Poisson-Boltzmann equation numerical methods

Here zm denotes the valency of the counterions with the highest absolute charge. The function G is dependent upon the zeta potential of the particle as well as the bulk concentrations, valencies and diffusivities of the ions. The equilibrium electrical potential can be obtained using the Runge-Kutta method to solve the Poisson-Boltzmann equation numerically, and then the integral in Eq. (52) can be evaluated. 

How can Equation (11.79) be solved Before computers were available only simple ihapes could be considered. For example, proteins were modelled as spheres or ellipses Tanford-Kirkwood theory) DNA as a uniformly charged cylinder and membranes as planes (Gouy-Chapman theory). With computers, numerical approaches can be used to solve the Poisson-Boltzmann equation. A variety of numerical methods can be employed, including finite element and boundary element methods, but we will restrict our discussion to the finite difference method first introduced for proteins by Warwicker and Watson [Warwicker and Watson 1982]. Several groups have implemented this method here we concentrate on the work of Honig s group, whose DelPhi program has been widely used. 

For studying the stability of colloidal particles in suspension (Chapter 13) or for determining the potential at the surface of particles (Chapter 12), one often needs expressions for potential distributions around small particles that have curved surfaces. Solving the Poisson-Boltzmann equation for curved geometries is not a simple matter, and one often needs elaborate numerical methods. The linearized Poisson-Boltzmann equation (i.e., the Poisson-Boltzmann equation in the Debye-Hiickel approximation) can, however, be solved for spherical electrical double layers relatively easily (see Section 12.3a), and one obtains, in place of Equation (37),... 

The solution of the linearized Poisson-Boltzmann equation around cylinders also requires numerical methods, although when cylindrical symmetry and the Debye-Hiickel approximation are assumed the equation can be solved. The solution, however, requires advanced mathematical techniques and we will not discuss it here. It is nevertheless useful to note the form of the solution. The potential for symmetrical electrolytes has been given by Dube (1943) and is written in terms of the charge density a as... 

In a second method, the free energy was calculated by adding the electric (eq 1), entropic (eq 3), and chemical contributions, and the interaction free energy was obtained by subtracting the free energy for infinite separation. The integrals were calculated using the numerical solution of the Poisson—Boltzmann equation for y>(z,Z) and o(Z). 

In all of the discussion above, comparisons have been made between various types of approximations, with the nonlinear Poisson-Boltzmann equation providing the standard with which to judge their validity. However, as already noted, the nonlinear Poisson-Boltzmann equation itself entails numerous approximations. In the language of liquid state theory, the Poisson-Boltzmann equation is a mean-field approximation in which all correlation between point ions in solution is neglected, and indeed the Poisson-Boltzmann results for sphere-sphere [48] and plate-plate [8,49] interactions have been derived as limiting cases of more rigorous approaches. For many years, researchers have examined the accuracy of the Poisson-Boltzmann theory using statistical mechanical methods, and it is... 

The performance of the approximate methods introduced above is satisfactory under conditions normally encountered in practice. The typical deviation in the thermodynamic properties based on the approximate results obtained from the corresponding exact values, evaluated by a direct numerical integration of the Poisson-Boltzmann equation, is on the order of 5% [23,29]. 

Strauss et al. [28] has developed a numerical method for the nonlinear Poisson-Boltzmann equation 4 > 25 mV for this spherical particle in a spherical cell geometry. Figure 11.5 is a plot of the osmotic pressure for a suspension of identical particles with 100 mV surface potential and KU = 3.3. In this figure, the configurational osmotic pressure is also given and is much smaller than that of the osmotic pressure due to the double layer. The osmotic pressure increases with increased volume fraction due to the further overlap of the double layers sur-roimding each particle. 

Potential energy descriptors proposed as an indicator of hydrophobicity [Oprea and Waller, 1997]. Originally, they were calculated using the finite difference approximation method the linearized Poisson-Boltzmann equation was solved numerically to compute the electrostatic contribution to solvation at each grid point. Desolvation energy field values were calculated as the difference between solvated (grid dielectric = 80) and in vacuo (grid dielectric = 1). 

