Monte Carlo methods for linear and non-linear Poisson-Boltzmann equation

TL;DR

This paper compares Monte Carlo methods for solving the linearized Poisson-Boltzmann equation on biomolecules and introduces a new probabilistic interpretation and Monte Carlo algorithm for the nonlinear version.

Contribution

It provides a comparative analysis of replacement methods for the linearized equation and introduces a novel probabilistic interpretation and Monte Carlo approach for the nonlinear Poisson-Boltzmann PDE.

Findings

Comparison of replacement methods with deterministic solver APBS

Development of a new probabilistic interpretation for the nonlinear PDE

Implementation and testing of a Monte Carlo algorithm for the nonlinear case

Abstract

The electrostatic potential in the neighborhood of a biomolecule can be computed thanks to the non-linear divergence-form elliptic Poisson-Boltzmann PDE. Dedicated Monte-Carlo methods have been developed to solve its linearized version (see e.g.Bossy et al 2009, Mascagni & Simonov 2004}). These algorithms combine walk on spheres techniques and appropriate replacements at the boundary of the molecule. In the first part of this article we compare recent replacement methods for this linearized equation on real size biomolecules, that also require efficient computational geometry algorithms. We compare our results with the deterministic solver APBS. In the second part, we prove a new probabilistic interpretation of the nonlinear Poisson-Boltzmann PDE. A Monte Carlo algorithm is also derived and tested on a simple test case.