Accurate boundary-integral formulations for the calculation of electrostatic forces with an implicit-solvent model

TL;DR

This paper introduces an accurate boundary-integral method for calculating electrostatic forces using the Poisson-Boltzmann equation, outperforming finite difference methods and evaluating four force calculation approaches for molecular interactions.

Contribution

It presents an exact electric field calculation on the molecular surface and compares four force computation methods, highlighting the most accurate approach for implicit-solvent models.

Findings

Boundary element method outperforms finite difference method in force accuracy.

Maxwell stress tensor-based force calculation is most precise.

Energy variation-based approach provides comparable results in realistic scenarios.

Abstract

An accurate force calculation with the Poisson-Boltzmann equation is challenging, as it requires the electric field on the molecular surface. Here, we present a calculation of the electric field on the solute-solvent interface that is exact for piece-wise linear variations of the potential and analyze four different alternatives to compute the force using a boundary element method. We performed a verification exercise for two cases: the isolated and two interacting molecules. Our results suggest that the boundary element method outperforms the finite difference method, as the latter needs a much finer mesh than in solvation energy calculations to get acceptable accuracy in the force, whereas the same surface mesh than a standard energy calculation is appropriate for the boundary element method. Among the four evaluated alternatives of force calculation, we saw that the most accurate one is based on the Maxwell stress tensor. However, for a realistic application, like the barnase-barstar complex, the approach based on variations of the energy functional, which is less accurate, gives equivalent results. This analysis is useful towards using the Poisson-Boltzmann equation for force calculations in applications where high accuracy is key, for example, to feed molecular dynamics models or to enable the study of the interaction between large molecular structures, like viruses adsorbed onto substrates.