Fenley Andrew T, Gordon John C, Onufriev Alexey
Department of Physics, Virginia Tech, Blacksburg, Virginia 24061, USA.
J Chem Phys. 2008 Aug 21;129(7):075101. doi: 10.1063/1.2956497.
Analytical approximations to fundamental equations of continuum electrostatics on simple shapes can lead to computationally inexpensive prescriptions for calculating electrostatic properties of realistic molecules. Here, we derive a closed-form analytical approximation to the Poisson equation for an arbitrary distribution of point charges and a spherical dielectric boundary. The simple, parameter-free formula defines continuous electrostatic potential everywhere in space and is obtained from the exact infinite-series (Kirkwood) solution by an approximate summation method that avoids truncating the infinite series. We show that keeping all the terms proves critical for the accuracy of this approximation, which is fully controllable for the sphere. The accuracy is assessed by comparisons with the exact solution for two unit charges placed inside a spherical boundary separating the solute of dielectric 1 and the solvent of dielectric 80. The largest errors occur when the source charges are closest to the dielectric boundary and the test charge is closest to either of the sources. For the source charges placed within 2 A from the boundary, and the test surface located on the boundary, the root-mean-square error of the approximate potential is less than 0.1 kcal/mol/mid R:emid R: (per unit test charge). The maximum error is 0.4 kcal/mol/mid R:emid R:. These results correspond to the simplest first-order formula. A strategy for adopting the proposed method for realistic biomolecular shapes is detailed. An extensive testing and performance analysis on real molecular structures are described in Part II that immediately follows this work as a separate publication. Part II also contains an application example.
对简单形状的连续介质静电学基本方程进行解析近似，可以得到计算实际分子静电性质的计算成本较低的方法。在这里，我们推导了一个关于点电荷任意分布和球形介电边界的泊松方程的封闭形式解析近似。这个简单的、无参数的公式定义了空间中各处的连续静电势，它是通过一种近似求和方法从精确的无穷级数（柯克伍德）解中得到的，该方法避免了截断无穷级数。我们表明，保留所有项对于这种近似的准确性至关重要，对于球体来说，这种准确性是完全可控的。通过与放置在介电常数为1的溶质和介电常数为80的溶剂之间的球形边界内的两个单位电荷的精确解进行比较来评估准确性。当源电荷最接近介电边界且测试电荷最接近任何一个源时，误差最大。对于距离边界2 Å以内的源电荷，以及位于边界上的测试表面，近似势的均方根误差小于0.1 kcal/mol/|R|（每单位测试电荷）。最大误差为0.4 kcal/mol/|R|。这些结果对应于最简单的一阶公式。详细介绍了将所提出的方法应用于实际生物分子形状的策略。在紧随本文作为单独出版物的第二部分中描述了对真实分子结构的广泛测试和性能分析。第二部分还包含一个应用示例。
