Calculation of the solvation free energy of neutral and ionic molecules in diverse solvents

The solvation free energy density (SFED) model was modified to extend its applicability and predictability. The parametrization process was performed with a large, diverse set of solvation free energies that included highly polar and ionic molecules. The mean absolute error for 1200 solvation free energies of the 379 neutral molecules in 9 organic solvents and water was 0.40 kcal/mol, and for 90 hydration free energies of ions was 1.7 kcal/mol. Overall, the calculated solvation free energies of a wide range of solute functional groups in diverse solvents were consistent with experimental data.     

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.     

Universal solvation model based on conductor‐like screening model

Atomic surface tensions are parameterized for use with solvation models in which the electrostatic part of the calculation is based on the conductor‐like screening model (COSMO) and the semiempirical molecular orbital methods AM1, PM3, and MNDO/d. The convergence of the calculated polarization free energies with respect to the numerical parameters of the electrostatic calculations is first examined. The accuracy and precision of the calculated values are improved significantly by adjusting two parameters that control the segmentation of the solvent‐accessible surface that is used for the calculations. The accuracy of COSMO calculations is further improved by adopting an optimized set of empirical electrostatic atomic radii. Finally, the electrostatic calculation is combined with SM5‐type atomic surface tension functionals that are used to compute the nonelectrostatic portions of the solvation free energy. All parameterizations are carried out using rigid (R) gas‐phase geometries; this combination (SM5‐type surface tensions, COSMO electrostatics, and rigid geometries) is called SM5CR. Six air–water and 76 water–solvent partition coefficients are added to the training set of air–solvent data points previously used to parameterize the SM5 suite of solvation models, thereby bringing the total number of data points in the training set to 2266. The model yields free energies of solvation and transfer with mean unsigned errors of 0.63, 0.59, and 0.61 kcal/mol for AM1, PM3, and MNDO/d, respectively, over all 2217 data points for neutral solutes in the training set and mean unsigned errors of 3.0, 2.7, and 3.1 kcal/mol, respectively, for 49 data points for the ions. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 340–366, 2000     

Calculation of solvation free energies of charged solutes using mixed cluster/continuum models

We derive a consistent approach for predicting the solvation free energies of charged solutes in the presence of implicit and explicit solvents. We find that some published methodologies make systematic errors in the computed free energies because of the incorrect accounting of the standard state corrections for water molecules or water clusters present in the thermodynamic cycle. This problem can be avoided by using the same standard state for each species involved in the reaction under consideration. We analyze two different thermodynamic cycles for calculating the solvation free energies of ionic solutes: (1) the cluster cycle with an n water cluster as a reagent and (2) the monomer cycle with n distinct water molecules as reagents. The use of the cluster cycle gives solvation free energies that are in excellent agreement with the experimental values obtained from studies of ion-water clusters. The mean absolute errors are 0.8 kcal/mol for H(+) and 2.0 kcal/mol for Cu(2+). Conversely, calculations using the monomer cycle lead to mean absolute errors that are >10 kcal/mol for H(+) and >30 kcal/mol for Cu(2+). The presence of hydrogen-bonded clusters of similar size on the left- and right-hand sides of the reaction cycle results in the cancellation of the systematic errors in the calculated free energies. Using the cluster cycle with 1 solvation shell leads to errors of 5 kcal/mol for H(+) (6 waters) and 27 kcal/mol for Cu(2+) (6 waters), whereas using 2 solvation shells leads to accuracies of 2 kcal/mol for Cu(2+) (18 waters) and 1 kcal/mol for H(+) (10 waters).     

Theoretical studies of the tautomeric equilibria for five-member N-heterocycles in the gas phase and in solution

