Online resource for theoretical study of hydration of biopolymers

An online resource has been developed for the theoretical study of hydration of biopolymers by the RISM (Reference Interaction Site Model) method, deriving from the integral equation theory of liquids. The online resource is based upon original software developed by the authors and includes all steps in studying a biopolymer with a given spatial structure and force field. It prepares the input data and carries out the RISM calculation yielding the atom-atom correlation functions of the biopolymer with water as solvent. From these functions the algorithm finds atomic partial contributions to the hydration free energy using various free energy expressions from integral equation theory. The calculated results are automatically recorded in a database, and become available on the website as tables of partial thermodynamic quantities. In addition, the website displays an interactive 3D model of a given molecule, the atoms of which can be painted in different colors in accordance with their partial contributions to the thermodynamic quantity chosen by the user. The user can interactively choose atoms on this molecule and their correlation functions will be displayed. The aim of our work was to develop and present a publicly-accessible resource on the basis of original software which could be used for scientific and educational purposes.     

Calculation of the hydration energy of polyvalent metal ions by the RISM method

The energy of hydration of Group Ia, IIa, and IIIb metals has been calculated on the basis of the integral equation theory of molecular liquids. Calculations were performed in the RISM approximation with hypernetted chain closure, partial inclusion of the orientation effect, and semiempirical corrections for excluded volume and for the existence of hydrogen bonding with the solute molecule. The analysis of calculation data made it possible to apply a correction to the computation method used, which ensured better convergence between calculation results and experimental data.     

A semiempirical model for estimating the hydration free energy of neutral nonpolar compounds

An improved semiempirical model for determining the hydration free energy of neutral nonpolar compounds is presented. The model is based on a combination of the RISM approach of the integral equation theory and empirical correlations. It is demonstrated that the developed model has high predictive ability for alkanes, alkenes, and dienes (present only in the test set of compounds). It is concluded that this semiempirical model can be applied in estimating the hydration free energy of more complicated structures based on saturated and nonsaturated aliphatic hydrocarbons.     

Hydration of ionic species studied by the reference interaction site model with a repulsive bridge correction

We have tested the reference interaction site model (RISM) for the case of the hypernetted chain (HNC) and the partially linearized hypernetted chain (PLHNC) closures improved by a repulsive bridge correction (RBC) for ionic hydrated species. We have analyzed the efficiency of the RISM/HNC+RBC and RISM/PLHNC+RBC techniques for decomposition of the electrostatic and the nonpolar hydration energies on the energetic and the enthalpic parts for polyatomic ions when the repulsive bridge correction is treated as a thermodynamic perturbation, and investigate the repulsive bridge effect on the electrostatic potential induced by solvent on solute atoms. For a number of univalent and bivalent atomic ions, molecular cations, and anions, the method provides hydration energies deviating only by several percents from the experimental data. In most cases, the enthalpic contributions to the free energies are also close to the experimental results. The above models are able to satisfactory predict the hydration energies as well as the electrostatic potential around the ionic species. For univalent atomic ions, they also provide qualitative estimates of the Samoilov activation energies.     

An integral equation theory for inhomogeneous molecular fluids: the reference interaction site model approach

An integral equation theory which is applicable to inhomogeneous molecular liquids is proposed. The "inhomogeneous reference interaction site model (RISM)" equation derived here is a natural extension of the RISM equation to inhomogeneous systems. This theory makes it possible to calculate the pair correlation function between two molecules which are located at different density regions. We also propose approximations concerning the closure relation and the intramolecular susceptibility of inhomogeneous molecular liquids. As a preliminary application of the theory, the hydration structure around an ion is investigated. Lithium, sodium, and potassium cations are chosen as the solute. Using the Percus trick, the local density of solvent around an ion is expressed in terms of the solute-solvent pair correlation function calculated from the RISM theory. We then analyze the hydration structure around an ion through the triplet correlation function which is defined with the inhomogeneous pair correlation function and the local density of the solvent. The results of the triplet correlation functions for cations indicate that the thermal fluctuation of the hydration shell is closely related to the size of the solute ion. The triplet correlation function from the present theory is also compared with that from the Kirkwood superposition approximation, which substitutes the inhomogeneous pair correlation by the homogeneous one. For the lithium ion, the behavior of the triplet correlation functions from the present theory shows marked differences from the one calculated within the Kirkwood approximation.     

