Implicit Solvation - Poisson-Boltzmann

Poisson-Boltzmann

Although this equation has solid theoretical justification, it is computationally expensive to calculate without approximations. The Poisson-Boltzmann equation (PB) describes the electrostatic environment of a solute in a solvent containing ions. It can be written in cgs units as:

\vec{\nabla}\cdot\left = -4\pi\rho^{f}(\vec{r}) - 4\pi\sum_{i}c_{i}^{\infty}z_{i}q\lambda(\vec{r})e^{\frac{-z_{i}q\Psi(\vec{r})}{kT}}

or (in mks):

\vec{\nabla}\cdot\left = -\rho^{f}(\vec{r}) - \sum_{i}c_{i}^{\infty}z_{i}q\lambda(\vec{r})e^{\frac{-z_{i}q\Psi(\vec{r})}{kT}}

where represents the position-dependent dielectric, represents the electrostatic potential, represents the charge density of the solute, represents the concentration of the ion i at a distance of infinity from the solute, is the valence of the ion, q is the charge of a proton, k is the Boltzmann constant, T is the temperature, and is a factor for the position-dependent accessibility of position r to the ions in solution (often set to uniformly 1). If the potential is not large, the equation can be linearized to be solved more efficiently.

A number of numerical Poisson-Boltzmann equation solvers of varying generality and efficiency have been developed, including one application with a specialized computer hardware platform. However, performance from PB solvers does not yet equal that from the more commonly used generalized Born approximation.

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