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

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.

Electric Double Layer and Orientational Ordering of Water Dipoles in Narrow Channels within a Modified Langevin Poisson-Boltzmann Model

“…The model value 3.1 D can be additionally decreased by taking into account structural correlations between water dipoles [ 60 , 78 ]. The ion-ion and ion-water correlations were taken into account also in the mean-field models of References [ 8 , 65 , 66 ].…”

“…The problem was also considered within a discrete lattice statistics model taking into account the asymmetric size of ions and orientational ordering of water dipoles [ 44 ]. Recently, ion-ion and ion-water correlations were also considered in a mean-field approach [ 65 , 66 ].…”

Maxwell Equations Without a Polarization Field, Using a Paradigm from Biophysics

“…(4) molecular models of ionic solutions that include water as a molecule. It is best to use models that are successful in predicting the activity of solutions of diverse composition and content and include water and ions as molecules of unequal nonzero size [52]. (5) classical models of impedance, dielectric, and molecular spectroscopy [16,[46][47][48][49][50][51].…”

“…They allow the design of circuits in our analog and digital electronic technology [101][102][103][104]. They allow the understanding of selectivity [52,[105][106][107] and current voltage relations of several important biological channel proteins in a wide range of solutions [52,[108][109][110]. In other cases-for example the description of ionic solutions with many components-the toy models can be too unrealistic to be useful.…”

“…Many of the atoms in a protein are assigned permanent charge greater than 0.2 in the force fields used in molecular dynamics, where is the elementary charge, and these charges tend to cluster in locations most important for biological function, just as they cluster at high density near the electrodes of batteries and other electrochemical systems. Enormous densities of charge (> 10M, sometimes much larger) are found in and near channels of proteins [24,52,111,112] and in the 'catalytic active sites' [113] of enzymes. Such densities are also found near nucleic acids, DNA and (all types of) RNA and binding sites of proteins in general.…”