The special features of the thermodynamic characteristics of hydration of univalent ions according to the reference interaction site model

The dependence of the Gibbs energy of hydration of univalent ions was calculated on the basis of integral equations for a molecular liquid. Calculations were performed in the approximation of the reference interaction site model (RISM). The relation between the energy of hydration of ions and local electrostatic field microinhomogeneities in solution was studied. It was shown that the best variant was the RISM with the hypernetted chain closure, partial inclusion of orientation effects, and semiempirical corrections for the excluded volume and the presence of H-bonds with solutes. Predictions based on the selected variant of the model were in close agreement with the experimental data. The analysis performed indicated directions for the improvement of the RISM method.     

WATsite: Hydration site prediction program with PyMOL interface

Water molecules that mediate protein–ligand interactions or are released from the binding site on ligand binding can contribute both enthalpically and entropically to the free energy of ligand binding. To elucidate the thermodynamic profile of individual water molecules and their potential contribution to ligand binding, a hydration site analysis program WATsite was developed together with an easy‐to‐use graphical user interface based on PyMOL. WATsite identifies hydration sites from a molecular dynamics simulation trajectory with explicit water molecules. The free energy profile of each hydration site is estimated by computing the enthalpy and entropy of the water molecule occupying a hydration site throughout the simulation. The results of the hydration site analysis can be displayed in PyMOL. A key feature of WATsite is that it is able to estimate the protein desolvation free energy for any user specified ligand. The WATsite program and its PyMOL plugin are available free of charge from http://people.pnhs.purdue.edu/~mlill/software . © 2014 Wiley Periodicals, Inc.     

Thermodynamics of Solution of Some Alkyl Acetates in Water

Summary.  The temperature dependence of the solubility of methyl acetate (MeOAc), ethyl acetate (EtOAc), 1-propyl acetate (1-PrOAc), 1-butyl acetate (1-BuOAc), 2-methyl-1-propyl acetate (iso-BuOAc), 2-butyl acetate (sec-BuOAc), and 2-methyl-2-propyl acetate (tert-BuOAc) in water in the temperature range from 298.2 to 318.2 K was determined. The experimental solubility data together with some literature values are presented as a function of temperature and given in analytical form. The solubilities of the investigated compounds at 298.2 K were correlated with the number of carbon atoms in the solute molecule. The standard thermodynamic functions for solubility, i.e. Gibbs free energy, enthalpy, heat capacity, and entropy, were calculated. The Gibbs free energy is positive and increases linearly with the number of carbon atoms, whereas the enthalpy and entropy are negative. The standard thermodynamic functions were converted to thermodynamic quantities for hydration. The Gibbs free energies of hydration of alkyl acetates, which are negative, are compared with the Gibbs free energy of hydration of some n-alkanes, 1-alcohols, and 1-alkylamines. The standard thermodynamic functions of hydration were analyzed using a modified version of the theoretical approach developed by Lee and Graziano [1]. Received October 16, 2000. Accepted May 14, 2001     

Thermodynamics of Solution of Some Alkyl Acetates in Water

 The temperature dependence of the solubility of methyl acetate (MeOAc), ethyl acetate (EtOAc), 1-propyl acetate (1-PrOAc), 1-butyl acetate (1-BuOAc), 2-methyl-1-propyl acetate (iso-BuOAc), 2-butyl acetate (sec-BuOAc), and 2-methyl-2-propyl acetate (tert-BuOAc) in water in the temperature range from 298.2 to 318.2 K was determined. The experimental solubility data together with some literature values are presented as a function of temperature and given in analytical form. The solubilities of the investigated compounds at 298.2 K were correlated with the number of carbon atoms in the solute molecule. The standard thermodynamic functions for solubility, i.e. Gibbs free energy, enthalpy, heat capacity, and entropy, were calculated. The Gibbs free energy is positive and increases linearly with the number of carbon atoms, whereas the enthalpy and entropy are negative. The standard thermodynamic functions were converted to thermodynamic quantities for hydration. The Gibbs free energies of hydration of alkyl acetates, which are negative, are compared with the Gibbs free energy of hydration of some n-alkanes, 1-alcohols, and 1-alkylamines. The standard thermodynamic functions of hydration were analyzed using a modified version of the theoretical approach developed by Lee and Graziano [1].     

A comparative analysis of models for hydration free energy calculations using the RISM theory of integral equations

Rigorous calculations and a detailed analysis of the free energies of hydration were performed using the RISM (reference interaction site model) approach for 96 compounds of various chemical natures. A comparison of all the existing models for calculation of hydration free energy, including models that use corrections and semiempirical parameters, was performed for the first time. The applicability ranges of all the models under consideration were determined. It was shown that the PWC model based on jointly using the RISM and chemoinformatics approaches gave most accurate hydration free energy values. This model allows the free energy of hydration to be predicted with high accuracy and, at the same time, does not require the use of substantial computer resources.     

