Ionic diffusion through protein channels: From molecular description to continuum equations

Ionic permeation through protein channels is a process of considerable physiological importance. Permeation occurs on a microsecond time scale, so direct molecular simulations with femtosecond time increments are not possible and a coarser description is unavoidable. Standard continuum formulations based on macroscopic conservation laws, such as the Poisson-Nernst-Planck equations, however, cannot be assumed valid in narrow channels. Thus the problem at hand is the description of ionic diffusion through narrow regions by averaged continuum equations. In this paper we propose a mathematical averaging procedure that, starting from a Langevin model of ionic motion, yields a coupled system of Poisson and Nernst-Planck type equations, containing conditional and unconditional charge densities. The proposed system of equations includes molecular details such as excluded volume effects and the dielectric force on a discrete ion that are absent in the PNP system.