Finite Difference Solutions to the Poisson-Boltzmann

Finite Difference Solutions to the Poisson-Boltzmann Equation 

As indicated in the preceding sections, evaluation of electrostatic energies requires knowledge of electrostatic potential for a given charge distribution. The Poisson equation relates spatial variation of the potential ) at position r to the density of the charge distribution p, in a medium with a dielectric constant e  

Equation [18] is valid when the polarizability of the dielectric is proportional to the electrostatic field strength. The operator V in the Cartesian coordinate system has the form (didx, didy, d/dz). When one deals with a system composed of a macromolecule immersed in an aqueous medium containing a dissolved electrolyte, the partial charges of each atom of the macromolecule can be described as fixed charges charges of the dissolved electrolyte can be described as mobile charges with density determined by a Boltzmann s distribution, and Eq. [18] can be written in the following form, known as the Poisson-Boltzmann equation  

Eor more details about this equation, see Refs. 56-59. 

The linearized Poisson-Boltzmann equation usually must be solved numerically, such as via the finite difference method. The principle of this method is as follows. Consider a small cube of side length h centered at a certain point, say r (see Eigure 2). Integrating Eq. [20] over the volume occupied by the cube and applying Gauss theorem (lyV-A)dv = n do), approximating continuous functions by distinct values at indicated points inside and outside the cube, and finally approximating derivatives by the ratio of the differences, we 

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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. 

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