Total Solvation Free Energy Research Articles

Solv: An Alternative Continuum Model Implementation Based on Fixed Atomic Charges, Scaled Particle Theory, and the Atom-Atom Potential Method.

An alternative continuum model implementation is reported. The electrostatic contribution to the solvation Gibbs free energy utilizes the noniterative conductor-like screening model of Vyboishchikov and Voityuk (DOI: 10.1002/jcc.26531) based on the fixed partial atomic charges. The nonelectrostatic solute-solvent dispersion-repulsion energy is calculated through the Caillet-Claverie atom-atom potential method employing the grid-based approach. The nonelectrostatic cavitation energy is computed within the scaled particle theory (SPT) formalism with the solute hard-sphere radius obtained via the Pierotti-Claverie (PC) scheme, from the solute molecular surface (SPT-S) or volume (SPT-V). The solvent hard-sphere radius is derived through the fitting to the experimental total solvation free energies of 2530 neutral species in 92 solvents. Application of the model to reproduce both the absolute and relative (reaction net) solvation free energies indicates that the SPT-V approach based on the CM5 charges is the best performer. The method is suggested for the solvation free energy calculation in the nonaqueous solvents.

A new insight into encapsulation process of a drug molecule in the polymer/surfactant system: a molecular simulation study

Drug delivery plays a substantial role in a more effective treatment of diseases of the central nervous system; therefore, the selection of an appropriate drug carrier system is very important to enhance the effectiveness of drugs. Due to the effect of surfactant on improvement of polymer performance in drug-carrying systems, the present study was devoted to investigating the influence of Polysorbate 80 (Pst80) surfactant on poly(n-butylcyanoacrylate)(PBCA)/Tacrine and Chitosan/Tacrine drug-carrying systems from molecular point of view. Interaction energy, structural characterization, Flory–Huggins interaction parameter, and solvation free energy were investigated for both systems by employing molecular dynamics simulations. According to the interaction energy and Flory–Huggins parameter results, Pst80 can be a more suitable choice for targeted releasing of drug in PBCA/Tacrine system compared with Chitosan/Tacrine system because Pst80 firmly surrounded the drug carrier PBCA and Tacrine. Additionally, the solvation free energy results demonstrated more solubility of PBCA/Pst80/Tacrine in water medium compared with that of Chitosan/Pst80/Tacrine. By consideration on different solvation free energy contributions, it was concluded that using a polymer with both hydrophilic and hydrophobic parts, presence of functional groups with heavy atoms on both polymer and surfactant and similarity in chemical nature of hydrophobic parts of both polymer and surfactant can be useful approaches to reduce the total solvation free energy. Preparation of an appropriate solubility of polymer/drug in water/surfactant medium is essential to enhance drug delivery system efficiency and reduce waste of drug in human body, which can be achieved by designing a drug-carrying system with the minimum solvation free energy. This study confirms the significant role of molecular dynamics simulation for a detailed study of polymer/surfactant/dug systems and clarifies its effective role for designing novel drug delivery systems, along with saving time and cost.

Pressure Effects on Protein Hydration Water Thermodynamics.

In this molecular dynamics simulation study, we analyze the impact of increasing hydrostatic pressure on the solvation of protein surfaces. Apart from the increasing volume work required for the formation of the protein solute cavity at high hydrostatic pressures, no significant additional trend is observed for solvation free energy contributions due to the protein-water interactions analyzed here. The latter is the result of approximately canceling pressure-induced changes of enthalpic and entropic solvation free energy contributions, which can be traced back to changes in the hydration of hydrophilic and hydrophobic groups of the protein. The 3D-2PT analysis used here allows for the visualization of local solvation free energy contributions in three dimensions with high spatial resolution. Local solvation free energy contributions per water molecule at hydrophobic surfaces are small but mainly favorable for transfer processes from the gas phase. This does not change considerably with increasing pressure, while the number of hydrating water molecules increases due to increased packing of the hydration shell. The number of hydrating water molecules also increases for hydrophilic protein surfaces, but solvation contributions per water molecule become less favorable with pressure in this case. As a consequence, contributions to the total solvation free energy from interactions between water molecules and hydrophobic surfaces become increasingly relevant at high hydrostatic pressures. Our results provide novel insights into solvent-mediated contributions to the thermodynamic driving force of pressure denaturation of proteins.

