Compensation Between the Entropic and Enthalpic Changes Arising from Changes in Solvent–Solvent Interactions in Mixed Solvents

A theorem presented by Professor Ben-Naim (J Phys Chem 82:874–885, 1978) states that the standard state enthalpy and entropy changes arising from changes in the solvent structure that are induced by solvation of a solute cancel exactly in the standard state Gibbs energy. In this paper this is explored by consideration of the thermodynamics of transfer of electrolytes in mixed solvents, using previously developed models of the solvation process. Two cases are considered. One is random solvation, where curvatures in plots of the transfer enthalpies and entropies, which arise from changes in solvent–solvent interactions, exactly compensate in the transfer Gibbs (free) energies, which are sensibly linear with solvent composition. The second type of system are those with strong preferential solvation where it is found that the transfer Gibbs energies can be accounted for quantitatively in terms of changes in the solute–solvent interactions, with no contribution from changes in solvent–solvent interactions. The results are entirely consistent with the Ben-Naim theorem.     

Solvent density inhomogeneities and solvation free energies in supercritical diatomic fluids: a density functional approach

Density functional theory is used to explore the solvation properties of a spherical solute immersed in a supercritical diatomic fluid. The solute is modeled as a hard core Yukawa particle surrounded by a diatomic Lennard-Jones fluid represented by two fused tangent spheres using an interaction site approximation. The authors' approach is particularly suitable for thoroughly exploring the effect of different interaction parameters, such as solute-solvent interaction strength and range, solvent-solvent long-range interactions, and particle size, on the local solvent structure and the solvation free energy under supercritical conditions. Their results indicate that the behavior of the local coordination number in homonuclear diatomic fluids follows trends similar to those reported in previous studies for monatomic fluids. The local density augmentation is particularly sensitive to changes in solute size and is affected to a lesser degree by variations in the solute-solvent interaction strength and range. The associated solvation free energies exhibit a nonmonotonous behavior as a function of density for systems with weak solute-solvent interactions. The authors' results suggest that solute-solvent interaction anisotropies have a major influence on the nature and extent of local solvent density inhomogeneities and on the value of the solvation free energies in supercritical solutions of heteronuclear molecules.     

Free energy of solvation from molecular dynamics simulation applying Voronoi-Delaunay triangulation to the cavity creation

The free energy of solvation for a large number of representative solutes in various solvents has been calculated from the polarizable continuum model coupled to molecular dynamics computer simulation. A new algorithm based on the Voronoi-Delaunay triangulation of atom-atom contact points between the solute and the solvent molecules is presented for the estimation of the solvent-accessible surface surrounding the solute. The volume of the inscribed cavity is used to rescale the cavitational contribution to the solvation free energy for each atom of the solute atom within scaled particle theory. The computation of the electrostatic free energy of solvation is performed using the Voronoi-Delaunay surface around the solute as the boundary for the polarizable continuum model. Additional short-range contributions to the solvation free energy are included directly from the solute-solvent force field for the van der Waals-type interactions. Calculated solvation free energies for neutral molecules dissolved in benzene, water, CCl4, and octanol are compared with experimental data. We found an excellent correlation between the experimental and computed free energies of solvation for all the solvents. In addition, the employed algorithm for the cavity creation by Voronoi-Delaunay triangulation is compared with the GEPOL algorithm and is shown to predict more accurate free energies of solvation, especially in solvents composed by molecules with nonspherical molecular shapes.     

