Poisson-Boltzmann equation electrostatic free energies

SASA), a concept introduced by Lee and Richards [9], and the electrostatic free energy contribution on the basis of the Poisson-Boltzmann (PB) equation of macroscopic electrostatics, an idea that goes back to Born [10], Debye and Htickel [11], Kirkwood [12], and Onsager [13]. The combination of these two approximations forms the SASA/PB implicit solvent model. In the next section we analyze the microscopic significance of the nonpolar and electrostatic free energy contributions and describe the SASA/PB implicit solvent model. 

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. 

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 ... 

After the solution of the Poisson—Boltzmann equation is obtained, the total double layer free energy per unit area is obtained by adding the electrostatic... 

A crucial parameter-free test of the theory is provided by its application to micelle formation from ionic surfactants in dilute solution [47]. There, if we accept that the Poisson-Boltzmann equation provides a sufficiently reasonable description of electrostatic interactions, the surface free energy of an aggregate of radius R and aggregation number N can be calculated horn the electrostatic free energy analytically. The whole surface free energy can be decomposed into two terms, one electrostatic, and another due to short-range molecular interactions that, from dimensional considerations, must be proportional to area per surfactant molecule, i.e. 

Glendinning, A.B. Russel, W.B. The electrostatic repulsion between charged spheres from exact solutions to the linearized Poisson-Boltzmann equation. J. Colloid Interface Sci. 1983, 93, 95-111 Carnie, S.L. Chan, D.Y.C. Interaction free energy between identical spherical colloidal... 

Chen SW, Honig B. Monovalent and divalent salt effects on electrostatic free energies defined by the nonlinear Poisson-Boltzmann equation application to DNA binding reactions. J Phys Chem B 1997 101 9113-9118. 

The need for computationally facile models for dynamical applications requires further trade-offs between accuracy and speed. Descending from the PB model down the approximations tree. Figure 7.1, one arrives at the generalized Born (GB) model that has been developed as a computationally efficient approximation to numerical solutions of the PB equation. The analytical GB method is an approximate, relative to the PB model, way to calculate the electrostatic part of the solvation free energy, AGei, see [18] for a review. The methodology has become particularly popular in MD applications [10,19-23], due to its relative simplicity and computational efficiency, compared to the more standard numerical solution of the Poisson-Boltzmann equation. 

In 1923, E Hiickel and P Debye, winner of the 1936 Nobel Prize in Chemistry, adapted the Poisson-Boltzmann theory to explain the nonidealities of dilute solutions of strong electrolytes. To visualize Na ions surrounded by CD ions and C1 surrounded by Na at the same time, think of a NaCl crystal that is expanded uniformly. Now add fluctuations, Debye and Hhckel focused on one ion as a charged sphere, and used the linear approximation to the Poisson-Boltzmann equation to compute the electrostatic free energy of creating the nonuniform distribution of its surrounding counterions and co-ions. 

The first three terms in Eq. (2) are calculated using the Cornell force field [8] with no cutoff. The electrostatic solvation free energy is calculated by solving the Poisson-Boltzmann equation with the Delphi software [9, 10], which has been shown to constitute a good compromise between accuracy and computing time. 

This chapter provides an introductory overview of the approaches used to predict ionization states of titratable residues in proteins, based on the assumption that the difference in protonation behavior of a given group isolated in solution, for which the ionization constant is assumed to be known, and the protonation behavior in the protein environment is purely electrostatic in origin. Calculations of the relevant electrostatic free energies are based on the Poisson-Boltzmann (PB) model of the protein-solvent system and the finite difference solution to the corresponding Poisson-Boltzmaim equation. We also discuss some relevant pH-dependent properties that can be determined experimentally. The discussion is limited to models that treat the solvent and the solute as continuous dielectric media. Alternative approaches based on microscopic simulations, which can be useful for small molecules (e.g., see Refs. 19-24) are not covered here because they are, in general, too time intensive for proteins. The present treatment is intended to be simple and pedagogic. 

As shown above the size of the explicit water simulations can be rather large, even for a medium sized protein as in the case of the sea raven antifreeze protein (113 amino acid residues and 5391 water). Simulations of that size can require a large amount of computer memory and disk space. If one is interested in the stability of a particular antifreeze protein or in general any protein and not concerned with the protein-solvent interactions, then an alternative method is available. In this case the simulation of a protein in which the explicit waters are represent by a structureless continuum. In this continuum picture the solvent is represented by a dielectric constant. This replacement of the explicit solvent model by a continuum is due to Bom and was initially used to calculate the solvation free energy of ions. For complex systems like proteins one uses the Poisson-Boltzmann equation to solve the continuum electrostatic problem. In... 

Electrostatic Free Energies for the Poisson-Boltzmann Equation... 

ELECTROSTATIC FREE ENERGIES FOR THE POISSON-BOLTZMANN EQUATION... 

Equation (62) is rigorous (assuming Gibbs and Helmholtz free energies are the same for ions in a solid—solution interfacial system) but requires explicit electrostatic potential equations in order to obtain an explicit activity coefficient equation. Since rigorous, explicit electrostatic potential equations for solute and surface site ions have yet to be derived, the approximate electrostatic potential equations, which are solutions to the linearized Poisson—Boltzmann equation, were used here and by Debye—Hiickel to give... 

S.W. Chen andB. Honig,/. Pfjys. Chem.B, 101,9113 (1997).MonovalentandDivalentSalt Effects on Electrostatic Free Energies Defined by the Nonlinear Poisson-Boltzmann Equation Application to DNA Binding Reactions. 

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