Poisson-Boltzmann solver

Finally, it is worth emphasizing that for classical solutes s 0 and Eq. (11.17) represents an exact solution to the continuum electrostatics problem. To emphasize this point, we have performed numerical comparisons of MM/PCM calculations versus results obtained from the "adaptive Poisson-Boltzmann solver" (APBS) [5], which represents a recent implementation of the three-dimensional finite-difference approach. (The solvent s ionic strength was set to zero in the APBS calculations.) Results for amino acids, plotted in Fig. 11.2, show sub-kcal/mol differences in most cases, and differences of < 0.1 kcal/mol for the "X = DAS" version of SS(V)PE... 

An issue with all of these discretization schemes—except possibly the genuine isodensity surface that is not considered in this work—is the fact that the solvation energy is a discontinuous function of the atomic coordinates, because discretization points appear and disappear as the overlap between atomic spheres changes. (In principle, the energy also loses rotational invariance upon discretization, but we fund that this problem is not serious [42]). The discontinuity problem, which is shared by finite-difference Poisson-Boltzmann solvers, has recently been resolved in the context of PCMs, with the development of intrinsically smooth discretization algorithms [42, 70, 76, 87]. These are discussed in Section 11.4.1. 

An issue with the PCM formalism introduced in Section 11.2.2.1 is that the electrostatic energy is in general a discontinuous function as the solute atoms are displaced, because the number and size of the surface tesserae may change as a function of solute geometry. A similar problem is suffered by finite-difference Poisson-Boltzmann solvers, and the "solution" in those cases (in order to achieve stable forces for MD simulations, for example) is tight thresholding and/or some kind of interpolation between grid points [80-83]. 

Geng, W, Krasny, R. A treecode-accelerated boundtiry integral Poisson-Boltzmann solver for solvated biomolecules. J. Comput. Phys. 247, 62-78 (2013). doi 10.1016/j.jcp.2013.03. 056... 

Adaptive Poisson-Boltzmann Solver (APBS). 1999-2003. Washington University, St. Louis,... 

Wang, J. U. N., and R. A. Y. Luo. 2010. Assessment of linear finite-difference Poisson-Boltzmann solvers. Journal of Computational Chemistry 31, no. 8 1689-1698. 

Since pairwise electrostatic interaction energies can be calculated from protein structures using Poisson-Boltzmann Equation (PBE) solvers [71,72], we can attempt to forge a unique link between protein structure and protein titration curves... 

Table 1 presents a list of the major software that currently solves the Poisson-Boltzmann equation for biomolecular systems. A variety of such programs exist, ranging from multipurpose computational biology packages (e.g., CHARMM, Jaguar, UHBD, and MacroDox) to specialized PB solvers (e.g., APBS, MEAD, and DelPhi). 

The recent development of high-resolution experimental techniques allows for the structural analysis of protein channels with unprecedented detail. However, the fundamental problem of relating the structure of ion channels to their function is a formidable task. This chapter describes some of the most popular simulation approaches used to model channel systems. Particle-based approaches such as Brownian and molecular dynamics will continue to play a major role in the study of protein channels and in validating the results obtained with the extremely fast continuum models. Research in the area of atomistic simulations will focus mainly on the force-field schemes used in the ionic dynamics simulation engines. In particular, polar interactions between the various components of the system need to be computed with algorithms that are more accurate than those currently used. The effects of the local polarization fields need to be accounted for explicitly and, at the same time, efficiently. Continuum models will remain attractive for their efficiency in depicting the electrostatic landscape of protein channels. Both Poisson-Boltzmann and Poisson-Nemst-Plank solvers will continue to be used to... 

Shi, X., and P. Koehl. 2008. The geometry behind numerical solvers of the Poisson-Boltzmann eqnation. Communications in Computational Physics 3, no. 5 1032-1050. 

Jaguar s solvation module uses a self-consistent reaction field (SCRF) model, which couples an accurate ab initio description of the charge distribution to a well-defined and realistic representation of the molecular cavity. The SCRF calculation is done by first calculating the gas phase wavefunction of the solute molecule, from which the electrostatic potential fitted charges are passed to an efficient finite element solver, which determines the reaction field by numerical solution of the Poisson-Boltzmann equations and represents the solvent as a layer of charges at the solute s molecular surface, which serves as the dielectric continuum boundary. The solvent point charges are incorporated in a subsequent wavefunction evaluation and the process is repeated until self-consistency is obtained. The cost is roughly twice that of a gas phase calculation. 

The Poisson-Boltzmann equation with our form for the solution will not allow the analytical solution to J r ) = 0, but r] is readily found via computer by inserting Eqs. [214]-[216] into Eq. [212], with f(x) determined for the appropriate geometry, and using a root solver. An illustrative example will be given below. 

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