1. Introduction
Plasma membranes in live cells are quasi-two-dimensional liquid mixtures of different lipids, cholesterol and transmembrane or anchored proteins. It is already well established that at the macroscopic length scale, the membranes in live cells are homogeneous, but at the mesoscale, which is larger than the range of the direct interactions between the membrane components, transient domains resembling liquid-ordered (Lo) and liquid disordered (Ld) phases appear [
1,
2,
3,
4,
5,
6,
7]. Numerous studies have confirmed that this heterogeneity follows from the fact that the thermodynamic state of the plasma membrane in live cells is close to the miscibility critical point [
2,
6]. Below the miscibility critical temperature, the separation into Lo and Ld phases would take place, and indeed, this was observed in membranes extracted from cells at temperatures lower than the temperature of the growing cell [
6,
7]. In the phase with mixed components, the size of the domains with the local composition different from the average composition increases when the thermodynamic state approaches the miscibility critical point, and depends on the ratio of the thermodynamic parameters to their critical values, not on the details of the interactions. For this reason, simplified models can be used to describe the universal critical phenomena [
3,
6,
8].
According to recent studies, the heterogeneity in lipid membranes plays an important role in biological functions [
6]. Some proteins are preferably soluble in the Lo phase, while other ones in the Ld phase, and the functions of many proteins depend on the local environment [
7]. When the Lo or Ld domains formed around certain membrane inclusions start to overlap, they induce effective interactions between the inclusions that are attractive or repulsive for the domains of the same or of different type, respectively [
6]. Such interactions are universal, i.e., are present between any selective objects immersed in any solvent that is close to the miscibility critical point, and are called the thermodynamic Casimir potential [
2,
8,
9,
10,
11,
12,
13,
14]. In the following, we use the term ’Casimir potential’ for the thermodynamic Casimir potential for simplicity of notation. However, the thermodynamic Casimir potential resulting from the confinement of density or concentration fluctuations near the critical point in fluids should not be mistaken with the original Casimir potential resulting from the confinement of quantum fluctuations.
The clustering of like proteins can be enhanced by the presence of ligands or antibodies, and the local concentration around the cluster differs from the average concentration in the whole membrane [
7]. In this context, the theoretical modeling of these highly dynamic compositional microdomains has remained quite challenging, since the mechanisms that regulate the shape, size, and also the lifetime of the spatially organized regions are influenced by lipid–lipid and lipid–protein interactions [
5,
15].
In general, attractive interactions of any origin lead to a macroscopic separation of dilute and dense phases at sufficiently low temperature. Effective attractions between the proteins, however, lead to the formation of clusters in different parts of the membrane, but a single droplet of the phase rich in proteins is not formed. One can argue that live cells are not in thermal equilibrium, and the coalescence of the clusters would take place on a larger time scale. Several other mechanisms that would inhibit further growth of the self-assembled clusters have been suggested [
6], but the issue remains controversial.
In this work, we suggest another possible reason for the inhibited further growth and coalescence of the clusters of membrane proteins. Macromolecules in cells are charged, and the electrostatic repulsion between proteins with like charges should counteract the aggregation. However, the ionic cloud around a charged object screens its charge. According to classical theories [
16,
17], the screening length (the size of the neutralizing ionic cloud) decreases with an increasing density of ions. Since the concentration of ions such as Na
+, K
+, Ca
2+, and Cl
− around macromolecules in cells is very large [
18], it is natural to assume that the screening length is much shorter than the range of the effective attractive interactions, and the screened electrostatic interactions can be neglected. Recently, however, it was discovered that in concentrated electrolytes, beyond about 0.5 M, the screening length starts to increase with an increasing density of ions [
19,
20,
21,
22]. The anomalous underscreening was experimentally observed in a number of concentrated electrolytes and studied by theory and simulations [
22,
23,
24,
25,
26,
27,
28,
29,
30]. The range of the electrostatic interactions between charged objects in concentrated electrolytes is very large; for example, the force between charged mica cylinders immersed in the 2 M NaCl
(aq) was measured up to 6 nm [
19]. The magnitude of these interactions at large distances, however, is small [
22]. The underscreening can be explained by ionic association into charge-neutral dimers or larger aggregates by which there remain fewer free ions capable of neutralizing the charged objects. The association becomes more efficient with an increasing density of ions, and the larger the density of ions, the less free ions remain in the system [
30,
31]. Importantly, the anomalous underscreening is universal in the sense that the screening length is proportional to the product of the density of ions and the Bjerrum length in various concentrated electrolytes [
20,
25]. The density of ions around macromolecules in live cells is very large—around DNA, it can be as large as 10 M [
18], therefore similar underscreening can be present in the case of charged membrane inclusions surrounded by a dense ionic cloud.