Breaking the polar‐nonpolar division in solvation free energy prediction

Implicit solvent models divide solvation free energies into polar and nonpolar additive contributions, whereas polar and nonpolar interactions are inseparable and nonadditive. We present a feature functional theory (FFT) framework to break this ad hoc division. The essential ideas of FFT are as follows: (i) representability assumption: there exists a microscopic feature vector that can uniquely characterize and distinguish one molecule from another; (ii) feature‐function relationship assumption: the macroscopic features, including solvation free energy, of a molecule is a functional of microscopic feature vectors; and (iii) similarity assumption: molecules with similar microscopic features have similar macroscopic properties, such as solvation free energies. Based on these assumptions, solvation free energy prediction is carried out in the following protocol. First, we construct a molecular microscopic feature vector that is efficient in characterizing the solvation process using quantum mechanics and Poisson–Boltzmann theory. Microscopic feature vectors are combined with macroscopic features, that is, physical observable, to form extended feature vectors. Additionally, we partition a solvation dataset into queries according to molecular compositions. Moreover, for each target molecule, we adopt a machine learning algorithm for its nearest neighbor search, based on the selected microscopic feature vectors. Finally, from the extended feature vectors of obtained nearest neighbors, we construct a functional of solvation free energy, which is employed to predict the solvation free energy of the target molecule. The proposed FFT model has been extensively validated via a large dataset of 668 molecules. The leave‐one‐out test gives an optimal root‐mean‐square error (RMSE) of 1.05 kcal/mol. FFT predictions of SAMPL0, SAMPL1, SAMPL2, SAMPL3, and SAMPL4 challenge sets deliver the RMSEs of 0.61, 1.86, 1.64, 0.86, and 1.14 kcal/mol, respectively. Using a test set of 94 molecules and its associated training set, the present approach was carefully compared with a classic solvation model based on weighted solvent accessible surface area. © 2017 Wiley Periodicals, Inc.     

Redox entropy of plastocyanin: developing a microscopic view of mesoscopic polar solvation

We report applications of analytical formalisms and molecular dynamics (MD) simulations to the calculation of redox entropy of plastocyanin metalloprotein in aqueous solution. The goal of our analysis is to establish critical components of the theory required to describe polar solvation at the mesoscopic scale. The analytical techniques include a microscopic formalism based on structure factors of the solvent dipolar orientations and density and continuum dielectric theories. The microscopic theory employs the atomistic structure of the protein with force-field atomic charges and solvent structure factors obtained from separate MD simulations of the homogeneous solvent. The MD simulations provide linear response solvation free energies and reorganization energies of electron transfer in the temperature range of 280-310 K. We found that continuum models universally underestimate solvation entropies, and a more favorable agreement is reported between the microscopic calculations and MD simulations. The analysis of simulations also suggests that difficulties of extending standard formalisms to protein solvation are related to the inhomogeneous structure of the solvation shell at the protein-water interface combining islands of highly structured water around ionized residues along with partial dewetting of hydrophobic patches. Quantitative theories of electrostatic protein hydration need to incorporate realistic density profile of water at the protein-water interface.     

Extension of a temperature-dependent aqueous solvation model to compounds containing nitrogen, fluorine, chlorine, bromine, and sulfur

Most methods for predicting free energies of solvation have been developed or validated exclusively for room temperature. Recently, we developed a model called SM6T for predicting aqueous solvation free energies as a function of temperature for solutes composed of C, H, or O, and here we present solvation model 8 with temperature dependence (SM8T) for predicting the temperature dependence of aqueous free energies of solvation for compounds containing H, C, N, O, F, S, Cl, and Br in the range 273-373 K. We also describe the database of experimental aqueous free energies of solvation used to parametrize the model. SM8T partitions the temperature dependence of the free energy of solvation into two components: the temperature dependence of the bulk electrostatic contribution to the free energy of solvation, which is computed using the generalized Born equation, and the temperature dependence of first-solvation-shell effects, which is modeled by terms proportional to the solvent-exposed surface areas of atoms in functional groups determined entirely by geometry. SM8T predicts the temperature dependence of aqueous free energies of solvation with a mean unsigned error of 0.08 kcal/mol over a database of 4403 measurements on 348 compounds at various temperatures. We also discuss the accuracy of SM8T for predicting the temperature dependence of aqueous free energies of solvation for ions and present free energies of solvation as a function of temperature for two sample ions.     

