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

Dexuan Xie

TL;DR

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.

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.

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

TL;DR

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.

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.
Paper Structure (7 sections, 3 theorems, 47 equations, 5 figures, 3 tables)

This paper contains 7 sections, 3 theorems, 47 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. A nonuniform size-modified PNP ion channel model
  3. Reformations of SMPNPIC
  4. A damped block iterative method for computing a SMPNPIC finite element solution
  5. An iterative method for solving each related nonlinear algebraic system
  6. SMPNPIC software package and numerical results
  7. Conclusions

Key Result

Theorem 3.1

The size-modified Nernst-Planck boundary value problem SMPNPIC_NP can be reformulated by the mathematical transformation out_transform as follows: where $c=(c_1,c_2,\ldots,c_n)$, $\widehat{{\cal D}}_i$ is a transformed diffusion function in the expression and $\bar{g}_i$ is a transformed boundary value function in the expression

Figures (5)

  • Figure 1: An illustration of the domain partition \ref{['DomainDecomp']} and surface partition \ref{['DomainDecomp2']}.
  • Figure 2: Two views of a crystallographic three-dimensional molecular structure of VDAC (PDB ID: 5XD0) in cartoon representations.
  • Figure 3: A comparison of the interface fitted tetrahedral box mesh $\Omega_h$ and tetrahedral solvent domain mesh $D_{s,h}$ in the case of Mesh 1 with those in the case of Mesh 2. Here the membrane region mesh $D_{m,h}$ and protein region mesh $D_{p,h}$ are colored in yellow and green, respectively, and the mesh data are given in Table \ref{['table:meshData']}.
  • Figure 4: A color mapping of the ionic concentrations generated by our SMPNPIC software package on a cross-section ($y=0$) of the solvent region $D_s$ for the VDAC (PDB ID: 5XD0) and the solution of four ionic species Cl$^-$, NO$_3^-$, Na$^+$, and K$^+$.
  • Figure 5: Mesh size influence on the ionic concentrations calculated by SMPNPIC.

Theorems & Definitions (3)

  • Theorem 3.1
  • Theorem 3.2
  • Theorem 5.1