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.
Highlights
Continuum partial differential equations (PDEs) 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 [10,17,21,95,121,176,177,178,179,180,181,182,183,184,185]
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 molecular dynamics (MD) simulations of various channel models [199,200,201] indicating that the steric (Figure 7R), correlation, dehydration (Figure 8L), and dielectric (Figure 8R) properties have made
We developed a cyclic model of Na+ /Ca2+ exchange mechanism in Na+ /Ca2+ exchanger (NCX) [70] using (60) to calculate five total potential states (TPS) of various Na+ and Ca2+ ions shown in Figure 12R, where TPS1 and TPS4 are stable and TPS2, TPS3, and TPS5 are unstable
Summary
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 [64,65,66,67,68,69,70,71,72,73,74,75,76]. 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. 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 not consistent with classical theories and may poorly estimate steric energies and other physical properties due to their coarse approximation of size effects.
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