phi\epsilon\_{0}=0,\tag{33}$$

$$E\_c p \cos \omega + \frac{\ln \mathcal{P}(\mathbf{x}, \omega) + \eta(\mathbf{x}) + 1}{\mathcal{P}} = 0,\tag{34}$$

$$
\langle \mathcal{P}(\mathfrak{x}, \omega) \rangle\_{\omega} - 1 = 0. \tag{35}
$$

Here, we write *β* = 1/*kT* and expand the dot product ∇*φ<sup>c</sup>* · **p** = *E<sup>c</sup> p* cos *ω* (see Figure 3). Solving Equations (32) and (33), we obtain the ion spatial distributions

$$n\_+(\mathbf{x}) = n\_0 \exp\left(-\beta e\_0 \phi - \lambda\_+\right),\tag{36}$$

$$n\_{-}(\mathbf{x}) = n\_{0} \exp\left(\beta e\_{0} \phi - \lambda\_{-}\right). \tag{37}$$

The boundary conditions state that *φ*(*x* → ∞) = 0 and *n*+,−(*x* → ∞) = *n*0, which renders *λ*<sup>+</sup> = *λ*<sup>−</sup> = 0. We may now turn our attention to the variation of permanent water dipoles orientation. Solving for P(*x*, *ω*), Equation (34) gives

$$\mathcal{P}(\mathbf{x},\omega) = \Lambda(\mathbf{x}) \exp\left(-\beta \mathbb{E}\_{\mathbf{c}} p \cos \omega\right),\tag{38}$$

where Λ(*x*) = exp(−*η*(*x*) − 1). Substituting the cavity field *E<sup>c</sup>* by *E* (Equation (4)) and dipole moment magnitude *p* by *p*<sup>0</sup> (Equation (5)) gives

$$\mathcal{P}(\mathbf{x},\omega) = \Lambda(\mathbf{x}) \exp\left(-\beta \frac{\Im E}{2} \left(\frac{2+n^2}{3}\right) p\_0 \cos\omega\right),\tag{39}$$

where *p*<sup>0</sup> is the magnitude of **p***<sup>e</sup>* . The final result is expressed using the constant *γ* defined in Equation (9):

$$\mathcal{P}(\mathbf{x},\omega) = \Lambda(\mathbf{x}) \exp\left(-\beta\gamma E p\_0 \cos\omega\right). \tag{40}$$

We can now evaluate the average internal dipole moment by integrating over mean orientations (considering Equation (23)),

$$\begin{split} p\langle\cos\omega\rangle &= p\_0 \left(\frac{2+n^2}{3}\right) \langle\cos\omega\rangle \\ &= \frac{\int\_0^\pi \left(\frac{2+n^2}{3}\right) p\_0 \cos\omega \exp\left(-\beta\gamma E p\_0 \cos\omega\right) d\Omega}{\int\_0^\pi \exp\left(-\beta\gamma E p\_0 \cos\omega\right) d\Omega} \\ &= -p\_0 \left(\frac{2+n^2}{3}\right) \mathcal{L}\left(\beta\gamma E p\_0\right). \end{split} \tag{41}$$

The orientational polarization of water is thus (see Equations (2) and (5)):

$$\begin{split} P(\mathbf{x}) &= \quad n\_{\mathrm{w}} p \langle \cos \omega \rangle \\ &= \quad -n\_{\mathrm{w}} p\_{0} \left( \frac{2 + n^{2}}{3} \right) \mathcal{L} \left( \beta \gamma E(\mathbf{x}) p\_{0} \right). \end{split} \tag{42}$$

If we insert the above result and the ion distribution functions (Equations (36) and (37)) into the average microscopic charge density equation *ρ*(*x*) = *ρ*free(*x*) − *dP*/*dx* [61,77], where *ρ*free is the contribution of the net ion charges Equations (26), (36) and (37) and *P*(*x*) is the polarization due to partially oriented water dipoles, we get the expression for *ρ*(*x*) in a one-dimensional case:

$$\rho(\mathbf{x}) = -2e\_0 n\_0 \sinh\left(\beta e\_0 \phi(\mathbf{x})\right) + n\_w p\_0 \left(\frac{2 + n^2}{3}\right) \frac{d}{d\mathbf{x}} \left(\mathcal{L}\left(\beta \gamma E(\mathbf{x}) p\_0\right)\right). \tag{43}$$

Inserting the above expression for average microscopic volume charge density *ρ*(*x*) into the Poisson's equation,

$$\phi''(\mathbf{x}) = -\frac{\rho(\mathbf{x})}{n^2 \varepsilon\_0},\tag{44}$$

we get the modified LPB differential equation for the electric potential *φ*(*x*):

$$\boldsymbol{\phi}^{\prime\prime}(\mathbf{x}) = \frac{1}{n^2 \varepsilon\_0} \left[ 2e\_0 n\_0 \sinh\left(\beta e\_0 \boldsymbol{\phi}(\mathbf{x})\right) - n\_w p\_0 \left(\frac{2 + n^2}{3}\right) \frac{d}{d\mathbf{x}} \left(\mathcal{L}\left(\beta \gamma \boldsymbol{\Sigma}(\mathbf{x}) p\_0\right)\right) \right],\tag{45}$$

where *φ* <sup>00</sup>(*x*) is the second derivative of the electric potential *φ*(*x*) with respect to *x* and *E*(*x*) = −*φ* 0 (*x*). Equation (45) can be factorized via a product rule if we take into account that the Langevin function is odd and its derivative is L 0 (*u*) = 1/*u* <sup>2</sup> <sup>−</sup> 1/ sinh<sup>2</sup> *u* in the following form [7]:

$$\frac{d}{d\mathbf{x}} \left[ \varepsilon\_0 \varepsilon\_r(\mathbf{x}) \phi'(\mathbf{x}) \right] = 2e\_0 n\_0 \sinh \left( \beta e\_0 \phi(\mathbf{x}) \right), \tag{46}$$

$$
\varepsilon\_r(\mathbf{x}) = n^2 + n\_w \frac{p\_0}{\varepsilon\_0} \left(\frac{2 + n^2}{3}\right) \frac{\mathcal{L}(\beta \gamma E(\mathbf{x}) p\_0)}{E(\mathbf{x})},\tag{47}
$$

where *εr*(*x*) is the relative permittivity (Equation (11)). This modified Langevin Poisson-Boltzmann (LPB) differential equation (Equation (46)) is subject to two boundary conditions. The first boundary condition arises from the electro-neutrality of the system, which assumes that the total net charge of the system is zero, hence

$$
\int \rho\_{\text{free}}(\mathbf{x}) \, dV - \sigma \mathbf{S} = \mathbf{0},
\tag{48}
$$

where *σ* is the negative membrane surface charge density, *S* is the total membrane surface area and *ρ*free(*x*) = −2*e*0*n*<sup>0</sup> sinh (*βe*0*φ*(*x*)) is the macroscopic (net) volume charge density of coions and counterions. Since the macroscopic volume charge density is only dependent on *x* (Equation (43)) and the differential *dV* = *S dx*, Equation (48) may be rewritten

$$\int\_0^\infty 2e\_0 n\_0 \sinh\left(\beta e\_0 \phi(\mathbf{x})\right) d\mathbf{x} = -\sigma. \tag{49}$$

If we integrate Equation (45) once over the whole system, we get

$$\phi'(\mathbf{x}=0) = -\frac{1}{n^2 \varepsilon\_0} \left[ \sigma + n\_w p\_0 \left( \frac{2 + n^2}{3} \right) \cdot \mathcal{L}(\beta \gamma \varepsilon \, p\_0|\_{\mathbf{x}=0}) \right]. \tag{50}$$

The second boundary condition states that the electric potential far away from the charged surface (in the bulk) is constant *φ* 0 (*x* → ∞) = 0, rendering L(*βγEp*0|*x*→∞) = 0. The modified LPB equation (Equation (46)) was derived in one dimension, but can be rewritten in a more general form to apply to an arbitrary three-dimensional geometry. In three dimensions, the steps are analogous and discussed in detail in a previous work [58], where a three-dimensional Lagrangian was derived for a model of finite-sized ions. With this in mind, the modified LPB equation (Equation (46)) can be rewritten:

$$\nabla \cdot \left[ \varepsilon\_0 n^2 \nabla \phi(\mathbf{r}) \right] + n\_{\text{w}} p\_0 \left( \frac{2 + n^2}{3} \right) \nabla \cdot \left( \mathbf{n}\_{\phi} \mathcal{L} \left( \beta \gamma E p\_0 \right) \right) = 2 \varepsilon\_0 n\_0 \sinh \left( \beta \varepsilon\_0 \phi(\mathbf{r}) \right), \tag{51}$$

where **n***<sup>φ</sup>* = ∇*φ*/|∇*φ*| = ∇*φ*/*E*. We may factor the last equation, so that

$$\nabla \cdot \left[ \varepsilon\_0 \left( n^2 + \frac{n\_{w} p\_0}{\varepsilon\_0} \left( \frac{2 + n^2}{3} \right) \frac{\mathcal{L} \left( \beta \gamma E p\_0 \right)}{E} \right) \nabla \phi(\mathbf{r}) \right] = 2 \varepsilon\_0 n\_0 \sinh \left( \beta \varepsilon\_0 \phi(\mathbf{r}) \right). \tag{52}$$

This modified LPB equation can be written even more compactly, considering the definition of spatially dependent permittivity *εr*(**r**) given by Equation (47) (for details, see Reference [58]):

$$\nabla \cdot \left[ \varepsilon\_0 \varepsilon\_r(\mathbf{r}) \nabla \phi(\mathbf{r}) \right] = 2 \varepsilon\_0 n\_0 \sinh \left( \beta \varepsilon\_0 \phi(\mathbf{r}) \right), \tag{53}$$

$$
\varepsilon\_r(\mathbf{r}) = n^2 + n\_w \frac{p\_0}{\varepsilon\_0} \left(\frac{2 + n^2}{3}\right) \frac{\mathcal{L}(\boldsymbol{\beta} \gamma \boldsymbol{E}(\mathbf{r}) p\_0)}{\boldsymbol{E}(\mathbf{r})}.\tag{54}
$$

Here, *ρ*free(**r**) is the macroscopic (net) volume charge density of coions and counterions. A corresponding three-dimensional variant of the boundary condition (Equation (50)) is

$$\nabla \phi(\mathbf{r} = \mathbf{r}\_{\text{surf}}) = -\frac{1}{n^2 \varepsilon\_0} \left[ \sigma \mathbf{n}\_{\phi} + \mathbf{n}\_{\phi} n\_{\text{w}} p\_0 \left( \frac{2 + n^2}{3} \right) \cdot \mathcal{L}(\beta \gamma E(\mathbf{r}) p\_0(\mathbf{r}) |\_{\mathbf{r} = \text{surf}}) \right]. \tag{55}$$

Rearranging, it follows that

$$\nabla \phi(\mathbf{r} = \mathbf{r}\_{\text{surf}}) \left[ 1 + \frac{\mathbf{n}\_{\Phi}}{\nabla \phi(\mathbf{r} = \mathbf{r}\_{\text{surf}})} \frac{n\_{\text{w}} p\_0}{n^2 \varepsilon\_0} \left( \frac{2 + n^2}{3} \right) \cdot \mathcal{L}(\beta \gamma E(\mathbf{r}) p\_0(\mathbf{r}) |\_{\mathbf{r} = \text{surf}}) \right] = -\frac{\sigma}{n^2 \varepsilon\_0} \mathbf{n}\_{\Phi}.\tag{56}$$

Evaluating the second expression on the left hand side of the last equation gives

$$\frac{\mathbf{n}\_{\phi}}{\nabla\phi(\mathbf{r} = \mathbf{r}\_{\text{surf}})} = \frac{\nabla\phi(\mathbf{r} = \mathbf{r}\_{\text{surf}})}{|\nabla\phi(\mathbf{r} = \mathbf{r}\_{\text{surf}})|} \frac{1}{\nabla\phi(\mathbf{r} = \mathbf{r}\_{\text{surf}})} = \frac{1}{E(\mathbf{r} = \mathbf{r}\_{\text{surf}})}.\tag{57}$$

Combining this simplification with Equation (42), Equation (55) becomes

$$
\nabla \phi(\mathbf{r} = \mathbf{r}\_{\text{surf}}) \varepsilon\_r(\mathbf{r} = \mathbf{r}\_{\text{surf}}) = -\frac{\sigma \mathbf{n}\_{\phi}}{\varepsilon\_0}.\tag{58}
$$

Here we also take into account the expression for *ε<sup>r</sup>* (Equation (47)). We see that the term inside the square brackets on the left-hand side of Equation (56) is precisely the definition of the relative permittivity on the surface of charged membrane *εr*(**r** = **r**surf) (Equation (54)), yielding the general result

$$
\nabla \phi(\mathbf{r} = \mathbf{r}\_{\text{surf}}) = -\frac{\sigma \mathbf{n}\_{\phi}}{\varepsilon\_0 \varepsilon\_r (\mathbf{r} = \mathbf{r}\_{\text{surf}})}.\tag{59}
$$

#### **4. Results**

Figure 4 shows the dependency of the calculated macroscopic (net) volume charge density of the electrolyte solution inside the nanotubes (*ρ*free(*r*)) on the radial distance from the geometrical axis of the tube. It can be seen in the figure that for larger radii of the inner cross-sections of the nanotubes (*R*), the value *ρ*free at the geometrical axis of the tube is zero, which means that the number densities of counterions and coions there are equal and the electric potential is constant, that is, zero in our case (see the right panel in Figure 5).

**Figure 4.** Macroscopic (net) volume charge density of coions and counterions (*ρ*free) as a function of the radial distance from the geometrical axis of tube (*r*) calculated for 4 values of the inner tube diameter *R*: 0.5 nm, 1.0 nm, 2.5 nm and 5.0 nm. The bulk concentrations of counterions and coions *<sup>n</sup>*0/*N<sup>A</sup>* <sup>=</sup> 0.15 mol/L and *<sup>σ</sup>* <sup>=</sup> <sup>−</sup>0.25 As/m<sup>2</sup> , *T* = 298 K, constant concentration of water *nw*/*N<sup>A</sup>* = 55 mol/L, optical refractive index *n* = 1.33 and magnitude of external dipole moment of water *p*<sup>0</sup> = 3.1 Debye, where *N<sup>A</sup>* is the Avogadro number.

**Figure 5.** Space dependence of electric potential in the cross-section of the tube interior calculated for 2 values of the inner tube diameter *R*: 1.0 nm and 5.0 nm. The values of the model parameters are the same as given at Figure 4.

On the contrary, for smaller values of the nanotube radius *R*, the value of *ρ*free at geometrical axis of the nanotube is not zero (Figure 4). Accordingly, for small values of the radius of the inner nanotube also the gradient of the electric field (Figure 6) and the electric potential at the nanotube geometrical axis are not zero (left panel in Figure 5). Hence, the bulk condition of the equal number densities of counterions and coions is fulfilled outside the interior of the nanotube.

**Figure 6.** The magnitude of electric field strength as a function of the radial distance from the geometrical axis of tube (*r*), calculated for 4 values of the inner tube diameter *R*: 0.5 nm, 1.0 nm, 2.5 nm and 5.0 nm. The values of the model parameters are the same as given at Figure 4. The narrow nanotube before and after entrance of the nanoparticles.

Figure 7 shows the dependency of the average orientation hcos (*ω*)i*<sup>ω</sup>* and the relative permittivity *ε<sup>r</sup>* on the radial distance from the geometrical axis of tube (*r*), calculated for four different values of nanotube inner radius *R*. It can be seen that for small radii, *R*, the average orientational of water dipoles is relatively strong also in the vicinity of geometrical axis of the tube, while for larger R the average orientation of water dipoles is strong only in the region near the charged inner surface of the tube.

**Figure 7.** Average orientation hcos (*ω*)i*<sup>ω</sup>* and relative permittivity *ε<sup>r</sup>* as a function of the radial distance from the geometrical axis of tube (*r*), calculated for 4 values of the inner tube diameter *R*: 0.5 nm, 1.0 nm, 2.5 nm and 5.0 nm. The values of the model parameters are the same as given at Figure 4.

#### **5. Discussion and Conclusions**

In this paper, we derived a modified Langevin Poisson-Boltzmann (LPB) model and the modified LPB equation to theoretically describe the electric double layer (EDL) for a monovalent electrolyte solution inside very narrow nanotubes with a negatively charged inner surface. In the modified LPB approach, the electronic polarization of the water is taken into account by assuming a permanent dipole embedded in the center of the sphere with a volume equal to the average volume of a water molecule. The effect of a polarizing environment is reproduced by introduction of the cavity field in the saturation regime [7,61,76]. In past EDL studies, treatments of cavity fields and structural correlations between water dipoles were limited to cases of relatively small electric field strengths, far away from the saturation limit of polarization and orientational ordering of water molecules [73–75]. High magnitudes of electric field strength were later added in several works [44,61,63,78]. A commonly oversimplified assumption when theoretically describing the EDL is the assumption of a surface charge density-independent relative permittivity in the inner (Stern) layer. Due to orientational ordering of water dipoles, the relative permittivity of the Stern layer depends on the electric field strength, that is, on the surface charge density (*σ*) of the electrode [51,79–82]. Furthermore, fitting the model curves with a range of free parameters to the experimental points [83] cannot prove that the Stern layer capacitance and permittivity is *σ*-independent. The decrease in the relative permittivity close to the charged surface (electrode) is obviously partially the consequence of orientational ordering of water dipoles close to the saturation regime or in the saturation regime as shown theoretically in References [6,27,44,54,58,59,61–64,80,82].

Within a recently presented phenomenological approach it is claimed that close to the charged surface, almost all water molecules belong to water shells around the ions, while the free water molecules are excluded [83]. The results of simulations clearly refute this fact [84] by showing increased water ordering in the direction towards the charged surface (including the region close to the charged surface) (Figure 7, upper panel) even for high salt concentrations [84], in quantitative agreement with mean-field theoretical predictions [7,82]. For example, for a magnitude of 0.16 As/m<sup>2</sup> surface charge density, there is practically no difference in the orientational ordering and space distribution of water dipoles close to the charged surface between water with and without NaCl (of concentration 500 mmol/L) [84]. In general, for magnitudes of surface charge density up to around 0.3 As/m<sup>2</sup> , where the mean-field approach can still be justified [7,82], there is only a weak quantitative influence of salt on the profile of orientational ordering of water dipoles in Stern and diffuse layers, but not qualitative [84]. Note that the multi-layering of water predicted in simulations [84] cannot be predicted within our mean-field approach [44,61] as well also not in the oversimplified phenomenological models [83].

