Non-extensitivity and criticality of atomic hydropathicity around a voltage-gated sodium channel's pore: a modeling study

Abstract

Voltage-gated sodium channels (NavChs) are pore-forming membrane proteins that regulate the transport of sodium ions through the cell membrane. Understanding the structure and function of NavChs is of major biophysical, as well as clinical, importance given their key role in cellular pathophysiology. In this work, we provide a computational framework for modeling system-size-dependent, i.e., cumulative, atomic properties around a NavCh's pore. We illustrate our methodologies on the bacterial NavAb channel captured in a closed-pore state where we demonstrate that the atomic environment around its pore exhibits a bi-phasic spatial organization dictated by the structural separation of the pore domains (PDs) from the voltage-sensing domains (VSDs). Accordingly, a mathematical model describing packing of atoms around NavAb's pore is constructed that allows-under certain conservation conditions-for a power-law approximation of the cumulative hydropathic dipole field effect acting along NavAb's pore. This verified the non-extensitivity hypothesis for the closed-pore NavAb channel and revealed a long-range hydropathic interactions law regulating atom-packing around the NavAb's selectivity filter. Our model predicts a PDs-VSDs coupling energy of [Formula: see text] kcal/mol corresponding to a global maximum of the atom-packing energy profile. Crucially, we demonstrate for the first time how critical phenomena can emerge in a single-channel structure as a consequence of the non-extensive character of its atomic porous environment.

Keywords: Criticality; Hydropathicity; NavAb; Non-extensitivity; Scaling; Voltage-gated sodium channels.

Fig. 1

Atom-packing around NavAb’s pore. a Contour map of the normalized traces $\overline{N}(\mathrm{p},l_{\alpha }(\mathrm{p}\left)\right)$ for p ∈ Q and α = 1,2,..,K_α = 800 (see SI, Section S2 for construction of Q). Black lines R(p), D(p), and L(p) depict pore’s geometrical characteristics. Magenta dashed line ν(p) serves as a geometrical representation of the geometrical crossover from the PDs to the VSDs (see SI, Section S4). Line ξ(p) depicts the trace of inflection points, and lines s(p) and o(p) depict the ending and beginning of the lag and asymptote domain, respectively. b Traces of normalized model parameters $Ā\left(\mathrm{p}\right)$, $\overline{t}\left(\mathrm{p}\right)$, $\overline{os}\left(\mathrm{p}\right)$ and of summary model parameter $\tilde{q}\left(\mathrm{p}\right)$ for p ∈ Q. c Mean absolute fitting error (MAFE) of the mathematical atom-packing model approximation on $\overline{N}(\mathrm{p},l_{\alpha }(\mathrm{p}\left)\right)$ for p ∈ Q. Vertical lines roughly indicate the pore regions of the selectivity filter (SF), of the central cavity (CC) and of the activation gate (AG) (see Fig. 2 in [41]). All normalizations were performed with respect to the maximum values of the corresponding traces. ES stands for extracellular side. IS stands for intracellular side. Shaded area around model-parameter traces indicate confidence intervals

Fig. 2

Non-extensitivity of atomic hydropathicity around the NavAb’s pore. a Trace of a statistical representation of the normalized atomic number, ${\langle \overline{N}(\mathrm{p},l_{\alpha }(\mathrm{p}\left)\right)\rangle }_{\alpha }$, and of its best-fitted model, 〈n(p,l_α(p))〉_α, are plotted in log-scale. 〈n(p,l_α(p))〉_α corresponds to the Richards model. b Trace of a statistical representation of the hydropathic imbalance magnitude, 〈I(p,l_α(p))〉_α, is plotted with its linear fittings for 〈α_s〉 < α ≤ 〈α_ξ〉 and α > 〈α_ξ〉, respectively, in log-scale. Up- and downregulation tendencies were quantified in terms of the linear fittings $\gamma \cdot \log \left[\alpha \right]+\mathrm{\beta }$ with γ_pre = 1.41 and β_pre = − 8.4, γ_post = − 2.76 and β_post = 16.43. The corresponding Pearson coefficients are PC_pre = 0.99 and PC_post = − 0.97, respectively, indicating the “goodness” of the regulation. Error bars in a and b represent 95% interval values. For clarity, the size of error bars in b is reduced by a factor of 0.5. Calculation of ${\langle \overline{N}(\mathrm{p},l_{\alpha }(\mathrm{p}\left)\right)\rangle }_{\alpha }$, 〈n(p,l_α(p))〉_α, 〈I(p,l_α(p))〉_α, 〈α_s〉, 〈α_ξ〉, 〈α_ν〉, and 〈α_o〉 was performed according to the statistical scheme presented in SI, Section S5. All normalizations were performed with respect to the maximum values of the corresponding traces

Fig. 3

Non-extensive modeling of inter-atomic hydropathic interactions around NavAb’s pore. a Cartoon illustration of two opposite-facing PD structural units. The radii s(p) and ξ(p) centered at the middle of p_z ∈ [− 18.0,− 16.5] roughly account for the scales at which the lag domain and the pre-inflection domains end, i.e., for the sizes of the lag and pre-inflection domain, respectively. Residues T175, L176, E177, S178, and W179 forming the selectivity filter are colored according to their hydropathic score based on the Kapcha-Rossky scale. p_z ∈ [− 18.0,− 16.5] is directly neighbored by atomic components forming the M181 and S178 residue side chains. b Trace of the hydropathic inter-atomic interaction strength (HIIS), I(p,l_α(p)), for a randomly chosen pore point from the pore region p_z ∈ [− 18.0,− 16.5]. The best-fitting pre- and post-inflection power-law approximations of HIIS are also plotted with their mean absolute relative fitting errors being 0.089 ± 0.004 and 0.21 ± 0.008, respectively. c Trace of the corresponding atom-packing energy (AE), $\left|\right|{\overrightarrow{\mathit{h}}}_z(\mathrm{p},l_{\alpha }(\mathrm{p}\left)\right)\left|\right|/l_{\alpha }\left(\mathrm{p}\right)$. Pre- and post-inflection modeling approximations of AE are also plotted with their mean absolute relative modeling errors being 0.096 ± 0.005 and 0.21 ± 0.009, respectively. Model extrapolation toward l_α(p) ≤ s(p) results in a mean absolute relative fitting error of 1.77 ± 0.47. ν(p) and o(p) account for the scales at which the PDs-VSDs geometrical crossover occurs and the asymptote domain begins, respectively