Tautomeric equilibria have been studied for five-member N-heterocycles and their methyl derivatives in the gas phase and in different solvents with dielectric constants of epsilon = 4.7-78.4. The free energy changes differently for tautomers upon solvation as compared to the gas phase, resulting in a shift of the equilibrium constant in solution. Solvents with increasing dielectric constant produce more negative solute-solvent interaction energies and increasing internal energies. The methyl-substituted imidazole and pyrrazole form delicate equilibria between two tautomeric forms. Depending on the solvent, the methyl-substituted triazoles and tetrazole have one or two major tautomers in solution. When estimating the relative solvation free energies by means of an explicit solvent model and using the FEP/MC method, one observes that the preferred tautomers differ in several cases from those predicted by the continuum solvent model. The 1,2-prototropic shift, as an intramolecular tautomerization path, requires about 50 kcal/mol activation energy for imidazole in the gas phase, and this route is also disfavored in a solution. The calculated activation free energy along the intramolecular path is 48-50 kcal/mol in chloroform and water as compared to a literature value of 13.6 kcal/mol for pyrrazole in DMSO. A molecular dynamics computer experiment favors the formation of an imidazole chain in chloroform, making the 1,3-tautomerization feasible along an intermolecular path in nonprotic solvents. In aqueous solution, one strong N-H...Ow hydrogen bond is formed for each species, whereas all other nitrogens in the ring form weaker, N...HwOw type hydrogen bonds. The tetrahydrofuran solvent acts as a hydrogen bond acceptor and forms N-H...Oether bonds. Molecules of the dichloromethane solvent are in favorable dipole-dipole interactions with the solute. The results obtained are useful in the design of N-heterocyclic ligands forming specified hydrogen bonds with protein side chains.     

Predicting hydration free energies with a hybrid QM/MM approach: an evaluation of implicit and explicit solvation models in SAMPL4

The correct representation of solute-water interactions is essential for the accurate simulation of most biological phenomena. Several highly accurate quantum methods are available to deal with solvation by using both implicit and explicit solvents. So far, however, most evaluations of those methods were based on a single conformation, which neglects solute entropy. Here, we present the first test of a novel approach to determine hydration free energies that uses molecular mechanics (MM) to sample phase space and quantum mechanics (QM) to evaluate the potential energies. Free energies are determined by using re-weighting with the Non-Boltzmann Bennett (NBB) method. In this context, the method is referred to as QM-NBB. Based on snapshots from MM sampling and accounting for their correct Boltzmann weight, it is possible to obtain hydration free energies that incorporate the effect of solute entropy. We evaluate the performance of several QM implicit solvent models, as well as explicit solvent QM/MM for the blind subset of the SAMPL4 hydration free energy challenge. While classical free energy simulations with molecular dynamics give root mean square deviations (RMSD) of 2.8 and 2.3 kcal/mol, the hybrid approach yields an improved RMSD of 1.6 kcal/mol. By selecting an appropriate functional and basis set, the RMSD can be reduced to 1 kcal/mol for calculations based on a single conformation. Results for a selected set of challenging molecules imply that this RMSD can be further reduced by using NBB to reweight MM trajectories with the SMD implicit solvent model.     

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.     

Solvation of monovalent anions in acetonitrile and N,N-dimethylformamide: Parameterization of the IEF-PCM model

The present work reports the parameterization of the polarizable continuum model for predicting the free energies of solvation for monovalent anions in acetonitrile and N,N-dimethylformamide. The parameterization of the model for acetonitrile employed the experimental free energies of solvation for a set of 12 charged solutes, containing H, C, N, O, S, F, Cl, Br, and I atoms. For the N,N-dimethylformamide solutions, experimental solvation free energies for 11 monovalent anions were used. A mean absolute error of 0.7 kcal/mol in the solvation free energies has been achieved for the 12 anions in acetonitrile, whereas the mean absolute error for the 11 anions corresponds to 0.5 kcal/mol in N,N-dimethylformamide. These results indicate that the polarizable continuum model is a suitable methodology for the study of thermodynamic effects in solutions of monovalent anions in both solvents.     

Converging free energy estimates: MM-PB(GB)SA studies on the protein-protein complex Ras-Raf

