Continuum electrostatics

The basic setup of the continuum electrostatics problem has been outlined above. The ASC formalism is based on an ansatz in which 

The apparent surface charge, a, should be distinguished from the actual surface charge that forms at any dielectric boundary [85]. The latter is given by 

ns represents the outward-pointing surface normal vector located at the point s, so that the derivative in Eq. (11.6) represents the outward-pointing normal component of the electric field. (The notation s = s+ indicates that this derivative should be evaluated infinitesimally outside of the cavity.) The normal electric field is discontinuous at a dielectric boundary, and satisfies a jump boundary condition [7, 85], 

This comes from the fact that the electric displacement (= x electric field) is continuous across the dielectric boundary. 

Equation (11.7) can be used to eliminate the exterior derivative of (p from Eq. (11.6). Then, given some initial approximation for rp (perhaps just tpf, which is known once the solute s wave function has been computed), one could compute the surface charge, and thus the reaction-field potential, without the need to perform any calculations outside of the solute cavity. For a QM solute, this procedure must then be iterated to self-consistency. The original PCM of Miertus, Scrocco, and Tomasi [60, 61] used precisely this approach this model is now known as D-PCM. It is less desirable than more modern PC Ms, owing to the need to compute the normal electric field, which may be subject to increased numerical noise relative to later formulations that involve only electrostatic potentials [77]. Perhaps more significantly, the formulation of this model has conflated the apparent and actual surface charge distributions, and corresponds to a neglect of volume polarization [13]. 

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Wood, R.H. Continuum electrostatics in a computational universe with finite cut-off radii and periodic boundary conditions Correction to computed free energies of ionic solvation. J. Chem. Phys. 103 (1995) 6177-6187. 

In this chapter we provide an introductory overview of the imphcit solvent models commonly used in biomolecular simulations. A number of questions concerning the formulation and development of imphcit solvent models are addressed. In Section II, we begin by providing a rigorous fonmilation of imphcit solvent from statistical mechanics. In addition, the fundamental concept of the potential of mean force (PMF) is introduced. In Section III, a decomposition of the PMF in terms of nonpolar and electrostatic contributions is elaborated. Owing to its importance in biophysics. Section IV is devoted entirely to classical continuum electrostatics. For the sake of completeness, other computational... 

IV. CLASSICAL CONTINUUM ELECTROSTATICS A. The Poisson Equation for Macroscopic Media... 

The continuum electrostatic approximation is based on the assumption that the solvent polarization density of the solvent at a position r in space is linearly related to the total local electric field at that position. The Poisson equation for macroscopic continuum media... 

Continuum electrostatic approaches based on the Poisson equation have been used to address a wide variety of problems in biology. One particularly useful application is in the determination of the protonation state of titratable groups in proteins [46]. For... 

The concentration of salt in physiological systems is on the order of 150 mM, which corresponds to approximately 350 water molecules for each cation-anion pair. Eor this reason, investigations of salt effects in biological systems using detailed atomic models and molecular dynamic simulations become rapidly prohibitive, and mean-field treatments based on continuum electrostatics are advantageous. Such approximations, which were pioneered by Debye and Huckel [11], are valid at moderately low ionic concentration when core-core interactions between the mobile ions can be neglected. Briefly, the spatial density throughout the solvent is assumed to depend only on the local electrostatic poten-... 

Simple considerations show that the membrane potential cannot be treated with computer simulations, and continuum electrostatic methods may constimte the only practical approach to address such questions. The capacitance of a typical lipid membrane is on the order of 1 j.F/cm-, which corresponds to a thickness of approximately 25 A and a dielectric constant of 2 for the hydrophobic core of a bilayer. In the presence of a membrane potential the bulk solution remains electrically neutral and a small charge imbalance is distributed in the neighborhood of the interfaces. The membrane potential arises from... 

