Treatment of geometric singularities in implicit solvent models

Abstract

Geometric singularities, such as cusps and self-intersecting surfaces, are major obstacles to the accuracy, convergence, and stability of the numerical solution of the Poisson-Boltzmann (PB) equation. In earlier work, an interface technique based PB solver was developed using the matched interface and boundary (MIB) method, which explicitly enforces the flux jump condition at the solvent-solute interfaces and leads to highly accurate biomolecular electrostatics in continuum electric environments. However, such a PB solver, denoted as MIBPB-I, cannot maintain the designed second order convergence whenever there are geometric singularities, such as cusps and self-intersecting surfaces. Moreover, the matrix of the MIBPB-I is not optimally symmetrical, resulting in the convergence difficulty. The present work presents a new interface method based PB solver, denoted as MIBPB-II, to address the aforementioned problems. The present MIBPB-II solver is systematical and robust in treating geometric singularities and delivers second order convergence for arbitrarily complex molecular surfaces of proteins. A new procedure is introduced to make the MIBPB-II matrix optimally symmetrical and diagonally dominant. The MIBPB-II solver is extensively validated by the molecular surfaces of few-atom systems and a set of 24 proteins. Converged electrostatic potentials and solvation free energies are obtained at a coarse grid spacing of 0.5 A and are considerably more accurate than those obtained by the PBEQ and the APBS at finer grid spacings.

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

“…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.…”

Poisson–Boltzmann–Nernst–Planck model

“…The surface plot of differences is very sensitive and depends not only on theoretical models, but also on the numerical procedure. 19,41 Figure 4 illustrates the differences in electrostatic potentials, positive ion concentrations, and negative ion concentrations. Clearly, the differences of results calculated from two models are relatively small in most regions.…”

High-order fractional partial differential equation transform for molecular surface construction

“…Once the surface has been generated, the electrostatic potential can be evaluated according to the Poisson equation [15, 24, 67] $$-\nabla ·false(\epsilon \nabla \phi false)=\sum _j{q}_j\delta false(\mathrm{boldr}-{\mathbf{r}}_{j}false),$$ where q j are the (fractional) charges of atoms at position r j ( j = 1, 2, …, N a ). The dielectric function ε is defined as a piecewise constant function $$\epsilon false(\mathrm{boldr}false)=true\{\begin{array}{cc}left{\epsilon }_m,& left\mathbf{r}\in {\Omega }_m,\\ left{\epsilon }_s,& left\mathbf{r}\in {\Omega }_s,\end{array}$$ where ε m = 1 and ε s = 80 are the dielectric constants in the molecular and solvent regions, respectively, these two regions are separated by the molecular surface.…”

“…Equation (60) is a typical elliptic interface problem with discontinuous coefficients and singular sources. It is very difficult to construct second-order convergent methods for this equation in the biomolecular context due to the geometric complexity, complex interface, singular charge sources and geometric singularities [15, 24, 67]. In this work, we make use of the second-order convergent matched interface and boundary (MIB) method [15, 68, 69, 73, 77, 78] to solve Eq.…”