A Hybrid Solver of Size Modified Poisson-Boltzmann Equation by Domain Decomposition, Finite Element, and Finite Difference

TL;DR

This paper introduces a new hybrid solver for the size-modified Poisson-Boltzmann equation that combines domain decomposition, finite element, and finite difference methods, demonstrating high accuracy and stability in biomolecular electrostatics calculations.

Contribution

A novel SMPBE hybrid solver integrating domain decomposition, finite element, and finite difference methods, implemented as a multi-language software package validated on biomolecular models.

Findings

High performance on six protein models.

Numerical stability and convergence in biomolecular electrostatics.

Accurate calculation of solvation and binding free energies.

Abstract

The size-modified Poisson-Boltzmann equation (SMPBE) is one important variant of the popular dielectric model, the Poisson-Boltzmann equation (PBE), to reflect ionic size effects in the prediction of electrostatics for a biomolecule in an ionic solvent. In this paper, a new SMPBE hybrid solver is developed using a solution decomposition, the Schwartz's overlapped domain decomposition, finite element, and finite difference. It is then programmed as a software package in C, Fortran, and Python based on the state-of-the-art finite element library DOLFIN from the FEniCS project. This software package is well validated on a Born ball model with analytical solution and a dipole model with a known physical properties. Numerical results on six proteins with different net charges demonstrate its high performance. Finally, this new SMPBE hybrid solver is shown to be numerically stable and convergent in the calculation of electrostatic solvation free energy for 216 biomolecules and binding free energy for a DNA-drug complex.