關於離子通道之波頌-能斯特-普郎克方程組的研究

Abstract

這篇論文主要是在觀察波頌-能斯特-普朗克方程組的解的行為。首先我們用擾動方法做了兩個情形：第一、在低離子濃度的假設下，我們可以推得高門-霍金-凱茲公式，這個式子可以用來解釋神經脈衝的產生；第二、得到離子流動所引起的電流和膜電壓為線性關係的充分條件。另外還以數值方法模擬離子通過不同形狀的通道時，位置和離子濃度以及電流和電壓的關係圖。最後我們用能量變分方法推導出對應不同形狀的離子通道的新波頌-能斯特-普朗克方程組；藉由新的方程，我們可以做出通道形狀為指數函數的近似解。

In this paper, we use the perturbation methods and numerical simulations to observe the behavior of the solution of Poisson-Nernst-Planck equations. First, we do the low-concentration limit case to derive the Goldman-Hodgkin-Katz formula which can be used to explain the occurrence of the nerve impulse. In addition, we obtain a sufficient condition such that current and voltage hold a linear relationship. And for the channel with different wall shapes, we find the numerical solutions. Furthermore, by the energetic variational approach, we derive the modifier Nernst-Planck equation corresponding different geometries of channel. By the modifier equation, we find an approximate solution when the wall shape function of ion channel is an exponential function.