求解離子通道之二階Poisson Nernst-Planck簡化法

Abstract

本論文運用基本的Poisson Nernst-Planck(PNP)數學模型來模擬生物細胞的離子通道在穩定狀態的情形,然而Poisson方程式可經由庫倫定律和高斯定裡來獲得,至於Nernst-Planck方程則是等價於漂移擴散數學模型,過去很多的計算方法都已經被應用在PNP數學模型上,然而我們的主要目的是去簡化在[24]裡面的計算方法,並且保持二階收斂的效果,但我們有一些必須要去處理的問題,例如我們應用位能分解方法[5]去處理Delta函數,使用合適的介面和邊界(matched interface and boundary)方法 [24]來處理不同介質的問題,而對於初始值猜測則必須由Poisson Boltzmman 方程得到,最後我們由分子資料銀行(protein data bank)得到真實的短杆菌肽(Gramicidin A)通道的資訊並且建構它的幾何結構。

The Poisson-Nernst-Planck (PNP) model is a basic continuum model for simulating ionic flows in an open ion channel. It is one of commonly used models in theoretical and computational. The Poisson equation is derived from Coulomb's law in electrostatics and Gauss's theorem in calculus. The Nernst-Planck equation is equivalent to the convection-diffussion model. Many computation methods have been constructed for the solution of the PNP equations. However, we want to simplify the second order solver of proposed in the literature [24] but, we must to deal with some problems. For example, singular charges, nonlinear coupling and interface. First, we apply the decomposition method [5] proposed by Chern, Liu,and Wang to cope with the singular charges. Second, the matched interface and boundary (MIB) method [24] is used for the interface problem. Third, The initial guess are given by Poisson Boltzmann (PB) equation and two iterative schemes are utilized to deal with the coupled nonlinear equations. Finally, the real data of Gramicidin A (GA) channel protein is obtained from the protein data bank (PDB).