Absolute Hydration Free Energy Research Articles

Exploring the Limits of the Generalized CHARMM and AMBER Force Fields through Predictions of Hydration Free Energy of Small Molecules.

Accurate force field parameters, potential energy functions, and receptor-ligand models are essential for modeling the solvation and binding of drug-like molecules to a receptor. A large and ever-growing chemical space of medicinally relevant scaffolds has also required these factors, especially force field parameters, to be highly transferable. Generalized force fields such as the CHARMM General Force Field (CGenFF) and the generalized AMBER force field (GAFF) have accomplished this feat along with other contemporaneous ones like OPLS. Here, we analyze the limits in the parametrization of drug-like small molecules by CGenFF and GAFF in terms of the various functional groups represented within them. Specifically, we link the presence of specific functional groups to the error in the absolute hydration free energy of over 600 small molecules, predicted by alchemical free energy methods implemented in the CHARMM program. Our investigation reveals that molecules with (i) a nitro group in CGenFF and GAFF are, respectively, over- or undersolubilized in aqueous medium, (ii) amine groups are undersolubilized more so in CGenFF than in GAFF, and (iii) carboxyl groups are more oversolubilized in GAFF than in CGenFF. We present our analyses of the potential factors underlying these trends. We also showcase the use of a machine-learning-based approach combined with the SHapley Additive exPlanations framework to attribute these trends to specific functional groups, which can be easily adopted to explore the limits of other general force fields.

Microsolvation of phenol in water: structures, hydration free energy and enthalpy

ABSTRACT In this work, we have studied the microsolvation of phenol in water. We started by identifying initial configurations of phenol-water clusters using classical molecular dynamics. The configurations are optimised at the ωB97XD/aug-cc-pVDZ level of theory. To understand the interaction between phenol and the solvating water molecules, we performed a quantum theory of atoms in molecules (QTAIM) analysis. The results show that the structures of phenol-water clusters are similar to those of neutral water clusters. The QTAIM analysis shows that the structures of phenol-water clusters are stabilised by strong OH··· O hydrogen bondings, weak CH··· O hydrogen bondings, and OH bonding interactions. The located structures of phenol-water clusters have been used to calculate the absolute hydration free energy and enthalpy of phenol for temperatures between 20 and 400 K. The hydration energies are calculated using the cluster continuum solvation model. It has been found that the explicit solvation has negligible effects on the hydration free energy and enthalpy of phenol. Furthermore, the hydration free energy of phenol is found to be linearly varying with increasing temperature, while the hydration enthalpy is found to be temperature independent. The estimated hydration free energy of phenol is slightly underestimated as compared to a previously reported experimental estimate.

Absolute Hydration Free Energy of Small Anions and the Aqueous pKa of Simple Acids.

Heats of formation and gas phase acidities for the simple acids and their deprotonated anions (A- = F-, Cl-, Br-, I-, OH-, SH-, SeH-, TeH-, OCl-, OBr-, and OI-) were calculated using the Feller-Peterson-Dixon (FPD) method with large basis sets including Douglass-Kroll scalar relativistic corrections. Hydration of the neutral and anionic species was predicted using the supermolecule-continuum approach, resulting in absolute hydration free energies that, when combined with calculated gas phase acidities, produce aqueous acidities and pKa values for these simple acids that are, in general, in excellent agreement with experimental literature values. Absolute hydration free energy values converged quickly in terms of the experimental values for neutral species, requiring only four explicit H2O molecules. HI is anomalous in that it fully dissociates ionically in a water tetramer and was treated without explicit water molecules. The hydration energies of anionic species converged more slowly and were modeled with up to 16 explicit H2O molecules. Calculated values for ΔHf and ΔGgas agree with experimental values within ca. 1.2 kcal/mol, and ΔGaq and ΔΔGhyd agree with experimental values within ca. 2 kcal/mol in most cases.

Efficient Alchemical Intermediate States in Free Energy Calculations Using λ-Enveloping Distribution Sampling.

Alchemical free energy calculations generally require intermediate states along a coupling parameter λ to establish sufficient phase space overlap for obtaining converged results. Such intermediate states can also be engineered to lower the energy barriers and, consequently, reduce the required sampling time. The recently introduced λ-enveloping distribution sampling (λ-EDS) scheme combines the properties of the minimum variance pathway and the EDS methods to improve sampling and allow for larger steps along the alchemical pathway compared to conventional approaches. This scheme also eliminates the need for soft-core potentials and retains the behavior of conventional λ-intermediate states as a limiting case. In this study, an automated procedure is developed to select the parameters of λ-EDS for optimal performance. The underlying theory is illustrated based on simulations of simple test systems (bond length changes in harmonic oscillators, mutations of dihedral angles, and charge creation in water), as well as on the calculation of the absolute hydration free energies of 12 small organic molecules.

Scaffold Hopping Transformations Using Auxiliary Restraints for Calculating Accurate Relative Binding Free Energies.

