  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.
