A Review of Mathematical Modeling, Simulation and Analysis of Membrane Channel Charge Transport

The molecular mechanism of ion channel gating and substrate modulation is elusive for many voltage gated ion channels, such as eukaryotic sodium ones. The understanding of channel functions is a pressing issue in molecular biophysics and biology. Mathematical modeling, computation and analysis of membrane channel charge transport have become an emergent field and give rise to significant contributions to our understanding of ion channel gating and function. This review summarizes recent progresses and outlines remaining challenges in mathematical modeling, simulation and analysis of ion channel charge transport. One of our focuses is the Poisson-Nernst-Planck (PNP) model and its generalizations. Specifically, the basic framework of the PNP system and some of its extensions, including size effects, ion-water interactions, coupling with density functional theory and relation to fluid flow models. A reduced theory, the PoissonBoltzmann-Nernst-Planck (PBNP) model, and a differential geometry based ion transport model are also discussed. For proton channel, a multiscale and multiphysics Poisson-Boltzmann-Kohn-Sham (PBKS) model is presented. We show that all of these ion channel models can be cast into a unified variational multiscale framework with a macroscopic continuum domain of the solvent and a microscopic discrete domain of the solute. The main strategy is to construct a total energy functional of a charge transport system to encompass the polar and nonpolar free energies of solvation and chemical potential related energies. Using the Euler-Lagrange variation, the coupled PNP equations and other transport equations are derived, whose solutions lead to the minimization of the total free energy and explicit profiles of electrostatic potential and densities of charge species. Current computational algorithms and tools for numerical simulations and results from mathematical analysis of ion channel systems are also surveyed.