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International
Tables for Crystallography Volume F Crystallography of biological macromolecues Edited by M. G. Rossmann and E. Arnold © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. F. ch. 22.3, p. 555
Section 22.3.2.5. Numerical methods
aE. R. Johnson Research Foundation, Department of Biochemistry and Biophysics, University of Pennsylvania, Philadelphia, PA 19104-6059, USA |
A variety of numerical methods exist for calculating electrostatic potentials of macromolecules. These include numerical solution of self-consistent field electrostatic equations, which has been used in conjunction with the protein dipole–Langevin dipole method (Lee et al., 1993![[link]](/graphics/greenarr.gif)
). Numerical solution of the Poisson–Boltzmann equation requires the solution of a three-dimensional partial differential equation, which can be nonlinear. Many numerical techniques, some developed in engineering fields to solve differential equations, have been applied to the PB equation. These include finite-difference methods (Bruccoleri et al., 1996; Gilson et al., 1988; Nicholls & Honig, 1991; Warwicker & Watson, 1982), finite-element methods (Rashin, 1990; Yoon & Lenhoff, 1992; Zauhar & Morgan, 1985), multigridding (Holst & Saied, 1993; Oberoi & Allewell, 1993), conjugate-gradient methods (Davis & McCammon, 1989) and fast multipole methods (Bharadwaj et al., 1994; Davis, 1994). Methods for treating the nonlinear PB equation include under-relaxation (Jayaram, Sharp & Honig, 1989) and powerful inexact Newton methods (Holst et al., 1994). The nonlinear PB equation can also be solved via a self-consistent field approach, in which one calculates the potential using equation (22.3.2.5), then the mobile charge density is calculated using equation (22.3.2.3), and the procedure is repeated until convergence is reached (Pack & Klein, 1984; Pack et al., 1986). The method allows one to include more elaborate models for the ion distribution, for example incorporating the finite size of the ions (Pack et al., 1993). Approximate methods based on spherical approximations (Born-type models) have also been used (Schaeffer & Frommel, 1990; Still et al., 1990). Considerable numerical progress has been made in finite methods, and accurate rapid algorithms are available. The reader is referred to the original references for numerical details.
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