Monte Carlo methods for linear and non-linear Poisson-Boltzmann equation
For computational biophysics, this work provides a novel Monte Carlo approach for the nonlinear Poisson-Boltzmann equation, though the linear comparison is incremental.
The paper compares Monte Carlo methods for the linearized Poisson-Boltzmann equation on real biomolecules, showing agreement with the deterministic solver APBS, and introduces a new probabilistic interpretation and Monte Carlo algorithm for the nonlinear Poisson-Boltzmann equation, tested on a simple case.
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.