Molecular dynamics simulations of the effect of sizing agent on the interface property in carbon fiber reinforced vinyl ester resin composite

The role of sizing agent in the formation of interphase and the effect on the interfacial properties of carbon fiber (CF) reinforced vinyl ester resin (VE) composite were studied by molecular dynamics (MD) simulations. A novel unsaturated sizing agent N-(4′4-diaminodiphenyl methane)-2-hydroxypropyl methacrylate (DMHM) was used in this study. Epoxy sizing agent (E44) and modified epoxy sizing agent (ME44) were used for comparison. Simulation results showed that DMHM exhibited the highest interfacial bonding strength with CF surface due to the strong polarity of the sizing agent molecule. This result was confirmed by the interfacial shear strength test. The swelling effect of sizing agent in liquid VE was evaluated by solubility in styrene and compatibility with VE component. All three sizing agents had good solubility in styrene according to the calculation of total solvation free energy, while E44 and DMHM exhibited good compatibility with VE molecule based on the mixing energy calculation. The swelling property of sizing agent was confirmed by nano-indentation test. The results of this study explain the higher interfacial shear strength (62.25 MPa) between VE and CF sized by DMHM.

Combining the polarizable Drude force field with a continuum electrostatic Poisson-Boltzmann implicit solvation model.

In this work, we have combined the polarizable force field based on the classical Drude oscillator with a continuum Poisson-Boltzmann/solvent-accessible surface area (PB/SASA) model. In practice, the positions of the Drude particles experiencing the solvent reaction field arising from the fixed charges and induced polarization of the solute must be optimized in a self-consistent manner. Here, we parameterized the model to reproduce experimental solvation free energies of a set of small molecules. The model reproduces well-experimental solvation free energies of 70 molecules, yielding a root mean square difference of 0.8 kcal/mol versus 2.5 kcal/mol for the CHARMM36 additive force field. The polarization work associated with the solute transfer from the gas-phase to the polar solvent, a term neglected in the framework of additive force fields, was found to make a large contribution to the total solvation free energy, comparable to the polar solute-solvent solvation contribution. The Drude PB/SASA also reproduces well the electronic polarization from the explicit solvent simulations of a small protein, BPTI. Model validation was based on comparisons with the experimental relative binding free energies of 371 single alanine mutations. With the Drude PB/SASA model the root mean square deviation between the predicted and experimental relative binding free energies is 3.35 kcal/mol, lower than 5.11 kcal/mol computed with the CHARMM36 additive force field. Overall, the results indicate that the main limitation of the Drude PB/SASA model is the inability of the SASA term to accurately capture non-polar solvation effects. © 2018 Wiley Periodicals, Inc.

A QM/MM study on the correlation between the polarisations of and electrons in a hydrated benzene

In a recent work, we performed free-energy analyses for hydration of benzene by conducting QM/MM-ER simulations, where the total solvation free energy was decomposed into contributions and . is the solvation free energy of the solute with a fixed electron density and is the residual free energy due to the electron density polarisation in solution. We, further, decomposed the free energies due to electron density fluctuations in aromatic solutes in aqueous solutions into contributions from and orbitals. We note, however, that the decompositions will not be validated when the polarisations of orbitals seriously couple with those of orbitals. In this paper, we study a correlation matrix between polarisations of and orbitals through QM/MM simulations to assess the coupling strength among the orbitals. We found that the electron density polarisation is dominated by the polarisation arising from transfers between the orbitals lying in the HOMO–LUMO region. Thus, the polarisation of electrons hardly couples with that from orbitals, which justifies our decomposition analyses.

A Computational Understanding of Solvent Effect on the Structure, Electronic, Thermochemical, and Spectroscopic Properties of Ni(η2-C6H4)(H2PCH2CH2PH2) Complex

Here we report the density functional calculations of the molecular parameters including the energy, geometries, electric dipole moments, vibrational IR frequencies, and 1H and 13C NMR chemical shifts of Ni(η2-C6H4)(H2PCH2CH2PH2) (a benzyne complex). Based on the polarizable continuum model (PCM), the effect of polarity of the solvent on these parameters was explored. The wavenumbers of υ(C1–C2) as well as the 1H and 13C NMR chemical shift values of complex in various solvents were calculated and correlated with the Kirkwood–Bauer–Magat equation (KBM), the solvent acceptor numbers (ANs), and the linear solvation energy relationship (LSER). The bonding interaction between the benzyne and Ni(H2PCH2CH2PH2) fragment was analyzed by means of the energy decomposition analysis (EDA). The character of the Ni–C bonds of the molecules was analyzed by natural bond orbital (NBO) analysis. Also, Monte Carlo simulations were used for the calculation of the total energy and solvation free energy of the complex in water.

