Electrostatics Poisson-Boltzmann equation

Tanford, C., Kirkwood, J. G. Theory of protein titration curves. I. General equations for impenetrable spheres. J. Am. Chem. Soc. 79 (1957) 5333-5339. 6. Garrett, A. J. M., Poladian, L. Refined derivation, exact solutions, and singular limits of the Poisson-Boltzmann equation. Ann. Phys. 188 (1988) 386-435. Sharp, K. A., Honig, B. Electrostatic interactions in macromolecules. Theory and applications. Ann. Rev. Biophys. Chem. 19 (1990) 301-332. 

Sharp, K. A., Honig, B. Calculating total electrostatic energies with the nonlinear Poisson-Boltzmann equation. J. Phys. Chem. 94 (1990) 7684-7692. Zhou, H.-X. Macromolecular electrostatic energy within the nonlinear Poisson-Boltzmann equation. J. Chem. Phys. 100 (1994) 3152-3162. 

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)). 

If there are ions in the solution, they will try to change their location according to the electrostatic potential in the system. Their distribution can be described according to Boltzmarm. Including these effects and applying some mathematics leads to the final linearized Poisson-Boltzmann equation (Eq. (43)). 

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]. 

Calculation of Electrostatic Potential by the Poisson-Boltzmann Equation... 

In addition to the nearest-neighbor interaction, each ion experiences the electrostatic potential generated by the other ions. In the literature this has generally been equated with the macroscopic potential 0 calculated from the Poisson-Boltzmann equation. This corresponds to a mean-field approximation (vide infra), in which correlations between the ions are neglected. This approximation should be the better the low the concentrations of the ions. 

For small deviations from electroneutrality, the charge density at x is proportional to -(x)/kT9 where < is the difference of the electrostatic potential from its (constant) value when there is no charge density (the density of a species of charge z is proportional to 1 - zkT on linearizing the Boltzmann exponential). Then the Poisson equation [Eq. (44)] becomes the linearized Poisson-Boltzmann equation ... 

Here, b is the distance between the nearest unit charges along the cylinder (b = 0.34nm for the ssDNA and b = 0.17nm for the dsDNA), (+) and (—) are related to cations and anions, respectively, and a = rss for the ssDNA and a rds for the dsDNA. The expressions (5) and (6) have been obtained using the equations for the electrostatic potential derived in [64, 65], where a linearization of the Poisson-Boltzmann equation near the Donnan potential in the hexagonal DNA cell was implemented. 

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... 

The electrostatic methods just discussed suitable for nonelectrolytic solvent. However, both the GB and Poisson approaches may be extended to salt solutions, the former by introducing a Debye-Huckel parameter67 and the latter by generalizing the Poisson equation to the Poisson-Boltzmann equation.68 The Debye-Huckel modification of the GB model is valid to much higher salt concentrations than the original Debye-Huckel theory because the model includes the finite size of the solute molecules. 

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))... 

A more detailed view of the dynamies of a ehromatin chain was achieved in a recent Brownian dynamics simulation by Beard and Schlick [65]. Like in previous work, the DNA is treated as a segmented elastic chain however, the nueleosomes are modeled as flat cylinders with the DNA attached to the cylinder surface at the positions known from the crystallographic structure of the nucleosome. Moreover, the electrostatic interactions are treated in a very detailed manner the charge distribution on the nucleosome core particle is obtained from a solution to the non-linear Poisson-Boltzmann equation in the surrounding solvent, and the total electrostatic energy is computed through the Debye-Hiickel approximation over all charges on the nucleosome and the linker DNA. 

Recent advances in computational power have made it possible to use efficient approximations of the Poisson-Boltzmann equation also to estimate the electrostatic component of protein-ligand interactions [56]. 

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 Poisson-Boltzmann equation. The slab model is based on a solution of the linearized Poisson-Boltzmann equation that is valid only for low electrostatic surface potentials. As... 

Rahaman and Hatton [152] developed a thermodynamic model for the prediction of the sizes of the protein filled and unfilled RMs as a function of system parameters such as ionic strength, protein charge, and size, Wq and protein concentration for both phase transfer and injection techniques. The important assumptions considered include (i) reverse micellar population is bidisperse, (ii) charge distribution is uniform, (iii) electrostatic interactions within a micelle and between a protein and micellar interface are represented by nonlinear Poisson-Boltzmann equation, (iv) the equilibrium micellar radii are assumed to be those that minimize the system free energy, and (v) water transferred between the two phases is too small to change chemical potential. 

The electrostatic force (F ) between two charged plates separated by an electrolyte solution can be determined from an existing imphcit solution to the nonlinear Poisson-Boltzmann equation and can be expressed in the form [187]... 

The charge density x on any electrolyte lamina parallel to the electrode and a distance x from it can be obtained by the application of electrostatics (Poisson s equation) and the Boltzmann distribution. Similarly, one can write for the intrinsic semiconductor, Poisson s equation... 

The purpose of the present chapter is to introduce some of the basic concepts essential for understanding electrostatic and electrical double-layer pheneomena that are important in problems such as the protein/ion-exchange surface pictured above. The scope of the chapter is of course considerably limited, and we restrict it to concepts such as the nature of surface charges in simple systems, the structure of the resulting electrical double layer, the derivation of the Poisson-Boltzmann equation for electrostatic potential distribution in the double layer and some of its approximate solutions, and the electrostatic interaction forces for simple geometric situations. Nonetheless, these concepts lay the foundation on which the edifice needed for more complicated problems is built. 

Figure 6.10 Electrostatic double-layer force between a sphere of R = 3 /um radius and a flat surface in water containing 1 mM monovalent salt. The force was calculated using the nonlinear Poisson-Boltzmann equation and the Derjaguin approximation for constant potentials (tpi = 80 mV, ip2 = 50 mV) and for constant surface charge (i/2/Ad so that at large distances both lead to the same potential.

The ionic groups on the micellar surface and the counterions will give rise to a nonuniform electrostatic potential according to the Poisson equation. If furthermore the electrostatic effects dominate the counterion distribution the ion concentration is determined by following a Boltzmann distribution. These approximations lead to the Poisson-Boltzmann equation. 

From the solution of the Poisson-Boltzmann equation one can calculate the electrostatic contribution to the free energy. It is illustrative to divide G into two parts302. The first is concerned with the free energy of the electric field and is given by ... 

The micellar surface has a high charge density and the stability of the aggregate is heavily dependent on the binding of counterions to the surface. From the solution of the Poisson-Boltzmann equation one finds that a large fraction (0.4—0.7) of the counterions is in the nearest vicinity of the micellar surface300. These ions could be associated with the Stern layer, but it seems simpler not to make a distinction between the ions of the Stern layer and those more diffusely bound. They are all part of the counterions and their distribution is primarily determined by electrostatic effects. 

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