Biological molecules perform their functions surrounded by water and mobile ions, which strongly influence molecular structure and behavior. The electrostatic interactions between a molecule and solvent are particularly difficult to model theoretically, due to the forces' long range and the collective response of many thousands of solvent molecules. The dominant modeling approaches represent the two extremes of the trade-off between molecular realism and computational efficiency: all-atom molecular dynamics in explicit solvent, and macroscopic continuum theory (the Poisson or Poisson--Boltzmann equation). We present the first fast-solver implementation of an advanced nonlocal continuum theory that combines key advantages of both approaches. In particular, molecular realism is included by limiting solvent dielectric response on short length scales, using a model for nonlocal dielectric response allows the resulting problem (a linear integro-differential Poisson equation) to be reformulated as a system of coupled boundary-integral equations using double reciprocity. Whereas previous studies using the nonlocal theory had been limited to small model problems, owing to computational cost, our work opens the door to studying much larger problems including rational drug design, protein engineering, and nanofluidics.

This paper presents a fast boundary-element method (BEM) algorithm that is well suited for solving electrostatics problems that arise in traditional and bio-microelectromechanical systems (bio-MEMS) design. The algorithm, FFTSVD, is Green's-function-independent for low-frequency kernels and efficient for inhomogeneous problems. FFTSVD is a multiscale algorithm that decomposes the problem domain using an octree and uses sampling to calculate low-rank approximations to dominant source distributions and responses. Long-range interactions at each length scale are computed using the FFT. Computational results illustrate that the FFTSVD algorithm performs better than precorrected-FFT (pFFT)-style algorithms or the multipole-style algorithms in FastCap.

Computer simulation is an important tool for improving our understanding of biomolecule electrostatics, in part to aid in drug design. However, the numerical techniques used in these simulation tools do not exploit fast solver approaches widely used in analyzing integrated circuit interconnects. In this paper we describe one popular formulation used to analyze biomolecule electrostatics, present an integral formulation of the problem, and apply the precorrected-FFT method to accelerate the solution of the integral equations.