Mathematical models of ion transport through cell membrane channels

We discuss various models of ion transport through cell membrane channels. Recent experimental data shows that sizes of some ion channels are compared to those of ions and that only few ions may be simultaneously in any single channel. Theoretical description of ion transport in such channels should therefore take into account stochastic fluctuations and interactions between ions and between ions and channel proteins. This is not satisfied by macroscopic continuum models based on the Poisson-Nernst-Planck equations. More realistic descriptions of ion transport are offered by microscopic molecular and Brownian dynamics. We present a derivation of the Poisson-Nernst-Planck equations. We also review some recent models such as single-file diffusion and Markov chains of interacting ions (boundary driven lattice gases).Such models take into account discrete and stochastic nature of ion transport and specifically interactions between ions in ion channels.