  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 Poisson- Boltzmann-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. Current computational algorithms and tools for
numerical simulations and results from mathematical analysis of ion channel
systems are also surveyed.