Estimating protein-protein interaction energies is a very challenging task for current simulation protocols. Here, absolute binding free energies are reported for the complex H-Ras/C-Raf1 using the MM-PB(GB)SA approach, testing the internal consistency and model dependence of the results. Averaging gas-phase energies (MM), solvation free energies as determined by Generalized Born models (GB/SA), and entropic contributions calculated by normal mode analysis for snapshots obtained from 10 ns explicit-solvent molecular dynamics in general results in an overestimation of the binding affinity when a solvent-accessible surface area-dependent model is used to estimate the nonpolar solvation contribution. Applying the sum of a cavity solvation free energy and explicitly modeled solute-solvent van der Waals interaction energies instead provides less negative estimates for the nonpolar solvation contribution. When the polar contribution to the solvation free energy is determined by solving the Poisson-Boltzmann equation (PB) instead, the calculated binding affinity strongly depends on the atomic radii set chosen. For three GB models investigated, different absolute deviations from PB energies were found for the unbound proteins and the complex. As an alternative to normal-mode calculations, quasiharmonic analyses have been performed to estimate entropic contributions due to changes of solute flexibility upon binding. However, such entropy estimates do not converge after 10 ns of simulation time, indicating that sampling issues may limit the applicability of this approach. Finally, binding free energies estimated from snapshots of the unbound proteins extracted from the complex trajectory result in an underestimate of binding affinity. This points to the need to exercise caution in applying the computationally cheaper "one-trajectory-alternative" to systems where there may be significant changes in flexibility and structure due to binding. The best estimate for the binding free energy of Ras-Raf obtained in this study of -8.3 kcal mol(-1) is in good agreement with the experimental result of -9.6 kcal mol(-1), however, further probing the transferability of the applied protocol that led to this result is necessary.     

Prediction of SAMPL2 aqueous solvation free energies and tautomeric ratios using the SM8, SM8AD,and SMD solvation models

We applied the solvation models SM8, SM8AD, and SMD in combination with the Minnesota M06-2X density functional to predict vacuum-water transfer free energies (Task 1) and tautomeric ratios in aqueous solution (Task 2) for the SAMPL2 test set. The bulk-electrostatic contribution to the free energy of solvation is treated as follows: SM8 employs the generalized Born model with the Coulomb field approximation, SM8AD employs the generalized Born approximation with asymmetric descreening, and SMD solves the nonhomogeneous Poisson equation. The non-bulk-electrostatic contribution arising from short-range interactions between the solute and solvent molecules in the first solvation shell is treated as a sum of terms that are products of geometry-dependent atomic surface tensions and solvent-accessible surface areas of the individual atoms of the solute. On average, three models tested in the present work perform similarly. In particular, we achieved mean unsigned errors of 1.3 (SM8), 2.0 (SM8AD), and 2.6 kcal/mol (SMD) for the aqueous free energies of 30 out of 31 compounds with known reference data involved in Task 1 and mean unsigned errors of 2.7 (SM8), 1.8 (SM8AD), and 2.4 kcal/mol (SMD) in the free energy differences (tautomeric ratios) for 21 tautomeric pairs in aqueous solution involved in Task 2.     

Comparative assessment of computational methods for the determination of solvation free energies in alcohol‐based molecules

The determination of differences in solvation free energies between related drug molecules remains an important challenge in computational drug optimization, when fast and accurate calculation of differences in binding free energy are required. In this study, we have evaluated the performance of five commonly used polarized continuum model (PCM) methodologies in the determination of solvation free energies for 53 typical alcohol and alkane small molecules. In addition, the performance of these PCM methods, of a thermodynamic integration (TI) protocol and of the Poisson–Boltzmann (PB) and generalized Born (GB) methods, were tested in the determination of solvation free energies changes for 28 common alkane‐alcohol transformations, by the substitution of an hydrogen atom for a hydroxyl substituent. The results show that the solvation model D (SMD) performs better among the PCM‐based approaches in estimating solvation free energies for alcohol molecules, and solvation free energy changes for alkane‐alcohol transformations, with an average error below 1 kcal/mol for both quantities. However, for the determination of solvation free energy changes on alkane‐alcohol transformation, PB and TI yielded better results. TI was particularly accurate in the treatment of hydroxyl groups additions to aromatic rings (0.53 kcal/mol), a common transformation when optimizing drug‐binding in computer‐aided drug design. © 2013 Wiley Periodicals, Inc.     

