Continuous dielectric media

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. 

Now we consider the following equations appropriate for continuous dielectric media ... 

If the solute were simply a collection of point charges surrounded by a continuous dielectric medium with the bulk dielectric constant of the solvent, the self-energy and the strength of charge-charge interactions in the solute would be reduced by a factor of . This is called dielectric screening. However, the solute itself occupies a finite volume, and solvent is excluded from this volume. This reduces the dielectric screening and is called... 

Equations (87)-(89) apply in aqueous solutions of two electrolytes in which the interaction potentials are conformal. For example, the assumptions utilized in the extensions of the Debye-Hiickel theory (e.g. water is considered as a continuous dielectric medium of dielectric constant D, that the cation-anion repulsive potential is that of hard spheres, and that all the... 

Usually, the most general nonspecific effects of dipole-orientational and electronic polarization of the medium are discussed, and the results of the theory of relaxational shifts developed under the approximation of a continuous dielectric medium may be used.(86 88) The shift of the frequency of the emitted light with time is a function of the dielectric constant e0, the refractive index n, and the relaxation time xR ... 

Of the three models that have been proposed to explain the properties of excess electrons in liquid helium, two are considered in detail (1) The electron is localized in a cavity in the liquid (2) The electron is a quasi-free particle. The pseudopotential method is helpful in studying both of these models. The most useful treatment of electron binding in polar solvents is based on a model with the solution as a continuous dielectric medium in which the additional electron induces a polarization field. This model can be used for studies with the hydrated electron. 

The electric field flux within a fluid volume of charge density pe in a continuous dielectric medium, such as water, can be described in terms of the permittivity... 

This estimate of the solvation energy is highly approximate, as it assumes that the solvent can be treated as a continuous dielectric medium on a microscopic scale. Nevertheless, it gives a rough... 

C) and this is the main reason for its ability to solubilize ionic solids, because the electrostatic attraction between anions and cations decreases 78.4 times in water. Polar liquids have much higher er values than non-polar molecules, and hydrogen-bonded liquids have exceptionally high values. It should be noted that e, is a macroscopic parameter derived for a continuous dielectric medium and cannot be used for interactions over molecular distances. 

Bayliss85 has made a quantitative approach to the study of solvent effects on the absorption spectra. Treating the solvent as a continuous dielectric medium, an expression has been developed for its effect on the Franck" Condon absorption of light, in terms of the polarization fprces of the solvent. The same result was obtained by employing methods based on quantum theory and classical dispersion theory. Bayliss85 has derived the following expression for the frequency shift, Av, caused by the solvent... 

The apparent oscillator strength is proportional to the integrated intensity under the molar absorption curve. To derive the formula, Chako followed the elassieal dispersion theory with the Lorentz-Lorenz relation (also known as the Clausius-Mosotti relation), assuming that the solute molecule is located at the center of the spherical cavity in the continuous dielectric medium of the solvent. Hence, the factor derived by Chako is also called the Lorentz-Lorenz correction. Similar derivation was also presented by Kortiim. The same formula was also derived by Polo and Wilson from a viewpoint different from Chako. 

The concept of reaction field, originally formulated by Onsager [194], has been proved to be fruitful in the quantum chemical treatment of polar subsystems (solutes) embedded in polarizable environment (solvent) [195]. Simple cavity models, where the solvent is represented by a continuous dielectric medium and the solute is sitting in a cavity inside this dielectric, has numerous application in the framework of semiempirical [196-200] and ab initio [201-205] methods. The utility of this concept in the modelisation of biochemical processes was pointed out by Tapia and his coworkers [206]. 

As in the primitive model of electrolyte solutions, the macroions and small ions are usually represented by charged HSs and the solvent is modeled as a continuous dielectric medium. The pair potential My(r) between two species i and J is given by... 

As far as representing the real experimental system, the model adopted considers only a homogenous planar region of charge-bearing groups at a surface. Electrolyte species are assumed to behave as point charges in a continuous dielectric medium distributed in the mean potential created by the surface and by the ions located in their mean position. Thus, correlation... 

During the last 40 years it has been possible to witness an important evolution on the way the environment around a solute molecule is described. The reaction field approach, the effect a continuous dielectric medium has on the charge distribution of a molecule that polarizes back the dielectric and generates a reaction potential, is a standard scheme to consider the solvent effects on many molecular properties. Most modem continuum models obtain through a self-consistent cycle the wave function of the molecule affected by the reaction potential thus the self-consistent reaction field acronym (SCRF). Solvatochromic effects have been more or less successfully explained using from Onsager s to more refined models like Nancy SCRF [44], Tomasi s polarizable continuum model (PCM) [45], Cramer and Tmhlar s SMx models [46]. 

A simplification that one can use instead of treating the solvent explicitly is to incorporate the solvent contribution by modeling it as a continuous dielectric medium. Such implicit solvation methods are widely used to reduce the computational expense of a simulation. A popular choice is the generalized Born method a review of this method and case studies of its application are contained in Bashford et al. ... 

Ultimately physical theories should be expressed in quantitative terms for testing and use, but because of the eomplexity of liquid systems this can only be accomplished by making severe approximations. For example, it is often neeessary to treat the solvent as a continuous homogeneous medium eharaeterized by bulk properties such as dielectric constant and density, whereas we know that the solvent is a molecular assemblage with short-range structure. This is the basis of the current inability of physical theories to account satisfactorily for the full scope of solvent effects on rates, although they certainly can provide valuable insights and they undoubtedly capture some of the essential features and even cause-effect relationships in solution kinetics. Section 8.3 discusses physical theories in more detail. 

If we now transfer our two interacting particles from the vacuum (whose dielectric constant is unity by definition) to a hypothetical continuous isotropic medium of dielectric constant e > 1, the electrostatic attractive forces will be attenuated because of the medium s capability of separating charge. Quantitative theories of this effect tend to be approximate, in part because the medium is not a structureless continuum and also because the bulk dielectric constant may be an inappropriate measure on the molecular scale. Eurther discussion of the influence of dielectric constant is given in Section 8.3. 

The continuum model of solvation has evolved from these beginnings. The solvent is treated as a continuous polarizable medium, usually assumed to be homogeneous and isotropic, with a uniform dielectric constant e.11-16 The solute molecule creates and occupies a cavity within this medium. The free energy of solvation is usually considered to be composed of three primary components ... 

The ions are regarded as rigid balls moving in a liquid bath. It is assumed that the macroscopic laws of motion in a viscous medium hold, and that the electrostatic interaction is determined by the theory of continuous dielectrics. This assumption implies that the moving particles are large compared to the molecular structure of the liquid. The most successful results of continuous theories can be found in any textbook of physical chemistry Stokes , law for viscous motion, Einstein s derivation of the dependence of viscosity on the concentration... 

---