Hybrid boundary element and finite difference method for solving the nonlinear Poisson–Boltzmann equation

Abstract

A hybrid approach for solving the nonlinear Poisson-Boltzmann equation (PBE) is presented. Under this approach, the electrostatic potential is separated into (1) a linear component satisfying the linear PBE and solved using a fast boundary element method and (2) a correction term accounting for nonlinear effects and optionally, the presence of an ion-exclusion layer. Because the correction potential contains no singularities (in particular, it is smooth at charge sites) it can be accurately and efficiently solved using a finite difference method. The motivation for and formulation of such a decomposition are presented together with the numerical method for calculating the linear and correction potentials. For comparison, we also develop an integral equation representation of the solution to the nonlinear PBE. When implemented upon regular lattice grids, the hybrid scheme is found to outperform the integral equation method when treating nonlinear PBE problems. Results are presented for a spherical cavity containing a central charge, where the objective is to compare computed 1D nonlinear PBE solutions against ones obtained with alternate numerical solution methods. This is followed by examination of the electrostatic properties of nucleic acid structures.

Computational Methods for Biomolecular Electrostatics

“…The most common numerical techniques for solving the PB equation are based on discretization of the domain of interest into small regions. Those methods include finite difference (Davis and McCammon, 1989;Nicholls and Honig, 1991;Holst and Saied, 1993;Holst and Saied, 1995;Baker et al, 2001), finite element Friesner, 1997a, 1997b;Baker et al, 2000;Holst et al, 2000;Baker et al, 2001;Dyshlovenko, 2002), and boundary element methods (Zauhar and Morgan, 1988;Juffer et al, 1991;Allison and Huber, 1995;Bordner and Huber, 2003;Boschitsch and Fenley, 2004), all of which continue to be developed to further improve the accuracy and efficiency of electrostatics calculations in the numerous biomolecular applications described below. The major software packages that can be used to solve the PB equation are listed in Table 1.…”

Electrodiffusion: A continuum modeling framework for biomolecular systems with realistic spatiotemporal resolution

“…However, including the solvent charge density would require volume integrals of the charge distribution over the entire domain. According to Boschitsch and Fenley,32 once the volume integrals appear in the BEM, the computation times incurred by a conventional, or even some multipole-accelerated integral equation methods, increase significantly and tend to be higher than even a finite difference scheme of comparable accuracy. A hybrid finite difference/BEM approach and a decomposition strategy were adopted and tested to solve the nonlinear PBE in their work.…”

“…A hybrid finite difference/BEM approach and a decomposition strategy were adopted and tested to solve the nonlinear PBE in their work. 32 So, rather than use boundary elements alone to solve the Poisson equation, we use the hybrid BE/FEM method described below to avoid the volume integrals.…”

AQUASOL: An efficient solver for the dipolar Poisson–Boltzmann–Langevin equation

“…21 These methods rely on the linearization of the DebyeHückel term that accounts for counterions around the solute; it is known however that this approximation is not valid for highly charged systems. 22 Boschitsch and Fenley 23 proposed a correction to the BME methods to account for the nonlinearity in the PB equation. Clearly, we need a more general framework for solving PB equations, especially in light of the recent theoretical modifications.…”