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.

翻译：本文提出了一种用于求解非均匀尺寸修正泊松-能斯特-普朗克离子通道（SMPNPIC）模型的高效有限元迭代方法，并开发了相应的SMPNPIC程序包，适用于具有三维晶体结构的离子通道蛋白及含多种离子组分的离子溶剂。特别地，我们通过新颖的数学技术构建并重新表述了SMPNPIC模型，使得该方法的每次迭代仅涉及线性边值问题和非线性代数系统，从而规避了SMPNPIC模型因强非线性、强非对称性及强微分方程耦合带来的数值求解困难。为进一步提升方法效率，我们采用了一种高效修正牛顿迭代法来求解每个相关的非线性代数系统。针对电压依赖性阴离子通道（VDAC）及四种离子组分混合溶液的数值实验表明，该方法具有收敛性，程序包具有高性能，并凸显了考虑非均匀离子尺寸效应的重要性。实验结果还通过VDAC的阴离子选择性特性部分验证了SMPNPIC模型的有效性。