分子在溶液中靜電勢能之演算法

Abstract

在此篇論文中，我們利用數值方法研究了在三維空間中的波瓦松-波茲曼方程式，並以此來計算分子在溶液中的靜電勢能。在研究過程中有三點困難之處：第一，分子在溶液中形成的邊界，內外的電介質係數為不連續且差異懸殊。第二，分子的電荷會產生出位能的奇異性質。第三，在溶液中所得到的靜電勢能反應是非線性的。 於是我們採取了以下幾個步驟一一來解決這些困難的地方：首先，對第一點而言，我們採取藕合界面方法來處理橢圓界面問題；其次，針對第二點，自由空間中的電位能可以用來將靜電勢能中的奇異部份取出；最後，關於第三點非線性的部份，我們使用阻尼牛頓法以達到二階收斂。 在數值的觀察上，我們利用水分子做探測器以得到其收斂速度以及計算結果，並測試了恐水性的蛋白質(代碼:1crn)和親水性的蛋白質(代碼:1DNG)。顯示此方法確實對於分子周圍的電位能與電場均可以達到二階的準確度。

In this paper, we study the Poisson-Boltzmann equation (PBE) in three dimensions numerically for computing the electrostatic potential for molecules in solvent. There are three numerical difficulties:(i) discontinuity of the dielectric coefficients large contrast across the boundaries of the macromolecules and solvent, (ii) potential singularity arisen from point charges of the macromolecules, (iii) nonlinearities of the solvent response. We take the following steps to resolve these difficulties. For (i), we adopt the coupling interface method, which can deal with elliptic interface problems with large jumps of elliptic coefficients across interfaces. For (ii), the point charge potential in free space is used to remove the singularity of the potential. For (iii), we implement the damped Newton's method to achieve quadratic convergence. Numerical investigation for convergence rate and computation solution are performed for test probe and for a hydrophobic protein (PDB ID:1crn) and a hydrophilic protein (PDB ID:1DNG). It is shown the method is second-order accurate even for both the electric potential and the electric field around the molecular boundary.