In Section III we described an approximation to the nonpolar free energy contribution based on the concept of the solvent-accessible surface area (SASA) [see Eq. (15)]. In the SASA/PB implicit solvent model, the nonpolar free energy contribution is complemented by a macroscopic continuum electrostatic calculation based on the PB equation, thus yielding an approximation to the total free energy, AVP = A different implicit... 

Another variant that may mrn out to be the method of choice performs the alchemical free energy simulation with a spherical model surrounded by continuum solvent, neglecting portions of the macromolecule that lie outside the spherical region. The reaction field due to the outer continuum is easily included, because the model is spherical. Additional steps are used to change the dielectric constant of that portion of the macromolecule that lies in the outer region from its usual low value to the bulk solvent value (before the alchemical simulation) and back to its usual low value (after the alchemical simulation) the free energy for these steps can be obtained from continuum electrostatics [58]. 

Several remedies have been suggested for improving the PB based pKa prediction methods. Most of them are based on strategies that combine conformational flexibility with the PB calculation. You and Bashford included multiple conformers by systematically scanning the side chain torsion angles [107], Alexov and Gunner used Monte-Carlo protocol to sample positions of hydroxyl and other polar protons [1], This method, referred to as the multi-conformation continuum electrostatic (MCCE), was later extended to include rotamers for residues that have strong electrostatic... 

Bashford D, Karplus M (1990) pKa s of ionizable groups in proteins Atomic detail from a continuum electrostatic model. Biochemistry 29 10219-10225. 

Georgescu RE, Alexov EG, Gunner MR (2002) Combining conformational flexibility and continuum electrostatics for calculating pKas in proteins. Biophys J 83 1731—1748. 

Schaefer M, Karplus M (1996) A Comprehensive Analytical Treatment of Continuum Electrostatics. JPhys Chem 100(5) 1578-1600. 

You TJ, Bashford D (1995) Conformation and hydrogen ion titration of proteins A continuum electrostatic model with conformational flexibility. Biophys.I 69 1721-1733. 

Hunenberger, P. H. McCammon, J. A., Ewald artifacts in computer simulations of ionic solvation and ion-ion interactions a continuum electrostatics study, J. Chem. Phys. 1999,110, 1856-1872... 

To exploit the concept of PMF to represent solvent in free energy calculations, practical approximations must be constructed. A common approach is to treat the two components Z H/"P(X) and Z lYelec(X) separately. Approximations for the nonpolar term are usually derived from geometric considerations, as in scaled particle theory, for example [62], The electrostatic contribution is usually derived from continuum electrostatics. We consider these two contributions in turn. 

Continuum electrostatics approximations in which the solvent is represented as a featureless dielectric medium are an increasingly popular approach for the electrostatic... 

The most important model parameter in PBFE and MM/PBSA is the dielectric constant used for the solutes. Most studies have taken an empirical approach, viewing the dielectric constant as an adjustable parameter. While this seems plausible, it is prudent to analyze the physical problem in more detail, because, in some cases, the experimental data can be fit by models that are distinctly unphysical, despite some plausible features. We therefore come back to the simplest possible PBFE calculation the important problem of proton binding, or pKa shifts. We discuss a nonem-pirical model that attempts to avoid parameter fitting and that gives insights into the limitations of simplified continuum electrostatic free energy methods. 

Nina, M. Beglov, D. Roux, B., Atomic radii for continuum electrostatics calculations based on molecular dynamics free energy simulations, J. Phys. Chem. B 1997, 101, 5239-5248... 

Baptista, A.M. Martel, P.J. Soares, C.M., Simulation of electron-proton coupling with a Monte-Carlo method application to cytochrome C3 using continuum electrostatics, Biophys. J. 1999, 76, 2978-2998... 

Simonson, T., Electrostatic free energy calculations for macromolecules a hybrid molecular dynamics/continuum electrostatics approach, J. Phys. Chem. B 2000, 104, 6509-6513. 

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