Fast direct methods for molecular electrostatics

Electrostatic interactions are vital for many aspects of biomolecular structure and function, including stability, activity, and specificity. Thus, a central problem in biophysical modeling is the electrostatic analysis of biomolecular systems. Here, we consider the numerical solution of the linearized Poisson-Boltzman equation (LPBE), a widely used model for the electrostatic potential of a large, solvated biomolecule. By combining boundary integral techniques with new multilevel matrix compression algorithms, we develop a fast direct solver for the LPBE that is accurate, robust, and can be more efficient than current methods by several orders of magnitude. The fast direct solver is general and applies to a wide range of integral operators based on non-oscillatory Green's functions, including those for the Laplace, low-frequency Helmholtz, Stokes, and LPBEs. The core algorithm uses the interpolative decomposition to compress the matrix discretizations of such operators, producing highly efficient representations that facilitate fast inversion. For boundary integral equations in 2D, the solver has complexity O(N), where N is the number of discretization elements; in 3D, it incurs an O(N 3/2) cost for precomputation followed by O(N log N) solves. As is typical of direct methods, each solve can be performed extremely rapidly, though the cost of precomputation can be high. Thus, the solver is particularly suited to problems where the precomputation time can be amortized, e.g., systems with ill-conditioned matrices or involving multiple right-hand sides. We demonstrate our solver on a number of examples and discuss various useful extensions. Furthermore, we apply our methods to the calculation of protein pKa values, which requires the computation of all pairwise titrating site energies. This corresponds to solving the LPBE on the same molecular geometry with many different boundary conditions on the protein surface, each manifesting as a different right-hand side, and hence presents a prime candidate for acceleration using our direct solver. Preliminary results are favorable and show the viability of our techniques for molecular electrostatics. Such fast direct methods could well have broad impact on many areas of computational science and engineering. We describe further applications in biology, chemistry, and physics, and outline some directions for future work