Multiscale models and approximation algorithms for protein electrostatics

Electrostatic forces play many important roles in molecular biology, but are hard to model due to the complicated interactions between biomolecules and the surrounding solvent, a fluid composed of water and dissolved ions. Continuum model have been surprisingly successful for simple biological questions, but fail for important problems such as understanding the effects of protein mutations. In this paper we highlight the advantages of boundary-integral methods for these problems, and our use of boundary integrals to design and test more accurate theories. Examples include a multiscale model based on nonlocal continuum theory, and a nonlinear boundary condition that captures atomic-scale effects at biomolecular surfaces.Comment: 12 pages, 6 figure

Analytical Nonlocal Electrostatics Using Eigenfunction Expansions of Boundary-Integral Operators

In this paper, we present an analytical solution to nonlocal continuum electrostatics for an arbitrary charge distribution in a spherical solute. Our approach relies on two key steps: (1) re-formulating the PDE problem using boundary-integral equations, and (2) diagonalizing the boundary-integral operators using the fact their eigenfunctions are the surface spherical harmonics. To introduce this uncommon approach for analytical calculations in separable geometries, we rederive Kirkwood's classic results for a protein surrounded concentrically by a pure-water ion-exclusion layer and then a dilute electrolyte (modeled with the linearized Poisson--Boltzmann equation). Our main result, however, is an analytical method for calculating the reaction potential in a protein embedded in a nonlocal-dielectric solvent, the Lorentz model studied by Dogonadze and Kornyshev. The analytical method enables biophysicists to study the new nonlocal theory in a simple, computationally fast way; an open-source MATLAB implementation is included as supplemental information.Comment: 19 pages, 7 figure