Extending the Solvation-Layer Interface Condition Continum Electrostatic Model to a Linearized Poisson-Boltzmann Solvent.

We extend the linearized Poisson-Boltzmann (LPB) continuum electrostatic model for molecular solvation to address charge-hydration asymmetry. Our new solvation-layer interface condition (SLIC)/LPB corrects for first-shell response by perturbing the traditional continuum-theory interface conditions at the protein-solvent and the Stern-layer interfaces. We also present a GPU-accelerated treecode implementation capable of simulating large proteins, and our results demonstrate that the new model exhibits significant accuracy improvements over traditional LPB models, while reducing the number of fitting parameters from dozens (atomic radii) to just five parameters, which have physical meanings related to first-shell water behavior at an uncharged interface. In particular, atom radii in the SLIC model are not optimized but uniformly scaled from their Lennard-Jones radii. Compared to explicit-solvent free-energy calculations of individual atoms in small molecules, SLIC/LPB is significantly more accurate than standard parametrizations (RMS error 0.55 kcal/mol for SLIC, compared to RMS error of 3.05 kcal/mol for standard LPB). On parametrizing the electrostatic model with a simple nonpolar component for total molecular solvation free energies, our model predicts octanol/water transfer free energies with an RMS error 1.07 kcal/mol. A more detailed assessment illustrates that standard continuum electrostatic models reproduce total charging free energies via a compensation of significant errors in atomic self-energies; this finding offers a window into improving the accuracy of Generalized-Born theories and other coarse-grained models. Most remarkably, the SLIC model also reproduces positive charging free energies for atoms in hydrophobic groups, whereas standard PB models are unable to generate positive charging free energies regardless of the parametrized radii. The GPU-accelerated solver is freely available online, as is a MATLAB implementation.

Molecule–Surface Recognition between Heterocyclic Aromatic Compounds and Kaolinite in Toluene Investigated by Molecular Theory of Solvation and Thermodynamic and Kinetic Experiments

Molecular recognition interactions between kaolinite and a series of heterocyclic aromatic compounds (HAC) representative of the N- and S-containing moieties in petroleum asphaltene macromolecules are investigated using the three-dimensional reference interaction site model with the Kovalenko–Hirata closure approximation (3D-RISM-KH) theory of solvation and experimental techniques in toluene solvent. The statistical-mechanical 3D-RISM-KH molecular theory of solvation predicts the adsorption configuration and thermodynamics from the 3D site density distribution functions and total solvation free energy, respectively, for adsorption of HAC and toluene on kaolinite. Spectrophotometric measurements show that, among the HAC studied, only acridine and phenanthridine adsorb quantitatively on kaolinite. For these pyridinic HAC, the adsorption results fitted to the Langmuir isotherm in the monolayer domain suggest a uniform monolayer of HAC molecules. The 3D-RISM-KH studies predict that the aluminum hydroxide surf...

Solvation Free Energies of Alanine Peptides: The Effect of Flexibility

The electrostatic (ΔGel), van der Waals cavity-formation (ΔGvdw), and total (ΔG) solvation free energies for 10 alanine peptides ranging in length (n) from 1 to 10 monomers were calculated. The free energies were computed both with fixed, extended conformations of the peptides and again for some of the peptides without constraints. The solvation free energies, ΔGel, and components ΔGvdw, and ΔG, were found to be linear in n, with the slopes of the best-fit lines being γel, γvdw, and γ, respectively. Both γel and γ were negative for fixed and flexible peptides, and γvdw was negative for fixed peptides. That γvdw was negative was surprising, as experimental data on alkanes, theoretical models, and MD computations on small molecules and model systems generally suggest that γvdw should be positive. A negative γvdw seemingly contradicts the notion that ΔGvdw drives the initial collapse of the protein when it folds by favoring conformations with small surface areas. When we computed ΔGvdw for the flexible peptides, thereby allowing the peptides to assume natural ensembles of more compact conformations, γvdw was positive. Because most proteins do not assume extended conformations, a ΔGvdw that increases with increasing surface area may be typical for globular proteins. An alternative hypothesis is that the collapse is driven by intramolecular interactions. We find few intramolecular H-bonds but show that the intramolecular van der Waals interaction energy is more favorable for the flexible than for the extended peptides, seemingly favoring this hypothesis. The large fluctuations in the vdw energy may make attributing the collapse of the peptide to this intramolecular energy difficult.

