Applying Poisson-Boltzmann Methods

The purpose of this section is to illustrate the application of PB methods to various molecular and biomolecular problems. Where feasible, the problems are described in sufficient detail to allow readers to further investigate these systems with any of the software packages listed in Table 1. Additionally, these examples are included in the APBS software package distribution (http //agave.wustl.edu/apbs/) and are freely available for download. 

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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. 

Another methodological approach, which has become more attractive in the last few years for estimating binding free energies of protein-protein complexes, is the MM-PBSA method (Molecular Mechanics/Poisson-Boltzmann Surface Area). This method is a fully atomistic approach that combines molecular mechanics and continuum solvent, and has several appealing features as the possibility of being applied to a variety of systems not suitable for FEP such as very large protein-protein complexes.69,129-137... 

A similar approximation method can be applied for the case of infinitely long cylindrical particles of radius a in a general electrolyte composed of N ionic species with valence z, and bulk concentration n, (/ = 1, 2,. . . , N). The cylindrical Poisson-Boltzmann equation is... 

This chapter deals with a method for obtaining the exact solution to the linearized Poisson-Boltzmann equation on the basis of Schwartz s method [1] without recourse to Derjaguin s approximation [2]. Then we apply this method to derive series expansion representations for the double-layer interaction between spheres [3-13] and those between two parallel cylinders [14, 15]. 

A hybrid approach of the extended scaled particle theory (SPT) and the Poisson-Boltzmann (PB) equation for the solvation free energy of non-polar and polar solutes has been proposed by us. This new method is applied for the hydration free energy of the protein, avian pancreatic polypeptide (36 residues). The contributions form the cavity formation and the attractive interaction between the solute and the solvent to the solvation free energy compensate each other. The electrostatic conffibution is much larger than other terms in this hyelration free energy, because hydrophilic residues are ionized in water. This work is the first step toward further applications of our new method to free energy difference calculation appeared in the stability analysis of protein. 

Implicit solvation models developed for condensed phases represent the solvent by a continuous electric field, and are based on the Poisson equation, which is valid when a surrounding dielectric medium responds linearly to the charge distribution of the solute. The Poisson equation is actually a special case of the Poisson-Boltzmann (PB) equation PB electrostatics applies when electrolytes are present in solution while the Poisson equation applies when no ions are present. Solving the Poisson equation for an arbitrary equation requires numerical methods, and many researchers have developed an alternative way to approximate the Poisson equation that can be solved analytically, known as the... 

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. 

Implicit solvation models developed for condensed phases represent the solvent by a continuous electric field, and are based on the Poisson equation, which is valid when a surrounding dielectric medium responds linearly to the charge distribution of the solute. The Poisson equation is actually a special case of the Poisson-Boltzmann (PB) equation PB electrostatics applies when electrolytes are present in solution, while the Poisson equation applies when no ions are present. Solving the Poisson equation for an arbitrary equation requires numerical methods, and many researchers have developed an alternative way to approximate the Poisson equation that can be solved analytically, known as the Generalized Born (GB) approach. The most common implicit models used for small molecules are the Conductor-like Screening Model (COSMO) [96,97], the Dielectric Polarized Continuum Model (DPCM) [98], the Conductor-like modification to the Polarized Continuum Model (CPCM) [99], the Integral Equation Formalism implementation of PCM (lEF-PCM) [100] PB models and the GB SMx models of Cramer and Truhlar [52,57,101,102]. The newest Miimesota solvation models are the SMD (universal Solvation Model based on solute electron Density [57]) and the SMLVE method, which combines the surface and volume polarization for electrostatic interactions model (SVPE) [103-105] with semiempirical terms that account for local electrostatics [106]. Further details on these methods can be found in Chapter 11 of reference 52. 

The linearized Poisson-Boltzmann equation usually must be solved numerically, such as via the finite difference method. The principle of this method is as follows. Consider a small cube of side length h centered at a certain point, say r (see Eigure 2). Integrating Eq. [20] over the volume occupied by the cube and applying Gauss theorem (lyV-A)dv = n do), approximating continuous functions by distinct values at indicated points inside and outside the cube, and finally approximating derivatives by the ratio of the differences, we... 

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