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

TL;DR

This paper introduces an efficient finite element iterative method and software package for solving a complex nonuniform size-modified Poisson-Nernst-Planck ion channel model, enabling accurate simulations of ion transport in biological channels.

Contribution

The paper develops a novel reformulation and an efficient Newton-based iterative solver for the SMPNPIC model, improving computational performance and handling strong nonlinearities.

Findings

Method converges reliably for complex ion channel models

Software demonstrates high computational efficiency

Results validate the importance of nonuniform ion size effects

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.