Aqueous solvation free energies of ions and ion-water clusters based on an accurate value for the absolute aqueous solvation free energy of the proton

Thermochemical cycles that involve pKa, gas-phase acidities, aqueous solvation free energies of neutral species, and gas-phase clustering free energies have been used with the cluster pair approximation to determine the absolute aqueous solvation free energy of the proton. The best value obtained in this work is in good agreement with the value reported by Tissandier et al. (Tissandier, M. D.; Cowen, K. A.; Feng, W. Y.; Gundlach, E.; Cohen, M. J.; Earhart, A. D.; Coe, J. V. J. Phys. Chem. A 1998, 102, 7787), who applied the cluster pair approximation to a less diverse and smaller data set of ions. We agree with previous workers who advocated the value of -265.9 kcal/mol for the absolute aqueous solvation free energy of the proton. Considering the uncertainties associated with the experimental gas-phase free energies of ions that are required to use the cluster pair approximation as well as analyses of various subsets of data, we estimate an uncertainty for the absolute aqueous solvation free energy of the proton of no less than 2 kcal/mol. Using a value of -265.9 kcal/mol for the absolute aqueous solvation free energy of the proton, we expand and update our previous compilation of absolute aqueous solvation free energies; this new data set contains conventional and absolute aqueous solvation free energies for 121 unclustered ions (not including the proton) and 147 conventional and absolute aqueous solvation free energies for 51 clustered ions containing from 1 to 6 water molecules. When tested against the same set of ions that was recently used to develop the SM6 continuum solvation model, SM6 retains its previously determined high accuracy; indeed, in most cases the mean unsigned error improves when it is tested against the more accurate reference data.     

Optimized parameters for continuum solvation calculations with carbohydrates

Continuum solvent models have been shown to be an efficient method for the calculation of the energetics of biomolecules in solution. However, for these methods to produce accurate results, an appropriate set of atomic radii or volumes is needed. While these have been developed for proteins and nucleic acids, the same is not true of carbohydrates. Here, a set of optimized parameters for continuum solvation calculations of carbohydrates in conjunction with the Carbohydrate Solution Force Field are presented. Explicit solvent free-energy perturbation simulations were performed on a set of hexapyranose sugars and used to fit atomic radii for Poisson-Boltzmann and generalized-Born calculations, and to fit atomic volumes for use with the analytical continuum electrostatics model. The solvation energetics computed with the optimized radii and a Poisson-Boltzmann model show remarkable agreement with explicit solvent simulation, with a root-mean-square error of 1.19 kcal/mol over a large test set of sugars in many conformations. The generalized-Born model gives slightly poorer agreement, but still correlates very strongly, with an error of 1.69 kcal/mol. The analytical continuum electrostatics model correlates well with the explicit solvent results, but gives a larger error of 4.71 kcal/mol. The remarkable agreement between the solvation free energies computed in explicit and implicit solvent provides strong motivation for the use of implicit solvent models in the simulation of carbohydrate-containing systems.     

Implicit solvent models: Combining an analytical formulation of continuum electrostatics with simple models of the hydrophobic effect

The recent development of approximate analytical formulations of continuum electrostatics opens the possibility of efficient and accurate implicit solvent models for biomolecular simulations. One such formulation (ACE, Schaefer & Karplus, J. Phys. Chem., 1996, 100:1578) is used to compute the electrostatic contribution to solvation and conformational free energies of a set of small solutes and three proteins. Results are compared to finite-difference solutions of the Poisson equation (FDPB) and explicit solvent simulations and experimental data where available. Small molecule solvation free energies agree with FDPB within 1–1.5 kcal/mol, which is comparable to differences in FDPB due to different surface treatments or different force field parameterizations. Side chain conformation free energies of aspartate and asparagine are in qualitative agreement with explicit solvent simulations, while 74 conformations of a surface loop in the protein Ras are accurately ranked compared to FDPB. Preliminary results for solvation free energies of small alkane and polar solutes suggest that a recent Gaussian model could be used in combination with analytical continuum electrostatics to treat nonpolar interactions. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 322–335, 1999     