Besides the saturation in polarization/water dipole orientation at high magnitudes of the electric field strength, the important thing to consider in the EDL studies is also the saturation in the counterion concentration near the charged surface due to the finite size of ions. These steric effects were first predicted in the Wicke-Eigen's model (also called the Bikerman's model) and their modifications [3,5,22,25,27,35,85,86]. For finite sized ions, the dielectric permittivity profile in the vicinity of a charged surface is modulated by the depletion of water dipoles at the charged surface due to accumulated counterions [58,82]. In the modified LPB model [7,59], described in the present paper, these steric effects were not taken into account.

The described decrease in the relative permittivity relative to its bulk value in the present paper is the consequence of strong orientational ordering of the water dipoles in the vicinity of the charged surface (Figure 7). Contrary to our results it is claimed in Reference [87] that the relative permittivity is increased in direction to the charged surface. As pointed out in publications of different authors the predicted increase of relative permittivity near the charged surface in Reference [87] is unphysical [6,59] and defies the common wisdom in electrochemistry [56]. In addition, the experiments report just the converse as predicted in Reference [87], that is, the experiments indicated the decrease of relative permittivity near the charged surface [88,89]. The predicted substantial increase of relative permittivity in the inner part of the double layer near the charged surface in Reference [87] is due to arise in

the dipole density near the surface as pointed out in Reference [56]. This unphysical result [6] is the consequence of inconsistency of so-called dipolar PB theory presented in Reference [87] as indicated in Reference [59]. Namely, it was shown [59] that the dipolar PB theory for point-like ions in Reference [87] assumes an orientationally averaged Boltzmann factor for spatial distribution function for water dipoles, which is however not compatible with the assumption of point-like ions. Energy dependent spatial distribution of water dipoles cannot be taken into account simultaneously with the assumption of point-like ions, but only if the finite size of molecules in the electrolyte solution are taken into account [35,61]. This means that the dipolar PB model presented in Reference [87] is not a self-consistent model and consequently predicts unphysical results which are not compatible with experimental results even qualitatively, as noticed in References [6,56,59] and other publications.

The other important difference between our modified LPB model and the theoretical model presented in Reference [87] is that our value for (external) water dipole moment 3.1 D [7,44,51,61] is considerably smaller than the corresponding value 4.86 D used in Reference [87]. The value 3.1 D is closer to the experimental values of the effective dipole moment of water molecules in clusters (2.7 D) and in bulk solution (2.4–2.6 D) (see for example Reference [68]). The value 4.86 D is so large in order to compensate for the cavity field [6,61,74,75,78] that is not taken into account in Reference [87], as noticed also in Reference [6], but is considered in the present modified LPB model. The model value 3.1 D can be additionally decreased by taking into account structural correlations between water dipoles [60,78]. The ion-ion and ion-water correlations were taken into account also in the mean-field models of References [8,65,66].

It has been shown that for finite-sized ions the drop in the number density of water near a charged surface results in an additional decrease of permittivity [7,58]. A further generalization of the modified LPB model with steric effects taken into account within a lattice-statistics model of a modified LPB is found in References [44,51,82]. By taking into account asymmetric finite size of ions the modified LPB equation was generalized to (modified Langevin Eigen-Wicke model) [44,51,82]:

$$\frac{d}{d\mathfrak{x}} \left[ \varepsilon\_0 \varepsilon\_r(\mathbf{x}) \frac{d\phi}{d\mathfrak{x}} \right] = 2\varepsilon\_0 n\_s n\_0 \frac{\sinh\left(\beta e\_0 \phi\right)}{\mathcal{D}\_A(\phi, E)},\tag{60}$$

where *εr*(*x*) is the spatial dependence of relative permittivity:

$$\varepsilon\_{r}(\mathbf{x}) = n^{2} + n\_{0w} n\_{s} \frac{p\_{0}}{\varepsilon\_{0}} \left(\frac{2 + n^{2}}{3}\right) \left(\frac{\mathcal{F}(\gamma p\_{0} E \beta)}{\mathcal{D}\_{A}(\phi, E)E}\right) \tag{61}$$

and

$$\mathcal{D}\_A(\phi) = \mathfrak{a}\_+ n\_0 e^{-\varepsilon\_0 \phi \beta} + \mathfrak{a}\_- n\_0 e^{+\varepsilon\_0 \phi \beta} + \frac{n\_{0w}}{\gamma p\_0 E \beta} \sinh\left(\gamma p\_0 E \beta\right). \tag{62}$$

Here, the parameters *α*<sup>+</sup> and *α*<sup>−</sup> are the number of lattice sites occupied by a single positive and negative hydrated ion, respectively, where a single water molecule is assumed to occupy just one lattice site. The reduced number density of lattice sites *ns*/*N<sup>A</sup>* = 55 mol/L is equal to the concentration of pure water [44,51,82]. The symbol *n*0*<sup>w</sup>* stands for the bulk number density of water molecules. The function F(*u*) is defined as F(*u*) = L(*u*) sinh (*u*)/*u*, where L(*u*) is the Langevin function.

The results of the present paper are important when considering electric fields within artificial as well as biological channels containing an electrolyte. Much attention has recently been given to understanding tunneling nanotubes (TNTs), small tubular structures that drive cell communication and spreading of pathogens [12]. Not yet fully understood, it is thought that these tubular structures initiate from local membrane bending facilitated by laterally distributed proteins or anisotropic membrane nanodomains. Further research is needed to clarify the role of EDL in the inception of these structures, since cytoplasmatic proteins and other elements are electrically charged. When such motor proteins are complemented by protruding cytoskeletal forces provided by the polymerization of f-actin, TNT formation is crucial in determining cell morphology, sometimes even leading to endovesiculation of the red blood cell membrane [90–92]. Recently, within a molecular mean-field approach and taking

into account the asymmetric size of ions, polarization of water, and ion-ion and ion-water correlations, the ionic and water flows through biological ion channels was theoretically considered [65,66].

To conclude, in the present paper we started from a mean-field Helmholtz free energy functional, presented a thorough derivation of the modified LPB equation and model by minimization of the system free energy for the case of planar geometry. A special emphasis was devoted to orientational ordering of water dipoles, taken into account in the expression for the free energy by rotational entropy. Our approach provides a distinct analytical description of the interplay between mean-field electrostatic and entropic effects arising from the mixing entropy of ions and rotational entropy of water dipoles in EDL.

The derived modified LPB equation in planar geometry is then generalized for arbitrary geometry and then used to calculate numerically the average orientation of water dipoles, relative permittivity *εr* , magnitude of electric field strength, electric potential and the macroscopic (net) volume charge density of coions and counterions for a cylindrical geometry (in dependence on radial distance from the center of the tube).

Among other things it is indicated that in the saturation regime close to the charged surface, where the magnitude of electric field is very large (Figure 6), strong orientational water dipole ordering (Figure 7, upper panel) may result in a strong local decrease of permittivity (Figure 7, lower panel). The relative permittivity of the electrolyte solution decreases with increasing magnitude of the electric field strength.

Most interesting, we have shown that in the case of very narrow nanotubes the macroscopic (net) volume charge density of coions and counterions (*ρ*free) at geometrical axis of the nanotube is not zero (Figure 4). In addition, in narrow nanotubes the water dipoles are partially oriented also close to the axis of the nanotube (Figure 7, upper panel), as schematically shown in (Figure 8). The potential importance of this phenomena for the transport through the narrow channels with the charged inner surface, specific only for very narrow nanotubes, should be investigated in the future. The channels in biological membranes can be an interesting example of such systems.

**Figure 8.** A schematic figure of a radial arrangement of water dipoles inside a very narrow cylindrical nanotube. The inner surface of the tube is negatively charged.

#### **6. Materials and Methods**

To solve Equation (52), a partial differential equation, we have used Comsol Multiphysics and its electrostatics stationary solver. The mesh consists of 4946 elements, the boundary condition (Equation (59)) was applied on the 2D cross-section of the nanotube and the geometry was solved for 10293 DoFs. The numerical results were solved using a stationary nonlinear solver (Automatic (Newton)), which implements a damped Newton's approach, with a minimum damping factor of 10−<sup>6</sup> .

**Author Contributions:** M.D.: writing original draft preparation, equation derivation, writing-review and editing, visualization; E.G.: equation derivation, software, methodology, numerical calculations, calculations and preparation of the figures, editing; V.K.-I.: equation derivation, resources, editing, conceptualization, supervision, funding acquisition; A.I.: equation derivation, methodology, resources, editing, conceptualization, supervision, validation, funding acquisition. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the Slovenian Research Agency (ARRS), grant numbers P2-0232, P3-0388, J3-9262 and J1-9162.

**Conflicts of Interest:** The authors declare no conflict of interest.

### **Abbreviations**

The following abbreviations are used in this manuscript:


#### **References**


© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Application of a Statistical and Linear Response Theory to Multi-Ion Na**<sup>+</sup> **Conduction in NaChBac**

**William A. T. Gibby 1,\*, Olena A. Fedorenko 2,3, Carlo Guardiani 1,4, Miraslau L. Barabash <sup>1</sup> , Thomas Mumby <sup>1</sup> , Stephen K. Roberts <sup>3</sup> , Dmitry G. Luchinsky 1,5 and Peter V. E. McClintock 1,\***


**Abstract:** Biological ion channels are fundamental to maintaining life. In this manuscript we apply our recently developed statistical and linear response theory to investigate Na<sup>+</sup> conduction through the prokaryotic Na<sup>+</sup> channel NaChBac. This work is extended theoretically by the derivation of ionic conductivity and current in an electrochemical gradient, thus enabling us to compare to a range of whole-cell data sets performed on this channel. Furthermore, we also compare the magnitudes of the currents and populations at each binding site to previously published single-channel recordings and molecular dynamics simulations respectively. In doing so, we find excellent agreement between theory and data, with predicted energy barriers at each of the four binding sites of ∼4, 2.9, 3.6, and 4*kT*.

**Keywords:** ion channel; statistical theory; linear response; ionic transport; NaChBac

#### **1. Introduction**

Biological channels are natural nanopores that passively transport ions across cellular membranes. These channels are of enormous physiological and pharmacological importance, and so investigation of their transport properties is an area of great interest and research. For example, Na<sup>+</sup> channels play a key role in the generation of the action potential [1–3]. Furthermore, artificial nanopores are primarily designed for their transport functionality which can be informed by our understanding of biological channels.

A primary function of these channels is their ability to discriminate effectively between ions, whilst still conducting them at high rates. An example is NaChBac from *Bacillus halodurans*, which is the first bacterial voltage-gated sodium channel (Nav) to have been characterised, and thus is a prokaryotic prototype for investigating the structure–function relationship of Nav channels [4]. It conducts ions at rates of 10<sup>7</sup> s <sup>−</sup><sup>1</sup> despite having permeability ratios favouring Na<sup>+</sup> over K<sup>+</sup> and over Ca++. Recently we reported these values to be *at least* 10:1 and 5:1 respectively [5]. In fact from the reversal potential the Na+/K<sup>+</sup> permeability ratio is found to be 25:1, which is closer in agreement but still less than [6] who found the ratio to be 170:1. This contrasts with potassium channels such as KcsA where selectivity is reversed, favouring K<sup>+</sup> over Na<sup>+</sup> at 1000:1 [7]. The channel itself is formed from several coupled subsystems, but we focus on the selectivity filter (SF), which is the primary region responsible for selectivity between ions. The SF can readily be mutated to generate a range of conducting (and non-conducting) channel types which exhibit different selectivity and conductivity properties compared to those exhibited by the wild-type (WT) channel (see [5]).

**Citation:** Gibby, W.A.T.; Fedorenko, O.A.; Guardiani, C.; Barabash, M.L.; Mumby, T.; Roberts, S.K.; Luchinsky, D.G.; McClintock, P.V.E. Application of a Statistical and Linear Response Theory to Multi-Ion Na<sup>+</sup> Conduction in NaChBac. *Entropy* **2021**, *23*, 249. https://doi.org/10.3390/e23020249

Academic Editor: Antonio M. Scarfone

Received: 11 December 2020 Accepted: 11 February 2021 Published: 21 February 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

The SF has the amino acid (Here: T = threonine, L = Lecucine, E = glutamte, S = serine, W = tryptophan, A = alanine and x highlights where the sequence is not conserved and can be several possible amino acids.) sequence TLESWAS, and thus shares the TxExW sequence with eukaryotic calcium channels [6]. Unfortunately, a crystal structure of NaChBac is not available. However, Guardiani et al. [8–10] applied homology structural modelling to produce a structure of NaChBac that we will use in this publication. We conduct a variety of different Molecular Dynamics (MD) simulations (see Figure 1) to explore its properties. During simulation the SF was found to have an average radius *R<sup>c</sup>* ∼ 2.8 Å, length *<sup>L</sup><sup>c</sup>* <sup>∼</sup> 12 Å and <sup>4</sup> binding sites for conducting Na<sup>+</sup> ions labelled S1–4 from the intra- to the extra-cellular side respectively. The conduction mechanism was found to involve knock-on between at least two, if not three, ions. Each binding site has a volume, as estimated in Table 1, whose sum gives the total volume of the pore *Vc*. The first two sites are formed at the backbone carbonyls of the threonine and leucine residues respectively. S1 is wider than the average pore radius with diameter 3.06 Å, but S2 has the average pore radius of 2.8 Å. As a result, these two sites accommodate the primary hydration shell with around 5–6 waters per ion, and thus prevent bare ion-protein interaction. S3 is of approximately the same size as S2, but the ion only interacts with four waters because it also interacts directly with the glutamate ring. The fourth site is formed on the extracellular side from the side chain of the serine residues and a sodium ion here has a 40% probability of interacting with one or two serines and a 60% probability of being fully hydrated by water. This is in stark contrast to the narrower potassium channels where K<sup>+</sup> ions are almost fully dehydrated as they permeate the pore. The Na<sup>+</sup> occupancies at each site have been determined by molecular simulation using 0.5 M bulk solutions. Both S1 and S4 have energy minima that are higher in energy than S2,3 and so are less likely to be occupied. In fact the average occupancy of S1,4 is only around half that of the most occupied site S2 (see Figure 7c).

**Figure 1.** Structure of NaChBac [8] visualised using chimera [11]. (**a**) Yellow ribbons denote the protein spanning a lipid membrane (orange strands) between two aqueous ionic solutions. The selectivity filter (SF) is located within the box and highlighted by the red ribbons. The charged glutamates in the SF are highlighted green, and Na<sup>+</sup> (purple), and Cl<sup>−</sup> (blue) ions alongside water molecules are included. (**b**) Structure of the SF for NaChBac with each amino acid highlighted and labelled by colour. The positions of the binding sites are included and labelled S1–S4 from the intrato the extra-cellular side respectively. In (**c**) we show the lattice model used to define the system.

**Table 1.** Table of averaged radii and length of each binding site, obtained through the homology based structural model of NaChBac from [8]. The corresponding surface areas and volumes were estimated by assuming that each site was spheroidal in shape. The binding site is identified from a minima in the potential of mean force (PMF), and its length is estimated from the distance between maxima in the PMF. The radius is estimated from the average calculated radius in this region. These lengths and radii are given in the table.


These results are consistent with the results of MD simulations that have been performed on a variety of similar bacterial NaV channels. Chakrabarti et al. [12] conducted a 21.6 µs-long MD simulation of NavAb, observing a variable number of ions in the pore, mainly two or three (rarely four) and spontaneous and reversible ionic diffusion along the pore axis. Ulmschneider et al. [13] simulated the open state of the pore domain of NavMs with a voltage applied, and calculated the conductance which at ∼33*pS* was in agreement with experimental results.

The SF has a nominal charge of −4*e* arising from the fixed gluatamte ring. However, determining the exact charge contribution from these pores is challenging due to the potential partial charges from remaining uncharged amino acids and the protonation that may occur at physiological pH levels. That latter is suspected to be true in voltage-gated Ca++ channels which share a ring of glutamates [14,15]. As a result, protonation of the glutamate ring in Navs has been studied fairly extensively [5,16–19]. Corry and Thomas [17] investigated the pore when only a single glutamate residue was protonated. The slightly protonated pore showed little difference in the potential of mean force vs. the normal pore. However, the doublyprotonated state showed a larger barrier for permeation to the pore, and reduced affinity for ion binding. Boiteux et al. [18] found a slight difference in the average number of Na<sup>+</sup> ions in the SF at 2.3 and 2.0 in the fully deprotanated and slightly-protonated states, respectively; however, both states were conducting. In simulations with two protonated residues, the authors observed the existence of a non-conducting state forming as a result of stable hydrogen bonds between the glutamates. As the number of protonated residues increased to three and four, Chloride Cl<sup>−</sup> ions started to bind and the pore became non-conductive for Na+. A similar study with shorter biased simulations suggested that protonation of a single Glu residue would diminish the conductance [16]. Meanwhile, a recent [19] study found that, at physiological pH, the pore may exist in the full deprotonation state but that it could also exist in the single or double-protonation states as well. Furthermore, the calculated pKa value decreases with each additional bound ion, implying that the presence of ions inside the pore leads to protonation of the SF. Thus, in [5] we introduced the notion of an effective charge describing the total charge in the pore as felt by the conducting ion, and its values were estimated by fitting Brownian dynamics simulations to experimental data for wild-type (WT) NaChBac and for a large selection of mutants. In our earlier work we studied NaChBac and its mutants theoretically and by Brownian dynamics simulation [5,20].

In earlier publications [5,21], we reported studies of Na<sup>+</sup> and Ca++ permeation in NaChBac, using Brownian dynamics models. The key result of modelling was that ionic conduction is analogous to electron transport in a quantum dot. As a function of the value of fixed charge, we observed a set of resonant conduction peaks separated by regions of blockade where the ions could not enter/leave the pore. This phenomenon is called *ionic Coulomb blockade* (ICB) [22], by analogy with (electronic) Coulomb blockade in quantum dots, for which the physics and the governing equations are essentially the same. Each resonant peak corresponds to an *n* → *n* + 1 barrier-less transition, which is of the knock-on kind

when *n* > 0 [23], and the regions of blockade are when the charge carrier cannot pass. The occurrence of ICB has also been confirmed experimentally in artificial nanopores [24,25]. Although the ICB model explained immediately the role of the fixed charge, and accounted convincingly for the effect of mutations in which the fixed charge is altered, it is only a good approximation when electrostatic forces are dominant, that is, for divalent and trivalent ions. Furthermore, it does not contain affinities in the pore or excess chemical potentials in the bulk and so it cannot describe selectivity between ions of the same charge. It is also not connected to the results of Molecular simulation (MD) or the structure, and it cannot describe the absolute magnitude of the permeating current.

