Solving the Poisson-Boltzmann Equation

Because of the complicated nature of biomolecular geometries and charge distributions, the PB equation (PBE) is usually solved numerically by a variety of computational methods. These methods typically discretize the (exact) continuous solution to the PBE via a finite-dimensional set of basis functions. In the case of the linearized PBE, the resulting discretized equations transform the partial differential equation into a linear matrix-vector form that can be solved directly. However, the nonlinear equations obtained from the full PBE require more specialized techniques, such as Newton methods, to determine the solution to the discretized algebraic equation.  

Generic simplicial discretizations offer a much more flexible alternative to Cartesian mesh finite difference techniques. Boundary element (BE) 

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Nicholls, A., Honig, B. A rapid finite difference algorithm, utilizing successive over-relaxation to solve the Poisson-Boltzmann equation. J. Comp. Chem. 12 (1991) 435-445. 

How can Equation (11.79) be solved Before computers were available only simple ihapes could be considered. For example, proteins were modelled as spheres or ellipses Tanford-Kirkwood theory) DNA as a uniformly charged cylinder and membranes as planes (Gouy-Chapman theory). With computers, numerical approaches can be used to solve the Poisson-Boltzmann equation. A variety of numerical methods can be employed, including finite element and boundary element methods, but we will restrict our discussion to the finite difference method first introduced for proteins by Warwicker and Watson [Warwicker and Watson 1982]. Several groups have implemented this method here we concentrate on the work of Honig s group, whose DelPhi program has been widely used. 

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... 

Interactions between cationic micelles and uni- and divalent anions have been treated quantitatively by solving the Poisson-Boltzmann equation in spherical symmetry and considering both Coulombic and specific attractive forces. Predicted rate-surfactant profiles are similar to those based on the ion-exchange and mass-action models (Section 3), but fit the data better for reactions in solutions containing divalent anions (Bunton, C. A. and Moffatt, J. R. (1985) J. Phys. Chem. 1985, 89, 4166 1986,90, 538). 

The inner potentials have to be calculated by solving the Poisson-Boltzmann equations for the potentials this is done in Appendix A. 

Beyond the IHP is a layer of charge bound at the surface by electrostatic forces only. This layer is known as the diffuse layer, or the Gouy-Chapman layer. The innermost plane of the diffuse layer is known as the outer Helmholtz plane (OHP). The relationship between the charge in the diffuse layer, o2, the electrolyte concentration in the bulk of solution, c, and potential at the OHP, 2> can be found from solving the Poisson-Boltzmann equation with appropriate boundary conditions (for 1 1 electrolytes (13))... 

The next step is to determine the electrical charge and potential distribution in this diffuse region. This is done by using relevant electrostatic and statistical mechanical theories. For a charged planar surface, this problem was solved by Gouy (in 1910) and Chapman (in 1913) by solving the Poisson-Boltzmann equation, the so called Gouy-Chapman (G-C) model. 

The potential in Eq. 31 is readily obtained by solving the Poisson-Boltzmann equation... 

For studying the stability of colloidal particles in suspension (Chapter 13) or for determining the potential at the surface of particles (Chapter 12), one often needs expressions for potential distributions around small particles that have curved surfaces. Solving the Poisson-Boltzmann equation for curved geometries is not a simple matter, and one often needs elaborate numerical methods. The linearized Poisson-Boltzmann equation (i.e., the Poisson-Boltzmann equation in the Debye-Hiickel approximation) can, however, be solved for spherical electrical double layers relatively easily (see Section 12.3a), and one obtains, in place of Equation (37),... 

Solve the Poisson-Boltzmann equation for a spherically symmetric double layer surrounding a particle of radius Rs to obtain Equation (38) for the potential distribution in the double layer. Note that the required boundary conditions in this case are at r = Rs, and p - 0 as r -> oo. (Hint Transform p(r) to a new function y(r) = r J/(r) before solving the LPB equation.)... 

In order to determine the force in a specific situation, the potential must first be calculated. This is done by solving the Poisson-Boltzmann equation. In a second step, the force per unit area is calculated. It does not matter for which point we calculate II, the value must be the same for every . 

Algorithm, Utilizing Successive Over-Relaxation to Solve the Poisson-Boltzmann Equation. 

To calculate the free enagy of the system, one must solve the Poisson-Boltzmann equation to obtain the electrical potential and the ionic distribution. Because the Wigner-... 

To solve the Poisson-Boltzmann equation, boundary conditions are required. For a constant surface potential tf o on the surface of the large particles,... 

Consequently, one has to solve the Poisson—Boltzmann equation, which for a uni-univalent electrolyte has the form... 

Here zm denotes the valency of the counterions with the highest absolute charge. The function G is dependent upon the zeta potential of the particle as well as the bulk concentrations, valencies and diffusivities of the ions. The equilibrium electrical potential can be obtained using the Runge-Kutta method to solve the Poisson-Boltzmann equation numerically, and then the integral in Eq. (52) can be evaluated. 

It is calculated by solving the Poisson-Boltzmann equation with the software Delphi v.4 or other equivalent software.107 108 200 245,246 Normally, it is obtained by... 

Without going into much mathematical detail, the capacity of the space charge region is derived by solving the Poisson-Boltzmann equation as... 

Debye-Huckel approximation — In calculating the potential distribution around a charge in a solution of a strong -> electrolyte, - Debye and -> Hiickel made the assumption that the electrical energy is small compared to the thermal energy ( zjei (kT), and they solved the -> Poisson-Boltzmann equation V2f = - jT- gc° eexp( y) by expanding the exponential... 

A. Nicholls and B. Honig, /. Comput. Chem., 12, 435 (1991). A Rapid Finite Difference Algorithm, Utilizing Successive Over-Relaxation to Solve the Poisson-Boltzmann Equation. 

The ion and electrical potential distributions in the electrical double layer can be determined by solving the Poisson-Boltzmann equation [2,3]. According to the theory of electrostatics, the relationship between the eleetrieal potential ij/ and the local net charge density per unit volume at any point in the solution is deseribed by the Poisson equation ... 

Solving the Poisson-Boltzmann equation with proper boundary conditions will determine the local electrical double layer potential field y/ and hence, via Eq.(3), the local net charge density distribution. 

Equation 5.178 demonstrates that for two identically charged surfaces n, is always positive, i.e., corresponds to repulsion between the surfaces. In general, we have 0 < m < 1, because the coions are repelled from the film due to the interaction with the film surfaces. To find the exact dependence of riel oil tho film thickness, h, we solve the Poisson-Boltzmann equation for the distribution of the electrostatic potential inside the film. The solution provides the following connection between riel 2nd h for symmetric electrolytes " i ... 

The radial potential distribution inside the capillary, (r), is then obtained by solving the Poisson-Boltzmann equation for cylindrical symmetry (30). The resulting potential depends on a single adjustable constant which is fixed by the boundary condition on the potential which relates the p gential gradient at r=l/2Dp to the surface charge density, J c. Then we define... 

An alternative way of solving the Poisson Boltzmann equation is the finite element method, which uses nonuniform and not rectangular grids. For example, the grid may be made finer around an active site to accurately evaluate ligand binding, and coarser elsewhere. This achieves comparable accuracy with the finite difference methods, but with a smaller number of grid points. Unfortunately, the finite element method has not been used extensively in applications only implementations of the method have been reported to date [45 47],... 

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