A fast empirical GAFF compatible partial atomic charge assignment scheme for modeling interactions of small molecules with biomolecular targets

We report here a new and fast approach [Transferable Partial Atomic Charge Model (TPACM4)-upto four bonds] for deriving the partial atomic charges of small molecules for use in protein/DNA-ligand docking and scoring. We have created a look-up table of 5302 atom types to cover the chemical space of C, H, O, N, S, P, F, Cl, and Br atoms in small molecules together with their quantum mechanical RESP fit charges. The atom types defined span diverse plausible chemical environments of each atom in a molecule. The partial charge on any atom in a given molecule is then assigned by a reference to the look-up table. We tested the sensitivity of the TPACM4 partial charges in estimates of hydrogen bond dimers energies, solvation free energies and protein-ligand binding free energies. An average error ±1.11 kcal/mol and a correlation coefficient of 0.90 is obtained in the calculated protein-ligand binding free energies vis-à-vis an RMS error of ±1.02 kcal/mol and a correlation coefficient of 0.92 obtained with RESP fit charges in comparison to experiment. Similar accuracies are realized in predictions of hydrogen bond energies and solvation free energies of small molecules. For a molecule containing 50-55 atoms, the method takes on the order of milliseconds on a single processor machine to assign partial atomic charges. The TPACM4 programme has been web-enabled and made freely accessible at http://www.scfbio-iitd.res.in/software/drugdesign/charge.jsp.     

Quantum Monte Carlo calculations of the dissociation energy of the water dimer

We report diffusion quantum Monte Carlo (DMC) calculations of the equilibrium dissociation energy D(e) of the water dimer. The dissociation energy measured experimentally, D(0), can be estimated from D(e) by adding a correction for vibrational effects. Using the measured dissociation energy and the modern value of the vibrational energy Mas et al., [J. Chem. Phys. 113, 6687 (2000)] leads to D(e)=5.00+/-0.7 kcal mol(-1), although the result Curtiss et al., [J. Chem. Phys. 71, 2703 (1979)] D(e)=5.44+/-0.7 kcal mol(-1), which uses an earlier estimate of the vibrational energy, has been widely quoted. High-level coupled cluster calculations Klopper et al., [Phys. Chem. Chem. Phys. 2, 2227 (2000)] have yielded D(e)=5.02+/-0.05 kcal mol(-1). In an attempt to shed new light on this old problem, we have performed all-electron DMC calculations on the water monomer and dimer using Slater-Jastrow wave functions with both Hartree-Fock approximation (HF) and B3LYP density functional theory single-particle orbitals. We obtain equilibrium dissociation energies for the dimer of 5.02+/-0.18 kcal mol(-1) (HF orbitals) and 5.21+/-0.18 kcal mol(-1) (B3LYP orbitals), in good agreement with the coupled cluster results.     

Calculating solvation energies by means of a fluctuating charge model combined with continuum solvent model

Continuum solvent models have shown to be very efficient for calculating solvation energy of biomolecules in solution. However, in order to produce accurate results, besides atomic radii or volumes, an appropriate set of partial charges of the molecule is needed. Here, a set of partial charges produced by a fluctuating charge model-the atom-bond electronegativity equalization method model (ABEEMσπ) fused into molecular mechanics is used to fit for the analytical continuum electrostatics model of generalized-Born calculations. Because the partial atomic charges provided by the ABEEMσπ model can well reflect the polarization effect of the solute induced by the continuum solvent in solution, accurate and rapid calculations of the solvation energies have been performed for series of compounds involving 105 small neutral molecules, twenty kinds of dipeptides and several protein fragments. The solvation energies of small neutral molecules computed with the combination of the GB model with the fluctuating charge protocol (ABEEMσπ∕GB) show remarkable agreement with the experimental results, with a correlation coefficient of 0.97, a slope of 0.95, and a bias of 0.34 kcal∕mol. Furthermore, for twenty kinds of dipeptides and several protein fragments, the results obtained from the analytical ABEEMσπ∕GB model calculations correlate well with those from ab initio and Poisson-Boltzmann calculations. The remarkable agreement between the solvation energies computed with the ABEEMσπ∕GB model and PB model provides strong motivation for the use of ABEEMσπ∕GB solvent model in the simulation of biochemical systems.     