To provide a more accurate description, we needed a more fundamental model. We therefore developed a kinetic model [20], to investigate Na<sup>+</sup> vs. K<sup>+</sup> selectivity. This model was based on a simplified two site model of NaChBac and it was made self-consistent through the form of its transition rates. These were chosen such that the kinetic model and an earlier statistical and linear response theory had the same form of conductivity at low voltages. However, this did not include the complete structure or any comparison to results from MD simulation. It also did not include the binding site conductivities, or account for the correlations between ions at different binding sites. These two properties are expected to be important for fully describing the permeation properties and making quantitative predictions of the function of biological channels because it is known that small mutations in structure can lead to significant changes in function, for example, [5,26,27]. This was shown in [28], where we introduced a statistical and linear response theory fully accounting for structure and the properties of each binding site, and used it to analyse a point mutation in KcsA exploring the reasoning behind its drop in conductivity and occupancy.

In the present paper, we apply this recently developed statistical and linear response theory [28] to NaChBac with a more accurate model based on the structure introduced in [8]. The theory will include all four binding sites and their estimated volumes and surface areas, and the excess chemical potentials at each site. Furthermore, we extend this theory by deriving the conductivity at linear response in the presence of an electrochemical gradient. The theory is successfully compared to experimental single-channel and wholecell recordings (some of which published in [5,20]), and results from MD simulations [8]. Finally, the theory allows us to make quantitative predictions of the current-concentration and current-voltage relations, and the effective open probability of the channel; as a function of the energy profile, experimental bulk concentration and structure of the pore.

In what follows, with SI units *e* is the unit charge, *T* the temperature, *z* the ionic valence, and *k* Boltzmann's constant.

#### **2. Experimental Methods and Data**

To apply the theory to NaChBac, and to compare with experimental recordings and make predictions, we consider two experiments. For further details of the experimental methods, including generation of the mutant channels and their expression, as well as details of the electro-physiological experiments, we refer to [5], and here we only present a concise summary. The first of these data sets is single-channel current-voltage recordings originally published in [20]. In these experiments identical bath and pipette solutions containing (in mM: 137 NaCl, 10 HEPES and 10 glucose, pH 7.4 adjusted with 3.6 mM NaOH) were used. Single-channel recordings are possible because Na<sup>+</sup> is the preferred substrate with sufficiently high conductance to provide a single-channel current amplitude which significantly exceeded noise (i.e., a favorable signal-to-noise ratio). In Figure 2a we plot the current-voltage curve, and in (b,c) we provide a current-time trace made at +100 mV. Trace (c) begins at the end of trace (b). There are at least three active channels passing currents with the magnitudes shown by the dashed lines.

**Figure 2.** (**a**) Single channel currents recorded from NaChBac (originally published in [20]). (**b**,**c**) The original recording made at +100 mV in the 140 mM NaCl solution; the trace contains contributions from at least three active channels; and (**c**) represents a continuation in time of trace (**b**). The dashed lines show the amplitude level per channel, the numbers on the ordinate denoting the number of open channels.

In the second series of experiments, we performed whole-cell current measurements through NaChBac, in different Na+/K<sup>+</sup> concentrations (see Figure 3). The black and purple curves in (a) (and the curve in (c)), that is, with 0M and 0.14M of NaCl solutions in the bath solution respectively (or 0.1M and 0M of KCl), were published in [5]. An identical experiment on a mutant was performed and described in [20]. In each case, the pipette solution contained (in mM) 120 Cs-methanesulfonate, 20 Na-gluconate, 5 CsCl, 10 EGTA, and 20 HEPES, pH 7.4 adjusted with 1.8 CsOH, meanwhile the bath solution contained (in mM); 137 NaCl, 10 HEPES and 10 glucose, pH 7.4 (adjusted with 3.6 mM NaOH). Permeability to K<sup>+</sup> was investigated by incrementally replacing the NaCl bath solution with an equivalent KCl solution such that the total ionic concentration was fixed at 140 mM. Total current across the cell was then normalized and, because one can assume that the total number of channels, their type and their open probability is conserved in each cell for the duration of the recording, it can effectively be modeled as a single channel. This normalization was with respect to the absolute value of peak current and is shown in Figure 3a. In (b) we show the current-concentration behaviour at −10 mV, which corresponds to the peak current. The reversal potential is plotted in (c); in cases where inward current was not detected, estimated values were determined from the voltage at which outward current could be detected. Finally, in (d) and (e) we provide the corresponding current-time traces.

Since NaChBac is highly impermeable to K<sup>+</sup> and Cl<sup>−</sup> we have neglected the presence of these ions in the pore and in our theory we shall simply consider a single ion species, that is, Na<sup>+</sup> inside the pore.

#### *Comparison of NaChBac Structures*

In this subsection we shall compare the structure of NaChBac from the homology model which was used in [8], and the Cryo-EM structures 6vx3.pdb and 6vwx.pdb from [4].

In Figure 4 we provide an overlay of the homology model (yellow ribbons) and the 6vx3.pdb structure (green ribbons), using all of the backbone atoms. (a) provides the overlay of the whole pore and (b) provides a snap-shot of the selectivity filter (SF). From visual inspection there is clearly good agreement between the structures. In the pore the root-mean-square distance between structures (computed using the backbone atoms) is 17.47 Å and 7.14 Å in the SF.

**Figure 3.** (**a**) Mean peak whole cell voltage-current relationships from cells expressing NaChBac channels, obtained in the bath solution with decreasing Na<sup>+</sup> content ranging from 140 mM to 0 mM (with NaCl being replaced with equimolar KCl). The peak currents were determined from time vs. current traces (examples shown in parts (**d**,**e**). Peak currents are normalized to the peak current recorded from the same cell in 140 mM NaCl-containing solution in the absence of K+; error bars represent the standard error of the mean (SEM), determined from at least 4 independent cells. In (**b**) we show mean reversal potentials (±SEM) determined from data plotted in part (**a**). In cases where inward current was not detected, the reversal potential was assumed to be the voltage at which outward current could be detected. In (**c**) we plot the mean (±SEM) peak whole cell current (determined from data plotted in part a) as a function of Na concentration. Parts (**d**,**e**) are examples of time-dependent NaChBac currents recorded in 140 mM NaCl (**d**) and 126 mM NaCl and 14 mM KCl (**e**).

**Figure 4.** Comparison of NaChBac structures from the homology model (yellow) introduced in [8] and the Cryo-Em structure in green (6vx3.pdb) from [4]. (**a**) represents the whole pore and (**b**) is a snapshot of the (half) selectivity filter.

To further explore these structures we considered the pore radius which can be compared using the HOLE program. In Figure 5 we show a comparison between structures. The homology model is more open than the Cryo-EM structures (6vx3 and 6vwx) both at the level of the cytosolic mouth (minimum centered on *z* = −15 Å) and in the region of the SF (around *z* = 0–12 Å). This is confirmed by volume filling representations of the pores which show a bottleneck close to the cytosolic mouth of 6vx3. The SF of 6vwx is narrower because the SF is occupied by two Na<sup>+</sup> ions, and these attract the side chains of the glutamates and the backbone carbonyls of the leucines, moving them towards the centre of the pore. Hence, there are two distinct minima in the pore radius which cannot be spotted in the radius profile of the homology model because this structure was obviously empty. However, the fact that the SF in 6vx3 (whose SF is empty) is also narrower than that of the model suggests that the structural differences might reflect different functional states in the channel cycle. In fact in the paper [4], Gao comments on the narrow radius of the cytosolic mouth, and on the arrangement of the Voltage Sensor Domain, suggesting that these structures might represent an inactivated conformation of the pore. By contrast, our homology model was built using the fully open conformation of NavMs from *Magnetococcus sp.* (PDB ID: 4F4L) as a template. As a result, our homology model probably represents an open conformation of NaChBac. This choice was deliberately taken on the assumption that an open conformation would be more suitable for the computational study of permeation and selectivity. In summary, the good agreement in overlayed structures, along with the choice to use an open conformation of NavMs as a template, makes us confident our model is a reliable system for the study of the selectivity and permeation of NaChBac.

**Figure 5.** Comparison of average pore radius in the homology model structure (red) [8] and Cryo-EM structures 6vx3.pdb (black) and 6vwx.pdb (pink) [4]. The green and blue dashed lines denote the ionic Na<sup>+</sup> and hydrated Na<sup>+</sup> radii, respectively, and the purple dashed lines at *z* = 0, 13 Å highlight the selectivity filter region.

#### **3. Theory**

To model the SF we consider a system comprised of a pore thermally and diffusively coupled at either entrance to bulk reservoirs. This system and the effective grand canonical ensemble was considered and rigorously derived for multi-ion species in [28], and here we only present the necessary details needed to describe a single-species system. This pore is represented as a 1-dimensional lattice with 4 sites that may be occupied by a single ion at

most. These are labelled S1–4 starting from the intracellular side in (c) of Figure 1. This figure also provides in (a) an overview of the system and (b) a snapshot of the SF which is highlighted by the red ribbons in (a). Clearly each configuration of Na<sup>+</sup> ions in the pore represent a distinct state of the system with total state space {*nj*}. In this system ions inside the pore interact electrostatically with each other and charges on the surface of the pore via E. Furthermore, they also interact locally at each binding site, *m*, via short-range contributions *µ*¯ *c <sup>m</sup>* and may experience an applied potential *φ c <sup>m</sup>*. Thus, with only Na<sup>+</sup> in the pore we can write the following distribution function, *P*({*nj*}),

$$P(\{n\_{\vec{j}}\}) = \mathcal{Z}^{-1} \frac{(\mathbf{x}\_{Na}^{b})^{n\_{\text{Na}}}}{n\_{0}! n\_{\text{Na}}!} \exp[- (\mathcal{E}(\{n\_{\vec{j}}\}) - \sum\_{m} n\_{\text{Nam}} (\Delta \vec{\mu}\_{\text{Nam}} + e \varepsilon \Delta \phi\_{m}^{b}))/kT]. \tag{1}$$

We have introduced ∆ to denote the difference between bulk and site *m* in the pore such that ∆*µ*¯ *b <sup>m</sup>* = *µ*¯ *<sup>b</sup>* <sup>−</sup> *<sup>µ</sup>*¯ *c <sup>m</sup>* and ∆*φ b <sup>m</sup>* = *φ <sup>b</sup>* <sup>−</sup> *<sup>φ</sup> c <sup>m</sup>*. In these cases *µ*¯ and *φ* denote the excess chemical potential and applied voltage in the bulk or at site *m* respectively. The prefactor contains factorial terms due to the indistinguishably of ions *nNa* and empty sites *n*<sup>0</sup> in the pore, and *xNa* denotes the mole fraction. For clarity we will drop the Na subscript. The necessary statistical properties such as site or pore occupancy can be derived from the partition function Z or Grand potential Ω = −*kT* log(Z).

In [28] we demonstrated that the response to an applied electric field can be calculated following Kubo and Zwanzig [29–31]. We showed that the susceptibility density at each site can easily be derived and related to the conductivity at each site following the Generalised Einstein relation. The total conductivity through the pore is thus calculated by summing the reciprocals of the site-conductivity, in analogy to resistors in series. As a result all sites must be conducting for the total conductivity to be non-negligible. This effect partly explains the reduced conduction of a KcsA mutant [26], although we have to be mindful that the overall pore charge also decreases, increasing the overall energy barrier for conduction, and contributing to the reduced conductivity. We shall extend this derivation here by considering the response to an electrochemical gradient comprised of an electric potential gradient *δφ* and a concentration gradient *δc*. We shall assume that both bulk reservoirs are perturbed symmetrically so that the left (+) and right (−) electrochemical potentials, *µ b* , can be written,

$$
\mu^b = kT \log(\left(c \pm \delta c / 2\right) / c\_w) + \overline{\mu}^0 + ez\phi^0 \pm ez\delta\phi/2,\tag{2}
$$

where *c<sup>w</sup>* is the concentration of the solvent which is much larger than that of the ions at around ∼55M, and *c* is the concentration of the solute, *µ*¯ 0 is the equilibrium bulk excess potential which we assume to be unperturbed by the electrochemical gradient and *φ* 0 is the equilibrium electrical potential (which we will consider to be 0). In the following derivation we will write *c*/*c<sup>w</sup>* as the mole fraction *x*. Thus following [28] we can write the following free energy, *G*({*nj*}, *δφ*, *δc*), in the presence of this gradient by linearising *µ <sup>b</sup>* about small *δc*,

$$\begin{split} G(\{n\_{\vec{j}}\}, \delta\phi, \delta c) &= \mathcal{E}(\{n\_{\vec{j}}\}) - \sum\_{m=1}^{M} n\_{m} (kT \log(\mathbf{x}) + \Delta\bar{\mu}\_{m}^{0} \pm \frac{kT}{2c} \delta c \pm e \varepsilon \nu\_{m}^{b} \delta \phi) \\ &+ kT \ln(n\_{0})! + kT \ln n!. \end{split} \tag{3}$$

In this expression we have rewritten *δφ<sup>b</sup> <sup>m</sup>* = *ν b <sup>m</sup>δφ* where *ν b <sup>m</sup>* is a function representing the fraction of the voltage drop to move from either the left or right bulk to site *m* in the pore (see [28] for details). In a symmetrically distributed pore (which we assume), the average of *ν b <sup>m</sup>* is equal to 1/2. In this regime the probability distribution function can be written as

$$P(\{n\_{\bar{j}}\}, \delta\phi, \delta\mathfrak{c}) = Z^{-1} \frac{\mathfrak{x}^n}{n\_0! n!} \exp[- (\mathcal{E} - \sum\_m n\_m (\Delta\bar{\mu}\_m^0 \pm e\varepsilon\nu\_m^b \delta\phi \pm \frac{kT}{2c} \delta\mathfrak{c})/kT]. \tag{4}$$

Here the partition function *Z* is defined in the standard manner from the conservation of probability and distinguished from the equilibrium partition function Z. Both the free energy and distribution function can also be expressed in terms of the chemical gradient *η <sup>L</sup>* <sup>−</sup> *<sup>η</sup> <sup>R</sup>* because

$$kT\log(\mathbf{x}^L/\mathbf{x}^R) = \delta\eta = \frac{kT}{c}\delta c.\tag{5}$$

The distribution (4) can be linearised about both small *δφ* and *δc*. When calculating the average particle density at each site h*nm*i*δc*,*δφ*/*Vm*, where *V<sup>m</sup>* is the site volume, one can obtain relations for the susceptibilities due to the electrical gradient *χ δφ <sup>m</sup>* and the chemical or concentration gradient *χ δη <sup>m</sup>* . The former is defined in [28], since we assume a symmetrical pore the latter is defined as,

$$\chi\_m^{\delta\eta} = \frac{1}{2kT} \left( \left\langle n\_m \left( \sum\_m n\_m \right) \right\rangle - \left\langle \left( \sum\_m n\_m \right) \right\rangle \left\langle n\_m \right\rangle \right) \frac{1}{V\_m}.\tag{6}$$

It is worth noting that this expression is similar to *χ δφ <sup>m</sup>* and is proportional to the variance of particle number at site *m* plus the covariance between sites *m* and the remaining sites in the pore. These susceptibilities are also proportional to the electrical conductivity, *σm*, at each binding site, which can be defined from the Einstein relation as: *σ<sup>m</sup>* = *ze*2*Dmχ<sup>m</sup>* where *D<sup>m</sup>* and *χ<sup>m</sup>* correspond to the diffusivity and susceptibility at each site respectively. As a result, the total current across the pore can be calculated as [28]

$$I = \left(\sum\_{m} \frac{1}{\frac{A\_m}{L\_m} \mathcal{O}\_m} \right)^{-1} (\delta\phi + \delta\eta/e),\tag{7}$$

where we recall that *δφ* is the voltage gradient in *V*, *δη* is the chemical gradient in *kT*, and *A<sup>m</sup>* and *L<sup>m</sup>* are the surface area and length of site *m* respectively. Finally, the conductivity at each site is calculated from

$$
\sigma\_m = ze^2 D\_m \left( \chi\_m^{\delta\phi} + \chi\_m^{\delta x} \right),
\tag{8}
$$

which is a function of the equilibrium bulk chemical potential.

#### **4. Application to NaChBac**

In Figure 6a, we consider the free energy spectra for selected (most favoured) pore configurations of NaChBac calculated from Equation (3) (when *δφ* = 0 and *δc* = 0). We consider 0.14M NaCl solutions, and 0–3 ions inside the pore. In Equation (3) the total electrostatic energy, E, is calculated by approximating the pore as a capacitor of total charge *n<sup>f</sup>* and capacitance *C* taking the form E = *Uc*(*n<sup>f</sup>* + *n*) <sup>2</sup> where *U<sup>c</sup>* = *<sup>e</sup>* 2 2*C* [21,22]. Since the permitivitiy of water inside the pore is not known (though it must be less than the bulk value of 80) we consider *U<sup>c</sup>* = 10*kT*. This approximation is discussed in detail in [28]. The energy spectra are parabolic vs. *n<sup>f</sup>* , and each *n*-ion state has multiple configurations (15 in total) and we only highlight the most favoured. These states are determined by the values of ∆*µ*¯ *Nam*, and their exact values are determined from fitting to experimental data (see Section 4.1). Differences in this term lead to energy splitting between possible configurations because the site occupied, in addition to the total number of ions inside the pore, determines the energy, *conducting* states correspond to the degeneracies where the lowest energy levels intersect, cf. [23], and this was shown to be the case in KcsA [28]. In NaChBac, the circle highlighting the 2–3 resonant transition occurs at around *n<sup>f</sup>* ∼ −2.7. Importantly, this differs from *n<sup>f</sup>* = −2.5, suggesting that the the 3rd-ion faces an energy barrier to enter each site. If the concentration of the solutions was increased the energy barrier would decrease and the location of the resonant conduction would shift along the abscissa towards *n<sup>f</sup>* − 2.5. It is worth reiterating that *n<sup>f</sup>* here represents the total pore charge, and so differences from the fixed glutamate ring charge of −4*e* can be explained

from the additional contribution of all other charges and possible protonation inside the pore. Extended discussions of this point are provided in [5,16–19].

In Figure 6b we plot the energy spectra of the favoured 2 and 3-ion states, vs. *n<sup>f</sup>* but also vs. bulk concentration. From the explanation above it is clear that the latter affects the value of *n<sup>f</sup>* at which the two energy levels intersect. At low concentrations the energy barrier to add an ion to the pore is large. Thus, strong negative pore charge is required to reduce the barrier to attract the ion. Conversely at large concentrations the barrier is small and so less negative charge is needed. Thus one would expect the experimental current to be larger for measurements at higher concentrations, if these could be made.

