A treecode-accelerated boundary integral Poisson–Boltzmann solver for electrostatics of solvated biomolecules

We present a treecode-accelerated boundary integral (TABI) solver for electrostatics of solvated biomolecules described by the linear Poisson–Boltzmann equation. The method employs a well-conditioned boundary integral formulation for the electrostatic potential and its normal derivative on the molecular surface. The surface is triangulated and the integral equations are discretized by centroid collocation. The linear system is solved by GMRES iteration and the matrix–vector product is carried out by a Cartesian treecode which reduces the cost from O(N2)O(N2) to O(NlogN)O(NlogN), where N is the number of faces in the triangulation. The TABI solver is applied to compute the electrostatic solvation energy in two cases, the Kirkwood sphere and a solvated protein. We present the error, CPU time, and memory usage, and compare results for the Poisson–Boltzmann and Poisson equations. We show that the treecode approximation error can be made smaller than the discretization error, and we compare two versions of the treecode, one with uniform clusters and one with non-uniform clusters adapted to the molecular surface. For the protein test case, we compare TABI results with those obtained using the grid-based APBS code, and we also present parallel TABI simulations using up to eight processors. We find that the TABI solver exhibits good serial and parallel performance combined with relatively simple implementation, efficient memory usage, and geometric adaptability.