Computation of Electrostatic Forces on Solvated Molecules Using the Poisson-Boltzmann Equation

Numerical solutions of the Poisson-Boltzmann equation (PBE) have found wide application in the computation of electrostatic energies of hydrated molecules, including biological macromolecules. However, solving the PBE for electrostatic forces has proved more difficult, largely because of the challenge of computing the pressures exerted by a high dielectric aqueous solvent on the solute surface. This paper describes an accurate method for computing these forces. We begin by presenting a novel derivation of the forces acting in a system governed by the PBE. The resulting expression contains three distinct terms: the effect of electric fields on 'fixed" atomic charges; the dielectric boundary pressure, which accounts for the tendency of the high dielectric solvent to displace the low dielectric solute wherever an electric field exists; and the ionic boundary pressure, which accounts for the tendency of the dissolved electrolyte to move into regions of nonzero electrostatic potential. Techniques for extracting each of these three force contributions from finite difference solutions of the PBE for a solvated molecule are then described. Tests of the methods against both analytic and numeric results demonstrate their accuracy. Finally, the electrostatic forces acting on the two members of a salt bridge in the enzyme triose phosphate isomerase are analyzed. The dielectric boundary pressures are found to make substantial contributions to the atomic forces. In fact, their neglect leads to the unphysical situation of a significant net electrostatic force on the system. In contrast, the ionic boundary forces are usually extremely weak a t physiologic ionic strength.