Communications in Information and Systems

Volume 22 (2022)

Number 3

Special issue on bioinformatics and biophysics in honor of professor Michael Waterman on his 80th birthday

Guest Editors: Fengzhu Sun (University of Southern California), Guowei Wei (Michigan State University), Stephen S.-T. Yau (Tsinghua University), and Shan Zhao (University of Alabama)

Benchmarking electrostatic free energy of the nonlinear Poisson–Boltzmann model for the Kirkwood sphere

Pages: 305 – 315



Sylvia Amihere (Department of Mathematics, University of Alabama, Tuscaloosa, Al., U.S.A.)

Weihua Geng (Department of Mathematics, Southern Methodist University, Dallas, Texas, U.S.A.)

Shan Zhao (Department of Mathematics, University of Alabama, Tuscaloosa, Al., U.S.A.)


Various numerical packages have been developed to solve the Poisson- Boltzmann equation (PBE) for the electrostatic analysis of solvated biomolecules. A common benchmark test for the PBE solvers is the Kirkwood sphere, for which analytical potential and free energy are available for the linearized PBE. However, the Kirkwood sphere does not admit analytical solution for the nonlinear PBE involving a hyperbolic sine term. This paper proposes a simple numerical approach, so that the energy of the Kirkwood sphere for the nonlinear PBE can be calculated at a very high precision. This provides a new benchmark test for the future developments of nonlinear PBE solvers.


nonlinear Poisson–Boltzmann equation, Kirkwood sphere, matched interface and boundary (MIB), electrostatic free energy

Zhao’s research is partially supported by the National Science Foundation (NSF) of USA under grant DMS-2110914. Geng’s research is partially supported by the NSF under grant DMS-2110922.

Received 5 March 2022

Published 22 July 2022