Continuum solvation models in the linear interaction energy method

The linear interaction energy (LIE) method in combination with two different continuum solvent models has been applied to calculate protein-ligand binding free energies for a set of inhibitors against the malarial aspartic protease plasmepsin II. Ligand-water interaction energies are calculated from both Poisson-Boltzmann (PB) and Generalized Born (GB) continuum models using snapshots from explicit solvent simulations of the ligand and protein-ligand complex. These are compared to explicit solvent calculations, and we find close agreement between the explicit water and PB solvation models. The GB model overestimates the change in solvation energy, and this is caused by consistent underestimation of the effective Born radii in the protein-ligand complex. The explicit solvent LIE calculations and LIE-PB, with our standard parametrization, reproduce absolute experimental binding free energies with an average unsigned error of 0.5 and 0.7 kcal/mol, respectively. The LIE-GB method, however, requires a constant offset to approach the same level of accuracy.     

Correlation energy functionals dependent on an effective number of electrons: charged species and equilibrium geometries

Recently proposed spin-dependent and spin-independent correlation energy functionals [Perez-Jimenez et al., J. Chem. Phys. 116, 10571 (2002)] based on an effective number of electrons N are extended to deal with charged systems. By introducing the concept of an effective atomic number Z analogous to N, the spin-dependent functional in combination with Becke's exchange [Becke, Phys. Rev. A 38, 3098 (1988)] yields a mean absolute error (MAE) of 5.4 kcal/mol for the 88 ionization potentials and 58 electron affinities included in the extended G2 set, and a MAE of 4.1 kcal/mol for the 312 data comprising the above plus the 148 enthalpies of formation of the extended G2 set and the 18 total energies of the neutral atoms H through Ar. Geometry optimizations performed on the 53 molecules of the G2-1 test set with the above combination of exchange and correlation functionals yield MAEs of 0.017 A and 1.5 degrees for the 68 bond lengths and 29 angles analyzed as compared with the experimental estimates.     

Solvated interaction energy (SIE) for scoring protein-ligand binding affinities. 1. Exploring the parameter space

We present a binding free energy function that consists of force field terms supplemented by solvation terms. We used this function to calibrate the solvation model along with the binding interaction terms in a self-consistent manner. The motivation for this approach was that the solute dielectric-constant dependence of calculated hydration gas-to-water transfer free energies is markedly different from that of binding free energies (J. Comput. Chem. 2003, 24, 954). Hence, we sought to calibrate directly the solvation terms in the context of a binding calculation. The five parameters of the model were systematically scanned to best reproduce the absolute binding free energies for a set of 99 protein-ligand complexes. We obtained a mean unsigned error of 1.29 kcal/mol for the predicted absolute binding affinity in a parameter space that was fairly shallow near the optimum. The lowest errors were obtained with solute dielectric values of Din = 20 or higher and scaling of the intermolecular van der Waals interaction energy by factors ranging from 0.03 to 0.15. The high apparent Din and strong van der Waals scaling may reflect the anticorrelation of the change in solvated potential energy and configurational entropy, that is, enthalpy-entropy compensation in ligand binding (Biophys. J. 2004, 87, 3035-3049). Five variations of preparing the protein-ligand data set were explored in order to examine the effect of energy refinement and the presence of bound water on the calculated results. We find that retaining water in the final protein structure used for calculating the binding free energy is not necessary to obtain good results; that is the continuum solvation model is sufficient. Virtual screening enrichment studies on estrogen receptor and thymidine kinase showed a good ability of the binding free energy function to recover true hits in a collection of decoys.     

