Using Correlated Monte Carlo Sampling for Efficiently Solving the Linearized Poisson−Boltzmann Equation Over a Broad Range of Salt Concentration

Abstract

Dielectric continuum or implicit solvent models provide a significant reduction in computational cost when accounting for the salt-mediated electrostatic interactions of biomolecules immersed in an ionic environment. These models, in which the solvent and ions are replaced by a dielectric continuum, seek to capture the average statistical effects of the ionic solvent, while the solute is treated at the atomic level of detail. For decades, the solution of the three-dimensional Poisson-Boltzmann equation (PBE), which has become a standard implicit-solvent tool for assessing electrostatic effects in biomolecular systems, has been based on various deterministic numerical methods. Some deterministic PBE algorithms have drawbacks, which include a lack of properly assessing their accuracy, geometrical difficulties caused by discretization, and for some problems their cost in both memory and computation time. Our original stochastic method resolves some of these difficulties by solving the PBE using the Monte Carlo method (MCM). This new approach to the PBE is capable of efficiently solving complex, multi-domain and salt-dependent problems in biomolecular continuum electrostatics to high precision. Here we improve upon our novel stochastic approach by simultaneouly computating of electrostatic potential and solvation free energies at different ionic concentrations through correlated Monte Carlo (MC) sampling. By using carefully constructed correlated random walks in our algorithm, we can actually compute the solution to a standard system including the linearized PBE (LPBE) at all salt concentrations of interest, simultaneously. This approach not only accelerates our MCPBE algorithm, but seems to have cost and accuracy advantages over deterministic methods as well. We verify the effectiveness of this technique by applying it to two common electrostatic computations: the electrostatic potential and polar solvation free energy for calcium binding proteins that are compared with similar results obtained using mature deterministic PBE methods.

Influence of Grid Spacing in Poisson–Boltzmann Equation Binding Energy Estimation

“…Grids with common dimensions and alignments were generated for each separate binding partner and complex. The ACG predictions of ΔG el and ΔΔG el were corroborated by similar computations with APBS 19 and an in-house stochastic linear PB solver 5-7 (results not shown).…”

Kinetic Monte Carlo investigation of tetragonal strain on Onsager matrices

“…The correlated finite difference (CFD) method introduces an artificial correlation between the KMC trajectories for the positively and negatively strained diffusion systems to reduce the statistical error. [29,30] The relative statistical error of CFD is…”

Progress in developing Poisson-Boltzmann equation solvers

“…A stochastic approach utilizing Monte Carlo methods for solving the PBE was developed by Mascagni, Fenley and co-workers and was shown that the stochastic based linear 3D PBE solvers have very low memory demands [64, 121, 125, 150] . It was demonstrated that by applying series of numerical optimizations one can make the computational time of these Monte Carlo LPBE solvers competitive with deterministic methods.…”