Improving description of hydrogen bonds at the semiempirical level: water–water interactions as test case

Hydrogen bonding is not well described by available semiempirical theories. This is an important restriction because hydrogen bonds represent a key feature in many chemical and biochemical processes, besides being responsible for the singular properties of water. In this study, we describe a possible solution to this problem. The basic idea is to replace the nonphysical gaussian correction functions (GCF) appearing in the core–core repulsion terms of most MNDO‐based semiempirical methods by a simple function exhibiting the correct physical behavior in the whole range of intermolecular separation distances. The parameterized interaction function (PIF) is the sum of atom‐pair contributions, each one having five adjustable parameters. In this work, the approach is used to study water–water interactions. The parameters are optimized to reproduce a reference ab initio intermolecular energy surface for the water–water dimer obtained at the MP2/aug‐cc‐pVQZ level. OO, OH, and HH parameters are reported for the PM3 method. The results of PM3‐PIF calculations remarkably improve qualitatively and quantitatively those obtained at the standard PM3 level, both for water–dimer properties and for water clusters up to the hexamer. For example, the root‐mean‐square deviation of the PM3‐PIF interaction energies, with respect to ab initio values obtained using 700 points of the water dimer surface, is only 0.47 kcal/mol. This value is much smaller than that obtained using the standard PM3 method (4.2 kcal/mol). The PM3‐PIF water dimer energy minimum (−5.0 kcal/mol) is also much closer to ab initio data (−5.0±0.01 kcal/mol) than PM3 (−3.50 kcal/mol). The method is therefore promising for the development of new semiempirical approaches as well as for application of combined quantum mechanics and molecular mechanics techniques to investigate chemical processes in water. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 572–581, 2000     

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.     

Development of surface-SFED models for polar solvents

We developed surface grid-based solvation free energy density (Surface-SFED) models for 36 commonly used polar solvents. The parametrization was performed with a large and diverse set of experimental solvation free energies mainly consisting of combinations of polar solvent and multipolar solute. Therefore, the contribution of hydrogen bonds was dominant in the model. In order to increase the accuracy of the model, an elaborate version of a previous hydrogen bond acidity and basicity prediction model was introduced. We present two parametrizations for use with experimentally determined (Surface-SFED/HB(exp)) and empirical (Surface-SFED/HB(cal)) hydrogen bond acidity and basicity values. Our computational results agreed well with experimental results, and inaccuracy of empirical hydrogen bond acidity and basicity values was the main source of error in Surface-SFED/HB(cal). The mean absolute errors of Surface-SFED/HB(exp) and Surface-SFED/HB(cal) were 0.49 and 0.54 kcal/mol, respectively.     

Effect of the Solute Cavity on the Solvation Energy and its Derivatives within the Framework of the Gaussian Charge Scheme

The treatment of the solvation charges using Gaussian functions in the polarizable continuum model results in a smooth potential energy surface. These charges are placed on top of the surface of the solute cavity. In this article, we study the effect of the solute cavity (van der Waals-type or solvent-excluded surface-type) using the Gaussian charge scheme within the framework of the conductor-like polarizable continuum model on (a) the accuracy and computational cost of the self-consistent field (SCF) energy and its gradient and on (b) the calculation of free energies of solvation. For that purpose, we have considered a large set of systems ranging from few atoms to more than 200 atoms in different solvents. Our results at the DFT level using the B3LYP functional and the def2-TZVP basis set show that the choice of the solute cavity does neither affect the accuracy nor the cost of calculations for small systems (< 100 atoms). For larger systems, the use of a vdW-type cavity is recommended, as it prevents small oscillations in the gradient (present when using a SES-type cavity), which affect the convergence of the SCF energy gradient. Regarding the free energies of solvation, we consider a solvent-dependent probe sphere to construct the solvent-accessible surface area required to calculate the nonelectrostatic contribution to the free energy of solvation. For this part, our results for a large set of organic molecules in different solvents agree with available experimental data with an accuracy lower than 1 kcal/mol for both polar and nonpolar solvents.     

Single-ion solvation free energies and the normal hydrogen electrode potential in methanol, acetonitrile, and dimethyl sulfoxide

The division of thermodynamic solvation free energies of electrolytes into contributions from individual ionic constituents is conventionally accomplished by using the single-ion solvation free energy of one reference ion, conventionally the proton, to set the single-ion scales. Thus, the determination of the free energy of solvation of the proton in various solvents is a fundamental issue of central importance in solution chemistry. In the present article, relative solvation free energies of ions and ion-solvent clusters in methanol, acetonitrile, and dimethyl sulfoxide (DMSO) have been determined using a combination of experimental and theoretical gas-phase free energies of formation, solution-phase reduction potentials and acid dissociation constants, and gas-phase clustering free energies. Applying the cluster pair approximation to differences between these relative solvation free energies leads to values of -263.5, -260.2, and -273.3 kcal/mol for the absolute solvation free energy of the proton in methanol, acetonitrile, and DMSO, respectively. The final absolute proton solvation free energies are used to assign absolute values for the normal hydrogen electrode potential and the solvation free energies of other single ions in the solvents mentioned above.     

