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

Life as we know it depends critically on electrostatic interactions within and be- tween biological molecules such as proteins. One simple, but surprisingly efiective, model for studying these interactions treats a biomolecule of interest as a dielectric continuum of homo- geneous 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 mixed- dielectric Poisson problem, widely studied in the mathematics and electromagnetics communities. In this paper we describe some simple analytical approximations to a boundary-integral equa- tion 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.