Finite difference Poisson-Boltzmann methods

Rocchia W, Sridharan S, Nicholls A, Alexov E, Chiabrera A, Honig B (2002) Rapid grid-based construction of the molecular surface for both molecules and geometric objects applications to the finite difference Poisson-Boltzmann method, J Comp Chem, 23 128-137... 

K. Sharp,/. Comput. Chem., 12, 454 (1991). Incorporating Solvent and Ion Screening into Molecular Dynamics Using the Finite-Difference Poisson-Boltzmann Method. 

Numerical strategies for computing both the electronic and nuclear components of the ET rate are now rather advanced (see the chapter by Newton). For example, in both proteins and small molecules, finite-difference Poisson-Boltzmann methods are widely used for computing the outer-sphere component of the solvent reorganization energy Aq [9, 10, 11, 12] ... 

Vizcarra CL, Zhang NG, Marshall SA, Wingreen NS, Zeng C, Mayo SL (2008) An improved pairwise decomposable finite-difference Poisson-Boltzmann method for computational protein design. J Comput Chem 29 1153-1162... 

Prabhu, N.V., Zhu, P., Sharp, K.A. Implementation and testing of stable, fast implicit solvation in molecular dynamics using the smooth-permittivity finite difference Poisson-Boltzmann method. J. Comput. Chem. 2004,25(16), 2049-64, December. 

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. 

These FDPB-based methods might be further improved by using a position-dependent dielectric function that treats distinct regions of the protein differently (e.g., surface, interior, polar, nonpolar, charged, flexible, rigid, etc.), as has been mentioned by Warshel and others. These methods, as they mature, can be applied to questions of protein stability versus pH, the pH-dependent binding of inhibitors, and so on. The availability of these fast and automated methods makes the finite difference Poisson-Boltzmann method a useful predictive tool for the computational chemist. 

To solve the PB equation for arbitrary geometries requires some type of discretization, to convert the partial differential equation into a set of difference equations. Finite difference methods divide space into a cubic lattice, with the potential, charge density, and ion accessibility defined at the lattice points (or grid points ) and the permittivity defined on the branches (or grid lines ). Equation [1] becomes a system of simultaneous equations referred to as the finite difference Poisson-Boltzmann (FDPB) equation ... 

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

Relative to finite-difference Poisson-Boltzmann approaches, such methods have the advantage that only the two-dimensional cavity surface must be discretized. 

Fig. 1. Explanation of the principles of the finite-difference method for solution of the Poisson-Boltzmann equation...

Davis, M. E., McCammon, J. A. Solving the finite difference linearized Poisson-Boltzmann equation A comparison of relaxation and conjugate gradients methods.. J. Comp. Chem. 10 (1989) 386-394. 

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. 

The continuum treatment of electrostatics can also model salt effects by generalizing the Poisson equation (12) to the Poisson-Boltzmann equation. The finite difference approach to solving Eq. (12) extends naturally to treating the Poisson-Boltzmann equation [21], and the boundary element method can be extended as well [19]. 

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). 

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