If the hypothesis of anomalous underscreening in live cells is correct, there are two universal interactions between charged membrane inclusions with preferential solubility in one type of lipids, namely the Casimir and the underscreened electrostatic interactions. Notably, these interactions are of the opposite sign. Like proteins attract and repel each other with the Casimir and the electrostatic potential, respectively. In contrast, the Casimir and the electrostatic potentials between oppositely charged proteins with different preferential solubility are repulsive and attractive, respectively.
Let us consider the effect of the competition between these two types of interactions on a general level. The total interaction between charged selective objects depends on the ranges and strengths of the attractive and repulsive contributions to the potential. Let us first assume that the range of the Casimir potential is larger, but its magnitude is smaller. In this case, the total interaction takes a minimum at a rather large distance, and the overal shape of the potential resembles the shape of the interactions between atoms or molecules [
32]. In this case, the macroscopic separation of dilute and dense phases should be expected. Indeed, experiments for colloidal particles in near-critical water–lutidine mixture have shown the formation of dense droplets in three-dimensional space upon approaching the critical temperature of the solvent [
12].
If the range of the underscreened electrostatic interactions is larger and its amplitude is smaller, the resulting potential between like proteins may have the form of short-range attraction and long-range repulsion (SALR) [
32]. Such interactions with different shapes and ranges of the attractive and repulsive parts have already been investigated in two-dimensional (2D) or quasi-2D systems [
13,
33,
34,
35,
36,
37], and in all cases, the formation of almost monodisperse clusters was observed at low densities and relatively high temperatures. At low temperatures, a dilute dispersion of particles is observed at very low density, and for increasing density, the sequence of stable phases is: clusters forming a hexagonal lattice, stripes, liquid with hexagonally ordered bubbles, and finally dense liquid. Only the size of the aggregates changes upon variation of the shape of the interactions. The stripe and hexagonal arrangements of domains and transitions between them were observed in a vesicle adhering to a supported lipid bilayer [
38], in agreement with theoretical predictions for ordered patterns [
33,
34,
35,
36] .
Since in concentrated ionic solutions the magnitude of the underscreened electrostatic interactions is small and its range is very large, we may expect that the charged membrane inclusions preferring one type of lipid may interact with the SALR potential if the local concentration of ions is well above 1M. Note that this hypothesis is consistent with the formation of almost monodisperse protein clusters surrounded by the preferred domains in the membranes. Of course, further studies should clarify if the effective interactions between certain membrane proteins in certain conditions indeed have the SALR form. Experimental studies on charged inclusions with preferential solubility in one type of lipid in a near-critical lipid bilayer surrounded by concentrated ionic solution should shed more light on this issue.
The self-assembly of the particles interacting with the effective SALR potential can be significantly influenced by the presence of oppositely charged membrane inclusions. Instead of focusing on particular cases, we ask again a very general question about spontaneous pattern formation in a mixture of oppositely charged particles or macromolecules with a different preferential solubility in components of a quasi-2D critical liquid mixture that, in turn, is located in a concentrated ionic solution. Our main question is what types of patterns are energetically favorable for different strengths and ranges of the interactions between like and different particles. Based on the properties of one-component SALR systems, we expect that the ordered low-temperature patterns lose their long-range order upon heating stepwise. First, the periodic spacial distribution of the aggregates is destroyed, but the short-range order is maintained, i.e., the aggregates do not disintegrate and become mobile. Further heating leads to the random distribution of the particles or molecules.
The mixture of self-assembling particles or molecules has attracted increasing attention [
39,
40,
41,
42,
43,
44,
45], but the question of how the self-assembled patterns depend both on the relative ranges and strengths of the attractive and repulsive parts of the potential and on the concentration of species remains open.
It should be mentioned that there may be many other sources of the SALR potential in biological and soft-matter systems, including the basic DLVO potential that is also relevant to protein interactions [
15]. In addition, interactions other than the SALR potential can lead to microsegregation if there is a competition between two length scales in the pair potential, such as, for instance, with core-softened potentials with attractive cross-interaction [
46,
47] or in the case of competing amphiphilic and solvophobic interactions [
48,
49]. Thus, studies on pattern formation in mixtures with competing interactions may concern a large class of systems with mesoscopic inhomogeneity.
Pattern formation by adsorbed particles can be affected by the curvature of the surface, as shown for example, for adsorption on a sphere [
50,
51]. In this work, we limited ourselves to large biological membranes compared to the size of the aggregates, and we considered the triangular lattice model introduced and studied in Refs. [
43,
44]. Previous studies focused on the interactions between like particles with repulsion at large distances being significantly stronger than the attraction at short distances. In addition to the patterns present in a one-component system, the alternating stripes of two components, parallel chains of alternating clusters, zig-zag patterns, and clusters of one component filling holes in a second component were obtained for some regions of the plane of chemical potentials (or densities of the two components) [
43,
45].
