Poisson-Boltzmann continuum electrostatics

Since electrostatic effects dominate the thermodynamic cycle as shown in Figure 10-2, major development efforts have focused on the calculation of electrostatic energy for transferring the neutral and charged forms of the ionizable group from water with dielectric constant of about 80 to the protein with a low dielectric constant (see later discussions). This led to the development of continuum based models, where water and protein are described as uniform dielectric media, and enter into the linearized Poisson-Boltzmann (PB) electrostatic equation,... 

Another way of calculating the electrostatic component of solvation uses the Poisson-Boltzmann equations [22, 23]. This formalism, which is also frequently applied to biological macromolecules, treats the solvent as a high-dielectric continuum, whereas the solute is considered as an array of point charges in a constant, low-dielectric medium. Changes of the potential within a medium with the dielectric constant e can be related to the charge density p according to the Poisson equation (Eq. (41)). 

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 ... 

The continuum treatment of electrostatics can also model salt effects by generalizing the Poisson equation (12) to the Poisson-Boltzmann equation. The finite difference approach to solving Eq. (12) extends naturally to treating the Poisson-Boltzmann equation [21], and the boundary element method can be extended as well [19]. 

Note that it is fairly common in the literature for continuum solvation calculations to be reported as having been carried out using Poisson-Boltzmann electrostatics even when no electrolyte concentration is being considered, i.e., the Poisson equation is considered a special case of the PB equation and not named separately. 

Grochowski, P., and Trylska, J. (2008). Continuum molecular electrostatics, salt effects and counterion binding. A review of the Poisson-Boltzmann theory and its modifications. Biopolymers 89, 93—113. 

The most expensive part of a simulation of a system with explicit solvent is the computation of the long-range interactions because this scales as Consequently, a model that represents the solvent properties implicitly will considerably reduce the number of degrees of freedom of the system and thus also the computational cost. A variety of implicit water models has been developed for molecular simulations [56-60]. Explicit solvent can be replaced by a dipole-lattice model representation [60] or a continuum Poisson-Boltzmann approach [61], or less accurately, by a generalised Bom (GB) method [62] or semi-empirical model based on solvent accessible surface area [59]. Thermodynamic properties can often be well represented by such models, but dynamic properties suffer from the implicit representation. The molecular nature of the first hydration shell is important for some systems, and consequently, mixed models have been proposed, in which the solute is immersed in an explicit solvent sphere or shell surrounded by an implicit solvent continuum. A boundary potential is added that takes into account the influence of the van der Waals and the electrostatic interactions [63-67]. 

Im W, Beglov D, Roux B. 1998. Continuum solvation model computation of electrostatic forces from numerical solutions to the Poisson-Boltzmann equation. Comput. Phys. Commun. 111 59-75... 

Numerous approaches to handling molecular solute-continuum solvent electrostatic interactions, are described in detail in several recent reviews. - The methods most widely used and most often applied to Brownian dynamics simulations, however, fall in the category of finite difference solutions to the Poisson-Boltzmann equation. So, here we concentrate on that approach, providing a review of the basic theory along with the state-of-the-art methods in calculating potentials, energies, and forces. 

---