A finite element iterative solver for a PNP ion channel model with Neumann boundary condition and membrane surface charge

Abstract

In this paper, an effective finite element iterative algorithm is presented for solving a Poisson-Nernst-Planck ion channel (PNPic) model with Neumann boundary value condition and a membrane surface charge density. It is constructed by a solution decomposition scheme to avoid singularity problems caused by atomic charges, an alternating block iterative scheme to sharply reduce computation complexity and computer memory requirement, and a Slotboom variable transformation scheme to significantly enhance numerical stability, as well as a modified Newton iterative scheme to efficiently solve each related nonlinear finite element equation. This PNPic finite element solver is then implemented as a software package that works for an ion channel protein with a crystallographic structure in a mixture solution of multiple ionic species. Furthermore, a finite element scheme is presented to compute a volume integral of a potential/concentration function over a block of a solvent region. This work can greatly improve the accuracy of a visualization tool for depicting the distribution pattern of a three-dimensional potential/concentration function across membrane in a simple two-dimensional curve. Numerical results for a mouse voltage-dependent anion-channel isoform (mVDAC1) in a solution of up to four ionic species are reported. They demonstrate the convergence of the PNPic iterative solver, the performance of the software package, and the valuable usage of the visualization tool in the comparison study of different potential and concentration functions. They also validate that this PNPic model can well retain the anion selectivity property of mVDAC1.