**Figure 6.** Free energy of the favoured states, plotted with <sup>∆</sup>*µ*¯ *Na*,1−<sup>4</sup> ∼ 2.3, 3.4, 2.8, 2.4*kT*. In (**a**) it is plotted vs. *n<sup>f</sup>* with 0.14M NaCl bulk solutions and in (**b**) vs. both *n<sup>f</sup>* and bulk concentration. In (**a**) the blue curves correspond to the occupied *n* > 0 states of the pore, and black denotes the empty state. The purple circle highlights the location at which the two most favoured 2 and 3 ion states coincide, and we see that at *n<sup>f</sup>* = −2.5 there is a small energy barrier. As bulk concentration increases this energy barrier reduces and the purple circle would shift towards *n<sup>f</sup>* = −2.5. This is further clarified by (**b**) which shows only the 2 and 3 ions states.

To obtain the values of ∆*µ*¯ *Na*,1−<sup>4</sup> we performed fitting to two data sets, and this will be explained in the following subsection.

#### *4.1. Comparison to Single Channel Data and MD*

The values of ∆*µ*¯ *Na*,*<sup>m</sup>* used in Figure 6 are obtained by fitting, performed using the LSQCURVEFIT function in Matlab. We fit theory to the equilibrium site occupancies h*nNa*,*m*i calculated from simulation data [8] (see Figure 2c), and the current at 35 mV. Current is needed here so that we can ensure it is of the correct order of magnitude. We also note that the difference in bulk NaCl concentration between the current and occupancy data is taken into account during fitting. To minimise the number of free parameters we also assumed that the diffusivity in the pore was constant, and equal to a tenth of the bulk value at ∼1.33 × 10−<sup>10</sup> m<sup>2</sup> s −1 , and calculated ∆*µ*¯ *Na*,*m*, relative to *n<sup>f</sup>* = −2.5. The diffusivity is expected to be smaller within a confined pore due to the nature of the binding sites [32,33] and, although this value may appear small, it produces a barrier-less conduction rate through the pore of <sup>∼</sup>0.9 <sup>×</sup> <sup>10</sup><sup>8</sup> ions per second which is of the order of tens of pA. We choose *n<sup>f</sup>* = −2.5 because the electrostatic contribution to add a third ion is zero, that is, E(3) − E(2) = 0.

Both data sets are in excellent agreement with the theory, with currents only starting to differ at relatively large voltages when the experimental data deviate from Ohmic behaviour. Clearly beyond this regime, the system is far from equilibrium and our theory will need to be extended accordingly. After fitting we obtain ∆*µ*¯ *Na*,1−<sup>4</sup> ∼ 2.3, 3.4, 2.8, 2.4*kT* when *n<sup>f</sup>* = −2.5, with the sum of squared residuals being small at 10−<sup>4</sup> . When the concentration is 0.14M the ions face the following barriers to enter each site: ∼ 4.0, 2.9, 3.6, 4.0*kT*. These barriers are fairly similar to each other, although it is clear that S2 is the more favoured site and this is shown by its occupancy. As already discussed and observed in Figure 6, the energy barrier at each site reduces when the bulk concentration increases from 0.14M, resulting in a

larger ionic current. This is confirmed by predicted current-voltage dependencies for 0.25 and 0.5M solutions respectively as showing increases in current; and the current-concentration behaviour in Figure 7b. In this latter case the bulk solutions are assumed to be symmetrical, with the driving force originating from a 50 mV voltage drop. This curve clearly demonstrates increasing conduction with concentration and we note that the current is relatively small <10 pA and is continuing to increase even at 2M because the overall energy barrier to enter the pore is large. We expect that these predictions can be further refined if more experimental measurements can be made.

**Figure 7.** (**a**) Comparison of theoretical current *vs*. experimental data (squares) taken from [20] with symmetrical 0.14M NaCl solutions. (**b**) Predicted current-concentration curve at 50 mV across the pore. (**c**) Comparison of equilibrium occupancy at each site vs. simulation data with 0.5M NaCl solutions [8]. In doing this fitting we find that <sup>∆</sup>*µ*¯ *Na*,1−<sup>4</sup> ∼ 2.3, 3.4, 2.8, 2.4*kT*, corresponding to energy barriers of <sup>∼</sup> 4, 2.9, 3.6, 4*kT* at 0.14M and we find the pore diffusivity to be <sup>∼</sup>1.33 <sup>×</sup> <sup>10</sup>−10m<sup>2</sup> s −1 .

#### *4.2. Comparison to Whole Cell Data*

The theory can now be compared to the experimental whole-cell current-voltage recordings outlined earlier. In this experiment the data are normalised against the maximal current which is calculated when −10 mV is applied across the pore, and the bath solution contains 0.14M of Na<sup>+</sup> ions. We note that in Figure 8a this normalisation is with respect to the absolute value of this maximal value.

Under experimental conditions only the bath solution was varied. As a result, the theoretical equilibrium concentration and (chemical potential) used to calculate the conductivity *σ* and hence current varies slightly at each experimental point. This is because they are defined from the average concentration (or chemical potential) from both bulk solutions. Since the chemical gradient is calculated from the difference in bulk concentrations, we consider the lower limit of bulk concentration to be 0.1 mM rather than 0, to avoid the gradient diverging at low concentrations. Even at with the lowest concentration being 0.1 mM, the gradient is ∼ 5*kT* and so at the edge of applicability of our theory.

In Figure 8a, we plot the normalised current-voltage curves for the range of bath solutions. Overall we see good agreement between theory and data, but with two exceptions. NaChBac is a voltage-gated channel so that, at negative voltages, the number of open channels is reduced because the open probability decreases resulting in a smaller overall current [5,34]. Thus, at voltages below −10 mV our current diverges from the experimental data, and hence serves as a prediction of the normalised current in a single open channel. This prediction is given by the dashed lines, which we note increase in magnitude as voltage becomes more negative because the gradient increases. Furthermore, when the bath solution contains no Na<sup>+</sup> (black dashed curve) we observe poor agreement between theory and experiment and so highlight the curve with a dashed line. Finally, the inset curve shows the current closest to equilibrium.

The system is in equilibrium when the net current is zero, and this occurs when the applied voltage is equal to the reversal potential *φ Re*. This was measured experimentally and is compared to the theoretical current in (b). In the theory the reversal potential is calculated from,

$$e\phi^{\text{Re}} = kT\log(\mathbf{x}^L/\mathbf{x}^R),\tag{9}$$

where *L*, *R* again refer to the left and right pipette/bath solutions respectively. We see good agreement except when the bath solution contains no Na+. Even, our reduced concentration of 0.1 mM yields a reversal potential smaller than −35 mV. This is echoed by the current at this concentration which is not in good agreement with the experiment (see the black dashed curve in Figure 8a). A possible explanation for these disagreements is that, in the absence of Na<sup>+</sup> in the bath solution, K<sup>+</sup> ions enter the pore but do not conduct, consequently blocking the pore. Furthermore, at this concentration we are at the limits of applicability because the chemical gradient is still relatively large ∼5*kT*. We plan to discuss this in a future manuscript after further investigations.

In Figure 9a we estimate the effective open probability *P*eff. This is defined relative to the open probability at peak current *P*max, from the ratio of theoretical and experimental current for each of the given concentrations. We neglect the estimate in the absence of Na<sup>+</sup> because the theoretical current did not agree with experimental data. We observe that *P*eff takes values between 0 and 1.5 except for three concentrations all at +50 mV of applied voltage. At 0.126M, 0.1386M and 0.14M bath concentration the theoretical current was below the experimental values and in the latter two concentrations of different sign. This produced estimated effective open probabilities, *P*eff, taking the values of 2.5, −15 and −0.5 for the three concentrations respectively (only *P*eff ∼ −0.5 is shown). Apart from these points however we observe it to be broadly sigmoidal and being 0 at negative voltages as anticipated. We expect, that the actual open probability, *P*Open, can be calculated through the following definition,

$$P\_{\text{Open}}(V) = P\_{\text{eff}} \times P\_{\text{max}\prime} \tag{10}$$

if the open probability of the maximal current is known.

In Figure 9b we highlight the current-concentration (*I* − *C*) behaviour by plotting the *I* − *C* curve at the peak voltage (−10 mV). Note that, unlike Figure 8a, the current is normalised to the maximum current at 0.14M (and not to the absolute value). As expected the theoretical current agrees fairly well with the experimental one except at low concentrations (.5 mM). The curve takes a quasi-linear shape because the current comprises two terms: (1) the conductivity prefactor and (2) the electrochemical gradient. The second term is of the standard form, but our conductivity is a function of the equilibrium bulk chemical potential, which through our derivation must take the averaged concentration between the two bulks and thus slightly varies with bath concentration as well.

**Figure 8.** (**a**) Comparison of theoretical (solid line) to experimental (squares) data of normalised (to absolute value) whole-cell current in the presence of an electrochemical gradient, for a range of extra-cellular bulk solutions. The peak occurs at −10 mV, and below this voltage the current reduces due to the reduction in the open probability. Dashed lines predict the normalised currents if the open probability remained unchanged from the value at the peak current. (**b**) Theoretical (solid) and experimental (squares) of the reversal potential (*φ Re*) for a range of concentrations. Theory only differs when the right bulk is absent of Na+.

**Figure 9.** (**a**) Estimated open probability from the ratio of experimental to theoretical current. Below −40 mV the open probability is close to zero indicating that the channels are closed. (**b**) Comparison of normalised theoretical current (solid line) and experimental (squares) data vs. bulk concentration, at −10 mV of applied voltage.

#### **5. Conclusions and Summary**

In summary, we have taken the statistical and linear response theory, originally derived in [28] and applied to KcsA and a mutant, and applied it to investigate Na<sup>+</sup> conduction in NaChBac. Importantly, in order to compare with experimental and simulation data see Figures 2 and 3), we needed to extend the theory to take account of a chemical gradient. In doing so, we derived the conductivity at each site and the total through the pore in the presence of an electrochemical gradient. The main result of the paper is the quantitative predictions of pore function that we make as a function of the energy profile, experimental bulk conditions, and the pore structure.

In Figure 7 we compared the theoretical current-voltage and equilibrium site occupancies to experimental and simulation data. This comparison allowed us to extract the

following values of ∆*µ*¯ *Na*,1−<sup>4</sup> ∼ 2.3, 3.4, 2.8, 2.4*kT*. At the experimental concentration 0.14M, the 3rd ion faces an energy barrier to enter each site within the pore of ∼4, 2.9, 3.6, 4*kT*. Although these values are not barrier-less as observed in KcsA [28], they are not expected to be because the experimental current is smaller in NaChBac. Furthermore, these parameters lead to barrier heights consistent with [8,20]. Using these parameters we have predicted the current for higher concentrations, including the current-concentration behaviour with 50 mV of applied voltage and current-voltage dependencies for 0.25 and 0.5M solutions. As expected both show an increase of current as the bulk solution increases. We expect that with more experimental data, we could refined these parameters.

In Figures 8 and 9 we compared the theory to normalised whole-cell data, under the assumption that the normalisation effectively renders it a single-channel for the point of comparison. The theory was found to be in good agreement with experiment except for when the bath solution was devoid of Na+. A possible explanation is that in the absence of Na+, K<sup>+</sup> ions enter the pore but do not conduct, subsequently blocking the pore. Furthermore, at this concentration we are at the limits of applicability because the chemical gradient is still relatively large ∼5*kT*. We plan to investigate this in a future manuscript by introducing a far-from equilibrium kinetic model that accounts for both Na<sup>+</sup> and K<sup>+</sup> ions. Such a model was briefly introduced in [20]. However, it failed to account properly for the correlations between ions at different sites, and only considered a 2 site pore; and so further development is needed.

Finally, we expect our theory to be applicable to the study of mixed-valence, that is, Na+/Ca++ selectivity in NaChBac and related voltage gated Ca++ channels, alongside artificial nano-pores.

**Author Contributions:** Data curation, O.A.F.; Formal analysis, W.A.T.G.; Project administration, D.G.L. and P.V.E.M.; Supervision, D.G.L., P.V.E.M. and S.K.R.; Writing—original draft, W.A.T.G.; Writing—review & editing, W.A.T.G., O.A.F., C.G., M.L.B., T.M., S.K.R., D.G.L. and P.V.E.M. All authors have read and agreed to the published version of the manuscript.

**Funding:** The work was funded in part by a PhD Scholarship from the Faculty of Science and Technology of Lancaster University, the Engineering and Physical Sciences Research Council (grants EP/M016889/1 and EP/M015831/1), and by a Leverhulme Trust Research Project Grant RPG-2017- 134. C.G. is currently supported by a project that has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 803213).

**Data Availability Statement:** Data related to this research are openly available from the University of Lancaster Research Directory at (https://doi.org/10.17635/lancaster/researchdata/421 (accessed on 17 February 2021)). Gibby, W.A.T.; Fedorenko, O.A.; Guardiani, C.; Barabash, M.L.; Mumby, T.; Roberts, S.K.; Luchinsky, D.G.; McClintock, P.V.E. (2020) Data for Application of a Statistical and Linear Response Theory to Multi-Ion Na<sup>+</sup> Conduction in NaChBac [Dataset].

**Acknowledgments:** We acknowledge valuable discussions with Bob Eisenberg, Igor Khovanov and Aneta Stefanovska.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


## *Review* **Dynamics of Ion Channels via Non-Hermitian Quantum Mechanics**

**Tobias Gulden <sup>1</sup> and Alex Kamenev 2,3\***


**Abstract:** We study dynamics and thermodynamics of ion transport in narrow, water-filled channels, considered as effective 1D Coulomb systems. The long range nature of the inter-ion interactions comes about due to the dielectric constants mismatch between the water and the surrounding medium, confining the electric filed to stay mostly within the water-filled channel. Statistical mechanics of such Coulomb systems is dominated by entropic effects which may be accurately accounted for by mapping onto an effective quantum mechanics. In presence of multivalent ions the corresponding quantum mechanics appears to be non-Hermitian. In this review we discuss a framework for semiclassical calculations for the effective non-Hermitian Hamiltonians. Non-Hermiticity elevates WKB action integrals from the real line to closed cycles on a complex Riemann surfaces where direct calculations are not attainable. We circumvent this issue by applying tools from algebraic topology, such as the Picard-Fuchs equation. We discuss how its solutions relate to the thermodynamics and correlation functions of multivalent solutions within narrow, water-filled channels.

**Keywords:** non-Hermitian Hamiltonians; algebraic topology; semiclassical methods; nanopores; ion transport; statistical mechanics

**Citation:** Gulden, T.; Kamenev, A. Dynamics of Ion Channels via Non-Hermitian Quantum Mechanics. *Entropy* **2021**, *23*, 125. https:// doi.org/10.3390/e23010125

Received: 2 December 2020 Accepted: 15 January 2021 Published: 19 January 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

#### **1. Introduction**

Transport of ions through narrow channels plays a big role in many biological and technological systems. Many pathogens attack cells by forming nanopores in the cell membrane by using pore-forming toxins (PFTs) [1,2]. This punches holes in the cell membrane through which ions diffuse to the outside, effectively killing the cell. Physically this is similar to artificial nanopores in, e.g., silicon [3,4]. These are heavily used in genetic sequencing techniques where high-throughput of selective transport is the most important factor [5]. Other similar examples include free-standing silicon nanowires [6,7] and waterfilled nanotubes [8,9]. These systems play various different roles in biology and technology. However they all follow the same underlying physics of a quasi-1D statistical system formed by ions confined to move in a narrow water-filled tube inside a lipid membrane or solid medium [10–16]. What makes this system special is the large ratio between the dielectric constants of water, *κ*<sup>1</sup> ' 80, and the surrounding media (e.g., for lipids or silicon oxide *κ*<sup>2</sup> ' 2 − 4). Because of this, the electric field created by an ion within is confined to stay mostly inside the water-filled channel and does not leak into the surrounding medium. As several numerical simulations in three dimensions point out the flow field also follows almost entirely the channel direction [17–19]. This simple observation has profound consequences.

First, as was noticed by Parsegian [20], there is a potential barrier for an ion to enter the channel. This barrier is equal to the energy difference between an ion being inside and outside the channel. For a channel of radius *a* the electric field created by an ion of charge *e* in the middle of the channel is *E*<sup>0</sup> = 2*e*/(*κ*1*a* 2 ). The corresponding field energy integrated over the channel volume is *U*<sup>0</sup> = *κ*1 8*π E* 2 <sup>0</sup>*πa* <sup>2</sup>*L* = *kBT*(*λBL*)/(2*a* 2 ), where *L* is the length of

the channel and *λ<sup>B</sup>* = *e* <sup>2</sup>/(*κ*1*kBT*) <sup>≈</sup> 7Å is the Bjerrum length at ambient temperature [11]. For a typical channel with *L* ≈ 40Å and *a* ≈ 5Å the corresponding (self-)energy barrier exceeds ambient temperature *kBT* by a factor of 5 or 6. This means that such a channel would block the transport of ions. However, there are at least two mechanisms which can be employed to overcome this issue. One is placing charged radicals along the channel path. The other is entropic screening of the barrier by a collective effect of multiple cations and anions inside the channel. In this review we focus on this latter phenomena, while the former is addressed in References [13,14,21,22].

The second consequence of the mismatch of dielectric constants is that the mutual interactions between the ions within the channel acquire the form of the 1D Coulomb potential

$$\Phi(\mathbf{x}\_i - \mathbf{x}\_j) = eE\_0|\mathbf{x}\_i - \mathbf{x}\_j|\_{\prime} \tag{1}$$

where *x<sup>i</sup>* are 1D coordinates of the ions along the channel axis. As illustrated in Figure 1, the electric field lines emanating from a charge are bent to run along the channel. Only after a characteristic length *ξ* given by the implicit relation *ξ* <sup>2</sup> = *a* 2 *κ*1/(2*κ*2)ln(2*ξ*/*a*) the field lines start penetrating the lipid membrane and escaping the channel [11]. For a water-filled channel in a lipid membrane this gives *ξ* ≈ 7*a*. Hence, for a sufficiently short channel with *L* < *ξ* or (as considered in Section 3) a large concentration of salt ions where the characteristic distance between two ions is smaller than *ξ*, the interactions effectively follow the 1D Coulomb potential. The linear nature of the potential (1) leads to the curious observation that the energy barrier of transporting a charge through the channel can't be less than *U*0, irrespective of how many other ions are present in the channel [20]. Indeed, for the most favorable arrangement of alternating positive and negative ions, the electric field along the channel alternates between ±*E*0. This leads back to the value of *U*<sup>0</sup> for the electrostatic energy of adding a single ion to the channel in the presence of the other ions. This may seem as a predicament that collective screening can't lower the transport barrier. Such conclusion is premature, however. The resolution of this apparent paradox is that in a system of multiple particles at a finite temperature it is the *free energy* (rather than the *energy*) which determines the transport barrier. The difference between the two is given by the entropy, i.e., it is the entropy of the ion gas within the channel which provides the screening mechanism. The nature of entropic suppression of the transport barrier can be traced to the aforementioned independence of the energy *U*<sup>0</sup> of the positions of individual ions. This observation implies that there is a large number of microscopic configurations which are close in energy. This is the hallmark of a state with large entropy and thus lower free energy.

