Rapid Bounds on Electrostatic Energies Using Diagonal Approximations of Boundary-integral Equations

Abstract

Life as we know it depends critically on electrostatic interactions within and between biological molecules such as proteins. One simple, but surprisingly effective, model for studying these interactions treats a biomolecule of interest as a dielectric continuum of homogeneous low permittivity with some embedded distribution of charges, and the aqueous solvent around it as another homogeneous dielectric with higher permittivity. This gives rise to a mixeddielectric Poisson problem, widely studied in the mathematics and electromagnetics communities. In this paper we describe some simple analytical approximations to a boundary-integral equation formulation of the mixed-dielectric problem. Remarkably, the approximations (which we call BIBEE, for boundary-integral-based electrostatics estimation) give provable upper and lower bounds for the actual electrostatic energy. Because BIBEE methods preserve interactions between components of the charge distribution, they may represent one approach to rapidly approximate the Green's function for the geometry of interest.

A Nonlinear Boundary Condition for Continuum Models of Biomolecular Electrostatics

“…Our numerical calculations employed boundary-integral methods with simple model geometries, such as spheres for the monatomic ions and ellipsoids to model the amino acids. These ellipsoids represent simple shape approximations [29,30] and we expect that they will be useful for fast approximate calculations such as in implicit-solvent molecular dynamics [34,25,35,36]. Calculations for atomistic models of large molecules such as proteins will require fast, parallel boundary-element method solvers [24,28], and implementation of such software represents an area of ongoing work.…”

“…The few theories that directly address hydration asymmetry [23,19,20] are not actually Poisson models, but generalizations of the Born-ion problem (the spherically symmetric case of a sphere with a central charge). Recently we proposed the first successful asymmetric Poisson theory that can be solved for complex molecular geometries [14], by translating the existing models' physical insights into a boundary-integral equation (BIE) formulation of the Poisson problem [24,25]. This led to a modified BIE formulation in which we replaced the usual Maxwell boundary condition for the continuity of the normal flux with a nonlinear boundary condition (NLBC) [14].…”