Modeling hydrophobic solvation of nonspherical systems: Comparison of use of molecular surface area with accessible surface area

It is shown that the molecular surface and the accessible surface lead to exactly the same results when calculating solvation free energies and transfer free energies, from methods using the surface tension as a parameter if the exact geometric curvature is used with the accessible surface. However, the use of the exact curvature is not necessarily the best approach chemically. Other modifications, including an approximate curvature improves the approach. Such modifications are difficult to include in methods in which the molecular surface rather than the accessible surface is used to calculate solvent effects. A modification of a Gaussian curvature term is necessary if dissociation is to be accounted for properly. The inclusion of a Gaussian curvature term, in addition to the usual mean curvature term, reconciles the difference in magnitude of the microscopic and macroscopic surface tension in the case of the accessible surface area. © 1997 by John Wiley & Sons, Inc.     

Molecular simulation of interaction between passivated gold nanoparticles in supercritical CO2

Molecular dynamics simulations have been performed to study the potential of mean force (PMF) between passivated gold nanoparticles (NPs) in supercritical CO(2) (scCO(2)). The nanoparticle model consists of a 140 atom gold nanocore and a surface self-assembled monolayer, in which two kinds of fluorinated alkanethiols were considered. The molecular origin of the thermodynamics interaction and the solvation effect has been comprehensively studied. The simulation results demonstrate that increasing the solvent density and ligand length can enhance the repulsive feature of the free energy between the passivated Au nanoparticles in scCO(2), which is in good agreement with previous experimental results. The interaction forces between the two passivated NPs have been decomposed to reveal various contributions to the free energy. It was revealed that the interaction between capping ligands and the interaction between the capping ligands and scCO(2) solvent molecules cooperatively determine the total PMF. A thermodynamic entropy-energy analysis for each PMF contribution was used to explain the density dependence of PMF in scCO(2) fluid. Our simulation study is expected to provide a novel microscopic understanding of the effect of scCO(2) solvent on the interaction between passivated Au nanoparticles, which is helpful to the dispersion and preparation of functional metal nanoparticles in supercritical fluids.     

Grid inhomogeneous solvation theory: Hydration structure and thermodynamics of the miniature receptor cucurbit[7]uril

The displacement of perturbed water upon binding is believed to play a critical role in the thermodynamics of biomolecular recognition, but it is nontrivial to unambiguously define and answer questions about this process. We address this issue by introducing grid inhomogeneous solvation theory (GIST), which discretizes the equations of inhomogeneous solvation theory (IST) onto a three-dimensional grid situated in the region of interest around a solute molecule or complex. Snapshots from explicit solvent simulations are used to estimate localized solvation entropies, energies, and free energies associated with the grid boxes, or voxels, and properly summing these thermodynamic quantities over voxels yields information about hydration thermodynamics. GIST thus provides a smoothly varying representation of water properties as a function of position, rather than focusing on hydration sites where solvent is present at high density. It therefore accounts for full or partial displacement of water from sites that are highly occupied by water, as well as for partly occupied and water-depleted regions around the solute. GIST can also provide a well-defined estimate of the solvation free energy and therefore enables a rigorous end-states analysis of binding. For example, one may not only use a first GIST calculation to project the thermodynamic consequences of displacing water from the surface of a receptor by a ligand, but also account, in a second GIST calculation, for the thermodynamics of subsequent solvent reorganization around the bound complex. In the present study, a first GIST analysis of the molecular host cucurbit[7]uril is found to yield a rich picture of hydration structure and thermodynamics in and around this miniature receptor. One of the most striking results is the observation of a toroidal region of high water density at the center of the host's nonpolar cavity. Despite its high density, the water in this toroidal region is disfavored energetically and entropically, and hence may contribute to the known ability of this small receptor to bind guest molecules with unusually high affinities. Interestingly, the toroidal region of high water density persists even when all partial charges of the receptor are set to zero. Thus, localized regions of high solvent density can be generated in a binding site without strong, attractive solute-solvent interactions.     

