Highly accurate biomolecular electrostatics in continuum dielectric environments

Abstract

Implicit solvent models based on the Poisson-Boltzmann (PB) equation are frequently used to describe the interactions of a biomolecule with its dielectric continuum environment. A novel, highly accurate Poisson-Boltzmann solver is developed based on the matched interface and boundary (MIB) method, which rigorously enforces the continuity conditions of both the electrostatic potential and its flux at the molecular surface. The MIB based PB solver attains much better convergence rates as a function of mesh size compared to conventional finite difference and finite element based PB solvers. Consequently, highly accurate electrostatic potentials and solvation energies are obtained at coarse mesh sizes. In the context of biomolecular electrostatic calculations it is demonstrated that the MIB method generates substantially more accurate solutions of the PB equation than other established methods, thus providing a new level of reference values for such models. Initial results also indicate that the MIB method can significantly improve the quality of electrostatic surface potentials of biomolecules that are frequently used in the study of biomolecular interactions based on experimental structures.

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

“…As a result, we usually observe low accuracy of the solution potential at the surface and low convergence rate. 62 Possible solutions to these problems include reducing the mesh spacing to improve resolution as well as application of improved and robust solvers. While we have taken both options into account while developing AQUASOL ͑i.e., special care was taken to limit the AQUASOL memory usage to allow for large meshes, the discretization scheme allows for nonuniform meshes to give the possibility to increase resolution at the interface, and AQUA-SOL strives to fast convergence by adopting an inexact Newton solver͒, it remains that these are workarounds that treat the symptoms related to the use of Cartesian meshes rather than the problems at the root.…”

“…There has been recently significant interest from applied mathematicians to develop PB solvers with interface methods that specifically deal with the continuity and accuracy issues at the molecular surfaces. Methods such as the jump condition capturing finite difference scheme, 63,64 and the matched interface and boundary 62,65,66 seem very promising and we are currently investigating ways to incorporate them in AQUASOL. Finite element methods represent a viable alternative to the finite difference methods discussed above. They allow for non-Cartesian meshes that provide better approximation of the geometry of the solutes.…”

Minimal molecular surfaces and their applications

“…By defining the MMS as the solvent-solute dielectric interface, the electrostatic potentials of proteins can be attained via the numerical solution of the Poisson-Boltzmann equation. Twenty six proteins, most of them are adopted from a test set used in previous studies, 71,72 are employed. Two proteins, i.e., Cu/Zn superoxide dismutase (PDB ID: 1b4l) and acetylcholin esterase (PDB ID: lea5), are well-known for their important electrostatic effects.…”

Continuum molecular electrostatics, salt effects, and counterion binding—A review of the Poisson–Boltzmann theory and its modifications

“…Such an approach arises from the fact that the variation of the osmotic work in the functional yields the ionic charge density in the PBE. The second argument of the min function involves an integration constant B i , whose value is chosen in order to ensure that the cutoffs of the concentration (40) and of the osmotic energy (42) become effective at the same threshold value of the potential w,…”