For the present work, we assume that the repulsive part of the potential is weaker than the attractive part. In the case of one kind of self-assembling particles, it was observed that weaker repulsion at large distances leads to larger clusters, but otherwise, the sequence of ordered patterns remained the same [
35]. Now, we address the question of if the same phases are present in the mixture for different shapes of the interactions, or if new patterns can emerge when the relative strength of the repulsive and attractive parts of the potential is varied. We find that the pattern formation is much more complex, and different sequences of ordered patterns are obtained when the interaction at large distances is weakened but the range remains unchanged. This result is in a strong contrast to the one-component case, where the universal pattern formation was found in 2D and 3D cases [
33,
34,
35,
36].
In
Section 2, we introduce the model and describe the simulation method. The results are presented in
Section 3. In
Section 3.1 and
Section 3.2, the energetically favorable patterns in an open system that exchanges molecules with the bulk reservoir are shown. The results presented in
Section 3.1 were obtained by theoretical calculations. The effects of thermal motion at low temperature (kinetic energy much smaller than the potential energy) are obtained by GCMC simulations and discussed in
Section 3.2. Finally,
Section 3.3 presents the simulation results for fixed numbers of particles in the canonical ensemble for different proportions of the first and the second component.
Section 4 contains the summary and conclusions of this paper.
2. Model and Simulation Method
Biological membranes are three-dimensional (3D) objects, but their thickness in the direction perpendicular to the lipid bilayer is comparable with the size of the membrane proteins, and the patterns are formed in the lateral directions. Thus, the membranes can be considered quasi-2D objects. A simplified model of a quasi-2D object is, for example, a mixture confined in a slit with a thickness that is a bit larger than the thickness of the largest molecules. Two-dimensional and quasi-two-dimensional models of self-assembling systems have been developed and studied, and in all cases, the same qualitative behavior concerning pattern formation was obtained [
13,
33,
34,
35,
36,
37,
45]. Thus, we assume that for pattern formation by transmembrane proteins or other membrane inclusions, the third dimension plays a subdominant role and 2D models predict the lateral patterns sufficiently well. In one of these 2D models [
35,
36], the particles occupied the cells of a triangular lattice for the simplicity of simulation, but this approximation did not alter the sequence of spontaneously formed patterns. A similar lattice model was introduced for the mixture of two-types of SALR particles with cross-interaction of opposite sign [
43].
Following ref. [
43], we assume that the particles or macromolecules occupy cells of a triangular lattice, and each cell can contain no more than one particle. The linear size of the cell, equal to the diameter of the particles, is taken as the length unit (
Figure 1). The particles in the cells separated by the vector
connecting the centers of the cells interact with the potential (see
Figure 1)
where
refer to the first and the second component, and
where
and
are the energies of the attraction and repulsion between like particles for
and
, respectively, and the cross-interaction is of the opposite sign. The lattice model defined above has the key features of a binary mixture of oppositely charged particles or macromolecules with different solubility in a mixture close to the miscibility critical point.
The energy of a particular distribution of particles is equal to the sum of energies of all interacting pairs, and is given by the formula
where
or 0 if the cell
is occupied by the particle of the
i-th component or not, respectively. The sum
is made over all lattice cells, and summation convention for repeated indexes is used. If the quasi-2D liquid containing the particles (or macromolecules) is allowed to exchange particles with a bulk reservoir, there is an additional contribution to the grand thermodynamic potential equal to the chemical potential
that is associated with an insertion of a particle of type
i to the system. In this case, we consider the thermodynamic Hamiltonian
where
is equal to the number of particles of the
i-th component. We choose
for the energy unit, and define the parameter
. The chemical potentials will be measured in
units as well.
The most probable distribution of the particles, corresponding to the minimum of E or H for a fixed number of particles or for equilibrium with the reservoir, respectively, is not destroyed by thermal motion at . If the energy of thermal motion is with denoting the Boltzmann constant and T the temperature, the potential energy dominates over the kinetic energy, and the minimum of E or H gives a fair approximation of the equilibrium structure. For increasing temperature, the competition between the entropy and the energy leads first to an increasing number of defects, next ordered domains with different orientation appear, and finally, a random distribution of the particles is present at very high T. Before analyzing the more complex case of , one should determine those ordered patterns that are energetically favored in the first place, and this is the purpose of this paper.
In our previous studies, we assumed
, since this case was thoroughly studied for the one-component system [
35,
36]. In the present work, we considered the ratio
to determine how the strength of like-particle repulsion for
influences the pattern formation. Now, one can expect a more complex scenario for the phase portrait of the mixture, since this relatively low repulsion might allow for a subtle competition between clustering and phase separation processes.
