*4.1. Materials*

N-Dodecyl-β-D-maltoside (DDM) ULTROL® Grade was from Merck (Madrid, Spain). Hepes, succinic acid, N-methyl-D-glucamine (NMG), NaCl, KCl, tetraoctylammonium (TOA+) chloride, and dimethylsulfoxide (DMSO) were from Sigma-Aldrich (Madrid, Spain). Ni2+-Sepharose Fast Flow resin was from GE Healthcare (Madrid, Spain).

#### *4.2. KcsA Heterologous Expression and Purification*

The quadruple mutant of KcsA bearing a single Trp per monomer (W67 KcsA) was generated by mutating the rest of the native Trp residues 26, 68, 87, and 113 to Phe (W26, 68, 87, 113F). Wild-type (WT) and W67 channels were expressed in *E.coli* M15 (pRep4) and purified by Ni2+/His-tag affinity chromatography according to previous reports [34,58]. Proteins were purified in 20 mM HEPES buffer, pH 7.0, containing 5 mM DDM, 5 mM NMG, and different concentrations of KCl or NaCl. To prepare samples at pH 4.0, aliquots of the above were dialyzed against 10 mM succinic acid buffer, pH 4.0, containing 5 mM DDM, 5 mM NMG, and the corresponding amounts of NaCl or KCl. The tetrameric state of the protein was routinely checked by SDS-PAGE (12%) [59].

#### *4.3. Thermal Denaturation Assay and Cation Binding Analysis*

Thermal denaturation of WT and W67 KcsA channels was performed in a Varian Cary Eclipse spectrofluorometer by recording the temperature dependence of the protein intrinsic emission fluorescence at 340 nm after excitation at 280 nm, as previously described [37]. In these experiments, the protein was diluted to 1 μM protein concentration (in terms of monomers of KcsA) in either 20 mM Hepes buffer, pH 7.0, 5 mM DDM, and 10 mM NaCl (pH 7.0 buffer) or 10 mM succinic acid buffer, pH 4.0, 5 mM DDM, and 10 mM NaCl (pH 4.0 buffer), with additional amounts of NaCl, KCl, or TOA+, as required. The presence of 10 mM NaCl to start all titration experiments was required to maintain the tetrameric structure of the protein channels and to provide minimal stability to the proteins to start the thermal denaturation recordings.

When using TOA+, a concentrated stock (22 mM) was prepared in DMSO, and then aliquots from this stock were added to the samples and incubated for 30 min at room temperature. For titrations of either Na+ or K+ in the presence of TOA+, the same procedure as before was done, adding Na+ or K+ from a concentrated stock in the final step. The final amount of DMSO in the sample was always less than 1%, which, per se, has no effects on protein thermal stability.

The midpoint temperature of dissociation and unfolding of the tetramer (*<sup>t</sup>*m, in Celsius) was calculated from the thermal denaturation curves by fitting a two-state unfolding model to the data [38]. The dissociation constants of the KcsA–cation complexes (KDs) can be estimated from:

$$\frac{|\Delta \mathbf{T\_m}|}{T\_\mathbf{m}} = \frac{|\mathbf{T\_m} - (\mathbf{T\_m})\_0|}{T\_\mathbf{m}} = \frac{\mathbb{R}(\mathbf{T\_m})\_0}{\Delta \mathbf{H}\_0} \ln\left(1 + \frac{[\mathbf{L}]}{\mathbf{K\_D}}\right) \tag{1}$$

where T m and (T m)0 refer to the denaturation temperature (in Kelvin) for the protein in the presence and absence of ligands, respectively, R is the gas constant, and Δ H0 is the enthalpy change upon protein denaturation in the absence of ligands. The change in T m in the absence and presence of ligands ( Δ T m) is expressed in absolute terms (| Δ T m|) since the presence of a given ligand can induce either an increase or a decrease of the observable value.

