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

B. Nadler, Z. Schuss, A. Singer and B. Eisenberg
Department of Mathematics, Yale University, US

Keywords: ion channels, ionic diffusion, coarse graining

Abstract:
Diffusion of particles through a microscopic region, as in ionic permeation through protein channels, is an important phenomenon in many diverse fields. Permeation through ion channels occurs on a microsecond time scale, far longer than the femtosecond scale of atomic motion. Since direct molecular dynamics simulations are not possible for such long time scales, a coarser description is unavoidable. Standard continuum formulations based on macroscopic conservation laws, such as the Poisson-Nernst-Planck equations, cannot obviously be assumed valid in narrow channels. This leads to the more general coarse graining problem: The description of the diffusive motion of interacting particles in a confined region connected to a bath by averaged continuum equations. In this paper we propose a mathematical averaging procedure that, starting from a Langevin model of ionic motion, yields effective continuum equations and boundary conditions. Our main result is a coupled system of Poisson and Nernst-Planck type equations, containing conditional and unconditional charge densities, coupled to conditional potentials. The proposed system of equations differs from the standard PNP system used so far in two important aspects. First, the force term depends on conditional densities, and second, it contains the dielectric self force on a discrete ion near dielectric interfaces.

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