Electrostatic continuum method

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... 

Pratt and co-workers have proposed a quasichemical theory [118-122] in which the solvent is partitioned into inner-shell and outer-shell domains with the outer shell treated by a continuum electrostatic method. The cluster-continuum model, mixed discrete-continuum models, and the quasichemical theory are essentially three different names for the same approach to the problem [123], The quasichemical theory, the cluster-continuum model, other mixed discrete-continuum approaches, and the use of geometry-dependent atomic surface tensions provide different ways to account for the fact that the solvent does not retain its bulk properties right up to the solute-solvent boundary. Experience has shown that deviations from bulk behavior are mainly localized in the first solvation shell. Although these first-solvation-shell effects are sometimes classified into cavitation energy, dispersion, hydrophobic effects, hydrogen bonding, repulsion, and so forth, they clearly must also include the fact that the local dielectric constant (to the extent that such a quantity may even be defined) of the solvent is different near the solute than in the bulk (or near a different kind of solute or near a different part of the same solute). Furthermore... 

The LIE method has also been used with continuum electrostatic methods. Nicolotti et al. (27) have published a nice report on this use of the LIE method for screening benzamidine inhibitors of thrombin. Another recent study used the atom-bond electronegativity equalization method on force field with the generalized Born (GB) continuum electrostatics approach to predict binding free energies of HIV-1 protease inhibitors (28). 

Sampling of the biomolecular conformations is usually performed using MD simulations or Monte Carlo methods (61, 62). The protonation state of titrateable amino acids can be treated with constant pH dynamics, QM/MM calculations, or continuum electrostatics methods (61, 62). Formation of a protein-protein encounter complex is often studied using Brownian dynamics (63). Studies of protein-protein docking involve electrostatic potential analysis and, more recently, protein flexibility models, for example normal mode analysis (64). 

C. M. Cortis, J.-M. Langlois, M. D. Beachy, and R. A. Friesner,/. Chem. Phys., 105, 5472 (1996). Quantum Mechanical Geometry Optimization in Solution Using a Finite Element Continuum Electrostatics Method. 

C. J. Gibas, P. Jambeck, and S. Subramaniam, Methods, 20, 292 (2000). Continuum Electrostatic Methods Applied to pH-Dependent Properties of Antibody-Antigen Association. 

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... 

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. 

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... 

Chen SL, Zhao DX, Yang ZZ (2010) An estimation method of binding free energy in terms of ABEEMsigmapi/MM and continuum electrostatics fused into LIE method. J Comput Chem 32(2) 338-348... 

The methods discussed here merely represent the beginning of increasingly complex computational architectures, where more than one QM method may be combined with other QM methods, MM methods, continuum electrostatic approaches and coarse-grained models in a coherent multi-scale framework. With such developments, increasingly complex chemical and biological systems can be analyzed computationally in a routine and realistic fashion. 

The expansion of the electrostatic potential into spherical harmonics is at the basis of the first quantum-continuum solvation methods (Rinaldi and Rivail, 1973 Tapia and Goschinski, 1975 Hylton McCreery et al., 1976). The starting points are the seminal Kirkwood s and Onsager s papers (Kirkwood 1934 Onsager 1936) the first one introducing the concept of cavity in the dielectric, and of the multipole expansion of the electrostatic potential in that spherical cavity, the second one the definition of the solvent reaction field and of its effect on a point dipole in a spherical cavity. The choice of this specific geometrical shape is not accidental, since multipole expansions work at their best for spherical cavities (and, with a little additional effort, for other regular shapes, such as ellipsoids or cylinders). 

Many continuum solvation methods prefer to attack the electrostatic problem by resorting to grid integration of the Poisson equation. The use of 3D grids makes it convenient to extend the model we have considered till now, characterized by a constant value of e, and by the corresponding Poisson and Laplace equations, i.e. 

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