Poisson-Boltzmann equation solvation effects

Two remaining problems relating to the treatment of solvation include the slowness of Poisson-Boltzmann calculations, when these are used to treat electrostatic effects, and the difficulty of keeping buried, explicit solvent in equilibrium with the external solvent when, e.g., there are changes in nearby solute groups in an alchemical simulation. Faster methods for solving the Poisson-Boltzmann equation by means of parallel finite element techniques are becoming available, however.22 24... 

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. 

Solvation energies for other multipoles inside a spherical cavity, including corrections due to salt effects, can be found, for example in Ref. 29. Analytical solutions of the Poisson equation for some other cavities, such as ellipse or cylinder, are also known [2] but are of little use in solvation calculations of biomolecules. For cavities of general shape only numerical solution of the Poisson and Poisson-Boltzmann equations is possible. There are two well-established approaches to the numerical solution of these equations the finite difference and the finite element methods. 

Another approach to include solvation effect is to couple the MFCC calculation with the Poisson-Boltzmann (PB) equation... 

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. ... 

---