Generalized Born equation

The Gpoi term is calculated by the generalized Born equation (Eq. (15))... 

GB/S A Generalized-Born/Surface-Area. A method for simulating solvation implicitly, developed by W.C. Still s group at Columbia University. The solute-solvent electrostatic polarization is computed using the Generalized-Born equation. Nonpolar solvation effects such as solvent-solvent cavity formation and solute-solvent van der Waals interactions are computed using atomic solvation parameters, which are based on the solvent accessible surface area. Both water and chloroform solvation can be emulated. 

Another advantage of PB based pKa calculations is that effects of electrolytes are readily accounted for in the PB equation. The Coulombic contribution in conjunction with salt dependence to the abnormally depressed pAVs of histidine in staphylococcal nuclease has been experimentally tested [56], Recently, the methodology used in the PB calculations (Eqs. 10-11 and 10-12) has been combined with the generalized Born (GB) implicit solvent model [94] to offer pKa predictions at a reduced computational cost [52],... 

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. 

A number of popular methods for computing the GB radii were proposed. The most popular one is based on the Coulomb field approximation, in which the generalized Born radii are defined by the equation ... 

An important modification of the general Schrodinger equation (Eq. 2.10) is that based on the Born-Oppenheimer approximation[l ], which assumes stationary nuclei. Further approximations include the neglect of relativistic effects, where they are less important, and the reduction of the many-electron problem to an effective one-electron problem, i. e., the determination of the energy and movement... 

The generalized Born model is given by the following equation [37],... 

But in general, the Born equation in its original form is used quite often to estimate the electrostatic contribution to the free energy of transfer of large ions 4nd entities which consist of an ion with several solvent molecules ... 

This combination of Equations [5] and [16] is called the Generalized Born/Surface Area model (GB/SA), and it is currently available in the Macro-ModeP computer package. The speed of the molecular mechanics calculations is not significantly decreased by comparison to the gas phase situation, making this model well suited to large systems. Moreover, the model takes account of some first-hydration-shell effects through the positive surface tension as well as the volume polarization effects. A selection of data for aqueous solution is provided later (Table 2), and the model is compared to experiment and to other models. Nonaqueous solvents have been simulated by changing the dielectric constant in the appropriate equations, 8 but to take the surface tension to be independent of solvent does not seem well justified. 

The SMx Approach Generalized Born Electrostatics Augmented by First-Hydration-Shell Effects Each of the foregoing solvation models, when implemented at the semiempirical level, resembles closely its implementations employing ah initio molecular orbital theory—indeed, the ab initio versions often predate the semiempirical. On the other hand, the generalized Born model, discussed with respect to Equation [16] for the case of molecular mechanics,has certain properties that make it particularly appropriate i" to the semiempirical level. 27,202,203 Qur own SMx models, where SM denotes solvation model, take advantage of this, and we now review these models. 

The CDS parameters, on the other hand, are expected neither to be solvent-independent nor to be clearly related to any particular solvent bulk observable, especially insofar as they correct for errors in the NDDO wavefunc-tion and its impact on the ENP terms. The CDS parameters also make up empirically for the errors that inevitably occur when a continuous charge distribution is modeled by a set of atom-centered nuclear charges and for the approximate nature of the generalized Born approach to solving the Poisson equation. Hence, the CDS parameters must be parameterized separately against available experimental data for every solvent. This requirement presents an initial barrier to developing new solvent parameter sets, and at present, published SMx models are available for water only (although a hexadecane parameter seH will be available soon). 

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. 

Among the many approximate models for solvation free energy evaluation, the most frequently used is the generalized Born (GB) model. It evaluates the solvation energy using the following equation ... 

III.D) where the solute is treated quantum mechanically and the solvent molecules classically [186-197]. The second approach [185] may be implemented in an entirely classical framework (e.g., through the solution of the Poisson equation or the introduction of the generalized Born model in molecular mechanics) or in a quantum mechanical framework where the wavefunction of the solute is optimized self-con-sistently in the presence of the reaction field which represents the mutual polarization of the solute and the bulk solvent. Due to the complexity of solvation phenomena, both approaches contain a number of severe approximations, and if a quantum chemical description is employed at all, it is usually restricted to the solute molecule. When choosing such a quantum chemical description from the usual alternatives ab initio, DFT, or semiempirical methods) it should be kept in mind that ab initio or DFT calculations may provide an accuracy that is far beyond the overall accuracy of the underlying solvation model. For a balanced treatment it may be attractive to employ efficient semiempirical methods provided that they capture the essential physics of solvation. The performance and predictive power of such semiempirical solvation models may then be improved further by a specific parametrization. 

The generalized Born (GB) model has been developed as a fast substitute of fhe PB equation [28-31]. The GB model can be tailored to match PB results for elecfrosfafic solvafion energies obtained by either the MS or the vdW protocol. The errors of GB resulfs in reproducing the PB counterparts are at least of fhe order of fypical mufafional effects on binding affinities. Therefore caution should be exercised when applying the GB model to calculate mutational effects. 

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. 

Tolstikhin, Siegert-state expansion for nonstationary systems. III. Generalized Born-Fock equations and adiabatic approximation for transitions to the continuum, Phys. Rev. A 77 (2008) 032711. 

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. 

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