**Figure 1.** This is an illustration of the electric field lines emanating from an ion inside a water-filled channel of radius *a* which is surrounded by a medium with lower dielectric constant. Due to the mismatch in dielectric constants the field lines run mostly along the channel which means that another charge would feel an effective 1D Coulomb potential. The ratio is finite however, i.e., a distance *ξ* away from the ion the field lines start permeating the outside medium. If the channel is shorter than this critical length scale, *L* < *ξ*, or the typical spacing between charges is smaller than *ξ*, then all interactions are well-described by the 1D Coulomb potential.

Formalizing these observations is not entirely straightforward. As was first realized by Edwards and Lenard (EL) in 1962 [10] it requires mapping of the 1D statistical system onto an effective quantum mechanics with cosine potential. In fact, this is a particular case of the generic correspondence between D-dimensional statistical mechanics of the

Coulomb gas and (D-1)-dimensional sine-Gordon field theory [23]. The D=2 version of this mapping is well-known in the physics of the Berezinskii-Kosterlitz-Thouless transition. The less appreciated fact is that the Hermitian potential of the form 2 cos *θ* = *e <sup>i</sup><sup>θ</sup>* + *e* −*iθ* is a consequence of having a neutral plasma of monovalent ions with charge ±*e*. In the EL mapping the *e* <sup>±</sup>*i<sup>θ</sup>* operators shift the value of the electric field in the channel (the variable canonically conjugated to *θ*) by a quanta ±2*E*0, which corresponds to the electric field generated by a unit charge ±*e*.

What happens in the presence of a multivalent dissociated salt, such as, e.g., *CaCl*<sup>2</sup> which produces a plasma with positive charges +2*e* and twice as many negative charges −*e*? It is not difficult to see that the EL mapping leads to an effective Hamiltonian with the potential <sup>1</sup> 2 *e* <sup>2</sup>*i<sup>θ</sup>* + *e* −*iθ* . Such a Hamiltonian is non-Hermitian and thus admits a complexvalued spectrum. This may present a problem for the interpretation of the original statistical mechanics of the Coulomb plasma. For example, the free energy density (a manifestly real quantity) is given by the logarithm of the partition function which therefore needs to be real and positive. Fortunately the effective non-Hermitian quantum operator obeys the so-called PT -symmetry [24], which ensures that all eigenvalues are real or appear as complex-conjugate pairs. When calculating the partition function, which includes summing over all eigenvalues, the imaginary parts cancel and we obtain a real, physical result [25]. However, in general there exist complex eigenvalues (spontaneously broken PT -symmetry). This translates to an oscillatory character of certain correlation functions, reflecting short-range charge density wave correlations within the channel.

To model the transport of ions through the channel in this framework we use the concept of boundary charges which was developed in Reference [11]. From now on we assume that the channel is sufficiently short so that all field lines stay inside the channel. If there are no ions inside the channel (or the sum of all charges is zero), then there is no electric field emanating from the channel. If a single ion is added in the center of the channel, then half of its electric field lines are exiting the channel on the left and the other half on the right, cf. Figure 1. This is akin to having two image boundary charges *q*, *q* <sup>0</sup> = <sup>1</sup> 2 at the two ends of the channel (charges are measures in units of *e*). These charges are provided by polarization effects in the well-conducting reservoirs. There are only integer charges inside the channel. Hence, if the boundary charge at one end is *q* (the ion emits a fraction *q* of its field lines at one end), then the other boundary charge is *q* <sup>0</sup> = 1 − *q*. Reference [11] shows that moving a unit probe charge through the channel (while allowing the other ions to equilibrate) creates boundary charges which change from zero to one. Once the boundary charges reach an integer value they may either be released from the end points and join the bulk, or enter into the channel. This makes thermodynamic properties periodic functions of *q* with unit period. In Section 2 we show that the boundary charge *q* takes the role of the quasi-momentum in the effective quantum mechanics. Hence, the bandwidth of the lowest quantum-mechanical band translates directly to the transport barrier.

This review is devoted to the mathematical apparatus needed to treat the non-Hermitian operators appearing in the physics of multivalent 1D plasmas. However, we want to stress that these methods can be applied more broadly to a wide range of non-Hermitian systems. In particular we focus on semiclassical methods applicable for relatively large concentrations of the dissociated salts. Our central observation is that the corresponding (complex) semiclassical trajectories may be viewed as closed cycles on Riemann surfaces of non-zero genus. The action integrals along such cycles are given by solutions of the Picard-Fuchs differential equation, allowing for their analytic evaluation. As a result one obtains asymptotically exact thermodynamic and correlation functions of the 1D multivalent Coulomb plasmas. Of particular interest is the transport barrier, given by the width of the lowest Bloch band (i.e., energy difference between anti-periodic and periodic ground-states of the Schrödinger equation). We obtain analytic results for the transport barriers for various combinations of ion valencies as functions of salt concentration and temperature.

The structure of this paper is as follows: in Section 2 we discuss the EL mapping of statistical mechanics to an effective quantum mechanics with a cosine potential and its generalizations to the non-Hermitian cases. Section 3 is devoted to the semiclassical treatment of the corresponding non-Hermitian operators using the Picard-Fuchs equation. At the end of that section we go beyond the usual semiclassical formulas and describe how to obtain second- and higher-order corrections with little computational effort. We provide a brief summary and discussions in Section 4.

#### **2. Thermodynamic Description and Equivalent Quantum Mechanics**

In this section we discuss the relationship between statistical mechanics of the ion channel and (non-Hermitian) quantum mechanics. We start with a thermodynamic description of the ion channel in terms of the grand-canonical partition function. Then we review how to map the partition function onto a Feynman propagator and derive a Hamilton operator from there. This mapping was pioneered by Edwards and Lenard [10] and subsequently used in several works as starting point [11,25–27]. If the system consists of cations and anions with the same valency and concentration, then the resulting Hamilton operator is Hermitian. However, if the positive and negative charges have different valency, for example solutions of the divalent salts *MgCl*<sup>2</sup> or *CaCl*2, non-Hermitian terms appear. Hence, the spectrum of the resulting operator also contains complex eigenvalues. We discuss how reality and positivity of the partition function is ensured. In the end we comment on the case if charge neutrality is violated.

#### *2.1. Derivation of the Hamilton Operator*

As discussed in Section 1 charged ions inside the channel interact with the effective 1-dimensional Coulomb potential Φ(*x*) = −*eE*0|*x*|, where *E*<sup>0</sup> = 2*e*/*κ*1*a* 2 is the electric field strength generated by a single ion with charge *e* inside a channel of radius *a* and dielectric constant *κ*<sup>1</sup> [11]. The total interaction energy of all ions in the channel is given by

$$
\mathcal{U} = \frac{1}{2} \iint\_0^L d\mathbf{x} d\mathbf{x}' \rho(\mathbf{x}) \Phi(\mathbf{x} - \mathbf{x}') \rho(\mathbf{x}'). \tag{2}
$$

Here we write the charge density for point charges in terms of *δ*-functions,

$$\rho(\mathbf{x}) = \sum\_{j=1}^{N\_1 + N\_2} \sigma\_j \delta(\mathbf{x} - \mathbf{x}\_j) + q(\delta(\mathbf{x}) - \delta(\mathbf{x} - L)),\tag{3}$$

where *σ<sup>j</sup>* = *n*<sup>1</sup> for 1 ≤ *j* ≤ *N*<sup>1</sup> and *σ<sup>j</sup>* = −*n*<sup>2</sup> for *N*<sup>1</sup> + 1 ≤ *j* ≤ *N*<sup>1</sup> + *N*2. This charge density represents *N*<sup>1</sup> cations with valency *n*<sup>1</sup> and *N*<sup>2</sup> anions with valency −*n*2, and the two fractional boundary charges ±*q* at *x* = 0, *L*. The channel is open and can exchange particles with two 3D bulk reservoirs at the ends. Therefore the thermodynamic properties are given by the grandcanonical partition function,

$$\mathcal{Z} = \sum\_{N\_1, N\_2=0}^{\infty} \frac{f\_1^{N\_1} f\_2^{N\_2}}{N\_1! N\_2!} \prod\_{j=1}^{N\_1+N\_2} \int\_0^L d\mathbf{x}\_j e^{-\mathcal{U}/k\_B T},\tag{4}$$

where *f*1,2 are the fugacities of the two charge species. As shown in References [10,11] and in Appendix A, the partition function can be converted into a functional integral by introducing an auxiliary field *θ*(*x*) as conjugate to the charge density *ρ*(*x*). Through this process all integrals over the variables *x<sup>j</sup>* decouple, bringing them to the form ∑*N*[ *f* R *dx eiσθ*(*x*) ] *<sup>N</sup>*/*N*! = exp{ *f* R *dx eiσθ*(*x*)}. The interaction potential (2), being inverse of the 1D Laplace operator, leads to an additional term exp{(*kBT*/*eE*0) R *dx θ∂*<sup>2</sup> *x θ*}. As a result the partition function (4) is identically written in terms of the Feynman path integral with an "imaginary time" *x*, describing quantum mechanics with the Hamiltonian

$$\hat{H} = (i\partial\_{\theta} - q)^{2} - \left(\mathfrak{a}\_{1}e^{i\mathfrak{a}\_{1}\theta} + \mathfrak{a}\_{2}e^{-i\mathfrak{a}\_{2}\theta}\right),\tag{5}$$

where *α*1,2 = *f*1,2*kBT*/*eE*<sup>0</sup> are dimensionless ion concentrations. The Feynman integral is the expectation value of the evolution operator over the imaginary "time" *L*,

$$\mathcal{Z}\_L = \left\langle q \middle| \mathcal{X} e^{-\frac{\epsilon \mathcal{E}\_0}{k\_B T} \int\_0^L dx \, \hat{H}} \middle| q \right\rangle = \sum\_m |\langle q | m \rangle|^2 e^{-\frac{\epsilon \mathcal{E}\_0 L}{k\_B T} \varepsilon\_m(q)}\tag{6}$$

where X is the *x*-ordering operator. Here {*εm*(*q*)}*<sup>m</sup>* is the spectrum of the effective Hamiltonian *<sup>H</sup>*<sup>ˆ</sup> , and <sup>|</sup>*m*<sup>i</sup> <sup>=</sup> *<sup>ψ</sup>m*(*θ*) are its eigenvectors in the Hilbert space of periodic functions, *ψm*(*θ*) = *ψm*(*θ* + 2*π*). The matrix elements are h*q*|*m*i = R <sup>2</sup>*<sup>π</sup>* 0 *dθe iqθψm*(*θ*). The boundary charge *q* plays the role of the Bloch quasi-momentum and the spectrum is periodic in *q* with unit period.

Note that for *α*<sup>1</sup> = *α*<sup>2</sup> and *n*<sup>1</sup> = *n*<sup>2</sup> the potential in Equation (5) reduces to the cosine function and the Hamiltonian becomes the well-known Mathieu Hamiltonian [10]. However, if these conditions are violated the potential is non-Hermitian [25]. We discuss implications of this in the following section.

#### *2.2. Physical Observables*

The partition function in Equation (6) gives the thermodynamic properties of the ion gas. However, to be physically meaningful the partition function needs to be real and positive, while the spectrum of the non-Hermitian Hamiltonian (5) may contain non-real eigenvalues. This issue is resolved because the Hamiltonian obeys a symmetry akin to PT -symmetry. The combined action of the "parity operator" P : *θ* → −*θ* and "time reversal" T : *i* → −*i* leaves the Hamiltonian in Equation (5) unchanged. Bender et al. [24] proved that all eigenvalues of a PT -symmetric Hamiltonian are either real or appear in complex conjugated pairs. Hence, summing over all eigenvalues in Equation (6) gives a real result. In [25] is was shown that for positive values of concentrations *α*1,2 > 0 the lowest energy band *ε*0(*q*) is entirely real-valued, ensuring positivity of the partition function. The higher bands *εm*(*q*) are in general complex-valued.

Hence we obtain a physically meaningful partition function, and can connect it to thermodynamic observables. The pressure of the Coulomb gas is its free energy per unit length

$$P = k\_B T \frac{\partial \ln Z\_L}{\partial L} \stackrel{L \to \infty}{\longrightarrow} -eE\_0 \varepsilon\_0(q) \, , \tag{7}$$

which for a long channel is determined by the eigenvalue with the smallest real part, *ε*0(*q*). In equilibrium the system minimizes its free energy by choosing an appropriate boundary charge *q*. In [25,26] this minimum was found to generally be the non-polarized state of the channel, i.e., *q* = 0. Adiabatic charge transfer through the channel is associated with the boundary charge *q* sweeping through its full period. As a result, the (free) energy barrier for ion transport is

*U*<sup>0</sup> = *eE*0*L*(∆*ε*)<sup>0</sup> , (8)

where (∆*ε*)<sup>0</sup> is the width of the lowest Bloch band. Therefore the ground state energy and the width of the lowest Bloch band of the Hamiltonian (5) give the leading thermodynamic and transport properties of the (*n*1, *n*2) Coulomb gas. In Section 3 we discuss analytic results for the eigenvalues and the bandwidth.

#### *2.3. Charge Non-Neutrality*

In [10] it was shown that for arbitrary values of *α*1,2 the Hamiltonian (5) is always isospectral to a similar charge-neutral Hamiltonian. This can be seen by shifting the coordinate as *θ* → *θ* + *θ*0. Upon such transformation the dimensionless concentrations *α*1,2 renormalize as *α*<sup>1</sup> → *α*1*e in*1*θ*<sup>0</sup> and *<sup>α</sup>*<sup>2</sup> <sup>→</sup> *<sup>α</sup>*2*<sup>e</sup>* <sup>−</sup>*in*2*θ*<sup>0</sup> . Notice that the combination *α n*2 1 *α n*1 2 remains invariant. Hence, the family of Hamiltonians (5) with

$$
\alpha\_1^{n\_2} \alpha\_2^{n\_1} = \text{const} \tag{9}
$$

is isospectral [10,25]. Therefore one may choose one representative from each isospectral family. A convenient choice is taking the representative with charge neutrality in the bulk reservoirs, i.e., *n*1*α*<sup>1</sup> = *n*2*α*<sup>2</sup> ≡ *α*. The physical reason for this symmetry is that the interior region of the channel always preserves charge neutrality due to the large self-energy of charges. The edge regions screen charge imbalances of the reservoirs. Therefore, irrespective of the relative fugacities of cations and anions in the reservoirs, the thermodynamics of the long channel are equivalent to the one in contact with neutral reservoirs with an appropriate salt concentration *α*. This brings the Hamiltonian (5) to the form

$$
\hat{H} = a \left[ \not{p}^2 - \left( \frac{1}{n\_1} e^{i n\_1 \theta} + \frac{1}{n\_2} e^{-i n\_2 \theta} \right) \right] \,\tag{10}
$$

where we define the momentum operator as

$$\mathfrak{p} = \mathfrak{a}^{-1/2} (-i\partial\_{\theta} + q); \qquad [\theta, \mathfrak{p}] = i\mathfrak{a}^{-1/2}.\tag{11}$$

The commutation relation shows that *α* <sup>−</sup>1/2 plays the role of the effective Planck constant. Hence, a large concentration of charges corresponds to the semiclassical limit of the Hamiltonian (10). We further rescale the eigenvalues *ε* as

$$
\mu \equiv \frac{n\_1 n\_2}{n\_1 + n\_2} \frac{\varepsilon}{\alpha}. \tag{12}
$$

This keeps the classical minimum of the potential at *u* = −1, irrespective of the concentration *α* and the valencies *n*1, *n*2. In Section 3 we discuss the spectral properties of the Hamiltonian (10) in the semiclassical limit.

#### **3. Large Charge Concentration**

In Section 2 we mapped the grand-canonical partition function of the Coulomb gas onto an equivalent quantum system. The resulting Hamiltonian, Equation (10), contains one free parameter *α* which is proportional to the concentration of charged ions. In this section we analyze the spectral problem of this Hamiltonian in the limit of large *α*. As argued after Equation (11), this is the semiclassical limit of the equivalent quantum problem. We use the main semiclassical results, Bohr-Sommerfeld quantization and Gamow's formula, to calculate the eigenvalues and bandwidths of the Hamiltonian for several different cases of valencies (*n*1, *n*2). In the case of equal valencies, *n*<sup>1</sup> = *n*2, the Hamiltonian (10) is the well-known Mathieu Hamiltonian which we discuss in Section 3.1. It's spectral properties were calculated using several different approaches [10,11,25–28]. In this review we focus on an approach based on integration on a complex Riemann surface [25,26,28]. We choose this method because it can also be applied to the cases with different valencies, *n*<sup>1</sup> 6= *n*2, see Section 3.2. In that situation the Hamiltonian is non-Hermitian, and the required action integrals are not attainable by straightforward integration. Instead we show how to relate them to integrals along closed cycles on a Riemann surface. Then we use powerful tools from algebraic topology to derive a differential equation for the action integrals. This is known as the Picard-Fuchs equation. The required actions are a combination of the solutions of this differential equation. Through this procedure we bypass the use of direct integration methods. From the actions we obtain the eigenvalues and the bandwidths, which are directly related to the ion pressure and transport barrier for ions in the channel. In Section 3.3 we go one step further. We use the same concepts to calculate the second-order corrections in the WKB series. Most importantly we show that these can be expressed in terms of the already-calculated action and its derivatives, and

therefore can be obtained with minimal computational effort. This gives an improved semiclassical approximation of the eigenvalues. Relating this to the pressure in the ion channel we find that beyond the ideal-gas pressure and the Debye-Hueckel correction there is another correction which only depends on the geometry of the channel but not on the concentration of ions. We compare these results to numerical calculations.

#### *3.1. Equal Valency*

As mentioned in Section 2 the Hamiltonian in Equation (5) is Hermitian if the valencies of the two charges are equal, *n*<sup>1</sup> = *n*2. Indeed, in this case it reduces to the well-known Mathieu Hamiltonian,

$$\mathcal{H} = \alpha \left[ \not{p}^2 - 2 \cos \theta \right]. \tag{13}$$

In literature there exist several studies of the Coulomb gas with charges of equal valency. In [10] it was first noted that the Coulomb gas is mapped onto the Mathieu equation. In [27] the authors perform a semiclassical calculation on this equation via direct integration. From this they obtain the required actions and analytic approximations of the eigenvalues and bandwidths. [11] provides additional qualitative arguments which lead to the same results. However, as mentioned above, in this section we will follow the Riemann surface methods developed in [25] because in that framework one can also study the case of unequal valencies *n*<sup>1</sup> 6= *n*<sup>2</sup> in Section 3.2, and these concepts form the basis of our considerations for higher-order corrections in Section 3.3.

