Numerical solution of the nonlinear Poisson–Boltzmann equation: Developing more robust and efficient methods

Abstract

We present a robust and efficient numerical method for solution of the nonlinear Poisson-Boltzmann equation arising in molecular biophysics. The equation is discretized with the box method, and solution of the discrete equations is accomplished with a global inexact-Newton method, combined with linear multilevel techniques we have described in a paper appearing previously in this journal. A detailed analysis of the resulting method is presented, with comparisons to other methods that have been proposed in the literature, including the classical nonlinear multigrid method, the nonlinear conjugate gradient method, and nonlinear relaxation methods such as successive over-relaxation. Both theoretical and numerical evidence suggests that this method will converge in the case of molecules for which many of the existing methods will not. In addition, for problems which the other methods are able to solve, numerical experiments show that the new method is substantially more efficient, and the superiority of this method grows with the problem size. The method is easy to implement once a linear multilevel solver is available, and can also easily be used in conjunction with linear methods other than multigrid.

Computational Methods for Biomolecular Electrostatics

“…The most common numerical techniques for solving the PB equation are based on discretization of the domain of interest into small regions. Those methods include finite difference (Davis and McCammon, 1989;Nicholls and Honig, 1991;Holst and Saied, 1993;Holst and Saied, 1995;Baker et al, 2001), finite element Friesner, 1997a, 1997b;Baker et al, 2000;Holst et al, 2000;Baker et al, 2001;Dyshlovenko, 2002), and boundary element methods (Zauhar and Morgan, 1988;Juffer et al, 1991;Allison and Huber, 1995;Bordner and Huber, 2003;Boschitsch and Fenley, 2004), all of which continue to be developed to further improve the accuracy and efficiency of electrostatics calculations in the numerous biomolecular applications described below. The major software packages that can be used to solve the PB equation are listed in Table 1.…”

Accurate, conformation-dependent predictions of solvent effects on protein ionization constants

“…Focusing yielded grid resolutions Ͼ 2.4 grid units/Å (34). The linearized form of the PB equation was solved by the inexact Newton method with a multilevel solver algorithm to aid convergence (35). The solute was mapped on the grid with PARSE charge and radii parameters (13).…”

Electrostatic correlations near charged planar surfaces

“…We then solve the algebraic equation with inexact-Newton method for the modified PB equation. 27,28 A small trick is used during discretization for the "Bratu" type problem, i.e., we approximate the second-order derivative using a nonlinear denominator 29 which works well near bifurcation points.…”