Stationary solutions to the Poisson-Nernst-Planck equations with steric effects
Abstract
Ion transport, the movement of ions across a cellular membrane, plays a crucial role in a wide variety of biological processes and can be described by the Poisson-Nernst-Planck equations with steric effects (PNP-steric equations). In this paper, we shall show that under homogeneous Neumann boundary conditions, the steady-state PNP-steric equations are equivalent to a system of differential algebraic equations (DAEs). Analyzing this system of DAEs inspires us to propose an assumption on coupling constants, the so-called (H1) which will be introduced in Sec:model, such that if (H1) holds true, the steady-state PNP-steric equations admit a unique stationary C2 solution. Moreover, we shall point out the occurrence of bifurcation when (H1) is violated, which may relate to the opening and closing of the ion channels. When (H1) fails, we also suggest a simple criterion to check whether the system of DAE equations admits unique monotone C2 solutions; or unique monotone piecewise C2 solutions with vertical tangents; or triple piecewise C2 solutions. To the best of the authors' knowledge, this is the first time such DAE approach has been utilized to obtain a complete investigation for the steady-state PNP-steric equations of two counter-charged ion species