Analytical Nonlocal Electrostatics Using Eigenfunction Expansions of Boundary-Integral Operators

TL;DR

This paper develops an analytical method using eigenfunction expansions of boundary-integral operators to solve nonlocal electrostatics problems in spherical geometries, simplifying calculations for complex biological systems.

Contribution

It introduces a novel analytical approach leveraging eigenfunction expansions for boundary-integral operators in nonlocal electrostatics, enabling faster and more straightforward calculations.

Findings

Re-derived Kirkwood's classical results for spherical geometries.

Developed an analytical method for nonlocal dielectric response in proteins.

Provided an open-source MATLAB implementation for practical use.

Abstract

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.