Numerical Solutions to the Poisson-Boltzmann Equation

Beyond z-z electrolytes, or for nonplanar interfaces, analytical solutions to the PB equation are difficult or nonexistent. Thus, one often must resort to numerical solutions fortunately, nowadays, with the power of digital computers and the availability of advanced numerical algorithms, that does not represent a very difficult task. 

For a planar interface as in the GC theory but for unsymmetrical electrolytes, the numerical integration can be easily accomplished starting from Equation 3.23 because it is essentially a first-order differential equation. With the additional boundary condition 

FIGURE 3.7 Plots of potential distributions near a charged planar surface according to Equation 3.32. (a) The charge of co-ions (ions with the same charge as the surface) is held constant (b) the charge of counterions is held constant. (Reprinted from Chen, Z., and R.K. Singh., /. Colloid. Interface. Sci., 245, 301-306, 2002. With permission.) 

Several numerical methods have been devised and applied to solve the PB equation for different geometries (Phillips 1995 Das et al. 1997 Dyshlovenko 2002 Lamm 2003 Bhuiyan, Outhwaite, and Henderson 2007 Lima, Tavares, and Biscaia 2007 Smagala and Fawcett 2007 Lu et al. 2008 Shi and Koehl 2008 Henderson and Boda 2009 Wang and Luo 2010). We will mention some accomplishments. 

James and Williams (1985) employed a Galerkin finite elanent schane to provide flexible numerical solutions to the PB equation in one and two dimensions, using a Newton sequence for the solution of the set of nonlinear equations arising from the finite-element discretization procedure the procedure was shown to be applicable for different geometries. Dyshlovenko (2002) applied a finite-element solution of the PB equation with an adaptive mesh refinement procedure. The procedure allowed the gradual improvement of the solution and adjustment of the geometry of the problem. The approach was successfully applied to several problems 

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We compare the exact numerical solution to the Poisson-Boltzmann equation (6.6) and the approximate results, Eq. (6.37) for case 1 (low surface charge density case) and Eq. (6.50) for case 2 (high surface charge density case) in Fig. 6.3, in which the scaled surface potential jo = zeij/JkT is plotted as a function of the scaled... 

Wall, T.F., and Berkowitz, J. Numerical solution to the Poisson - Boltzmann equation for spherical polyelectrolyte molecules. Journal of Chemical Physics, 1957, 26, p. 114-122. 

Im W, Beglov D, Roux B. 1998. Continuum solvation model computation of electrostatic forces from numerical solutions to the Poisson-Boltzmann equation. Comput. Phys. Commun. 111 59-75... 

The most rigorous dielectric continuum methods employ numerical solutions to the Poisson-Boltzmann equation [55]. As these methods are computationally quite expensive, in particular in connection with calculations of derivatives, much work has been concentrated on the development of computationally less expensive approximate continuum models of sufficient accuracy. One of the most widely used of these is the Generalized Born Solvent Accessible Surface Area (GB/SA) model developed by Still and coworkers [56,57]. The model is implemented in the MacroModel program [17,28] and parameterized for water and chloroform. It may be used in conjunction with the force fields available in MacroModel, e.g., AMBER, MM2, MM3, MMFF, OPTS. It should be noted that the original parameterization of the GB/SA model is based on the OPLS force field. 

Solvation Model Electrostatic Forces From Numerical Solutions to the Poisson-Boltzmann Equation. 

Here V(r), e(r), and p(r) are the electrostatic potential, the dielectric constant and the protein electron density in point r, while / stands for the ionic strength. DelPhi, a versatile software package providing numerical solutions to the Poisson-Boltzmann equation, is widely used for the calculation and visualization of the protein electrostatic potential (DelPhi 2009). 

Figure 50 Diagram illustrating the thermodynamic cycle for the determination of the binding energy of two macromolecules using the numerical solution to the Poisson-Boltzmann equation.

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