An application of coupled reference interaction site model/molecular dynamics to the conformational analysis of the alanine dipeptide

We present an application of our recently proposed coupled reference interaction site model (RISM) molecular dynamics (MD) solvation free energy methodology [Freedman and Truong, Chem. Phys. Lett. 381, 362 (2003); J. Chem. Phys. 121, 2187 (2004)] to study the conformational stability of alanine dipeptide in aqueous solution. In this methodology, radial distribution functions obtained from a single MD simulation are substituted into a RISM expression for solvation free energy. Consequently, iterative solution of the RISM equation is not needed. The relative solvation free energies of seven different conformations of the alanine dipeptide in aqueous solution are calculated. Results from the coupled RISM/MD methodology are in good agreement with those from earlier simulations using the accurate free energy perturbation approach, showing that the alphaR conformation is most stabilized by solution. This study establishes a framework for applying this coupled RISM/MD method to larger biological systems.     

Applications of the RISM equation to diatomic fluids: the liquids nitrogen,oxygen and bromine

The reference interaction site model (RISM) integral equation is used to study the equilibrium pair correlation for one-component liquids composed of homonuclear diatomic molecules. The integral equation is first tested by comparing the results obtained from it with those of computer simulation calculations. An internal consistency test is developed which seems to provide an a priori measure of the accuracy of the RISM equation. Then the theory is used to interpret the neutron scattering structure factor data taken on the liquids nitrogen and oxygen. For both of these liquids, a satisfactory explanation of the data is obtained by assuming that the short ranged repulsive forces between molecules are mimicked by a two-site hard core model. However, this simple model does not provide a satisfactory explanation of the data taken from liquid bromine. But, it is shown that a slightly more sophisticated model for the short-ranged repulsion between Br₂ molecules does provide an adequate explanation. With the molecular models determined by fitting the neutron scattering data, the RISM equation provides a method for determining the atom, atom to center-of-mass, and center-of-mass to center-of-mass intermolecular distribution functions in the diatomic liquids. From these functions, the local structures in the three liquids are analyzed. While orientational pair correlations are nearly negligible in both the liquids nitrogen and oxygen, these correlations are fairly substantial in liquid bromine. Furthermore, even when the orientation of a molecule does not greatly influence the orientation of its neighbors, it is found that the orientation of a molecule does have an important effect on the location of its neighbors. Thus, the coupling of translational and orientational coordinates is significant, even in the liquids nitrogen and oxygen.     

RISM study of sulphur dioxide at the liquid-vapour coexistence line

Sulphur dioxide is studied by the reference interaction site model. The site-site potential was modelled by the Lennard-Jones potential functions and direct correlation functions were calculated for thermodynamic points close to the liquid-vapour coexistence line. By means of the correlation functions the pressure and mean potential energy have been calculated. The results were compared with molecular dynamics results and also with experimental data. It turned out that the RISM procedure is competitive with the molecular dynamics method only in a relatively narrow temperature range close to the midpoint between the triple point and the critical point. There the agreement between RISM and molecular dynamics method is quite encouraging and also the experimental values of pressure and mean potential energy and compressibility are satisfactorily reproduced. On either side of this region RISM does not reproduce properly the structure and energetics of the liquid SO₂ system.     

Parametrization strategy for the MolFESD concept: quantitative surface representation of local hydrophobicity

We derive a new model for the established concept of the molecular free energy surface density (MolFESD) yielding a more rigorous representation of local surface contributions to the overall hydrophobicity of a molecule. The model parametrization makes efficient use of both local and global information about solvation thermodynamics, as formulated earlier for the problem of predicting free energies of hydration. The free energy of transfer is separated into an interaction contribution and a term related to the cavity formation. Interaction and cavity components are obtained from the statistical three-dimensional (3D) free energy density and a linear combination of surface and volume terms, respectively. An appropriate molecular interaction field generated by the program Grid is used as an approximate representation of the interaction part of the 3D free energy density. We further compress the 3D density by means of a linear combination of localized surface functions allowing for the derivation of local hydrophobic contributions in the form of a free energy surface density. For a set of 400 compounds our model yields significant correlation (R(2) = 0.95, sigma = 0.57) between experimental and calculated log P values. The final model is applied to establish a correlation between partial free energies of transfer for a series of sucrose derivatives and their relative sweetness, as studied earlier in the group of the authors. We find considerable improvement regarding the rms error of the regression thus validating the presented approach.     

