Solvation models Poisson-Boltzmann methods

These workers also calculated the relative stability of the tautomers lOa-c in the gas phase by ab initio and density functional theory (DFT) methods and in solution using several continuum solvation models such as self-consistent reaction fields (SCRF) and the Poisson-Boltzmann method. These results showed good agreement between the experimental and theoretical approaches. 

The final class of methods that we shall consider for calculating the electrostatic compone of the solvation free energy are based upon the Poisson or the Poisson-Boltzmann equatior Ihese methods have been particularly useful for investigating the electrostatic properties biological macromolecules such as proteins and DNA. The solute is treated as a body of co stant low dielectric (usually between 2 and 4), and the solvent is modelled as a continuum high dielectric. The Poisson equation relates the variation in the potential (f> within a mediu of uniform dielectric constant e to the charge density p ... 

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... 

In order to set up an MD simulation based on the implicit solvation model, a set of atomic radii is required, which is an additional set of input parameters compared to the explicit solvent case. A number of implicit solvent models have been developed, and the effect of the solvent is treated as an average potential acting on the solute. The implicit solvent model based on a finite difference solution of the Poisson-Boltzmann (PB) theory provides a rigorous theoretical framework and captures the polar component of the free energy of solvation for a given biomolecule quite well.35 However, it is a computationally expensive method and, therefore, has limited application in MD simulations. 

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. 

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. 

In addition to the solvent contributions, the electrochemical potential can be modeled. Application of an external electric field within a metal/vacuum interface model has been used to investigate the impact of potential alteration on the adsorption process [111, 112]. Although this approach can model the effects of the electrical double layer, it does not consider the adsorbate-solvent, solvent-solvent, and solvent-metal interactions at the electrode-electrolyte interface. In another approach, N0rskov and co-workers model the electrochemical environment by changing the number of electrons and protons in a water bilayer on a Pt(lll) surface [113-115]. Jinnouchi and Anderson used the modified Poisson-Boltzmann theory and DFT to simulate the solute-solvent interaction to integrate a continuum approach to solvation and double layer affects within a DFT system [116-120]. These methods differ in the approximations made to represent the electrochemical interface, as the time and length scales needed for a fiilly quantum mechanical approach are unreachable. 

As shown above the size of the explicit water simulations can be rather large, even for a medium sized protein as in the case of the sea raven antifreeze protein (113 amino acid residues and 5391 water). Simulations of that size can require a large amount of computer memory and disk space. If one is interested in the stability of a particular antifreeze protein or in general any protein and not concerned with the protein-solvent interactions, then an alternative method is available. In this case the simulation of a protein in which the explicit waters are represent by a structureless continuum. In this continuum picture the solvent is represented by a dielectric constant. This replacement of the explicit solvent model by a continuum is due to Bom and was initially used to calculate the solvation free energy of ions. For complex systems like proteins one uses the Poisson-Boltzmann equation to solve the continuum electrostatic problem. In... 

Following earlier work by Wood et al., Luo and Tucker have relaxed the constant density restriction, and developed a continuum model in which the dielectric constant may be position-dependent. This dielectric function, s(7), is defined, at each point r in the fluid, in terms of the local density of the fluid at , pi(f), which is itself determined by the local values of the electric field and the compressibility. However, the local value of the electric field at 7 must be found from electrostatic equations (see Poisson-Boltzmann Type Equations Numerical Methods) which depend upon the dielectric function s(r) everywhere. Hence, all of the relevant equations must be solved self-consistently, and this is done using a numerical grid algorithm (see Poisson-Boltzmann Type Equations Numerical Methods). The result of such calculations are the density profile of the fluid around the solute and the position-dependent electric field, from which the free energy of solvation may be evaluated. The effects of solvent compression on solvation energetics can be quite substantial. Compression-induced enhancements to the solvation free energy of nearly 15 kcal mol" have been calculated for molecular ions in SC water at Tt = 1.01 and pr = 0.8. ... 

Operating fuel cell involves solvent environment, and it is important to study water formation reaction under solvent conditions to understand the solvation effects on overall reaction kinetics. Adding water molecules explicitly or entire water bilayer to unit cells of simulating catalyst is considered as explicit solvent model. This method, however, does not resemble fully saturated system, and water molecules are introduced during the simulation that directly participates in the ORR [148]. Poisson-Boltzmann implicit solvent model is a method to resemble solvent as a continuum rather than individual molecules in explicit models [160,162]. This method is computationally inexhaustive and accurate enough to reproduce reliable results for the atomistic energy calculations. 

Jinnouchi and Anderson, as well as Goddard and coworkers, have instead adopted a Poisson Boltzmann distribution of countercharge (Model 2b.4). These methods couple this distribution with an implicit continuum solvation model for the solvent (water). The continuum model extends the double layer consideration to the diffusion layer region. Jinnouchi and Anderson highlight that strongly bound water molecules must still be included... 

---