Poisson-Boltzmann, generalized born

Key words Molecular dynamics, Poisson-Boltzmann, Generalized Born, Linear interaction energy, Binding free energy, Periodic boundary conditions, Steered molecular dynamics... 

Molecular Mechanics Poisson-Boltzmann or Generalized Born Surface Area Methods... 

Influence of the molecular environment on the structure and dynamics of molecular subsystems will be outlined referring to the solvation free energy (Chapter 4). Implicit solvent models based on the Poisson-Boltzmann (PB) equation and the Generalized Born (GB) model is discussed in 5 and 6. The PB or GB models are used for studies of molecular electrostatic properties and allow proper assignments of positions of protons (hydrogen atoms) within the given (bio)molecular structure. 

Liu, H.-Y., Zou, X. Electrostatics of ligand binding Parametrization of the generalized born model and comparison with the Poisson-Boltzmann approach. J. Phys. Chem. B 2006, 110,... 

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. 

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. 

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. 

Most biomolecules are either charged or highly polarized therefore, electrostatic interactions are indispensable in their theoretical description. The energy of electrostatic interactions can be modeled by a number of theoretical approaches, including Poisson-Boltzmann (PB) theory [3, 4, 26, 27], polarizable continuum theory [20, 156], and the generalized Born approximation [8, 9]. In our work, we incorporate PB theory for the polar solvation free energy and optimize the electrostatic solvation energy in our variational procedure. 

H. Tjong and H. X. Zhou. GBr6NL A generalized Born method for accurately reproducing solvation energy of the nonlinear Poisson-Boltzmann equation./ Chem. Phys., 126 195102,2007. 

An entirely different approach to the treatment of electrostatic interactions is to eliminate them entirely in favor of implicit solvation techniques [28, 315] which either solve the Poisson-Boltzmann equation [144, 337] or employ the Generalized Born [291] model of excluded volumes. 

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. 

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