Finite Ion Size and Permanent Charge in a Poisson–Nernst–Planck Ion Channel Model: Existence and a Critical Potential Phenomenon

Abstract

We consider a quasi-one-dimensional Poisson–Nernst–Planck (PNP) model of ionic transport through a membrane channel, incorporating two oppositely charged ion species, nonzero permanent charge distributions, and nonuniform finite ion sizes modeled via hard-sphere potentials. First, we establish a rigorous existence and local uniqueness result for the steady-state PNP system using geometric singular perturbation theory, treating the finite ion sizes as small parameters. Second, we conduct a detailed asymptotic analysis of zero-current ionic flows for small permanent charge. This analysis reveals a nonlinear interplay between finite ion size effects and fixed charge, including the existence of a critical membrane potential at which the influence of permanent charge on the ionic flux reverses sign. These findings highlight novel nonlinear effects arising from finite ion size and permanent charge in ionic flows, and offer insights into biological ion channel behavior that are beyond the reach of current experimental techniques.