Department of Computer Science and Genome Center, University of California, Davis, California 95616, USA.
J Chem Phys. 2010 Feb 14;132(6):064101. doi: 10.1063/1.3298862.
The Poisson-Boltzmann (PB) formalism is among the most popular approaches to modeling the solvation of molecules. It assumes a continuum model for water, leading to a dielectric permittivity that only depends on position in space. In contrast, the dipolar Poisson-Boltzmann-Langevin (DPBL) formalism represents the solvent as a collection of orientable dipoles with nonuniform concentration; this leads to a nonlinear permittivity function that depends both on the position and on the local electric field at that position. The differences in the assumptions underlying these two models lead to significant differences in the equations they generate. The PB equation is a second order, elliptic, nonlinear partial differential equation (PDE). Its response coefficients correspond to the dielectric permittivity and are therefore constant within each subdomain of the system considered (i.e., inside and outside of the molecules considered). While the DPBL equation is also a second order, elliptic, nonlinear PDE, its response coefficients are nonlinear functions of the electrostatic potential. Many solvers have been developed for the PB equation; to our knowledge, none of these can be directly applied to the DPBL equation. The methods they use may adapt to the difference; their implementations however are PBE specific. We adapted the PBE solver originally developed by Holst and Saied [J. Comput. Chem. 16, 337 (1995)] to the problem of solving the DPBL equation. This solver uses a truncated Newton method with a multigrid preconditioner. Numerical evidences suggest that it converges for the DPBL equation and that the convergence is superlinear. It is found however to be slow and greedy in memory requirement for problems commonly encountered in computational biology and computational chemistry. To circumvent these problems, we propose two variants, a quasi-Newton solver based on a simplified, inexact Jacobian and an iterative self-consistent solver that is based directly on the PBE solver. While both methods are not guaranteed to converge, numerical evidences suggest that they do and that their convergence is also superlinear. Both variants are significantly faster than the solver based on the exact Jacobian, with a much smaller memory footprint. All three methods have been implemented in a new code named AQUASOL, which is freely available.
泊松-玻尔兹曼（PB）形式是最受欢迎的建模分子溶剂化方法之一。它假设水的连续体模型，导致介电常数仅取决于空间中的位置。相比之下，偶极泊松-玻尔兹曼-朗之万（DPBL）形式将溶剂表示为具有不均匀浓度的可取向偶极子的集合；这导致了一个非线性介电常数函数，它不仅取决于位置，还取决于该位置的局部电场。这两个模型的假设差异导致它们生成的方程有很大差异。PB 方程是一个二阶、椭圆、非线性偏微分方程（PDE）。它的响应系数对应于介电常数，因此在系统考虑的每个子域内（即，在考虑的分子内外）都是常数。虽然 DPBL 方程也是一个二阶、椭圆、非线性 PDE，但它的响应系数是静电势的非线性函数。已经开发了许多用于 PB 方程的求解器；据我们所知，这些求解器都不能直接应用于 DPBL 方程。它们使用的方法可能会适应这种差异；然而，它们的实现是特定于 PBE 的。我们将由 Holst 和 Saied 最初开发的 PBE 求解器[J. Comput. Chem. 16, 337 (1995)]改编为求解 DPBL 方程的问题。这个求解器使用截断牛顿法和多重网格预处理器。数值证据表明，它对于 DPBL 方程是收敛的，并且收敛是超线性的。然而，对于在计算生物学和计算化学中常见的问题，它的速度较慢且对内存的要求较高。为了克服这些问题，我们提出了两种变体，一种是基于简化、不精确雅可比的拟牛顿求解器，另一种是直接基于 PBE 求解器的迭代自洽求解器。虽然这两种方法都不能保证收敛，但数值证据表明它们确实收敛，而且收敛也是超线性的。这两种变体都比基于精确雅可比的求解器快得多，且内存占用更小。所有三种方法都已在一个名为 AQUASOL 的新代码中实现，该代码是免费提供的。