#### *4.4. Steady-State Fluorescence Measurements*

The steady-state fluorescence anisotropy of W67 KcsA was measured on a Horiba Jobin Yvon Fluorolog-3-21 or SLM 8000 spectrofluorometer and calculated as:

$$<\text{ r}>\_{\text{SS}} = \frac{\text{I}\_{\text{VV}} - \text{G} \cdot \text{I}\_{\text{VH}}}{\text{I}\_{\text{VV}} + 2 \,\text{G} \cdot \text{I}\_{\text{VH}}} \tag{2}$$

where IVV and IVH are the fluorescence intensities (blank subtracted) of the vertically and horizontally polarized emission when the sample is excited with vertically polarized light, respectively, and the G factor (G = IHV/IHH) is the instrument correction factor. The samples were measured at 340 nm using an excitation wavelength of 300 nm [34]. A final protein concentration of 5 μM in either the pH 7.0 or pH 4.0 buffer, with or without TOA+, was used throughout. Ten measurements were done for each sample to calculate the average steadystate anisotropy values ( ±standard deviation).

#### *4.5. Time-Resolved Fluorescence Measurements and Intersubunit Distance Calculations*

Time-resolved fluorescence and anisotropy measurements with picosecond resolution were obtained using the time-correlated single-photon timing (SPT) technique. The fluorescence decays were measured at 345 nm (λexc = 300 nm) using an emission polarizer set at the magic angle (54.7◦) relative to the vertically polarized excitation beam produced by a frequency-doubled Rhodamine 6 G laser [60].

The fluorescence and anisotropy decays were analyzed as described [34]. The W67–W67 intersubunit lateral distances were calculated from the time-resolved anisotropy measurements according to:

$$\mathbf{r}(\mathbf{t}) = \frac{\mathbf{r}(0)}{4} \left[ 1 + \mathbf{e}^{-4\mathbf{k}\_1 \mathbf{t}} + 2\mathbf{e}^{(-\frac{9}{4}\mathbf{k}\_1 \mathbf{t})} \right] \cdot \mathbf{e}^{-\frac{\mathbf{t}}{q\_\mathbf{B}}} \tag{3}$$

where r(0) is the initial anisotropy, *k*1 is the rate constant for homo-FRET between neighboring tryptophan residues, and ϕg is the rotational correlation time of the KcsA–DDM complex (43 ns) [34]. The intertryptophan lateral distance R can be directly calculated via *k*1:

$$k\_1 = \frac{1}{\tau} \left(\frac{R\_0}{R}\right)^6 \tag{4}$$

where τ is the intensity-weighted mean fluorescence lifetime, and R0 is the critical radius computed with an orientation factor κ2 = 2/3. The goodness of fit was evaluated by statistical criteria (random distribution of weighted residuals and autocorrelation plots and a reduced χ2 < 1.2).

#### *4.6. Calculation of the K+ Binding Affinity to KcsA from Changes in the Steady-State Anisotropy Values*

The binding of K+ to KcsA W67, monitored by the changes in the <r>SS, presented a sigmoidal behavior that was also used to calculate the binding affinity of the channel for this permeant cation. Since we were mostly interested in the K+ binding event related to the equilibrium between the nonconductive (KcsA(NC)) and conductive (pH 7.0) or inactivated (pH 4.0) SF conformations of the channel (KcsA•K+(C/I)), we analyzed this second binding event, which can be described by a two-state equilibrium:

$$\text{KcsA}\_{\text{(NC)}} + \text{K}^+ \leftrightarrow \text{KcsA} \bullet \text{K}^+ \_{\text{(C/I})}$$

The total concentration of protein corresponds to:

$$\left[\text{KcsA}\right]\_{\text{total}} = \left[\text{KcsA}\right]\_{\text{NC}} + \left[\text{KcsA}\bullet\text{K}^{+}\right]\_{\text{C}/\text{I}}\tag{5}$$

and the dissociation constant (KD) is therefore defined by:

$$\text{Kp} = \frac{[\text{KcsA}]\_{\text{NC}^\cdot} \text{[K}^+]}{[\text{KcsA} \bullet \text{K}^+]\_{\text{C}/\text{I}}} \tag{6}$$