#### 3.1.1. Construction of the Riemann Surface

In the semiclassical ansatz we look for wave functions of the form *ψ* = *e iα* 1/2*S* , where *S* is the action of the classical problem with the normalized Hamiltonian (13). The semiclassical trajectories satisfy the classical Hamilton equations of motion and thus conserve the (complex) energy *u* in Equation (12),

$$2\mu = p^2 - 2\cos\theta \,. \tag{14}$$

In this normalization *u* = ∓1 corresponds to the bottom (top) of the cosine potential. Our approach to calculate the action integrals *S* = H *γ p*(*θ*, *u*)*dθ* is based on complex algebraic topology. First we set *z* = *e <sup>i</sup><sup>θ</sup>* and consider (*z*, *p*) as complex variables. Energy conservation, Equation (14), defines a family of complex algebraic curves parametrized by *u* and satisfying

$$\mathcal{E}\_u: \qquad \mathcal{F}(p, z) = p^2 z - (z^2 + 2uz + 1) = 0. \tag{15}$$

For *u* 6= ±1 it can be checked that (*∂*F/*∂z*, *∂*F/*∂p*) does not vanish on E*u*, so each E*<sup>u</sup>* is nonsingular. Then F(*p*, *z*) implicitly defines a locally holomorphic map *p* = *p*(*z*). The exceptions to this occur at *z* = 0, ∞, *z*±, where *z*<sup>±</sup> = −*u* ± *i* √ 1 − *u* <sup>2</sup> are the roots of *p* <sup>2</sup> = 0 (i.e., classical turning points). In a vicinity of these four branch points *p*(*z*) behaves as

$$\begin{aligned} p &\sim z^{-1/2}, & (z \sim 0) \\ p &\sim z^{1/2}, & (z \sim \infty) \\ p &\sim (z - z\_{\pm})^{1/2}, & (z \sim z\_{\pm}) \end{aligned} \tag{16}$$

respectively, i.e., *p*(*z*) is locally double-valued. Note that we added the point at infinity to have an even number of branch points. This compactifies the complex plane and makes it topologically equivalent to a Riemann sphere, cf. Figure 2.

**Figure 2.** Construction of the Riemann surface of genus 1, as defined by Equation (15). (**a**) In the *z*-plane there are four branch points at 0, *z*±, ∞ which are pairwise connected by two branch cuts (gray). (**b**) Considering *z* = ∞ as a regular point the complex plane compactifies to a Riemann sphere with two cuts on the sphere. (**c**) The double-valued nature of the function *p*(*z*) is resolved by defining two copies of the Riemann sphere. The branch cuts are opened and the spheres are deformed into tubes (**d**) and glued together to form a torus (**e**). The arrows are used to signify the edges that are glued together. There are two fundamental cycles *γ*0, *γ*<sup>1</sup> which are topologically different and non-trivial, i.e., they can not be smoothly transformed into each other or a point. Reproduced with permission from Reference [26].

To avoid dealing with *p*(*z*) as a double-valued function we introduce a second copy of the complex *z*-plane and the corresponding Riemann sphere. On both sheets we define two branch cuts connecting the four branch points, between 0, ∞ and the turning points *z*<sup>±</sup> respectively. *p*(*z*) is analytically continued across the branch cuts, i.e., when crossing a branch cut we jump from the first sheet to the second and vice versa. Identifying the branch cuts as edges we can deform the two Riemann spheres into tubes and glue them together to form a torus. This construction is visualized in Figure 2. Thus, the complex algebraic curve E*<sup>u</sup>* in Equation (15) defines a torus which is a compact Riemann surface of genus *g* = 1. (Generically, every compact Riemann surface is topologically equivalent to a sphere with some number of handles *g*, or a (multi-)torus with *g* holes, called the genus of the surface).

#### 3.1.2. Integrals on the Riemann Surface and the Picard-Fuchs Equation

The action integrals can be understood as integrals over closed cycles *γ*, *S*(*u*) = H *γ λ*(*u*), where

$$
\lambda(u) = p(\theta) \, d\theta = p(z) \frac{dz}{i z} = \frac{(z^2 + 2uz + 1)^{1/2}}{i z^{3/2}} \, dz \tag{17}
$$

is the action 1-form which, by construction, is holomorphic on the Riemann surface.

To visualize the relevant trajectories we momentarily return to *θ* and consider it as complex. In this representation one has square-root branch cuts along the real axis, connecting the classical turning points along the classically allowed region. The integration trajectories run just above or below the real axis and connect the turning points. After combining them to form closed cycles one can push these cycles off the real axis and away from the turning points without altering the integrals (by Cauchy's theorem). We call these the classical cycle *γ*<sup>0</sup> and the instanton cycle *γ*1, as shown in Figure 3. Translating these two cycles to the complex *z*-plane yields the contours in the right panel of that figure.

**Figure 3. Left**: The classically allowed (forbidden) regions along the *θ*-axis at energy *u* are shown by the solid (dashed) gray line. Deforming the classical (instanton) orbits into the complex plane leads to the cycles *γ*0(*γ*1). **Right**: Cycles *γ*<sup>0</sup> (red) and *γ*1 (blue) in the complex *z*-plane for *u* = −0.9. Notice that the cycle *γ*<sup>1</sup> crosses the two cuts from the first sheet (solid line) to the second sheet (dashed line) and back. Reproduced with permission from Reference [25].

Cauchy's theorem is also valid on the Riemann surface since the action form (17) is, by construction, holomorphic on the torus. Therefore all closed cycles can be deformed without changing the integrals, and can be expressed as a combination of an integer number of these two basis cycles. This leads to our key idea how to calculate the action integrals: for this we employ a central theorem of algebraic topology, de Rham's theorem. It states that on a Riemann surface there are exactly as many linearly independent holomorphic 1-forms to integrate upon as there are independent closed cycles to integrate along. This is valid up to exact forms, i.e., 1-forms which integrate to 0 along any closed cycle, and boundaries, i.e., closed curves which can be continuously deformed to a point. Hence, there are exactly two independent holomorphic 1-forms on the Riemann surface. Any set of three 1-forms is linearly dependent modulo an exact form which integrates to 0 upon integration along any closed cycle. (A full explanation of the mathematical concepts is beyond the scope of this review. A detailed discussion of relevant and related concepts is in [29], basic definitions and additional background are in [30,31]. All concepts can also be found online at [32]. A simplified derivation specifically for complex-valued Riemann surfaces is in chapter 2 of [28]).

Equipped with this we look at a set which contains the action 1-form (17) and its first two derivatives with respect to energy *u*, {*λ*(*u*), *λ* 0 (*u*), *λ* <sup>00</sup>(*u*)}. Taking derivatives does not change the structure of branch points, therefore these are three 1-forms which are all defined on the same Riemann surface. Hence, we know that there must exist a linear combination of these which is an exact form. Reference [25] explains in detail how to find the linear combination and the exact form as

$$\left( (u^2 - 1)\partial\_u^2 + \frac{1}{4} \right) \lambda(u) = \frac{d}{dz} \left[ \frac{i}{2} \frac{1 - z^2}{z^{1/2} (z^2 + 2uz + 1)^{1/2}} \right] dz \,. \tag{18}$$

It is evident from Stokes' theorem that the right-hand-side integrates to 0 along any closed cycle on the Riemann surface. Hence, we obtain

$$\oint\_{\gamma} \left( (u^2 - 1)\partial\_{\mu}^2 + \frac{1}{4} \right) \lambda(u) = (u^2 - 1)S''(u) + \frac{1}{4} \, S(u) = 0 \,. \tag{19}$$

This differential equation for the action *S*(*u*) is called the Picard-Fuchs Equation [29]. Integration is performed along a closed cycle *γ*, which can be the classical or the instanton cycle, *γ*0,1 in Figure 3. Therefore both the classical and instanton actions *S*0,1(*u*) are solutions of the Picard-Fuchs Equation (19). This equation is a second-order ordinary differential

equation, therefore it admits two independent solutions. These can be found in the form *F*0(*u* 2 ) and *uF*1(*u* 2 ), where

$$\begin{aligned} F\_0(u^2) &= \,\_2F\_1\left(-\frac{1}{4}, -\frac{1}{4}; \frac{1}{2}; u^2\right), \\ F\_1(u^2) &= \,\_2F\_1\left(+\frac{1}{4}, +\frac{1}{4}; \frac{3}{2}; u^2\right). \end{aligned} \tag{20}$$

are hypergeometric functions [33,34]. These solutions form a basis out of which *S*0,1(*u*) must be composed, so we write

$$\begin{aligned} \mathcal{S}\_0(\boldsymbol{u}) &= \mathcal{C}\_{00}\mathcal{F}\_0(\boldsymbol{u}^2) + \mathcal{C}\_{01}\boldsymbol{u}\mathcal{F}\_1(\boldsymbol{u}^2), \\ \mathcal{S}\_1(\boldsymbol{u}) &= \mathcal{C}\_{10}\mathcal{F}\_0(\boldsymbol{u}^2) + \mathcal{C}\_{11}\boldsymbol{u}\mathcal{F}\_1(\boldsymbol{u}^2). \end{aligned} \tag{21}$$

To find the correct coefficients *Cjk*, *j*, *k* = 0, 1 it is sufficient to evaluate the periods at one specific value of *u*. Employing the fact that the hypergeometric functions (20) are normalized and analytic at *u* = 0, i.e., *F<sup>k</sup>* (*u* 2 ) = 1 + O(*u* 2 ), one notices that *Sj*(*u*) = *Cj*<sup>0</sup> + *uCj*<sup>1</sup> + O(*u* 2 ). Thus, to identify *Cjk* we expand the integrand *λ*(*u*) to first order in *u* and evaluate the integrals *Sj*(*u*) at *u* = 0. Straightforward calculation yields

$$\mathbf{C}\_{00} = e^{-i\pi/2}\mathbf{C}\_{10} = 8\pi^{-1/2}\Gamma(3/4)^2,\tag{22}$$

$$\mathbf{C}\_{01} = e^{+i\pi/2}\mathbf{C}\_{11} = \pi^{-1/2}\Gamma(1/4)^2.$$

The relations between *C*0*<sup>k</sup>* and *C*1*<sup>k</sup>* are not accidental. They originate from the fact that the cycle *γ*<sup>1</sup> transforms into *γ*<sup>0</sup> by substitution *z* 0 = *e* <sup>−</sup>*iπz* and *u* 0 = *e <sup>i</sup>πu*, and vice versa. This gives a global symmetry between the two periods,

$$S\_0(\mu) = e^{-i\pi/2} S\_1(e^{i\pi}\mu) \,. \tag{23}$$

Equations (20)–(23) fully determine the classical and instanton actions *S*0,1(*u*). We now proceed to relate them to physical observables.

#### 3.1.3. Semiclassical Results

We seek semiclassical results for the sequence of low-energy bands terminating at *u* = −1. Therefore we quantize the classical action *S*0(*u*) according to the Bohr-Sommerfeld rule to determine the normalized energies *u<sup>m</sup>* as solutions of the equation

$$S\_0(\mu\_m) = 2\pi \alpha^{-1/2} (m + 1/2) \,, \qquad m = 0, 1, \dots \tag{24}$$

We see that the cycle *γ*<sup>0</sup> contracts to a point when the energy goes to the bottom of the potential, *u* → −1. This corresponds to vanishing of the classical action, *S*0(*u* = −1) = 0. To obtain an approximate analytic expression for the lowest energy levels *ε<sup>m</sup>* = 2*αu<sup>m</sup>* we expand the classical action to first order near the bottom of the potential,

$$S\_0(\mu) = 2\pi(\mu+1). \tag{25}$$

Equations (24) and (25) combined imply *ε<sup>m</sup>* = −2*α* + 2*α* 1/2(*m* + 1/2). As a result the pressure (7) of a monovalent gas is

$$P = -eE\_0 \varepsilon\_0 = 2k\_B T f - \sqrt{k\_B T e E\_0 f}.\tag{26}$$

The two terms here are the pressure of the ideal gas with fugacity *f* and the mean-field Debye-Hueckel interaction correction [22].

The instanton action *S*1(*u*) determines the bandwidth (∆*u*)*<sup>m</sup>* according to Gamow's formula,

$$(\Delta u)\_{\mathfrak{m}} = \frac{\omega}{\pi \sqrt{\mathfrak{a}}} e^{i \mathfrak{a}^{1/2} \mathbb{S}\_1(u\_\mathfrak{m})/2} \,. \tag{27}$$

Here *ω* = 2 is the frequency of the harmonic-oscillator approximation of the potential near the classical minimum. We expand the instanton action near the classical minimum and at the quantized energies *u<sup>m</sup>* = −1 + *α* <sup>−</sup>1/2(*m* + 1/2) to obtain

$$S\_1(u\_m) = 16i + 2i \left(m + \frac{1}{2}\right) \ln\left(\frac{m + 1/2}{32e a^{1/2}}\right) \,. \tag{28}$$

Applying this to Gamow's Formula (27) leads to

$$(\Delta \varepsilon)\_m = 2a(\Delta u)\_m = \frac{4}{\pi} \left(\frac{32e}{m+1/2}\right)^{m+1/2} e^{-8a^{1/2} + (m/2 + 3/4)\ln a} \,\,\,\tag{29}$$

This coincides with the known asymptotic results for the Mathieu Equation [27,35,36]. As explained below Equation (3), adiabatic charge transport is associated with a change of the boundary charge *q* (i.e., quasi-momentum) across the interval 0 < *q* < 1 (i.e., the Brillouin zone). Therefore the *free energy* transport barrier is given by the width of the lowest Bloch band, (∆*ε*)0. One notices that increasing the concentration of salt ions leads to an exponential entropic suppression of the transport barrier, (∆*ε*)<sup>0</sup> ∝ *α* 3/4*e* −8 √ *α* .

#### *3.2. Multivalent Ions*

So far we worked with the Hermitian example of the Mathieu Hamiltonian, i.e., when both ion species are monovalent, *n*<sup>1</sup> = *n*<sup>2</sup> = 1. With that we could validate the Riemann surface method by comparing the results to literature. In this section we discuss four different cases with multivalent ions (assuming *n*<sup>1</sup> > *n*<sup>2</sup> without loss of generality). In such a scenario the Hamiltonian (5) is non-Hermitian. This leads to complex values in the spectrum, which we present in Section 3.2.1. Furthermore, in classical motion the coordinate and momentum acquire complex values. This results in a phase space (*θ*, *p*) with two complex dimensions (instead of two real dimensions). The classical (instanton) action is obtained by integrating the momentum *p*(*θ*) along the trajectory which connects two turning points and solves the classical equations of motion with real (imaginary) time. However, solving the equations of motion in complex phase space (*θ*, *p*) is non-trivial, if at all attainable. Therefore we go from an integral along the trajectory to an integral along a closed cycle in the plane of complex *z* = *e <sup>i</sup><sup>θ</sup>* which encloses the trajectory, similar to the mapping in Figure 3. With that we connect the non-Hermitian problem to the method that we validated in the previous section. We discuss this calculation for four different combinations of charge valencies in Section 3.2.2. In Section 3.2.3 we connect the results to the classical and instanton actions and physical observables.

#### 3.2.1. Spectrum of the Non-Hermitian Hamiltonian

Non-Hermiticity of the Hamiltonian (10) has a significant effect on its spectrum. Namely, not all eigenvalues are real. In Figure 4 we show numerical results for the eigenvalues at large concentration *α*, for four different combinations of the integers (*n*1, *n*2). Most importantly all non-real eigenvalues appear as complex conjugate pairs. This is a consequence of the PT -symmetry of the Hamiltonian and crucial to obtain a physically meaningful partition function, as discussed in Section 2. Furthermore we see sequences of narrow bands which emerge from *u* = −*ν* with *ν <sup>n</sup>*1+*n*<sup>2</sup> = 1. These sequences approximately follow the lines connecting *u* = −*ν* and *u* = 1, but avoid the special point *u* = 1. At some point all of these branches merge. Beyond this the nature of the spectrum changes drastically, instead of narrow bands and large gaps we see wide bands separated by small gaps. This feature is similar to the case of a periodic Hermitian potential: as long as the energy lies below the maximum of the potential there are narrow bands, while for energies

exceeding the maximum there are wide bands. Hence, we associate the point where the spectral branches meet with the top of the potential. (It is important to bear in mind that for a complex-valued potential there is no proper definition of a "maximum".) The energy variable *u* is normalized so that in the Hermitian (1, 1) case this point lies at *u* = 1. In the non-Hermitian cases we observe *u* ≈ 0.96 for (2, 1), *u* ≈ 1.09 for (3, 1), *u* ≈ 1.20 for (4, 1), and *u* ≈ 0.84 for (3, 2). These values are independent of *α*, so this must be a consequence of the underlying classical mechanics.

**Figure 4.** The bands of the non-Hermitian Hamiltonian in space of complex energy *u*. Blue stands for *q* = 0, while red stands for *q* = <sup>1</sup> 2 . The dotted circle marks |*u*| = 1. In all cases we see multiple branches of narrow bands with complex values which terminate near the unit circle. The dashed line is a guide to the eye which connects the termination points of the branches, *u* = −(1) 1/(*n*1+*n*2) , to *u* = 1. **Top left**: (*n*<sup>1</sup> , *n*2) = (2, 1), *α* = 200; **top right**: (3, 1), *α* = 300; **bottom left**: (4, 1), *α* = 400; **bottom right**: (3, 2), *α* = 400. Reproduced with permission from References [25,26].

To calculate the statistical partition function in Equation (6) the most important eigenvalues are those with small real part. Therefore we will focus on the narrow bands and treat them in semiclassical approximation.

