Mathematical analysis of Poisson–Nernst–Planck models with permanent charges and boundary layers: studies on individual fluxes

Abstract

This work focuses on a one-dimensional Poisson–Nernst–Planck system including small permanent charges for ionic flows with one cation and one anion through a membrane channel. Our main interest is to examine the qualitative properties of the individual fluxes with boundary layers that is more realistic for ion channel problem study. Our result shows that the individual fluxes depend sensitively on multiple system parameters such as permanent charges, channel geometry, boundary conditions (concentrations and potentials) and boundary layers. For the relatively simple setting and assumptions of the model in this work, we are able to characterize the different effects of the nonlinear interaction among these system parameters in detail and gain a better understanding of the internal dynamics of ionic flows through membrane channels. Our analysis indicates that the small positive permanent charge cannot strengthen the flux of cation while reduce that of anion. Critical electric potentials which play crucial roles in studying ionic flow properties are identified. Some can be estimated experimentally. Numerical simulations are further performed and numerical results are consistent with our analytical ones.