Treatment of electrostatic effects in proteins: multigrid-based Newton iterative method for solution of the full nonlinear Poisson-Boltzmann equation.

The nonlinear Poisson-Boltzmann equation (NPBE) provides a continuum description of the electrostatic field in an ionic medium around a macromolecule. Here, a novel approach to the solution of the full NPBE is developed. This robust and efficient algorithm combines multilevel techniques with a damped inexact Newton's method. The CPU time required for solution of the full NPBE, which is less than that for standard single-grid approaches in solving the corresponding linearized equation, is proportional to the number of unknowns enabling applications to very large macromolecular systems. Convergence of the method is demonstrated for a variety of protein systems. Comparison of the solutions to the linearized Poisson-Boltzmann equation shows that the damping of the electrostatic field around the charge is increased and that the potential scales logarithmically with charge. The inclusion of the full nonlinearity thus reduces the impact of highly charged residues on protein surfaces and provides a more realistic representation of electrostatic effects. This is demonstrated through calculation of potential around the active site regions of the 1,266-residue tryptophan synthase dimer and in the computation of rate constants from Brownian dynamics calculations in the superoxide dismutase-superoxide and antibody-antigen systems.