Finite Difference Poisson-Boltzmann calculations

Fig. 11.28 Focusing can improve the accuracy of finite difference Poisson-Boltzmann calculations.

Applications of Finite Difference Poisson-Boltzmann Calculations... 

Wang, J., Cai, Q., Xiang, Y., and Luo, R. (2012). Reducing grid dependence in finite-difference Poisson-Boltzmann calculations,. Chem. Theory Comput. 8, pp. 2741-2751. 

Laberge, M., Vanderkooi, J.M., Sharp, K.A. Effect of a protein electric field on the CO stretch fi equency. Finite difference Poisson-Boltzmann calculations on carbonmonoxycy-tochromes c. J. Phys. Chem. 100, 10793-10801 (1996)... 

One of the chief uses of the water models is in protein simulations. Proteins have many ionizable groups. To correctly model a protein, therefore, the pH environment of a protein has to be considered. In Chapter 5, Professors James M. Briggs and Jan Antosiewicz review molecular dynamics simulations of pH-dependent properties of proteins. Finite difference Poisson-Boltzmann calculations as well as experimental and theoretical approaches used for determining pKgS of proteins are presented. 

Table 1 Free Energies of Hydration (kcal/mol) for Organic Molecules Calculated with the Finite Difference Poisson-Boltzmann (FDPB) Method and Experimental Results...

Sharp KA (1998) Calculation of electron transfer reorganization energies using the finite difference Poisson-Boltzmann model. Biophys J 73 1241-1250... 

Use of the finite difference Poisson-Boltzmann method to calculate the self-and interaction energies of the ionizable groups in water and in the protein. 

K. A. Sharp, Biophys.., 73,1241 (1998). Calculation of Electron Transfer Reorganization Energies Using the Finite Difference Poisson-Boltzmann Model. 

Born then decided to neglect the explicitly detailed structure of water molecules and replace them with a continuous electrically polarizable medium. This approximation is the same as that made in any Poisson-Boltzmann calculation, but unlike the Debye-Hiickel approximation, the ions in Born s calculation retained finite size. Thus, the Born treatment had the possibility of seeing chemically relevant differences due to ionic size. It should be remarked that Fajans was also looking for the chemically interesting dependence on ionic size in his less successful calculations. 

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). 

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. 

Shen J, Wendopolski J. Electrostatic binding energy calculation using the finite difference solution to the linearized Poisson-Boltzmann equation assessment of its accuracy. J Comput Chem 1996 17 350-357. 

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. 

Here we briefly discuss the calculation of the electrostatic energy of a molecular system from a finite difference solution of the linearized Poisson-Boltzmann equation. Calculations of the molecular electrostatic energy from grid solutions of the full nonlinear Poisson-Boltzmann equation are more involved and are discussed in detail elsewhere. ... 

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... 

Figure 9.6. Linear correlation between the outer-sphere reorganization energy, calculated by finite-difference solution of the Poisson-Boltzmann equation, and the inverse of the donor-acceptor distance in PCI-Am (X = CHj, R = Am) complex in acetonitrile. (Reproduced from [239] with permission. Copyright (1997) by the American Chemical Society.)...

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... 

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