Date of Award

August 2020

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Department

Mathematics

First Advisor

Dexuan Xie

Committee Members

Dexuan Xie, Hans Volkmer, Chao Zhu, Peter Hinow, Lei Wang

Keywords

Dirichelt boundary condition, Electrostatic potential, Finite element method, Ion channel, Ionic concentration, Poisson-Nernst-Planck equation

Abstract

The modeling and simulation of ion channel proteins are essential to the study of many vital physiological processes within a biological cell because most ion channel properties are very difficult to address experimentally in biochemistry. They also generate a lot of new numerical issues to be addressed in applied and computational mathematics. In this dissertation, we mainly deal with some numerical issues that are arisen from the numerical solution of one important ion channel dielectric continuum model, Poisson-Nernst-Planck (PNP) ion channel model, based on the finite element approximation approach under different boundary conditions and unstructured tetrahedral meshes. In particular, we present the derivation of an improved PNP ion channel model using Dirichlet boundary value conditions and membrane surface charges, and obtain its variational formulations. To solve this PNP ion channel model numerically, we develop a fast finite element iterative method and program it as a software package by using effective numerical techniques. This work makes it possible for us to carry out numerical tests in order to study the affection of different boundary value conditions on the PNP numerical solutions.

To solve a PNP ion channel model by the finite element method, one important task is to generate an interface fitted unstructured tetrahedral mesh but it is very challenging to complete since the PNP ion channel model involves three physical regions -- a protein region, a membrane region, and a solvent region, and the interfaces between these three regions are very complex. To address this mesh challenge, in this dissertation, we develop a new algorithm for generating a triangular surface mesh of a simulation box domain and a new algorithm for constructing a tetrahedral mesh of the membrane region, such that we can easily split a mesh of the simulation box domain into three submeshes --- the meshes of protein, membrane, and solvent regions in high quality. Remarkably, our membrane mesh generation algorithm works for an ion channel protein with an irregular ion channel pore provided that a triangular mesh of the interface between the membrane and protein regions does not have any hole. Furthermore, we implement these two new mesh algorithms based on the state-of-the-art package FEniCS, and then adapt them to one commonly-used ion channel mesh generation package.

With our PNP ion channel program package, we study the impacts of boundary value conditions, membrane surface changes, and simulation box sizes on the quality of a PNP ion channel model. Such studies are done numerically by using crystallographic molecular structures of ion channel proteins in a solution of multiple ionic species. We visualize the three-dimensional electrostatic potential and ionic concentrations not only in color mapping on a cross-section of protein, membrane, or solvent region but also in two-dimensional curves with curve values being the average values of potential and concentration functions over a block partition of the solvent region along the membrane normal direction.

%We also test whether the channel conductance and charge selectivity obtained from the PNP ion channel model could be sensitive to some mutations of an ion channel protein.

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