A formulation of statistical mechanics of ordered systems

Expressions are derived for the thermodynamic functions (Gibbs free energy, Helmholtz free energy, etc.) of an ordered system in terms of the single-particle distribution function,p(x), and correlation functions. The thermodynamic functions are treated as functionals of the single-particle distribution function. By minimizing the Helmholtz free energy with respect top(x) under constraints of constantT, V andN, an integral equation is obtained from whichp(x) can be determined. The correlation function of the ordered state in the region near the coexistence surface between ordered and disordered state is expanded about the correlation function of the disordered state, and the series is truncated. Methods for calculating the thermodynamic functions and the single-particle distribution function are presented, and our result is discussed in relation to other treatments of phase coexistence in the literature.     

On the molecular origins of volumetric data

We use a statistical thermodynamic approach and a simple thermodynamic model of hydration to examine the molecular origins of the volumetric properties of solutes. In this model, solute-solvent interactions are treated as a binding reaction. The free energy of hydration of the noninteracting solute species coincides with the free energy of cavity formation, while the free energy of solute-solvent interactions is given by the binding polynomial. By differentiating the relationship for the free energy of hydration with respect to temperature and pressure, one obtains the complete set of equations describing the thermodynamic profile of hydration, including enthalpy, entropy, volume, compressibility, expansibility, and so forth. The model enables one to rigorously define in thermodynamic terms the hydration number and the related concept of hydration shell, which are both widely used as operational definitions in experimental studies. Hydration number, nh, is the effective number of water molecules solvating the solute and represents the derivative of the free energy of hydration with respect to the logarithm of water activity. One traditional way of studying hydration relies on the use of volumetric measurements. However, microscopic interpretation of macroscopic volumetric data is complicated and currently relies on empirical models that are not backed by theory. We use our derived model to link the microscopic determinants of the volumetric properties of a solute and its statistical thermodynamic parameters. In this treatment, the partial molar volume, V degrees, of a solute depends on the cavity volume, hydration number, and the properties of waters of hydration. In contrast, the partial molar isothermal compressibility, K degrees T, and expansibility, E degrees, observables, in addition to the intrinsic compressibility and expansibility of the cavity enclosing the solute, hydration number, and the properties of waters of hydration, contain previously unappreciated relaxation terms that originate from pressure- and temperature-induced perturbation of the equilibrium between the solvated solute species. If significant, the relaxation terms may bring about a new level of nonadditivity to compressibility and expansibility group contributions that goes beyond the overlap of the hydration shells of adjacent groups. We apply our theoretical results to numerical analyses of the volume and compressibility responses to changes in the distribution of solvated species of polar compounds.     

An extended rism equation for molecular polar fluids

The RISM integral equation is extended to molecules with charged sites via a renormalization of the Coulomb potentials and the introduction of appropriate closure relations. For a fluid of diatomics with atomic charges of ±0.2 e the equation yields site-site correlation functions in qualitative agreement with those from computer simulation.     

Hydration of methylamine and methylammonium ion: structural and thermodynamic properties from the data of the integral equation method in the RISM approximation

The structural and thermodynamic properties of hydration of methylamine and methyl-ammonium ion were investigated by the integral equations method in the RISM approximation. According to calculations, the average number of water molecules in the first hydration shell of CH₃ group is 14.4 for aqueous methylamine and 12.7 for aqueous methylammonium solution. The first hydration shells of the NH₂ group of methylamine and the NH₃ ⁺ group of methylammonium ion contain 6.9 and 5.6 water molecules, respectively. The average number of H-bonds formed by the NH₂ group is 2.4 and that formed by the NH₃ ⁺ group is 3. The results obtained show no H-bonding between the nitrogen atom of NH^{3 +} group of methylammonium and water molecules. The hydrogen atom of water participating in the hydrogen bonding with the nitrogen atom of methylamine now is a constituent of the NH₃ ⁺ group of methylammonium ion. The hydration free energies and the ionization constant calculated within the framework of the RISM theory are in good agreement with experimental data.     

Multigrid solver for the reference interaction site model of molecular liquids theory

In this article, we propose a new multigrid-based algorithm for solving integral equations of the reference interactions site model (RISM). We also investigate the relationship between the parameters of the algorithm and the numerical accuracy of the hydration free energy calculations by RISM. For this purpose, we analyzed the performance of the method for several numerical tests with polar and nonpolar compounds. The results of this analysis provide some guidelines for choosing an optimal set of parameters to minimize computational expenses. We compared the performance of the proposed multigrid-based method with the one-grid Picard iteration and nested Picard iteration methods. We show that the proposed method is over 30 times faster than the one-grid iteration method, and in the high accuracy regime, it is almost seven times faster than the nested Picard iteration method.