Macromolecular electrostatic energy within the nonlinear Poisson–Boltzmann equation

A fundamental problem in macromolecular electrostatics is the calculation of the electrostatic energy of a macromolecule solvated in an electrolyte solution, i.e., the work required to charge up the macromolecule in the presence of the electrolytic ions. Through numerical calculations with the nonlinear Poisson–Boltzmann (PB) equation, Sharp and Honig J. Phys. Chem. 94, 7684 (1990) observed that this energy can be obtained with equal accuracy from the charging integral and from their energy–density integral. Here we give an elementary analytical proof of the exact equivalence of the two different formulations of the energy. In order to calculate the macromolecular electrostatic energy, a boundary element method Biophys. J. 65, 954 (1993) previously developed for the linearized PB equation is modified to solve the nonlinear PB equation. Illustrative calculations show that for globular proteins under physiological ionic strengths, the electrostatic energy calculated from the linearized PB equation differs very little from that calculated from the nonlinear equation.