It could be assumed that the binding of a cation X+ to the protein is a cooperative process that is described by an empiric Hill function (sigmoid curve), so the free and bound molar fractions of KcsA as a function of cation concentration can be expressed as:

$$\mathbf{x}\_{\text{free}} = \mathbf{x}\_{\text{NC}} = \frac{[\text{KcsA}]\_{\text{NC}}}{[\text{KcsA}]\_{\text{total}}} = \frac{\mathbf{K}\_{\text{d}}^{\text{h}}}{\mathbf{K}\_{\text{d}}^{\text{h}} + \left[\mathbb{X}^{+}\right]^{\text{h}}} \tag{7}$$

$$\mathbf{x}\_{\text{bound}} = \mathbf{x}\_{\text{C/I}} = \frac{[\text{KcsA}]\_{\text{C/I}}}{[\text{KcsA}]\_{\text{total}}} = \frac{[\text{X}^+]^\text{h}}{\mathbf{K}\_\text{d}^\text{h} + [\text{X}^+]^\text{h}} \tag{8}$$

where *h* is the Hill coefficient (cooperativity index).

The steady-state fluorescence anisotropy of a sample, <r>SS, prepared at a given X+ concentration is a weighted average of its limiting values, i.e., the steady-state fluorescence anisotropy of the nonconductive and conductive/inactivated conformations of KcsA, <r>NC, and <r>C/I, respectively. However, at variance with the usual case, the weighting factors here are the relative fluorescence intensities emitted by each species and not their molar fractions directly [61]. Mathematically, this is described as:

$$\mathbf{x} < \mathbf{r}> \quad = \frac{\mathbf{x}\_{\text{NC}} \cdot \Phi\_{\text{NC}}}{\mathbf{x}\_{\text{NC}} \cdot \Phi\_{\text{NC}} + \mathbf{x}\_{\text{C}/\text{I}} \cdot \Phi\_{\text{C}/\text{I}}} \cdot < \mathbf{r}> \\ \mathbf{x} > \mathbf{x}\_{\text{NC}} + \frac{\mathbf{x}\_{\text{C}/\text{I}} \cdot \Phi\_{\text{C}/\text{I}}}{\mathbf{x}\_{\text{NC}} \cdot \Phi\_{\text{NC}} + \mathbf{x}\_{\text{C}/\text{I}} \cdot \Phi\_{\text{C}/\text{I}}} \cdot < \mathbf{r}>\_{\text{C}/\text{I}} \tag{9}$$

where ΦNC and ΦC/I are the quantum yields of the nonconductive and conductive (pH 7.0) or inactivated states (pH 4.0), respectively.

If we define a parameter Q as the relative change in KcsA quantum yield upon K+ binding (ΦC/I/ΦNC) and combine the equations from above, the general expression that can be used to fit <r>SS data obtained in equilibrium binding studies is, therefore:

$$<\mathbf{r}> = <\mathbf{r}>\_{\text{NC}} + (\mathbf{Q} < \mathbf{r}>\_{\text{C}/\text{I}} - <\mathbf{r}>\_{\text{NC}}) \cdot \frac{\left[\mathbf{X}^{+}\right]^{\text{h}}}{\mathbf{K}\_{\text{d}}{\text{h}} + \left[\mathbf{X}^{+}\right]^{\text{h}}} \tag{10}$$

During the fitting procedures, the Q parameter was fixed at a constant value of 0.95 and 0.8 for the experiments carried out at pH 7.0 and pH 4.0, respectively.

**Author Contributions:** Conceptualization: J.A.P., M.P., and J.M.G.-R.; formal analysis: A.M.G., M.L.R., A.C., and C.D.-G.; investigation: A.M.G., C.D.-G., and M.L.R.; writing—original draft: J.M.G.-R.; writing—review and editing: M.L.R., C.D.-G., A.C., M.P., J.A.P., and J.M.G.-R. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was partly supported by grants PGC2018-093505-B-I00 from the Spanish "Ministerio de Ciencia e Innovación"/FEDER, UE, and UIDB/04565/2020 from FCT, Portugal. C.D.-G. acknowledges support from the Medical Biochemistry and Biophysics Doctoral Programme (M2B-PhD) FCT (reference: SFRH/PD/BD/135154/2017).

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

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data is contained in the article.

**Acknowledgments:** The time-resolved fluorescence intensity and anisotropy measurements were performed by Aleksander Fedorov at the IBB in Lisbon (Portugal). Eva Martinez provided excellent technical help throughout this work.

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