3.2.2. Riemann Surface and Picard-Fuchs Equation

We use the rescaled energy variable *u* in Equation (12), substitute *z* = *e iθ* in the Hamiltonian (10), and write the classical energy-momentum relation as

$$
\mu \frac{n\_1 + n\_2}{n\_1 n\_2} = p^2 - \left(\frac{1}{n\_1} e^{i n\_1 \theta} + \frac{1}{n\_2} e^{-i n\_2 \theta}\right). \tag{30}
$$

The generalization for the complex algebraic curve in Equation (15) is the family of curves

$$\mathcal{E}\_{\mathfrak{U}} : \qquad \mathcal{F}(p, z) = n\_1 n\_2 p^2 z^{n\_2} - \left( n\_2 z^{n\_1 + n\_2} + (n\_1 + n\_2) \mu z^{n\_2} + n\_1 \right) = 0. \tag{31}$$

This defines implicitly a double-valued function *p*(*z*). It is easy to see that (*∂*F/*∂z*, *∂*F/*∂p*) does not vanish on E*<sup>u</sup>* unless *u* = −*e* 2*πim <sup>n</sup>*1+*n*<sup>2</sup> for an integer *m*. For the non-singular values of *u* the function *p*(*z*) is locally holomorphic except for the points *z* = 0, ∞, *z<sup>j</sup>* , where *z<sup>j</sup>* , *j* = 1, ..., *n*<sup>1</sup> + *n*<sup>2</sup> are the roots of *p* <sup>2</sup> = 0. The *z<sup>j</sup>* are the turning points of classical motion in complex coordinates. Near these special points *p*(*z*) behaves as

$$\begin{aligned} p &\sim z^{-n\_2/2}, & (z \sim 0) & (\text{32})\\ p &\sim z^{n\_1/2}, & (z \sim \infty) & \\ p &\sim (z - z\_j)^{1/2}. & (z \sim z\_j) \end{aligned} $$

The *z<sup>j</sup>* are *n*<sup>1</sup> + *n*<sup>2</sup> branch points. If *n*<sup>2</sup> (*n*1) is odd, then 0 (∞) is an additional branch point; for even *n*<sup>2</sup> (*n*1) there is a normal pole at 0 (∞). Hence, there are *n*<sup>1</sup> + *n*<sup>2</sup> + 1 branch points on the Riemann sphere if one of the integers is odd, and *n*<sup>1</sup> + *n*<sup>2</sup> + 2 branch points if both are odd. (Here we ignore the case that *n*1, *n*<sup>2</sup> are both even, because if both integers can be divided by the same number *n* we can define *z* 0 = *e inθ* to obtain a simpler algebraic curve.) In all cases there is an even number of branch points which can be connected pairwise to form branch cuts. For (*n*1, *n*2) = (2, 1) we obtain four branch points and two branch cuts and a Riemann surface of genus 1, as in Figure 2. For (*n*1, *n*2) = (3, 1),(4, 1),(3, 2) the asymptotic expansions (32) give six branch points. Consequently there are three branch cuts in the complex plane. Through a similar construction as in Figure 2 one obtains a Riemann surface which is topologically equivalent to a figure "8", i.e., a figure with two holes and genus 2 [26,28]. In the following we consider these four cases because there are no naturally occurring ions with larger charge. However, mathematically the algebraic curves for higher values of the integers can be constructed in the same way, yielding Riemann surfaces with larger genus.

In Figure 5 we show the structure of branch points in the *z*-plane for these four cases. On a Riemann surface with genus *g* = 1(2) there are two (four) independent closed cycles [29]. In Figure 5 we define three cycles for the (2, 1) case, and five cycles for (4, 1) and (3, 2). This is done for convenience and symmetry reasons. The superfluous cycle can be expressed by the other cycles. For (2, 1) the linear combination *γ*<sup>0</sup> − *γ*<sup>1</sup> − *γ*<sup>2</sup> does not contain any of the branch points and is contractible to a point. For (4, 1) the trivial cycle is *γ*<sup>0</sup> − *γ*<sup>1</sup> + *γ*<sup>2</sup> + *γ*<sup>3</sup> − *γ*<sup>4</sup> ∼= 0, and for (3, 2) we see that *γ*<sup>0</sup> + *γ*<sup>1</sup> − *γ*<sup>2</sup> − *γ*<sup>3</sup> + *γ*<sup>4</sup> ∼= 0. We choose to include the additional cycle because it gives an easy representation for the symmetry relation between the corresponding actions *Sj*(*u*), akin to Equation (23). By substituting *z* 0 = *e* −*iφ z* and *u* 0 = *e <sup>i</sup>φ<sup>u</sup>* the cycles transform *<sup>γ</sup><sup>j</sup>* <sup>→</sup> *<sup>γ</sup>j*+<sup>1</sup> . For the (2, 1) case the resulting symmetry relation is

$$S\_0(\mathfrak{u}) = e^{\pi i/3} S\_1(e^{-2\pi i/3} \mathfrak{u}) = e^{-\pi i/3} S\_2(e^{2\pi i/3} \mathfrak{u}) \,. \tag{33}$$

The analogous symmetry relations for the genus-2 cases are shown in Reference [26]. To calculate the actions *S*(*u*) = H *γ λ*(*u*) we continue in the same manner as in Section 3.1. The 1-form (cf. Equation (17)) with general *n*1, *n*<sup>2</sup> is

$$\lambda(u) = p(\theta)d\theta = p(z)\frac{dz}{iz} = \frac{\left(n\_2 z^{n\_1 + n\_2} + (n\_1 + n\_2)uz^{n\_2} + n\_1\right)^{1/2}}{i\sqrt{n\_1 n\_2} z^{1 + n\_2/2}}dz.\tag{34}$$

**Figure 5.** The integration cycles in the complex *z*-plane for the four non-Hermitian cases that are discussed in Section 3.2. In all images we set *u* = 0. Each color represents one closed cycle of integration. Solid lines denote the sections which lie on the principal sheet, dashed lines the parts on the second sheet. **Top left**: (2, 1); **top right**: (3, 1); **bottom left**: (4, 1); **bottom right**: (3, 2). Note the differences in the structure of the branch cuts: in the (2, 1) case all branch points are finite, while in the (1, 1) case in Figure 3 one branch point lies at ∞. Similar differences exist between the other three figures, whether the branch points are at finite values of *z* or at ∞, and whether the origin is a branch point or a pole. Reproduced with permission from Reference [26].

On a Riemann surface of genus *g* = 1(2) there are two (four) independent closed cycles. According to the de Rham theorem, this is equal to the number of linearly independent 1-forms, modulo exact forms. Therefore a set of the 1-form (34) and its first few derivatives, {*∂ k <sup>u</sup>λ*(*u*)} *K k*=0 , is linearly dependent if it contains the first *K* = 2(4) derivatives. We build a linear combination of these which equals an exact form (for details see [26]). The integral

of the exact form along a closed cycle gives zero. What is left is a linear combination of the action and its first derivatives, cf. Equation (19). In the (2, 1) case we find this Picard-Fuchs equation as

$$(\mu^3 + 1)S\_j''(\mu) + \frac{\mu}{4}S\_j(\mu) = 0. \tag{35}$$

This is a second-order differential equation. The Picard-Fuchs equations for the genus-2 cases are fourth-order ODEs which can be found in Reference [26]. Equation (35) admits two solutions *F*0(*u* 3 ) and *uF*1(*u* 3 ) which are given in terms of hypergeometric functions [26,34],

$$F\_0(u^3) \quad = \,\_2F\_1\left(-\frac{1}{6}, -\frac{1}{6}; \frac{2}{3}; -u^3\right). \tag{36}$$

$$F\_1(u^3) \quad = \,\_2F\_1\left(+\frac{1}{6}, +\frac{1}{6}; \frac{4}{3}; -u^3\right).$$

The actions are a linear combination of these, *Sj*(*u*) = *Cj*0*F*0(*u* 3 ) + *Cj*1*uF*1(*u* 3 ). Expanding the hypergeometric functions near the origin, *F*0,1(*u* 3 ) = 1 + O(*u* 3 ), one notices that *Sj*(*u*) = *Cj*<sup>0</sup> + *uCj*<sup>1</sup> + O(*u* 3 ) as *u* → 0. The constants *C*0*<sup>k</sup>* are therefore given by *C*<sup>00</sup> = *S*0(0) and *C*<sup>01</sup> = *S* 0 0 (0). Straightforward integration and the symmetry relation (33) yield

$$\mathbf{C}\_{00} = \mathbf{C}\_{10}e^{\pi i/3} = \mathbf{C}\_{20}e^{-\pi i/3} \quad = \frac{2^{11/6}3\pi^{3/2}}{\Gamma(\frac{1}{6})\Gamma(\frac{1}{3})},\tag{37}$$

$$\mathbf{C}\_{01} = \mathbf{C}\_{11}e^{-\pi i/3} = \mathbf{C}\_{21}e^{\pi i/3} \quad = \frac{3^{1/2}\Gamma(\frac{1}{6})\Gamma(\frac{1}{3})}{2^{11/6}\pi^{1/2}}.$$

The actions *Sj*(*u*) for (*n*1, *n*2) = (2, 1) are fully given by Equations (33), (36), and (37). The analogous expressions for the genus-2 cases with (*n*1, *n*2) = (3, 1),(4, 1),(3, 2) are given in Reference [26]. In the next section we discuss how to obtain semiclassical results for the physical observables.

#### 3.2.3. Semiclassical Results in the Non-Hermitian Cases

In this section we calculate the eigenenergies and bandwidths of the non-Hermitian Hamiltonian in Equation (10) with the Bohr-Sommerfeld quantization condition and Gamow's formula. To utilize these standard semiclassical results we need to calculate the classical and the instanton actions, *<sup>S</sup>cl*,*inst*(*u*) = H *γcl*,*inst λ*(*u*). The crucial part hereby is identifying the correct cycle of integration. In Section 3.1, when discussing the case of a Hermitian Hamiltonian, we identified these with trajectories which connect the classical turning points through the classically allowed or forbidden region respectively, cf. Figure 3. In the non-Hermitian case this is not so clear, because there exist more than two turning points, and in the space with complex coordinate, momentum, and energy the concept of classically allowed or forbidden regions doesn't apply. Instead, to identify the correct actions *Scl*,*inst*(*u*) we look at the analytic behavior of these actions near special values of the energy *u*.

The Bohr-Sommerfeld condition requires that the classical action goes to zero at the classical minimum of the potential. This happens when two turning points collide which causes the corresponding cycle of integration to collapse to a point. We can easily check that in all four cases in Figure 5 the cycle *γ*<sup>0</sup> collapses to a point as *u* → −1. The corresponding action goes to zero, *S*0(−1) = 0. Therefore we identify *S*0(*u*) as the classical action which quantizes into the branch of eigenstates that terminates at *u* = −1. For (*n*1, *n*2) = (2, 1) it follows immediately from the symmetry relation (33) that at the singular point *u* = *e iπ*/3 (*e* <sup>−</sup>*iπ*/3) the cycle *γ*<sup>1</sup> (*γ*2) collapses to a point and the action *S*1(*u*) (*S*2(*u*)) goes to zero. It should be thus identified with the classical action for the spectral branch terminating at *u* = *e <sup>i</sup>π*/3 (*e* <sup>−</sup>*iπ*/3). In the same manner the analogous symmetry relations for the genus-2

cases in Reference [26] allow us to identify the classical actions for all the spectral branches in Figure 4. Quantizing these classical actions according to the Bohr-Sommerfeld rule,

$$S\_{\hat{f}}(\mu\_m^{(j)}) = 2\pi \mathfrak{a}^{-1/2} (m + 1/2), \qquad m = 0, 1, \ldots,\tag{38}$$

one finds the semiclassical energies *u* (*j*) *<sup>m</sup>* determining the *q* = 0 edges of the narrow bands in the complex plane. These results are compared with numerical data in Figure 6. The excellent agreement holds all the way up to the point where all spectral branches coalesce. Beyond this point the semiclassical approximation breaks down, which manifests in e.g., the appearance of wide Bloch bands.

**Figure 6.** Narrow energy bands (red dots) in the upper half-plane of complex energy *u* for large *α*, cf. Figure 4. In all four cases, *Im S*0(*u*) = <sup>0</sup> along the real axis, where the thin lines mark <sup>|</sup>*S*0(*u*)<sup>|</sup> <sup>=</sup> <sup>2</sup>*πα*−1/2(*<sup>m</sup>* <sup>+</sup> 1/2), the quantization condition. The other black lines mark *Im S<sup>j</sup>* (*u*) = 0 for the other actions *S<sup>j</sup>* (*u*), and the thin lines mark |*S<sup>j</sup>* (*u*)<sup>|</sup> <sup>=</sup> <sup>2</sup>*πα*−1/2(*<sup>m</sup>* <sup>+</sup> 1/2). In all cases *S<sup>j</sup>* (*u*) corresponds to an action encircling two branch points. These points coalesce at a singular value of *u* on the unit circle (dashed) where the spectral branch ends. Near intersections of two lines neither quantization condition holds, cf. *u* ≈ 0.90 + 0.31*i* in (4, 1) and *u* ≈ 0.82 in (3, 2). Beyond this intersection the states are quantized according to the sum of the two corresponding actions, *S*<sup>1</sup> + *S*<sup>2</sup> in (4, 1) and *S*<sup>2</sup> + *S*<sup>3</sup> in (3, 2), marked in green. To the right all lines coalesce and beyond this point we observe wide bands with narrow gaps. The lower half-plane shows the mirror image (i.e., complex conjugate) of the upper half plane. **Top left**: (*n*<sup>1</sup> , *n*2) = (2, 1), *α* = 200; **top right**: (3, 1), *α* = 300; **bottom left**: (4, 1), *α* = 400; **bottom right**: (3, 2), *α* = 400. Reproduced with permission from Reference [26].

All graphs exhibit spectral branches along the lines where one of the actions *Sj*(*u*) is real, while the narrow bands lie at the points determined by the Bohr-Sommerfeld condition (38). For (2, 1) and (3, 1) there exists a total of three spectral sequences, for (4, 1) and (3, 2) five sequences due to a higher number of special energies. In the (4, 1) case the two complex-valued branches intersect at *u* ≈ 0.90 + 0.32*i*. Beyond this point the two sequences merge into one, for which the quantization condition is neither determined by *S*<sup>1</sup> nor *S*<sup>2</sup> individually, but instead by the sum *S*<sup>1</sup> + *S*<sup>2</sup> (shown in green). For (3, 2) the two lines for the complex-conjugate pair *S*<sup>2</sup> and *S*<sup>3</sup> collide at *u* ≈ 0.84, the other pair collides at *u* ≈ 0.98 where the semiclassical approximation breaks down. A closer look at the state at *u* ≈ 0.89 reveals that this cannot be explained by the quantization of *S*<sup>0</sup> along the real axis. However, it meets the Bohr-Sommerfeld condition (38) for *S*<sup>2</sup> + *S*<sup>3</sup> with *m* = 17. Thus we may conclude that the spectral branches can be derived from the

Bohr-Sommerfeld condition for one of the actions, or upon intersection of two branches by the sum of the two actions of these branches.

To calculate the width of these bands with Gamow's formula,

$$(\Delta u)\_{\mathfrak{m}} = \frac{\omega}{\pi \sqrt{\alpha}} e^{i\mathfrak{a}^{1/2} \mathcal{S}\_{\text{inst}}(u\_m)/2},\tag{39}$$

we need to identify the instanton actions. The classical frequency *ω* is determined from the harmonic oscillator approximation, i.e., by expanding the potential around *θ* = 0. In Hermitian quantum mechanics the instanton trajectory connects the two classical turning points through the classically forbidden region, cf. Figure 3. Hence, we identify the instanton cycle as the other possible cycle that connects the same two turning points. This is a combination of all other integration cycles *γ<sup>i</sup>* . The instanton actions that correspond to the classical actions *S*0(*u*) are

$$\begin{aligned} \mathcal{S}\_{\text{inst}}(u) &= & -\mathcal{S}\_1(u) + \mathcal{S}\_2(u), \quad &(2,1); \\ \mathcal{S}\_{\text{inst}}(u) &= & -\mathcal{S}\_1(u) - \mathcal{S}\_2(u) + \mathcal{S}\_3(u), \quad &(3,1); \\ \mathcal{S}\_{\text{inst}}(u) &= & -\mathcal{S}\_1(u) - \mathcal{S}\_2(u) + \mathcal{S}\_3(u) + \mathcal{S}\_4(u), \quad &(4,1); \\ \mathcal{S}\_{\text{inst}}(u) &= & -\mathcal{S}\_1(u) + \mathcal{S}\_2(u) - \mathcal{S}\_3(u) + \mathcal{S}\_4(u), \quad &(3,2). \end{aligned} \tag{40}$$

From the symmetry relation (33) between the actions and its analogons for the genus-2 cases it is easy to check that these combinations are purely imaginary, which makes the bandwidth in Equation (39) real, as required.

More can be said when considering the analytic structure of the classical and instanton action in a vicinity of *u* = −1. Therefore we use a concept called monodromy [29,32], which is visualized in Figure 7. We choose some *u* & −1 and allow *u* to wind around −1 (i.e., (*u* + 1) → (*u* + 1)*e* 2*πi* ). The two branch points inside the cycle *γ*<sup>0</sup> in Figure 5 are exchanged by this transformation via a counter-clockwise half-turn; the branch cut in effect rotates by 180◦ . For *γ*<sup>0</sup> this has no effect, the cut turns within it. Not so for *γ*1: if this cycle is never to intersect the branch points, it is continuously deformed and as a result of this monodromy transformation we obtain *γ*<sup>1</sup> → *γ*<sup>1</sup> + *γ*0, thus *S*<sup>1</sup> picks up a contribution of *S*0. This effect is visualized in Figure 7. While we have returned to the initial value of *u*, the period *S*<sup>1</sup> does not return to its original value and thus can't be analytic. This occurs for every monodromy cycle near *u* = −1. The only function which monotonically increases as the phase of its argument grows is the complex logarithm. Thus, *S*<sup>1</sup> must have a logarithmic dependence on 1 + *u*. One can check that

$$S\_1(u) = Q\_1(u) - \frac{i}{2\pi} S\_0(u) \ln(1+u) \tag{41}$$

yields the correct behavior, where *Q*1(*u*) and *S*0(*u*) are analytic functions of (1 + *u*). The same applies to the other cycle which is connected to the same branch cut. Therefore the instanton action *Sinst* in Equation (40) picks up a contribution of −2*S*0. Hence, we can derive the Bohr-Sommerfeld quantization condition (38) from the requirement that the monodromy transformation leaves the bandwidth (39) unchanged.

**Figure 7.** In a monodromy transformation the parameter *u* is smoothly changed around a critical value in parameter space and returned to its original value, e.g., (1 + *u*) → (1 + *u*)*e* 2*πi* . During the transformation the branch points (blue) move in the complex plane, and the same structure of branch points is recovered. However, if a special value of the parameter *u* is enclosed by the trajectory in parameter space, e.g., *u* = −1, then the two branch points which collide at *u* = −1 are exchanged. During the transformation the integration cycle (red) is not allowed to cross a branch point, hence they are pulled along with the branch points. To restore the original cycle a closed cycle enclosing the two branch points has to be added.

A comparison of the results for the bandwidth with numerical simulations is shown in Figure 8 for the four non-Hermitian cases and the Hermitian (1, 1) case. All cases show good agreement with the numerical data already for moderate values of the parameter *α*. (Note however, that for the genus-2 cases Gamow's formula had to be multiplied by an overall factor of 3/2 (in (3, 1) case) or 2 (in (4, 1) and (3, 2) cases), respectively. The origin of this preexponential factor is beyond the scope of this paper.)