Implicit nonpolar solvent models

We have systematically analyzed a new nonpolar solvent model that separates nonpolar solvation free energy into repulsive and attractive components. Our analysis shows that either molecular surfaces or volumes can be used to correlate with repulsive free energies of tested molecules in explicit solvent with correlation coefficients higher than 0.99. In addition, the attractive free energies in explicit solvent can also be reproduced with the new model with a correlation coefficient higher than 0.999. Given each component optimized, the new nonpolar solvent model is found to reproduce monomer nonpolar solvation free energies in explicit solvent very well. However, the overall accuracy of the nonpolar solvation free energies is lower than that of each component. In the more challenging dimer test cases, the agreement of the new model with explicit solvent is less impressive. Nevertheless, it is found that the new model works reasonably well for reproducing the relative nonpolar free energy landscapes near the global minimum of the dimer complexes.     

The Binding Energy Distribution Analysis Method (BEDAM) for the Estimation of Protein-Ligand Binding Affinities

The Binding Energy Distribution Analysis Method (BEDAM) for the computation of receptor-ligand standard binding free energies with implicit solvation is presented. The method is based on a well established statistical mechanics theory of molecular association. It is shown that, in the context of implicit solvation, the theory is homologous to the test particle method of solvation thermodynamics with the solute-solvent potential represented by the effective binding energy of the protein-ligand complex. Accordingly, in BEDAM the binding constant is computed by means of a weighted integral of the probability distribution of the binding energy obtained in the canonical ensemble in which the ligand is positioned in the binding site but the receptor and the ligand interact only with the solvent continuum. It is shown that the binding energy distribution encodes all of the physical effects of binding. The balance between binding enthalpy and entropy is seen in our formalism as a balance between favorable and unfavorable binding modes which are coupled through the normalization of the binding energy distribution function. An efficient computational protocol for the binding energy distribution based on the AGBNP2 implicit solvent model, parallel Hamiltonian replica exchange sampling and histogram reweighting is developed. Applications of the method to a set of known binders and non-binders of the L99A and L99A/M102Q mutants of T4 lysozyme receptor are illustrated. The method is able to discriminate without error binders from non-binders, and the computed standard binding free energies of the binders are found to be in good agreement with experimental measurements. Analysis of the results reveals that the binding affinities of these systems reflect the contributions from multiple conformations spanning a wide range of binding energies.     

Parameterization of the Hamiltonian Dielectric Solvent (HADES) Reaction‐Field Method for the Solvation Free Energies of Amino Acid Side‐Chain Analogs

Optimization of the Hamiltonian dielectric solvent (HADES) method for biomolecular simulations in a dielectric continuum is presented with the goal of calculating accurate absolute solvation free energies while retaining the model’s accuracy in predicting conformational free‐energy differences. The solvation free energies of neutral and polar amino acid side‐chain analogs calculated by using HADES, which may optionally include nonpolar contributions, were optimized against experimental data to reach a chemical accuracy of about 0.5 kcal mol^?1. The new parameters were evaluated for charged side‐chain analogs. The HADES results were compared with explicit‐solvent, generalized Born, Poisson–Boltzmann, and QM‐based methods. The potentials of mean force (PMFs) between pairs of side‐chain analogs obtained by using HADES and explicit‐solvent simulations were used to evaluate the effects of the improved parameters optimized for solvation free energies on intermolecular potentials.     

The effect of aqueous solvation upon alpha-helix formation for polyalanines

The incremental free energies of aqueous solution for acetyl(ala)NNH2 in its extended unfolded and alpha-helical conformations are compared using the SM5.2 solvation method of Cramer and Truhlar. A combination of density functional theory (DFT) at the B3LYP/D95(d,p) and AM1 has been employed using the ONIOM method. The incremental solvation energies of alpha-helical structures are very similar for both ONIOM and AM1 optimized structures as these structures do not significantly change upon solution. However, the conformations of the unfolded peptides change from extended beta-strand to polyproline II conformations upon aqueous solution. The incremental solvation free energy per residue of the polyproline II structure is about 2 kcal/mol/residue greater than that for the alpha-helix, representing an upper limit for the difference between the solvation energies. However, most of this difference disappears when the energy required to distort the optimized gas-phase extended beta-strand structure to the optimized polyproline II solution structure is included in the analysis, leaving an estimated difference in incremental solvation free energy of 0.3-0.5 kcal/mol favoring the unfolded structure. The solution structure sacrifices the stability derived from the intramolecular C5 H-bonds for more favorable interactions with the aqueous solvent.     