We follow the same strategy as in Ref. [
43]. The types of ordered patterns are found by Monte Carlo simulations at finite (but sufficiently low) temperatures for several points in the
plane, and
H was calculated for the unit cell of each periodic pattern using the periodic boundary conditions. The stability regions of different phases were determined by comparison with the thermodynamic Hamiltonian per lattice cell in the ordered phases. One advantage of the simulation methodology is that it provides information about the role of thermal fluctuations on the phase ordering of the mixture. In this context, in future studies, we will explore the evolution of the ordered patterns with temperature in some specific cases of interest.
The simulation procedure was described in detail in Ref. [
43]. In this scheme, we fix the temperature at low values and then produce scans in the chemical potential
of each species (
i =
), applying the parallel tempering method. This strategy involves running simulations in parallel for different samples of the system, each of them under fixed values of chemical potentials that are assigned after binning the interval of interest for the
’s. Then, starting from a lattice occupied at random by both species, particles are inserted or removed with probabilities that follow the Metropolis rule [
52]. Along the simulations and at regular time intervals, an exchange between neighboring copies at slightly different
’s is attempted, and then accepted or rejected according to a probability that involves the energy cost of such a move. We note that in a grand canonical ensemble (
), the number of particles (per unit area) for each species,
, fluctuates around an average value at equilibrium prescribed by the
’s.
The outcome of these parallel tempering runs will be the equilibrium average concentrations of each species in the whole membrane as a function of the chemical potential, a quantity that is an analog to an adsorption isotherm. Also the fluctuations of the particles concentrations are considered here, since they provide access to the compressibility of the mixture , defined as with being the variance of the type particle number density, i.e., .
In addition to the particle concentration and its fluctuations, the energy of the system given by Equation (
3) was also considered, together with the specific heat defined as
, which provides a thermodynamic signature for possible phase transitions.
Finally, we also performed simulations in the canonical ensemble (), aimed to describe the time evolution of the mixture at finite temperature and with conserved densities. Now, in addition to a constant temperature T, we also fixed the number of particles of both species during the simulation. In this case, the Monte Carlo moves consist of displacements of a randomly chosen particle to an empty neighboring cell. These moves are then accepted or rejected following the Metropolis rule.
4. Summary and Conclusions
We put forward a hypothesis that the further growth and coalescence of the clusters of membrane proteins are inhibited by underscreened electrostatic interactions. The hypothesis is based on the recent experimental discovery of long-range forces between charged objects in concentrated electrolytes [
19,
20,
21,
22] confirmed by simulations and theory [
23,
24,
25,
26,
27,
28,
29,
30]. Indeed, at large separations the electrostatic repulsion can dominate over the Casimir attraction associated with critical concentration fluctuations, if the latter is stronger but of a shorter range. On the other hand, the Casimir attraction dominates at short distances. Such effective short-range attraction–long-range repulsion (SALR) interactions can lead to the formation of well-separated clusters with optimal size if the number of proteins is not too large.
If the above hypothesis is correct, then the distribution of the membrane proteins can be governed by the competition between the Casimir and the electrostatic potentials. To study the outcomes of this competition on a very general level, we focused on a question of energetically favorable patterns in a binary mixture with the SALR potential between like particles and the cross-interaction of the opposite sign. To this end, we considered a triangular lattice model and focused on two cases. In the first case, the membrane proteins are in equilibrium with a reservoir characterized by the chemical potentials that control the density of the proteins. In the second case, the numbers of the proteins are fixed.
We found that the energetically favorable patterns depend significantly on the shape of the SALR potential, especially on the relative strength of its repulsive and attractive parts, and on the density and proportions of the membrane proteins. In our case of the repulsion to attraction ratio
, there are many more ordered patterns, and the ground state
is more complex than for the
studied in Ref. [
43]. This is in contrast to one-component SALR systems, where changes in the shape of the potential lead only to change in the size of the assemblies.
For a fixed number of particles, dispersed clusters are formed when the second component is absent. The addition of a small amount of the second component leads to the formation of a single two-component cluster with the proteins forming a regular pattern. The number of dispersed first-component clusters decreases, and the pattern in the two-component cluster changes with the increasing density of the second component. For comparable amounts of the two components, all particles are involved in the big single cluster (
Figure 7). Thus, the second component can work as a trap for the small clusters of the first component.
We conclude that the two universal competing forces can lead to surprisingly rich pattern formation in quasi-2D mixtures. The patterns can be controlled by the shape of the SALR potential, by equilibrium with the bulk reservoir, or by the numbers of the macromolecules in the membrane. Thermal motion destroys the long-range order, but a short-range partial order remains and is even more interesting, as shown by the simulation snapshots in
Figure 3,
Figure 4,
Figure 5 and
Figure 6.