Exact numerical results are used to validate the available approximate models described by Eqs. (16)-(19). The comparison is shown in Fig. 4 for particles with scaled radii Rk = 0.1 and Rk = 15. The interaction energy was determined for two identical spheres in a z z electrolyte solution. The approximate solutions are given by Eqs. (16)-(19) and the equation for the HHF model given in Table 3. For the exact numerical solution, the full Poisson-Boltzmann equation was discretized and solved by the finite volume method. The results have been plotted for two particle sizes kR = 0.1 (Fig. 4A) and k/ = 15 (Fig. 4B). 

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. 

Coalson RD, Beck TL. Poisson-Boltzmann type equations numerical methods. In Schleyer PvR, ed. Encyclopedia of Computational Chemistry. Vol. 3. New York Wiley, 1998 2086-2100. 

Solvation energies for other multipoles inside a spherical cavity, including corrections due to salt effects, can be found, for example in Ref. 29. Analytical solutions of the Poisson equation for some other cavities, such as ellipse or cylinder, are also known [2] but are of little use in solvation calculations of biomolecules. For cavities of general shape only numerical solution of the Poisson and Poisson-Boltzmann equations is possible. There are two well-established approaches to the numerical solution of these equations the finite difference and the finite element methods. 

Holst MJ, Saied F. Numerical solution of the nonlinear Poisson-Boltzmann equation developing more robust and efficient methods. J Comput Chem 1995 16 337-364. 

Numerous approaches to handling molecular solute-continuum solvent electrostatic interactions, are described in detail in several recent reviews. - The methods most widely used and most often applied to Brownian dynamics simulations, however, fall in the category of finite difference solutions to the Poisson-Boltzmann equation. So, here we concentrate on that approach, providing a review of the basic theory along with the state-of-the-art methods in calculating potentials, energies, and forces. 

The need for computationally facile models for dynamical applications requires further trade-offs between accuracy and speed. Descending from the PB model down the approximations tree. Figure 7.1, one arrives at the generalized Born (GB) model that has been developed as a computationally efficient approximation to numerical solutions of the PB equation. The analytical GB method is an approximate, relative to the PB model, way to calculate the electrostatic part of the solvation free energy, AGei, see [18] for a review. The methodology has become particularly popular in MD applications [10,19-23], due to its relative simplicity and computational efficiency, compared to the more standard numerical solution of the Poisson-Boltzmann equation. 

Numerically, it is now a common practice to calculate within the dielectric continuum formulation but employing cavities of realistic molecular shape determined by the van der Waals surface of the solute. The method is based upon finite-difference solution of the Poisson-Boltzmann equation for the electrostatic potential with the appropriate boundary conditions [214, 238, 239]. An important outcome of such studies is that even in complex systems there exists a strong linear correlation between the calculated outer-sphere reorganization energy and the inverse donor-acceptor distance, as anticipated by the Marcus formulation (see Fig. 9.6). More... 

Numerical Method Considering a symmetric electrolyte solution such as KCl solution in a parallel plate microchannel with a height of 2H, the electrical potential distribution is described by the one-dimensional Poisson-Boltzmann equation ... 

Eigure 2 compares the LPM results for the nonlinear Poisson-Boltzmann equation (Eq. 13) and the analytical solutions of the linearized equation (Eq. 35), together with a numerical solution using the multigrid method. The parameters are the bulk ionic molar concentration c o = — 4 M, = c oNa where Na is Avogadro s number, z = 1 is the dielectric constant of the... 

Dyshlovenko, P. E. 2002. Adaptive numerical method for Poisson-Boltzmann equation and its application. Computer Physics Communications 147, no. 1-2 335-338. doi 10.1016/ 80010-4655(02)00298-9. 

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