This article summarizes the development of a fast boundary element method for the linear Poisson-Boltzmann equation governing biomolecular electrostatics. Unlike previous fast boundary element implementations, the present treatment accommodates finite salt concentrations thus enabling the study of biomolecular electrostatics under realistic physiological conditions. This is achieved by using multipole expansions specifically designed for the exponentially decaying Green's function of the linear Poisson-Boltzmann equation. The particular formulation adopted in the boundary element treatment directly affects the numerical conditioning and thus convergence behavior of the method. Therefore, the formulation and reasons for its choice are first presented. Next, the multipole approximation and its use in the context of a fast boundary element method are described together with the iteration method employed to extract the surface distributions. The method is then subjected to a series of computational tests involving a sphere with interior charges. The purpose of these tests is to assess accuracy and verify the anticipated computational performance trends. Finally, the salt dependence of electrostatic properties of several biomolecular systems (alanine dipeptide, barnase, barstar, and coiled coil tetramer) is examined with the method and the results are compared with finite difference Poisson-Boltzmann codes.

A hybrid approach for solving the nonlinear Poisson-Boltzmann equation (PBE) is presented. Under this approach, the electrostatic potential is separated into (1) a linear component satisfying the linear PBE and solved using a fast boundary element method and (2) a correction term accounting for nonlinear effects and optionally, the presence of an ion-exclusion layer. Because the correction potential contains no singularities (in particular, it is smooth at charge sites) it can be accurately and efficiently solved using a finite difference method. The motivation for and formulation of such a decomposition are presented together with the numerical method for calculating the linear and correction potentials. For comparison, we also develop an integral equation representation of the solution to the nonlinear PBE. When implemented upon regular lattice grids, the hybrid scheme is found to outperform the integral equation method when treating nonlinear PBE problems. Results are presented for a spherical cavity containing a central charge, where the objective is to compare computed 1D nonlinear PBE solutions against ones obtained with alternate numerical solution methods. This is followed by examination of the electrostatic properties of nucleic acid structures.