Nonlocal Poisson-Fermi model for ionic solvent

Analytical solution of a linear nonlocal Poisson-Boltzmann equation with multiple charges in a spherical solute region surrounded by a water spherical shell

In this paper, an analytical solution of a linear nonlocal Poisson-Boltzmann equation (NPBE) test model with multiple charges in a spherical solute region surrounded by a water spherical shell is derived as a single series of Legendre polynomials and modified spherical Bessel functions. The classic Kirkwood ball model is then shown to be a special case of the NPBE test model so that its analytical solution is regained from a double series of associated Legendre polynomials (derived by Kirkwood in 1934) to a new single series of Legendre polynomials, sharply reducing its computational cost. As an application of these series solutions, a comparison study is done to demonstrate the differences between the Kirkwood and NPBE test models.

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

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.

Selectivity of the KcsA potassium channel: Analysis and computation

Ion channels regulate the flux of ions through cell membranes and play significant roles in many physiological functions. Most of the existing literature focuses on computational approaches based on molecular dynamics simulation or numerical solution of the modified Poisson-Nernst-Planck (PNP) system. In this paper, we present an analytical and computational study of a mathematical model of the KcsA potassium channel, including the effects of ion size (Bikerman model) and solvation energy (Born model). Under equilibrium conditions, we obtain an analytical solution of our modified PNP system, which is used to explain selectivity of KcsA of various ions (K^{+}, Na^{+}, Cl^{-}, Ca^{2+}, and Ba^{2+}) due to negative permanent charges inside the filter region and the effect of ion sizes. Our results show that K^{+} is always selected over Na^{+}, as smaller Na^{+} ions have larger solvation energy. As the amount of negative charges in the filter exceeds a critical value, divalent ions (Ca^{2+} and Ba^{2+}) can enter the filter region and block the KcsA channel. For the nonequilibrium cases, due to difficulties associated with a pure analytical or numerical approach, we use a hybrid analytical-numerical method to solve the modified PNP system. Our predictions of selectivity of KcsA channels and saturation phenomenon of the current-voltage (I-V) curve agree with experimental observations.

A Flux Ratio and a Universal Property of Permanent Charges Effects on Fluxes

In this work, we consider ionic flow through ion channels for an ionic mixture of a cation species (positively charged ions) and an anion species (negatively charged ions), and examine effects of a positive permanent charge on fluxes of the cation species and the anion species. For an ion species, and for any given boundary conditions and channel geometry,we introduce a ratio _(Q) = J(Q)/J(0) between the flux J(Q) of the ion species associated with a permanent charge Q and the flux J(0) associated with zero permanent charge. The flux ratio _(Q) is a suitable quantity for measuring an effect of the permanent charge Q: if _(Q) > 1, then the flux is enhanced by Q; if _ < 1, then the flux is reduced by Q. Based on analysis of Poisson-Nernst-Planck models for ionic flows, a universal property of permanent charge effects is obtained: for a positive permanent charge Q, if _1(Q) is the flux ratio for the cation species and _2(Q) is the flux ratio for the anion species, then _1(Q) < _2(Q), independent of boundary conditions and channel geometry. The statement is sharp in the sense that, at least for a given small positive Q, depending on boundary conditions and channel geometry, each of the followings indeed occurs: (i) _1(Q) < 1 < _2(Q); (ii) 1 < _1(Q) < _2(Q); (iii) _1(Q) < _2(Q) < 1. Analogous statements hold true for negative permanent charges with the inequalities reversed. It is also shown that the quantity _(Q) = |J(Q) − J(0)| may not be suitable for comparing the effects of permanent charges on cation flux and on anion flux. More precisely, for some positive permanent charge Q, if _1(Q) is associated with the cation species and _2(Q) is associated with the anion species, then, depending on boundary conditions and channel geometry, each of the followings is possible: (a) _1(Q) > _2(Q); (b) _1(Q) < _2(Q).

Low-voltage electrostatic modulation of ion diffusion through layered graphene-based nanoporous membranes

Ion transport in nanoconfinement differs from that in bulk and has been extensively researched across scientific and engineering disciplines^1-4. For many energy and water applications of nanoporous materials, concentration-driven ion diffusion is simultaneously subjected to a local electric field arising from surface charge or an externally applied potential. Due to the uniquely crowded intermolecular forces under severe nanoconfinement (<2 nm), the transport behaviours of ions can be influenced by the interfacial electrical double layer (EDL) induced by a surface potential, with complex implications, engendering unusual ion dynamics^5-7. However, it remains an experimental challenge to investigate how such a surface potential and its coupling with nanoconfinement manipulate ion diffusion. Here, we exploit the tunable nanoconfinement in layered graphene-based nanoporous membranes to show that sub-2 nm confined ion diffusion can be strongly modulated by the surface potential-induced EDL. Depending on the potential sign, the combination and concentration of ion pairs, diffusion rates can be reversibly modulated and anomalously enhanced by 4~7 times within 0.5 volts, across a salt concentration gradient up to seawater salinity. Modelling suggests that this anomalously enhanced diffusion is related to the strong ion-ion correlations under severe nanoconfinement, and cannot be explained by conventional theoretical predictions.

Nonlocal Poisson-Fermi double-layer models: Effects of nonuniform ion sizes on double-layer structure

This paper reports a nonuniform ionic size nonlocal Poisson-Fermi double-layer model (nuNPF) and a uniform ionic size nonlocal Poisson-Fermi double-layer model (uNPF) for an electrolyte mixture of multiple ionic species, variable voltages on electrodes, and variable induced charges on boundary segments. The finite element solvers of nuNPF and uNPF are developed and applied to typical double-layer tests defined on a rectangular box, a hollow sphere, and a hollow rectangle with a charged post. Numerical results show that nuNPF can significantly improve the quality of the ionic concentrations and electric fields generated from uNPF, implying that the effect of nonuniform ion sizes is a key consideration in modeling the double-layer structure.

Experimental charge density from electron microscopic maps

The charge density (CD) distribution of an atom is the difference per unit volume between the positive charge of its nucleus and the distribution of the negative charges carried by the electrons that are associated with it. The CDs of the atoms in macromolecules are responsible for their electrostatic potential (ESP) distributions, which can now be visualized using cryo-electron microscopy at high resolution. CD maps can be recovered from experimental ESP density maps using the negative Laplacian operation. CD maps are easier to interpret than ESP maps because they are less sensitive to long-range electrostatic effects. An ESP-to-CD conversion involves multiplication of amplitudes of structure factors as Fourier transforms of these maps in reciprocal space by 1/d² , where d is the resolution of reflections. In principle, it should be possible to determine the charges carried by the individual atoms in macromolecules by comparing experimental CD maps with experimental ESP maps.