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.     

Performance of SM8 on a test to predict small-molecule solvation free energies

The SM8 quantum mechanical aqueous continuum solvation model is applied to a 17-molecule test set proposed by Nicholls et al. (J. Med. Chem. 2008, 51, 769) to predict free energies of solvation. With the M06-2X density functional, the 6-31G(d) basis set, and CM4M charge model, the root-mean-square error (RMSE) of SM8 is 1.08 kcal mol(-1) for aqueous geometries and 1.14 kcal mol(-1) for gas-phase geometries. These errors compare favorably with optimal explicit and continuum models reported by Nicholls et al., having RMSEs of 1.33 and 1.87 kcal mol(-1), respectively. Other models examined by these workers had RMSEs of 1.5-2.6 kcal mol(-1). We also explore the use of other density functionals and charge models with SM8 and the RMSE increases to 1.21 kcal mol(-1) for mPW1/CM4 with gas-phase geometries, to 1.50 kcal mol(-1) for M06-2X/CM4 with gas-phase geometries, and to 1.27-1.64 kcal mol(-1) with three different models at B3LYP gas-phase geometries.     

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.     

Absolute solvation free energy of Li+ and Na+ ions in dimethyl sulfoxide solution: a theoretical ab initio and cluster-continuum model study

The solvation of the lithium and sodium ions in dimethyl sulfoxide solution was theoretically investigated using ab initio calculations coupled with the hybrid cluster-continuum model, a quasichemical theory of solvation. We have investigated clusters of ions with up to five dimethyl sulfoxide (DMSO) molecules, and the bulk solvent was described by a dielectric continuum model. Our results show that the lithium and sodium ions have four and five DMSO molecules into the first coordination shell, and the calculated solvation free energies are -135.5 and -108.6 kcal mol(-1), respectively. These data suggest a solvation free energy value of -273.2 kcal mol(-1) for the proton in dimethyl sulfoxide solution, a value that is more negative than the present uncertain experimental value. This and previous studies on the solvation of ions in water solution indicate that the tetraphenylarsonium tetraphenylborate assumption is flawed and the absolute value of the free energy of transfer of ions from water to DMSO solution is higher than the present experimental values.     

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.     

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.     

Electrostatics of proteins in dielectric solvent continua. I. Newton's third law marries qE forces

The authors reformulate and revise an electrostatic theory treating proteins surrounded by dielectric solvent continua [B. Egwolf and P. Tavan, J. Chem. Phys. 118, 2039 (2003)] to make the resulting reaction field (RF) forces compatible with Newton's third law. Such a compatibility is required for their use in molecular dynamics (MD) simulations, in which the proteins are modeled by all-atom molecular mechanics force fields. According to the original theory the RF forces, which are due to the electric field generated by the solvent polarization and act on the partial charges of a protein, i.e., the so-called qE forces, can be quite accurately computed from Gaussian RF dipoles localized at the protein atoms. Using a slightly different approximation scheme also the RF energies of given protein configurations are obtained. However, because the qE forces do not account for the dielectric boundary pressure exerted by the solvent continuum on the protein, they do not obey the principle that actio equals reactio as required by Newton's third law. Therefore, their use in MD simulations is severely hampered. An analysis of the original theory has led the authors now to a reformulation removing the main difficulties. By considering the RF energy, which represents the dominant electrostatic contribution to the free energy of solvation for a given protein configuration, they show that its negative configurational gradient yields mean RF forces obeying the reactio principle. Because the evaluation of these mean forces is computationally much more demanding than that of the qE forces, they derive a suggestion how the qE forces can be modified to obey Newton's third law. Various properties of the thus established theory, particularly issues of accuracy and of computational efficiency, are discussed. A sample application to a MD simulation of a peptide in solution is described in the following paper [M. Stork and P. Tavan, J. Chem. Phys., 126, 165106 (2007).     

