Mathematical Analysis of the BIBEE Approximation for Molecular Solvation: Exact Results for Spherical Inclusions

TL;DR

This paper provides a rigorous mathematical analysis of the BIBEE approximation for molecular electrostatics using spherical models, revealing its properties, limitations, and potential improvements for accurate solvation energy predictions.

Contribution

It offers the first exact analysis of BIBEE for spherical inclusions, clarifies its relationship with Generalized Born models, and proposes a modified BIBEE model with improved accuracy.

Findings

Eigenfunctions of the reaction-potential are preserved in BIBEE for spheres

Modified BIBEE predicts solvation energies within 4% of full Poisson calculations

Analysis introduces a new perspective separating material properties and geometry

Abstract

We analyze the mathematically rigorous BIBEE (boundary-integral based electrostatics estimation) approximation of the mixed-dielectric continuum model of molecular electrostatics, using the analytically solvable case of a spherical solute containing an arbitrary charge distribution. Our analysis, which builds on Kirkwood's solution using spherical harmonics, clarifies important aspects of the approximation and its relationship to Generalized Born models. First, our results suggest a new perspective for analyzing fast electrostatic models: the separation of variables between material properties (the dielectric constants) and geometry (the solute dielectric boundary and charge distribution). Second, we find that the eigenfunctions of the reaction-potential operator are exactly preserved in the BIBEE model for the sphere, which supports the use of this approximation for analyzing charge-charge interactions in molecular binding. Third, a comparison of BIBEE to the recent GB$\epsilon$ theory suggests a modified BIBEE model capable of predicting electrostatic solvation free energies to within 4% of a full numerical Poisson calculation. This modified model leads to a projection-framework understanding of BIBEE and suggests opportunities for future improvements.