Assessing the Accuracy of Inhomogeneous Fluid Solvation Theory in Predicting Hydration Free Energies of Simple Solutes

Accurate prediction of hydration free energies is a key objective of any free energy method that is applied to modeling and understanding interactions in the aqueous phase. Inhomogeneous fluid solvation theory (IFST) is a statistical mechanical method for calculating solvation free energies by quantifying the effect of a solute acting as a perturbation to bulk water. IFST has found wide application in understanding hydration phenomena in biological systems, but quantitative applications have not been comprehensively assessed. In this study, we report the hydration free energies of six simple solutes calculated using IFST and independently using free energy perturbation (FEP). This facilitates a validation of IFST that is independent of the accuracy of the force field. The results demonstrate that IFST shows good agreement with FEP, with an R2 coefficient of determination of 0.99 and a mean unsigned difference of 0.7 kcal/mol. However, sampling is a major issue that plagues IFST calculations and the results suggest that a histogram method may require prohibitively long simulations to achieve convergence of the entropies, for bin sizes which effectively capture the underlying probability distributions. Results also highlight the sensitivity of IFST to the reference interaction energy of a water molecule in bulk, with a difference of 0.01 kcal/mol changing the predicted hydration free energies by approximately 2.4 kcal/mol for the systems studied here. One of the major advantages of IFST over perturbation methods such as FEP is that the systems are spatially decomposed to consider the contribution of specific regions to the total solvation free energies. Visualizing these contributions can yield detailed insights into solvation thermodynamics. An insight from this work is the identification and explanation of regions with unfavorable free energy density relative to bulk water. These regions contribute unfavorably to the hydration free energy. Further work is necessary before IFST can be extended to yield accurate predictions of binding free energies, but the work presented here demonstrates its potential.

Development of a methodology to compute solvation free energies on the basis of the theory of energy representation for solutions represented with a polarizable force field

We have developed a method of molecular simulations utilizing a polarizable force field in combination with the theory of energy representation (ER) for the purpose of establishing an efficient and accurate methodology to compute solvation free energies. The standard version of the ER method is, however, based on the assumption that the solute-solvent interaction is pairwise additive for its construction. A crucial step in the present method is to introduce an intermediate state in the solvation process to treat separately the many-body interaction associated with the polarizable model. The intermediate state is chosen so that the solute-solvent interaction can be formally written in the pairwise form, though the solvent molecules are interacting with each other with polarizable charges dependent on the solvent configuration. It is, then, possible to extract the free energy contribution δμ due to the many-body interaction between solute and solvent from the total solvation free energy Δμ. It is shown that the free energy δμ can be computed by an extension of the recent development implemented in quantum mechanical∕molecular mechanical simulations. To assess the numerical robustness of the approach, we computed the solvation free energies of a water and a methanol molecule in water solvent, where two paths for the solvation processes were examined by introducing different intermediate states. The solvation free energies of a water molecule associated with the two paths were obtained as -5.3 and -5.8 kcal∕mol. Those of a methanol molecule were determined as -3.5 and -3.7 kcal∕mol. These results of the ER simulations were also compared with those computed by a numerically exact approach. It was demonstrated that the present approach produces the solvation free energies in comparable accuracies to simulations of thermodynamic integration (TI) method within a tenth of computational time used for the TI simulations.