**Figure 8.** Analytic (numerical) results for the logarithm of the bandwidth of the lowest band, ln(∆*ε*)0, as a function of *α* 1/2, for all five cases with Riemann surfaces of genus 1 or 2. (1, 1): solid line (stars), (2, 1): dashed line (diamonds), (3, 1): dotted line (circles), (4, 1): short-dashed line (triangles), and (3, 2): dash-dotted line (squares). Reproduced with permission from Reference [26].

To summarize, we find that in all cases the bandwidth is of the form

$$(\Delta \varepsilon)\_m = A \times \left(\frac{k}{m+1/2}\right)^{(m+1/2)} \times \exp\left(-b\sqrt{a} + (m/2 + 3/4)\ln a\right). \tag{42}$$

The pressure, which is calculated from the lowest eigenvalue, contains the ideal gas pressure and the Debye-Hueckel correction,

$$P = \mathbb{C}k\_B T f - c\sqrt{k\_B \text{TeE}\_0 f}. \tag{43}$$

Here *A*, *k* and *b*, and *C* and *c*, are numerical factors that can be calculated directly by expanding *S*<sup>0</sup> and *Sinst*:


These values quantify the thermodynamic properties of the ion channels for all five different combinations of charged ions which give a Riemann surface of genus 1 or 2. With a maximum valency of 4 these are also the physically relevant cases. Most importantly we show that the Coulomb gas with unequal valency *n*<sup>1</sup> 6= *n*<sup>2</sup> has the same qualitative behavior as the standard gas with ions of equal valency, *n*<sup>1</sup> = *n*2. In all cases the pressure consists of the ideal gas pressure and the Debye-Hueckel correction, see Equation (43). Crucially for transport through the ion channel, in all cases the bandwidth shows exponential decay with the square-root of the fugacity *α* and has a universal pre-exponential factor of *α* 3/4 . However the factor *b* in the exponent shrinks when the valency is increased, meaning that the transport barrier falls off slower with increased charge concentration when transporting ions with larger valency.

#### *3.3. Higher-Order Corrections from Exact Wkb Method*

The approximations for the eigenvalues of the non-Hermitian Hamiltonian can be improved further by considering second- and higher-order terms in the WKB series. The inspiration comes from the exact WKB method which was studied extensively in the context of resurgence theory [37,38]. We use this to get a better approximation for the eigenvalues, and with that the pressure of the Coulomb gas, at moderate values of the charge concentration *α* & 1. The key is that the *q* = 0 band edge, which gives the pressure in equilibrium, is determined by an infinite series in *α* −1 (i.e., ¯*h* 2 in usual quantum mechanics),

$$\sum\_{n=0}^{\infty} \frac{(-1)^n}{\mathfrak{a}^n} \oint\_{\gamma\_{cl}} \rho\_{2n}(\theta, u\_m) d\theta = \frac{2\pi (m + 1/2)}{\sqrt{\mathfrak{a}}}.\tag{44}$$

*ρ*0(*θ*, *u*) = *p*(*θ*, *u*) is the classical momentum, and the other terms can be found through a recursive relation [37]. Equation (44) is sometimes also referred to as the generalized Bohr-Sommerfeld quantization condition. Reference [38] shows a calculation of the exact WKB series at all orders for a class of Hermitian genus-1 cases which include the cosine potential, i.e., the (1, 1) case in our notation. Here we follow the ideas in [39] and chapter 5 of [28] which give a general procedure to calculate the terms order-by-order for any potential, and can also be applied to non-Hermitian Hamiltonians.

It is evident that truncation of Equation (44) at the *n* = 0 term leads to the usual Bohr-Sommerfeld quantization condition. To improve upon this we include the *n* = 1 term. The integrand is given by

$$\rho\_2(\theta, u)d\theta = \left(\frac{\partial^2\_{\theta}(\rho\_0(\theta, u)^2)}{48\rho\_0(\theta, u)^3} + \frac{5}{24}\partial\_{\theta}\frac{\rho'\_0(\theta, u)}{\rho\_0(\theta, u)^2}\right)d\theta,\tag{45}$$

where the prime denotes a derivative with respect to *θ* [37]. The second term is an exact form which integrates to zero. We drop this exact form, use the expression (30) for the classical momentum *p* = *ρ*0, and perform the coordinate transformation *z* = *e iθ* to write the second-order 1-form as

$$\mathfrak{P}\_2(z,u)dz = \frac{-n\_1 z^{n\_1} - n\_2 z^{-n\_2}}{48\left(u\frac{n\_1 + n\_2}{n\_1 n\_2} + \frac{1}{n\_1} z^{n\_1} + \frac{1}{n\_2} z^{-n\_2}\right)^{3/2} iz} dz. \tag{46}$$

A comparison with Equation (34) shows that the second-order 1-form *ρ*˜2(*z*, *u*)*dz* has the same branch points as the action 1-form *λ*(*u*). Therefore it is defined on the same Riemann surface. As discussed in the preceding sections, on the Riemann surfaces of genus *g* = 1(2) there exist two (four) linearly independent 1-forms, up to an exact form. We take {*∂ k <sup>u</sup>λ*(*u*)} *K k*=0 as this maximal independent set with *K* = 1(3). This forms a basis for the space of all 1-forms. Hence, the second-order correction can be written as a linear combination of these basis 1-forms, modulo an exact form. We find this linear combination in the same way as in the derivation of the Picard-Fuchs Equations (19) and (35) and integrate it along the classical cycle *γcl* to get

$$\oint\_{\gamma\_{\rm{ld}}} \not p\_2(z, u) dz = -a(S\_0'(u) + 2uS\_0''(u)), \quad \frac{(n\_1, n\_2)}{a} \begin{array}{c|c|c|c} \text{(3.1)} & \text{(2.1)} & \text{(3.1)} & \text{(4.1)} & \text{(5.1)}\\ \hline a & \text{1/48} & \text{1/18} & \text{3/32} & \text{2/15} & \text{3/10} \end{array} \tag{47}$$

These expressions fully define the second-order corrections in terms of the classical action and its derivatives with respect to *u*. These are easily obtained from the previous results, Equations (20)–(22), (36) and (37) (see Reference [26] for the genus-2 cases). Note that in the genus-1 cases the second derivative *S* 00 0 (*u*) can be replaced with *S*0(*u*) by using the Picard-Fuchs Equations (19) and (35).

Here we want to stress that calculation of the second-order (and any higher) correction is only as computationally demanding as deriving the Picard-Fuchs equation. It does not require solving the differential equation and matching boundary conditions because the correct classical action was already identified. Therefore this can also be used as a simple method to simply calculate the higher-order WKB terms if the classical action was obtained in a different manner. The improvement in the approximation of the lowest eigenvalue is shown in Figure 9.

**Figure 9.** Log-plot of the deviation of the first-order (dashed line) and second-order (solid line) WKB result from the exact numerical result for the lowest eigenvalue as a function of *α*. We show the five different cases: (1, 1) in black, (2, 1) in blue, (3, 1) in red, (4, 1) in orange, (3, 2) in purple. The error drops by several orders of magnitude when taking the second-order WKB term into account. The approximations converge to the exact result as *α* → ∞; however, already at moderate values of *α* & 1 the approximations give quite accurate results.

With the second-order result we can calculate the eigenvalues *u* up to order *α* −1 . Therefore we expand the classical action *S*0(*u*) for *u* & −1 to order (*u* + 1) <sup>2</sup> and solve for *u*. Taking the lowest eigenvalue *u*<sup>0</sup> and applying this to the formula for the pressure (7) gives

$$P = c\_0 k\_B T f - c\_1 \sqrt{e E\_0 k\_B T f} - c\_2 e E\_{0\prime} \tag{48}$$

with the following constants:


This gives the ideal gas pressure and the Debye-Hueckel correction from the usual Bohr-Sommerfeld condition. The second-order WKB term gives an additional correction which is independent of the fugacity but only depends on the geometric properties of the channel which are included in the definition of *E*0.

#### **4. Summary of Semiclassical Results**

In this review we discussed analytic calculations of the thermodynamic properties of an ion channel at large charge concentrations, with an extension to moderate concentrations. We started with discussing a standard mapping of a statistical system onto an effective quantum system [10,23]. When performing this mapping there is no guarantee that the resulting effective Hamiltonian is Hermitian and has a purely real spectrum. Physically one needs to obtain a real and positive partition function. This is e.g., guaranteed if the Hamiltonian obeys PT -symmetry and its lowest eigenvalue is purely real.

Translation between the quantum results and thermodynamic observables is straightforward. Most importantly, the pressure (i.e., free energy density) is given by the quantum mechanical ground-state energy. The adiabatic transport barrier is the width of the lowest Bloch band. The complex energies of excited states, c.f. Figures 4 and 6, describe higherorder correlation functions. Their imaginary part is responsible for spatial oscillations, while the real part yields an overall exponential decay. Such decaying oscillatory correlation functions reflect short-range charge density wave ionic order within the channel. As seen in Figures 4 and 6, the onset of complex eigenvalues happens at lower energies for ions with larger valencies, which implies stronger charge density fluctuations. In all cases we observe that an increase of the charge concentration leads to an exponential reduction of the transport barrier, however this decay is slower if the ion valencies are large. This is visualized in Figure 8.

The approximation with the effective 1D Coulomb potential, Equation (1), works best at large ion concentration. Electric field lines leak out of the channel after a characteristic length *ξ* which is given by *ξ* <sup>2</sup> = *a* 2 *κ*1/(2*κ*2)ln(2*ξ*/*a*), where *a* is the radius of the channel and *κ*1, *κ*<sup>2</sup> are the dielectric constants of water and the surrounding medium. Therefore the 1D Coulomb potential best approximates the situation where the characteristic distance between the ions is small. This is the case of large charge concentration, which is also the case when then semiclassical approximation is applicable.

Here we discuss a method how to perform semiclassical calculations without the need to solve the classical equations of motion and without direct integration. This is particularly useful in the non-Hermitian cases when the solutions to the equations of motions are hardly attainable. Instead we derive and solve the Picard-Fuchs differential equation, which is a tool from algebraic topology. The power of the Picard-Fuchs equation is that it is a coordinate-free expression, i.e., one does not need to know the classical trajectories. In the last part we extend our calculations to second- and higher-order terms in the WKB series. These provide a clearly improved approximation for the eigenvalues especially at moderate charge concentrations, see Figure 9.

The applicability of the Picard-Fuchs method extends far beyond the case of ion channels. It can be a powerful tool for Hermitian and non-Hermitian systems alike, as it can be applied to generic Hamiltonians. Especially the extension to second- and higher-order terms in the WKB series requires very little computational effort once the classical action has been calculated. Mappings of a generic statistical system onto an effective quantum system can lead to a non-Hermitian Hamiltonian for which semiclassical calculations with direct integration are difficult. We believe that the Picard-Fuchs method can be especially useful in these cases, as it allows us to circumvent the complications associated with direct integration like solving equations of motion with complex coordinates.

**Author Contributions:** Both authors contributed equally to all aspects of the manuscript. Both authors have read and agreed to the published version of the manuscript.

**Funding:** A.K. was supported by NSF grants DMR-2037654. T.G. acknowledges funding from the Institute of Science and Technology (IST) Austria, and from the European Union's Horizon 2020 research and innovation program under the Marie Skłodowska-Curie Grant Agreement No. 754411.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Acknowledgments:** We are indebted to Boris Shklovskii for introducing us to the problem, and Alexander Gorsky and Peter Koroteev for introducing us to the Picard-Fuchs methods. A very special thanks goes to Michael Janas for several years of excellent collaboration on these topics. TG thanks Michael Kreshchuk for introduction to the exact WKB method and great collaboration on related projects. Figures 3 and 4 are reproduced from Reference [25] with friendly permission by the Russian Academy of Sciences. Figures 2, 4, 5, 6, and 8 are reproduced from Reference [26] with friendly permission by IOP Publishing. ©IOP Publishing. All rights reserved.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A. Equivalence between Partition Function and Quantum Mechanics**

Here we present details of the mapping between the statistical system of charged ions and an effective single-particle quantum mechanical problem. Our goal is to evaluate the grandcanonical partition function of the Coulomb gas, Equation (4),

$$\mathcal{Z} = \sum\_{N\_1, N\_2=0}^{\infty} \frac{f\_1^{N\_1} f\_2^{N\_2}}{N\_1! N\_2!} \prod\_{j=1}^{N\_1+N\_2} \int\_0^L d\mathbf{x}\_j \, e^{-\mathcal{U}/T} \, \tag{A1}$$

where the gas potential energy *U* is given by Equations (2) and (3). To this end we first consider an auxiliary identity:

1 = Z D*ρ*(*x*) *δ ρ*(*x*) − *N*1+*N*<sup>2</sup> ∑ *j*=1 *σjδ*(*x* − *xj*) − *q*(*δ*(*x*) − *δ*(*x* − *L*)) ! = Z D*ρ*(*x*) Z D*θ*(*x*) *e* −*i* R*L* 0 *dx θ*(*x*) *ρ*(*x*)− *N*1+*N*2 ∑ *j*=1 *σjδ*(*x*−*x<sup>j</sup>* )−*q*(*δ*(*x*)−*δ*(*x*−*L*)) ! (A2) = ZZ D*ρ*(*x*)D*θ*(*x*) *e* −*i* R*L* 0 *dx θ*(*x*)*ρ*(*x*)− *N*1+*N*2 ∑ *j*=1 *σjθ*(*x<sup>j</sup>* )−*q*(*θ*(0)−*θ*(*L*)) ! .

Here *ρ*(*x*) is a continuous field for the charge density, and *θ*(*x*) its conjugate field. Substituting this identity into the expression for the partition function, one finds:

Z = ZZ<sup>∞</sup> −∞ *dθ*0*dθ<sup>L</sup>* (2*π*) 2 *e iq*(*θ*0−*θL*) ZZ D*ρ*(*x*)D*θ*(*x*)*e* <sup>−</sup> <sup>1</sup> 2*T* RR*L* 0 *dxdx*0*ρ*(*x*)Φ(*x*−*x* 0 )*ρ*(*x* 0 )−*i* R*L* 0 *dx θ*(*x*)*ρ*(*x*) × ∞ ∑ *N*1=0 1 *N*1! *f*<sup>1</sup> Z *L* 0 *dx eiσ*1*θ*(*x*) *N*1 × ∞ ∑ *N*2=0 1 *N*2! *f*<sup>2</sup> Z *L* 0 *dx eiσ*2*θ*(*x*) *N*<sup>2</sup> (A3) = ZZ<sup>∞</sup> −∞ *dθ*0*dθ<sup>L</sup>* (2*π*) 2 *e iq*(*θ*0−*θL*) Z D*θ*(*x*) *e* − *T* 2 RR*L* 0 *dxdx*0 *θ*(*x*)Φ−<sup>1</sup> (*x*−*x* 0 )*θ*(*x* 0 )+R*<sup>L</sup>* 0 *dx*(*f*<sup>1</sup> *e in*1 *<sup>θ</sup>*(*x*)+*f*2*e* −*in*2 *θ*(*x*) ) .

The integral over *θ*(*x*) runs over all functions with the boundary conditions *θ*(0) = *θ*<sup>0</sup> and *θ*(*L*) = *θL*. We also use that the valencies of the charges are *σ<sup>j</sup>* = *n*<sup>1</sup> for 1 ≤ *j* ≤ *N*<sup>1</sup> and *σ<sup>j</sup>* = −*n*<sup>2</sup> for *N*<sup>1</sup> + 1 ≤ *j* ≤ *N*<sup>1</sup> + *N*2. It is straightforward to verify that the inverse of the interaction operator is given by Φ−<sup>1</sup> (*x* − *x* 0 ) = −(2*eE*0) −1 *δ*(*x* − *x* 0 )*∂* 2 *x* , because the Coulomb potential in any dimension is a resolvent of the Poisson equation and therefore its inverse is the Laplacian. As a result, the functional integral on the r.h.s. of the last expression takes the form of the Feynman propagator

$$G(\theta\_0, \theta\_L; L) \equiv \int \mathcal{D}\theta(\mathbf{x}) \, e^{-\frac{\mathbf{x}\_T}{4} \int \mathrm{d}x \left[ (\partial\_\mathbf{i} \theta)^2 - \left( a\_1 e^{i n\_1 \theta(\mathbf{x})} + a\_2 e^{-i n\_2 \theta(\mathbf{x})} \right) \right]} \, \tag{A4}$$

where *x<sup>T</sup>* = *T*/(*eE*0) and *α*1,2 = 4 *f*1,2/*xT*. Expression (A4) represents the "quantum mechanical" probability to propagate from *θ*<sup>0</sup> to *θ<sup>L</sup>* during the (imaginary) "time" *L*. The corresponding stationary "Schrödinger equation" for the eigenfunction Ψ*m*(*θ*, *x*) = Ψ*m*(*θ*) exp{−2 *εmx*/*xT*} has the form:

$$-\frac{\partial^2 \Psi\_m(\theta)}{\partial \theta^2} - \left(\mathfrak{a}\_1 e^{i\mathfrak{v}\_1 \theta(\mathbf{x})} + \mathfrak{a}\_2 e^{-i\mathfrak{v}\_2 \theta(\mathbf{x})}\right) \Psi\_m(\theta) = \mathfrak{e}\_m \Psi\_m(\theta). \tag{A5}$$

In terms of the stationary eigenfunctions of this equation the propagator takes the form

$$G(\theta\_0, \theta\_L; L) = \sum\_m \Psi\_m(\theta\_0) \Psi\_m(\theta\_L) \, e^{-2\varepsilon\_{\mathfrak{M}} L/\chi\_T} \,. \tag{A6}$$

Finally the partition function (A3) is nothing but the Fourier transform of the propagator with respect to *θ*<sup>0</sup> and *θ<sup>L</sup>* and thus may be written as

$$\mathcal{Z} = \sum\_{m} \Psi\_{m}(q)\Psi\_{m}(q) \, e^{-2\varepsilon\_{m}L/\mathbf{x}\_{T}},\tag{A7}$$

where Ψ*m*(*q*) ≡ R *dθ*/(2*π*)Ψ*m*(*θ*) exp{*iθq*} = h*q*|*m*i is the quasi-momentum representation of the wavefunction in the *m*-th Bloch band with the energy *εm*. Instead of dealing with Bloch wavefunctions with the boundary condition Ψ*m*(*θ* + 2*π*) = *e <sup>i</sup>*2*πq*Ψ*m*(*θ*) one may perform a gauge transformation to deal with periodic wavefunctions and having *q* as the vector potential in the Schrödinger equation. This way we arrive at Equations (5) and (6) in the main text.

#### **References**


*Article*
