Liang J, Subramaniam S
National Center for Supercomputing Applications, Department of Molecular and Integrative Physiology, University of Illinois at Urbana-Champaign, Urbana 61801, USA.
Biophys J. 1997 Oct;73(4):1830-41. doi: 10.1016/S0006-3495(97)78213-4.
In continuum approaches to molecular electrostatics, the boundary element method (BEM) can provide accurate solutions to the Poisson-Boltzmann equation. However, the numerical aspects of this method pose significant problems. We describe our approach, applying an alpha shape-based method to generate a high-quality mesh, which represents the shape and topology of the molecule precisely. We also describe an analytical method for mapping points from the planar mesh to their exact locations on the surface of the molecule. We demonstrate that derivative boundary integral formulation has numerical advantages over the nonderivative formulation: the well-conditioned influence matrix can be maintained without deterioration of the condition number when the number of the mesh elements scales up. Singular integrand kernels are characteristics of the BEM. Their accurate integration is an important issue. We describe variable transformations that allow accurate numerical integration. The latter is the only plausible integral evaluation method when using curve-shaped boundary elements.
在分子静电学的连续介质方法中，边界元法（BEM）能够为泊松-玻尔兹曼方程提供精确解。然而，该方法的数值方面存在重大问题。我们描述了我们的方法，即应用基于阿尔法形状的方法来生成高质量网格，该网格能精确表示分子的形状和拓扑结构。我们还描述了一种将平面网格上的点映射到分子表面精确位置的解析方法。我们证明，导数边界积分公式比非导数公式具有数值优势：当网格单元数量增加时，条件良好的影响矩阵可以保持，条件数不会恶化。奇异被积核是边界元法的特征。它们的精确积分是一个重要问题。我们描述了允许精确数值积分的变量变换。当使用曲线形边界元时，后者是唯一合理的积分求值方法。
