Coupling finite and boundary element methods to solve the Poisson-Boltzmann equation for electrostatics in molecular solvation
Coupling finite and boundary element methods to solve the Poisson-Boltzmann equation for electrostatics in molecular solvation
复制标题
DOI：
10.1002/jcc.27262
复制
发表时间：
2023-12-21
影响因子：
3
通讯作者：
Cooper，Christopher D.
中科院分区：
文献类型：
--
作者：
Bosy，Michal;Scroggs，Matthew W.;Cooper，Christopher D.
Poisson-Boltzmann方程被广泛用于模拟分子系统中的静电。可用的软件包解决它使用有限差分，有限元和边界元方法，其中后者是有吸引力的，由于精确表示的分子表面和部分电荷，并在无穷大的边界条件的确切执行。然而，边界元法仅限于线性方程和材料特性的分段常数变化。在这项工作中，我们提出了一个计划，耦合有限元和边界元的线性Poisson-Boltzmann方程，其中有限元法应用于一个有限的solutegion和边界元法的externalsolventregion。作为概念验证练习，我们使用最简单的方法：使用质量矩阵和对角预处理的Johnson-Nédélec耦合，通过Python接口使用Bempp-cl和FeNiCSx库实现。我们展示了我们的实现通过计算极性分量的溶剂化自由能的一组分子使用常数和高斯变化的介电常数。作为验证，我们比较了广泛的结合能数据集的完善的有限差分求解器，并与高斯折射率的有限差分代码APBS（至0.5%）进行了比较。我们还显示了从蛋白质G B1（955个原子）到免疫球蛋白G（20，148个原子）的缩放结果。对于小问题，耦合方法是有效的，优于纯边界积分方法。对于高斯变折射率，这超出了边界元的单独适用性，我们能够在单个工作站上运行中型到大型的问题。更好的预处理技术的发展和使用分布式内存并行较大的系统仍然是未来工作的一个领域。我们希望这项工作将作为未来发展的灵感，考虑空间变化的场参数，以及隐式溶剂模型的分子静电的混合线性-非线性方案。
The Poisson–Boltzmann equation is widely used to model electrostatics in molecular systems. Available software packages solve it using finite difference, finite element, and boundary element methods, where the latter is attractive due to the accurate representation of the molecular surface and partial charges, and exact enforcement of the boundary conditions at infinity. However, the boundary element method is limited to linear equations and piecewise constant variations of the material properties. In this work, we present a scheme that couples finite and boundary elements for the linearised Poisson–Boltzmann equation, where the finite element method is applied in a confinedsoluteregion and the boundary element method in the externalsolventregion. As a proof‐of‐concept exercise, we use the simplest methods available: Johnson–Nédélec coupling with mass matrix and diagonal preconditioning, implemented using the Bempp‐cl and FEniCSx libraries via their Python interfaces. We showcase our implementation by computing the polar component of the solvation free energy of a set of molecules using a constant and a Gaussian‐varying permittivity. As validation, we compare against well‐established finite difference solvers for an extensive binding energy data set, and with the finite difference code APBS (to 0.5%) for Gaussian permittivities. We also show scaling results from protein G B1 (955 atoms) up to immunoglobulin G (20,148 atoms). For small problems, the coupled method was efficient, outperforming a purely boundary integral approach. For Gaussian‐varying permittivities, which are beyond the applicability of boundary elements alone, we were able to run medium to large‐sized problems on a single workstation. The development of better preconditioning techniques and the use of distributed memory parallelism for larger systems remains an area for future work. We hope this work will serve as inspiration for future developments that consider space‐varying field parameters, and mixed linear‐nonlinear schemes for molecular electrostatics with implicit solvent models.