Surveying implicit solvent models for estimating small molecule absolute hydration free energies

Implicit solvent models are powerful tools in accounting for the aqueous environment at a fraction of the computational expense of explicit solvent representations. Here, we compare the ability of common implicit solvent models (TC, OBC, OBC2, GBMV, GBMV2, GBSW, GBSW/MS, GBSW/MS2 and FACTS) to reproduce experimental absolute hydration free energies for a series of 499 small neutral molecules that are modeled using AMBER/GAFF parameters and AM1-BCC charges. Given optimized surface tension coefficients for scaling the surface area term in the nonpolar contribution, most implicit solvent models demonstrate reasonable agreement with extensive explicit solvent simulations (average difference 1.0-1.7 kcal/mol and R(2)=0.81-0.91) and with experimental hydration free energies (average unsigned errors=1.1-1.4 kcal/mol and R(2)=0.66-0.81). Chemical classes of compounds are identified that need further optimization of their ligand force field parameters and others that require improvement in the physical parameters of the implicit solvent models themselves. More sophisticated nonpolar models are also likely necessary to more effectively represent the underlying physics of solvation and take the quality of hydration free energies estimated from implicit solvent models to the next level.     

Prediction of solvation free energies from computed properties of solute molecular surfaces

We have shown that the solvation energies of a group of 12 solutes in 7 different solvents can be presented analytically in terms of quantities computed at the density functional B3P86/6‐31+G** level for the isolated solute molecules. These quantities include the molecular surface area and several properties of the electrostatic potential on the surface, e.g., the most positive and negative values, the average deviation of the potential, the positive and negative portions of the surface, and their average potentials. Overall, the average absolute deviation of the predicted from the experimental solvation free energies is 0.25 kcal/mol; the poorest results are obtained for the solute butanone, for which the average absolute deviation is 0.63 kcal/mol. The forms of the relationships reflect the natures of the solute–solvent interactions; for the solvents with low dielectric constants, these are primarily global, involving extended portions of the molecular surfaces, whereas for the more polar solvents, site‐specific interactions also play key roles. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 76: 643–647, 2000     

Assessment of Gaussian-3 and density-functional theories on the G3/05 test set of experimental energies

The G3/99 test set [L. A. Curtiss, K. Raghavachari, P. C. Redfern, and J. A. Pople, J. Chem. Phys. 112, 7374 (2000)] of thermochemical data for validation of quantum chemical methods is expanded to include 78 additional energies including 14 enthalpies of formation of the first- and second-row nonhydrogen molecules, 58 energies of molecules containing the third-row elements K, Ca, and Ga-Kr, and 6 hydrogen-bonded complexes. The criterion used for selecting the additional systems is the same as before, i.e., experimental uncertainties less than +/- 1 kcal/mol. This new set, referred to as the G3/05 test set, has a total of 454 energies. The G3 and G3X theories are found to have mean absolute deviations of 1.13 and 1.01 kcal/mol, respectively, when applied to the G3/05 test set. Both methods have larger errors for the nonhydrogen subset of 79 species for which they have mean absolute deviations of 2.10 and 1.64 kcal/mol, respectively. On all of the other types of energies the G3 and G3X methods are very reliable. The G3/05 test set is also used to assess density-functional methods including a series of new functionals. The most accurate functional for the G3/05 test set is B98 with a mean absolute deviation of 3.33 kcal/mol, compared to 4.14 kcal/mol for B3LYP. The latter functional has especially large errors for larger molecules with a mean absolute deviation of 9 kcal/mol for molecules having 28 or more valence electrons. For smaller molecules B3LYP does as well or better than B98 and the other functionals. It is found that many of the density-functional methods have significant errors for the larger molecules in the test set.     

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