Mathematical studies of Poisson–Nernst–Planck model for membrane channels: Finite ion size effects without electroneutrality boundary conditions

Abstract

We study a quasi-one-dimensional steady-state Poisson–Nernst–Planck model with a local hard-sphere potential for ionic flows of two oppositely charged ion species through a membrane channel. Of particular interest are qualitative properties of ionic flows in terms of individual fluxes J k without assuming electroneutrality boundary conditions . For the first order approximation J k 1 (in diameter of charged particle) of J k , our result shows that the quantities ∂ V J k 1 and ∂ V λ 2 J k 1 that play critical roles in characterizing ion size effects on ionic flows can be both negative depending sensitively on boundary concentrations and relative ion valences while they are always positive under electroneutrality conditions. This new observation indicates that, for fixed ion species, opposite finite size effects on ionic flows can occur determined further by boundary concentrations. Numerical simulation is performed to detect some critical potentials closely related to ion size effects. Numerical simulation results are consistent with our analytical predictions.