Influence of Grid Spacing in Poisson-Boltzmann Equation Binding Energy Estimation

Grid-based solvers of the Poisson-Boltzmann, PB, equation are routinely used to estimate electrostatic binding, ΔΔG_el, and solvation, ΔG_el, free energies. The accuracies of such estimates are subject to grid discretization errors from the finite difference approximation to the PB equation. Here, we show that the grid discretization errors in ΔΔG_el are more significant than those in ΔG_el, and can be divided into two parts: (i) errors associated with the relative positioning of the grid and (ii) systematic errors associated with grid spacing. The systematic error in particular is significant for methods, such as the molecular mechanics PB surface area, MM-PBSA, approach that predict electrostatic binding free energies by averaging over an ensemble of molecular conformations. Although averaging over multiple conformations can control for the error associated with grid placement, it will not eliminate the systematic error, which can only be controlled by reducing grid spacing. The present study indicates that the widely-used grid spacing of 0.5 Å produces unacceptable errors in ΔΔG_el, even though its predictions of ΔG_el are adequate for the cases considered here. Although both grid discretization errors generally increase with grid spacing, the relative sizes of these errors differ according to the solute-solvent dielectric boundary definition. The grid discretization errors are generally smaller on the Gaussian surface used in the present study than on either the solvent-excluded or van der Waals surfaces, which both contain more surface discontinuities (e.g., sharp edges and cusps). Additionally, all three molecular surfaces converge to very different estimates of ΔΔG_el.