A New Putative Cholesterol-Recognition Motif in Transmembrane Proteins

We report the finding of a new putative cholesterol-recognition motif in the transmembrane region of membrane proteins like the nicotinic acetylcholine receptor (AChR) and other members of the Cys-loop superfamily of neurotransmitters involved in rapid synaptic transmission, as well as in members of the important superfamily of G protein coupled receptors (GPCRs) involved in Alzheimer's disease. The motif is an inverted CRAC domain (cholesterol recognition/interaction amino acid consensus) which we have named CARC and is present in the transmembrane regions of the receptor subunits having extensive contact with the surrounding lipid, predominantly in the first (TM1) segment in the case of the AChR, and optimally suited to convey cholesterol-mediated signaling. Three cholesterol molecules could be docked on the transmembrane regions of the five human AChR subunits, rendering a total of 15 cholesterol molecules per AChR molecule, adding up to a total of about 200 kJ.mol-1 per receptor molecule, i.e. ∼40% of the total lipid solvation free energies for the whole AChR molecule. The novel motif is remarkably conserved among Cys-loop receptors along the phylogenetic scale, from prokaryotes to humans, its high degree of conservation suggesting that it could be responsible for some of the structural/functional properties of these transduction macromolecules.Reference: Baier, C.J., Fantini, J. and Barrantes, F.J. (2011). Sci. Rep. 1:00069.

Communication: Free-energy analysis of hydration effect on protein with explicit solvent: Equilibrium fluctuation of cytochrome c

The relationship between the protein conformation and the hydration effect is investigated for the equilibrium fluctuation of cytochrome c. To elucidate the hydration effect with explicit solvent, the solvation free energy of the protein immersed in water was calculated using the molecular dynamics simulation coupled with the method of energy representation. The variations of the protein intramolecular energy and the solvation free energy are found to compensate each other in the course of equilibrium structural fluctuation. The roles of the attractive and repulsive components in the protein-water interaction are further examined for the solvation free energy. The attractive component represented as the average sum of protein-water interaction energy is dominated by the electrostatic effect and is correlated to the solvation free energy through the linear-response-type relationship. No correlation with the (total) solvation free energy is seen, on the other hand, for the repulsive component expressed as the excluded-volume effect.

FAMBE-pH: a fast and accurate method to compute the total solvation free energies of proteins.

A fast and accurate method to compute the total solvation free energies of proteins as a function of pH is presented. The method makes use of a combination of approaches, some of which have already appeared in the literature; (i) the Poisson equation is solved with an optimized fast adaptive multigrid boundary element (FAMBE) method; (ii) the electrostatic free energies of the ionizable sites are calculated for their neutral and charged states by using a detailed model of atomic charges; (iii) a set of optimal atomic radii is used to define a precise dielectric surface interface; (iv) a multilevel adaptive tessellation of this dielectric surface interface is achieved by using multisized boundary elements; and (v) 1:1 salt effects are included. The equilibrium proton binding/release is calculated with the Tanford-Schellman integral if the proteins contain more than approximately 20-25 ionizable groups; for a smaller number of ionizable groups, the ionization partition function is calculated directly. The FAMBE method is tested as a function of pH (FAMBE-pH) with three proteins, namely, bovine pancreatic trypsin inhibitor (BPTI), hen egg white lysozyme (HEWL), and bovine pancreatic ribonuclease A (RNaseA). The results are (a) the FAMBE-pH method reproduces the observed pK a's of the ionizable groups of these proteins within an average absolute value of 0.4 p K units and a maximum error of 1.2 p K units and (b) comparison of the calculated total pH-dependent solvation free energy for BPTI, between the exact calculation of the ionization partition function and the Tanford-Schellman integral method, shows agreement within 1.2 kcal/mol. These results indicate that calculation of total solvation free energies with the FAMBE-pH method can provide an accurate prediction of protein conformational stability at a given fixed pH and, if coupled with molecular mechanics or molecular dynamics methods, can also be used for more realistic studies of protein folding, unfolding, and dynamics, as a function of pH.

Improving the Accuracy of the Linear Interaction Energy Method for Solvation Free Energies.

A linear response method for estimating the free energy of solvation is presented and validated using explicit solvent molecular dynamics, thermodynamic perturbation calculations, and experimental data. The electrostatic contribution to the solvation free energy is calculated using a linear response estimate, which is obtained by comparison to the free energy calculated using thermodynamic perturbation. Systematic deviations from the value of (1)/2 in the potential energy scaling factor are observed for some types of compounds, and these are taken into account by introducing specific coefficients for different chemical groups. The derived model reduces the rms error of the linear response estimate significantly from 1.6 to 0.3 kcal/mol on a training set of 221 molecules used to parametrize the model and from 3.7 to 1.3 kcal/mol on a test set of 355 molecules that were not used in the derivation of the model. The total solvation free energy is estimated by combining the derived model with an empirical size dependent term for predicting the nonpolar contribution. Using this model, the experimental hydration free energies for 192 molecules are reproduced with an rms error of 1.1 kcal/mol. The use of LIE in simplified binding free energy calculations to predict protein-ligand binding free energies is also discussed.

