Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487, USA.
Department of Physics and Astronomy, Clemson University, Clemson, SC 29634, USA.
Math Biosci Eng. 2021 Jan 21;18(2):1370-1405. doi: 10.3934/mbe.2021072.
Numerical treatment of singular charges is a grand challenge in solving the Poisson-Boltzmann (PB) equation for analyzing electrostatic interactions between the solute biomolecules and the surrounding solvent with ions. For diffuse interface PB models in which solute and solvent are separated by a smooth boundary, no effective algorithm for singular charges has been developed, because the fundamental solution with a space dependent dielectric function is intractable. In this work, a novel regularization formulation is proposed to capture the singularity analytically, which is the first of its kind for diffuse interface PB models. The success lies in a dual decomposition - besides decomposing the potential into Coulomb and reaction field components, the dielectric function is also split into a constant base plus space changing part. Using the constant dielectric base, the Coulomb potential is represented analytically via Green's functions. After removing the singularity, the reaction field potential satisfies a regularized PB equation with a smooth source. To validate the proposed regularization, a Gaussian convolution surface (GCS) is also introduced, which efficiently generates a diffuse interface for three-dimensional realistic biomolecules. The performance of the proposed regularization is examined by considering both analytical and GCS diffuse interfaces, and compared with the trilinear method. Moreover, the proposed GCS-regularization algorithm is validated by calculating electrostatic free energies for a set of proteins and by estimating salt affinities for seven protein complexes. The results are consistent with experimental data and estimates of sharp interface PB models.
处理奇异电荷是求解泊松-玻尔兹曼（PB）方程以分析溶质生物分子与周围溶剂中离子之间静电相互作用的一个重大挑战。对于用光滑边界分隔溶质和溶剂的弥散界面 PB 模型，由于具有空间相关介电函数的基本解难以处理，因此尚未开发出用于奇异电荷的有效算法。在这项工作中，提出了一种新的正则化公式来解析地捕获奇点，这是弥散界面 PB 模型中的首创。成功之处在于双重分解——除了将电势分解为库仑和反应场分量外，介电函数也分为常数基底加空间变化部分。利用常数介电基底，通过格林函数解析表示库仑势。去除奇点后，反应场势满足具有平滑源的正则化 PB 方程。为了验证所提出的正则化，还引入了高斯卷积表面（GCS），它可以有效地为三维真实生物分子生成弥散界面。通过考虑分析和 GCS 弥散界面，以及与三次线性方法进行比较，检验了所提出的正则化的性能。此外，通过计算一组蛋白质的静电自由能和估计七种蛋白质复合物的盐亲和力，验证了所提出的 GCS-正则化算法。结果与实验数据和尖锐界面 PB 模型的估计一致。