Fitting atomic correlation parameters for RECEP (rapid estimation of correlation energy from partial charges) method to estimate molecular correlation energies within chemical accuracy

The accuracy of the RECEP method [Chem Phys 1997, 224, 33 and Chem Phys Lett 1999, 307, 469] has been increased considerably by the use of fitted atomic correlation parameters. This method allows an extremely rapid, practically prompt calculation of the correlation energy of molecules after an HF‐SCF calculation. The G2 level correlation energy and HF‐SCF charge distribution of 41 closed‐shell neutral molecules (composed of H, C, N, O, and F atoms) of the G2 thermochemistry database were used to obtain the fitted RECEP atomic correlation parameters. Four different mathematical definitions of partial charges, as a multiple choice, were used to calculate the molecular correlation energies. The best results were obtained using the natural population analysis, although the other three are also recommended for use. For the 41 molecules, the G2 results were approached within a 1.8 kcal/mol standard deviation (the mean absolute difference was 1.5 kcal/mol). The RECEP atomic correlation parameters were also tested on a different, nonoverlapping set of other 24 molecules from the G2 thermochemistry database. The G2 results of these 24 molecules were approached within a 2.3 kcal/mol standard deviation (the mean absolute difference was 1.9 kcal/mol). This method is recommended to estimate total correlation energies of closed shell ground‐state neutral molecules at stationary (minimums and transition states) points on the potential surface. Extension of the work for charged molecules, radicals, and molecules containing other atoms is straightforward. Numerical example as a recipe is also provided. © 2000 John Wiley & Sons, Inc. J Comput Chem 22: 241–254, 2001     

Accuracy of free energies of hydration for organic molecules from 6-31g*-derived partial charges

Absolute free energies of hydration have been computed for 13 diverse organic molecules using partial charges derived from ab initio 6-31G* wave functions. Both Mulliken charges and charges fit to the electrostatic potential surface (EPS) were considered in conjunction with OPLS Lennard–Jones parameters for the organic molecules and the TIP4P model of water. Monte Carlo simulations with statistical perturbation theory yielded relative free energies of hydration. These were converted to absolute quantities through perturbations to reference molecules for which absolute free energies of hydration had been obtained previously in TIP4P water. The average errors in the computed absolute free energies of hydration are 1.1 kcal/mol for the 6-31G* EPS charges and 4.0 kcal/mol for the Mulliken charges. For the EPS charges, the largest individual errors are under 2 kcal/mol except for acetamide, in which case the error is 3.7 kcal/mol. The hydrogen bonding between the organic solutes and water has also been characterized. © John Wiley & Sons, Inc.     

Solvent effect on K to Na ion mutation: a Monte Carlo simulation study

Monte Carlo simulation studies of statistical perturbation theory (SPT) have been carried out to investigate the solvent effects on the relative free energies of solvation and the difference in partition coefficients (log P) for K⁺ to Na⁺ ion mutation in the several solvents. We compared the relative free energies for interconversion of K⁺ to Na⁺, in H₂O (TIP4P) in this study with those published works, that in H₂O (TIP4P) is −16.55 kcal/mol in this study, those of the published works are −17.6, −17.3 and −17.31 kcal/mol and that of the experiment is −17.6 kcal/mol, respectively. Comparing the relative free energies for interconversion of K⁺ to Na⁺, in CH₃OH in this study with those published works, that in CH₃OH is −18.08±0.28 kcal/mol in this study, that of molecular dynamic simulation is −19.6±0.4 kcal/mol and that of the experimental work is −17.3 kcal/mol, respectively. There is good agreement among the several studies if we consider both methods of obtaining the solvation (or hydration) free energies and the standard deviations. For the present K⁺ and Na⁺ ions, the relative free energies of solvation vs Born's function of solvents are decreased with increasing Born's function of solvent except for CH₃OH, THF and MEOME. There is also good agreement between the calculated structural properties in this study and the computer simulation, ab initio and experimental works.