Solvation and chemical engineering thermodynamics

With the continued advances in computational chemistry, solvation calculation from first principle methods is becoming a promising route for phase equilibrium modeling. However, except in a few instances, such an approach has not been widely adopted, which may be a consequence of the abstractness of solvation itself and also of the lack of a simple bridge between solvation and other thermodynamic properties. Here, we establish and summarize the relationship between solvation and other properties frequently used in phase equilibrium modeling. An important quantity, called the total solvation free energy, is introduced so that one can easily derive engineering thermodynamic models (such as an equation of state) from solvation models, or vice versa. The equations presented here are of general validity and would be useful for obtaining existing model parameters from solvation calculations and for developing new models stemming from the ideas of molecular solvation.

A novel quantum mechanical/molecular mechanical approach to the free energy calculation for isomerization of glycine in aqueous solution

The free energy change associated with the isomerization reaction of glycine in water solution has been studied by a hybrid quantum mechanical/molecular mechanical (QM/MM) approach combined with the theory of energy representation (QM/MM-ER) recently developed. The solvation free energies for both neutral and zwitterionic form of glycine have been determined by means of the QM/MM-ER simulation. The contributions of the electronic polarization and the fluctuation of the QM solute to the solvation free energy have been investigated. It has been found that the contribution of the density fluctuation of the zwitterionic solute is estimated as -4.2 kcal/mol in the total solvation free energy of -46.1 kcal/mol, while that of the neutral form is computed as -3.0 kcal/mol in the solvation free energy of -15.6 kcal/mol. The resultant free energy change associated with the isomerization of glycine in water has been obtained as -7.8 kcal/mol, in excellent agreement with the experimental data of -7.3 or -7.7 kcal/mol, implying the accuracy of the QM/MM-ER approach. The results have also been compared with those computed by other methodologies such as the polarizable continuum model and the classical molecular simulation. The efficiency and advantage of the QM/MM-ER method has been discussed.

Ion solvation thermodynamics from simulation with a polarizable force field.

Thermodynamic measurements of the solvation of salts and electrolytes are relatively straightforward, but it is not possible to separate total solvation free energies into distinct cation and anion contributions without reference to an additional extrathermodynamic assumption. The present work attempts to resolve this difficulty using molecular dynamics simulations with the AMOEBA polarizable force field and perturbation techniques to directly compute absolute solvation free energies for potassium, sodium, and chloride ions in liquid water and formamide. Corresponding calculations are also performed with two widely used nonpolarizable force fields. The simulations with the polarizable force field accurately reproduce in vacuo quantum mechanical results, experimental ion-cluster solvation enthalpies, and experimental solvation free energies for whole salts, while the other force fields do not. The results indicate that calculations with a polarizable force field can capture the thermodynamics of ion solvation and that the solvation free energies of the individual ions differ by several kilocalories from commonly cited values.

Implicit solvent models

Implicit solvent models for biomolecular simulations are reviewed and their underlying statistical mechanical basis is discussed. The fundamental quantity that implicit models seek to approximate is the solute potential of mean force, which determines the statistical weight of solute conformations, and which is obtained by averaging over the solvent degrees of freedom. It is possible to express the total free energy as the reversible work performed in two successive steps. First, the solute is inserted in the solvent with zero atomic partial charges; second, the atomic partial charges of the solute are switched from zero to their full values. Consequently, the total solvation free energy corresponds to a sum of non-polar and electrostatic contributions. These two contributions are often approximated by simple geometrical models (such as solvent exposed area models) and by macroscopic continuum electrostatics, respectively. One powerful route is to approximate the average solvent density distribution around the solute, i.e. the solute–solvent density correlation functions, as in statistical mechanical integral equations. Recent progress with semi-analytical approximations makes continuum electrostatics treatments very efficient. Still more efficient are fully empirical, knowledge-based models, whose relation to explicit solvent treatments is not fully resolved, however. Continuum models that treat both solute and solvent as dielectric continua are also discussed, and the relation between the solute fluctuations and its macroscopic dielectric constant(s) clarified.