The SGB/NP hydration free energy model based on the surface generalized born solvent reaction field and novel nonpolar hydration free energy estimators

The development and parameterization of a solvent potential of mean force designed to reproduce the hydration thermodynamics of small molecules and macromolecules aimed toward applications in conformation prediction and ligand binding free energy prediction is presented. The model, named SGB/NP, is based on a parameterization of the Surface Generalized Born continuum dielectric electrostatic model using explicit solvent free energy perturbation calculations and a newly developed nonpolar hydration free energy estimator motivated by the results of explicit solvent simulations of the thermodynamics of hydration of hydrocarbons. The nonpolar model contains, in addition to the more commonly used solvent accessible surface area term, a component corresponding to the attractive solute-solvent interactions. This term is found to be important to improve the accuracy of the model, particularly for cyclic and hydrogen bonding compounds. The model is parameterized against the experimental hydration free energies of a set of small organic molecules. The model reproduces the experimental hydration free energies of small organic molecules with an accuracy comparable or superior to similar models employing more computationally demanding estimators and/or a more extensive set of parameters.     

Solvation of hydrophobes in water and simple liquids

The solvation of nonpolar molecules in water and that in simple liquids are compared and contrasted. First, solvation thermodynamics is reviewed in a way that focuses on how the enthalpy and entropy of solvation depend on the choice of microscopic volume change v in the solvation process--including special choices v being zero (fixed-volume condition) and v being the partial molecular volume of a solute molecule (fixed-pressure condition)--and how the solvation quantities are related with temperature derivatives of the solvation free energy. Second, the solvation free energy and the solvation enthalpy of a Lennard-Jones (LJ) atom in model water are calculated in the parameter space representing the solute size and the strength of the solute-solvent interaction, and the results are compared with those for an LJ atom in the LJ solvent. The solvation diagrams showing domains of different types of solvation in the parameter space are obtained both for the constant-volume condition and for the constant-pressure condition. Similarities between water and the simple liquid are found when the constant-volume solvation is considered while a significant difference manifests itself in the fixed-pressure solvation. The domain of solvation of hydrophobic character in the parameter space is large in the constant-volume solvation both for water and for the simple liquid. When switched to the constant-pressure condition accompanying a microscopic volume change, the hydrophobic domain remains large in water but it becomes significantly small in the simple liquid. The contrasting results are due to the smallness of the thermal pressure coefficient of water at low temperatures.     

Morphometric approach to thermodynamic quantities of solvation of complex molecules: extension to multicomponent solvent

The morphometric approach (MA) is a powerful tool for calculating a solvation free energy (SFE) and related quantities of solvation thermodynamics of complex molecules. Here, we extend it to a solvent consisting of m components. In the integral equation theories, the SFE is expressed as the sum of m terms each of which comprises solute-component j correlation functions (j = 1,..., m). The MA is applied to each term in a formally separate manner: The term is expressed as a linear combination of the four geometric measures, excluded volume, solvent-accessible surface area, and integrated mean and Gaussian curvatures of the accessible surface, which are calculated for component j. The total number of the geometric measures or the coefficients in the linear combinations is 4m. The coefficients are determined in simple geometries, i.e., for spherical solutes with various diameters in the same multicomponent solvent. The SFE of the spherical solutes are calculated using the radial-symmetric integral equation theory. The extended version of the MA is illustrated for a protein modeled as a set of fused hard spheres immersed in a binary mixture of hard spheres. Several mixtures of different molecular-diameter ratios and compositions and 30 structures of the protein with a variety of radii of gyration are considered for the illustration purpose. The SFE calculated by the MA is compared with that by the direct application of the three-dimensional integral equation theory (3D-IET) to the protein. The deviations of the MA values from the 3D-IET values are less than 1.5%. The computation time required is over four orders of magnitude shorter than that in the 3D-IET. The MA thus developed is expected to be best suited to analyses concerning the effects of cosolvents such as urea on the structural stability of a protein.