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.

本文提出了一种有效的有限元迭代算法，用于求解具有Neumann边值条件和膜表面电荷密度的Poisson-Nernst-Planck离子通道（PNPic）模型。它由解决方案分解方案（以避免原子电荷引起的奇异性问题），交替块迭代方案（可显着降低计算复杂性和计算机内存需求）以及Slotboom变量转换方案（可显着增强数值稳定性）以及改进的Newton构成。迭代方案，可以有效地求解每个相关的非线性有限元方程。然后，将该PNPic有限元求解器实现为一个软件包，该软件包可在多种离子物种的混合溶液中用于具有晶体结构的离子通道蛋白。此外，提出了一种有限元方案来计算溶剂区域块上势/浓度函数的体积积分。这项工作可以极大地提高可视化工具的准确性，该可视化工具用于以简单的二维曲线描绘跨膜的三维电位/浓度函数的分布模式。报告了在多达四个离子物种的溶液中小鼠电压依赖性阴离子通道同种型（mVDAC1）的数值结果。他们展示了PNPic迭代求解器的收敛性，软件包的性能以及可视化工具在不同势能和浓度函数的比较研究中的宝贵用法。他们还验证了该PNPic模型可以很好地保留mVDAC1的阴离子选择性。