Poisson-Boltzmann equation energy models

Oosawa (1971) developed a simple mathematical model, using an approximate treatment, to describe the distribution of counterions. We shall use it here as it offers a clear qualitative description of the phenomenon, uncluttered by heavy mathematics associated with the Poisson-Boltzmann equation. Oosawa assumed that there were two phases, one occupied by the polyions, and the other external to them. He also assumed that each contained a uniform distribution of counterions. This is an approximation to the situation where distribution is governed by the Poisson distribution (Atkins, 1978). If the proportion of site-bound ions is negligible, the distribution of counterions between these phases is then given by the Boltzmann distribution, which relates the population ratio of two groups of atoms or ions to the energy difference between them. Thus, for monovalent counterions... 

A more detailed view of the dynamies of a ehromatin chain was achieved in a recent Brownian dynamics simulation by Beard and Schlick [65]. Like in previous work, the DNA is treated as a segmented elastic chain however, the nueleosomes are modeled as flat cylinders with the DNA attached to the cylinder surface at the positions known from the crystallographic structure of the nucleosome. Moreover, the electrostatic interactions are treated in a very detailed manner the charge distribution on the nucleosome core particle is obtained from a solution to the non-linear Poisson-Boltzmann equation in the surrounding solvent, and the total electrostatic energy is computed through the Debye-Hiickel approximation over all charges on the nucleosome and the linker DNA. 

The geometry. It is clear that the geometry of the system is much simplified in the slab model. Another possibility is to model the protein as a sphere and the stationary phase as a planar surface. For such systems, numerical solutions of the Poisson-Boltzmann equations are required [33]. However, by using the Equation 15.67 in combination with a Derjaguin approximation, it is possible to find an approximate expression for the interaction energy at the point where it has a minimum. The following expression is obtained [31] ... 

Rahaman and Hatton [152] developed a thermodynamic model for the prediction of the sizes of the protein filled and unfilled RMs as a function of system parameters such as ionic strength, protein charge, and size, Wq and protein concentration for both phase transfer and injection techniques. The important assumptions considered include (i) reverse micellar population is bidisperse, (ii) charge distribution is uniform, (iii) electrostatic interactions within a micelle and between a protein and micellar interface are represented by nonlinear Poisson-Boltzmann equation, (iv) the equilibrium micellar radii are assumed to be those that minimize the system free energy, and (v) water transferred between the two phases is too small to change chemical potential. 

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). 

Fig. 4 Comparison of the model improvements on the Derjaguin approximation to the exact numerical computational results of the full Poisson-Boltzmann equation for two spheres with the scaled radius Rk — 0.1 and Rk — 15 and constant surface potential [j/ ez/(k-gT) = 1. The scaled energy, G h), on the vertical axis is defined by G(h) = (/,)/jsM...

A computationally efficient analytical method has been developed for the crucial calculation of Born radii, which is required for each atom of the solute that carries a (partial) charge, and the Gpoi term has been parameterized to fit atomic polarization energies obtained by Poisson-Boltzmann equation [57]. The GB/SA model is thus fully analytical and affords first and second derivatives allowing for solvation effects to be included in energy minimizations, molecular dynamics, etc. The Gpoi term is most important for polar molecules and describes the polarization of the solvent by the solute. As force fields in general are not polarizable, it does not account for the polarization of the solute by the solvent. This is clearly an important limitation of this type of calculations. 

Qui et al. have compared experimental and calculated hydration free energies for a set of 35 small organic molecules with diverse functional groups by using the OPLS force field and the GB/SA hydration model [57], These calculations resulted in a mean absolute error of 0.9 kcal/mol. It is of interest to note that the results obtained with the GB/SA model were very similar to those obtained by the corresponding calculations using the full Poisson-Boltzmann equation. 

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. 

Zhou, R., Krilov, G., Berne, B.J. Comment on can a continuum solvent model reproduce the free energy landscape of a beta-hairpin folding in water The Poisson-Boltzmann equation. J. Phys. Chem. B 2004,108, 7528-30. 

Equations 12.14 and 12.16 are solved for the surface function S and electrostatic potential 0, respectively. These coupled "Laplace-Beltrami and Poisson-Boltzmann" equations are the governing equation for the DG-based solvation model in the Eulerian representation. The Lagrangian representation of the DG-based solvation model has also been derived [72]. Both the Eulerian and Lagrangian solvation models have been shown [71, 72] to be essentially equivalent and provide very good predictions of solvation energies for a diverse range of compounds. 

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