An Efficient Finite Element Solver for a Nonuniform size-modified Poisson-Nernst-Planck Ion Channel Model

Abstract

This paper presents an efficient finite element iterative method for solving a nonuniform size-modified Poisson-Nernst-Planck ion channel (SMPNPIC) model, along with a SMPNPIC program package that works for an ion channel protein with a three-dimensional crystallographic structure and an ionic solvent with multiple ionic species. In particular, the SMPNPIC model is constructed and then reformulated by novel mathematical techniques so that each iteration of the method only involves linear boundary value problems and nonlinear algebraic systems, circumventing the numerical difficulties caused by the strong nonlinearities, strong asymmetries, and strong differential equation coupling of the SMPNPIC model. To further improve the method's efficiency, an efficient modified Newton iterative method is adapted to the numerical solution of each related nonlinear algebraic system. Numerical results for a voltage-dependent anion channel (VDAC) and a mixture solution of four ionic species demonstrate the method's convergence, the package's high performance, and the importance of considering nonuniform ion size effects. They also partially validate the SMPNPIC model by the anion selectivity property of VDAC.