In silico screening of drug-target interactions is a key part of the drug discovery process. Changes in the drug scaffold via contraction or expansion of rings, the breaking of rings, and the introduction of cyclic structures from acyclic structures are commonly applied by medicinal chemists to improve binding affinity and enhance favorable properties of candidate compounds. These processes, commonly referred to as scaffold hopping, are challenging to model computationally. Although relative binding free energy (RBFE) calculations have shown success in predicting binding affinity changes caused by perturbing R-groups attached to a common scaffold, applications of RBFE calculations to modeling scaffold hopping are relatively limited. Scaffold hopping inevitably involves breaking and forming bond interactions of quadratic functional forms, which is highly challenging. A novel method for handling ring opening/closure/contraction/expansion and linker contraction/expansion is presented here. To the best of our knowledge, RBFE calculations on linker contraction/expansion have not been previously reported. The method uses auxiliary restraints to hold the atoms at the ends of a bond in place during the breaking and forming of the bonds. The broad applicability of the method was demonstrated by examining perturbations involving small-molecule macrocycles and mutations of proline in proteins. High accuracy was obtained using the method for most of the perturbations studied. The rigor of the method was isolated from the force field by validating the method using relative and absolute hydration free energy calculations compared to standard simulation results. Unlike other methods that rely on λ-dependent functional forms for bond interactions, the method presented here can be employed using modern molecular dynamics software without modification of codes or force field functions.

Implementation of the QUBE Force Field in SOMD for High-Throughput Alchemical Free-Energy Calculations.

The quantum mechanical bespoke (QUBE) force-field approach has been developed to facilitate the automated derivation of potential energy function parameters for modeling protein-ligand binding. To date, the approach has been validated in the context of Monte Carlo simulations of protein-ligand complexes. We describe here the implementation of the QUBE force field in the alchemical free-energy calculation molecular dynamics simulation package SOMD. The implementation is validated by demonstrating the reproducibility of absolute hydration free energies computed with the QUBE force field across the SOMD and GROMACS software packages. We further demonstrate, by way of a case study involving two series of non-nucleoside inhibitors of HIV-1 reverse transcriptase, that the availability of QUBE in a modern simulation package that makes efficient use of graphics processing unit acceleration will facilitate high-throughput alchemical free-energy calculations.

Absolute ion hydration free energy scale and the surface potential of water via quantum simulation

With a goal of determining an absolute free energy scale for ion hydration, quasi-chemical theory and ab initio quantum mechanical simulations are employed to obtain an accurate value for the bulk hydration free energy of the Na+ ion. The free energy is partitioned into three parts: 1) the inner-shell or chemical contribution that includes direct interactions of the ion with nearby waters, 2) the packing free energy that is the work to produce a cavity of size λ in water, and 3) the long-range contribution that involves all interactions outside the inner shell. The interfacial potential contribution to the free energy resides in the long-range term. By averaging cation and anion data for that contribution, cumulant terms of all odd orders in the electrostatic potential are removed. The computed total is then the bulk hydration free energy. Comparison with the experimentally derived real hydration free energy produces an effective surface potential of water in the range -0.4 to -0.5 V. The result is consistent with a variety of experiments concerning acid-base chemistry, ion distributions near hydrophobic interfaces, and electric fields near the surface of water droplets.

Hybrid Alchemical Free Energy/Machine-Learning Methodology for the Computation of Hydration Free Energies.

A methodology that combines alchemical free energy calculations (FEP) with machine learning (ML) has been developed to compute accurate absolute hydration free energies. The hybrid FEP/ML methodology was trained on a subset of the FreeSolv database and retrospectively shown to outperform most submissions from the SAMPL4 competition. Compared to pure machine-learning approaches, FEP/ML yields more precise estimates of free energies of hydration and requires a fraction of the training set size to outperform standalone FEP calculations. The ML-derived correction terms are further shown to be transferable to a range of related FEP simulation protocols. The approach may be used to inexpensively improve the accuracy of FEP calculations and to flag molecules which will benefit the most from bespoke force field parametrization efforts.

The H•/H- Redox Couple and Absolute Hydration Energy of H.

A supermolecule-continuum approach with water clusters up to n = 16 H2O molecules has been used to predict the absolute hydration free energies at 298 K (ΔGhyd) of both hydrogen (H•) and hydride (H-) to be 4.6 ± 1 and -78 ± 3 kcal/mol, respectively. These values are combined with a high accuracy prediction of the gas-phase electron affinity (ΔGgas,298K = -16.9 kcal/mol) to determine the aqueous electron affinity of H• of 99.5 ± 3 kcal/mol, which yields a reduction potential for H• vs SHE of -0.03 ± 0.15 V. This value is in agreement within 0.2 V with most estimates obtained using a wide variety of approaches. These results can be used to improve the absolute hydricity scale in water which provides additional insights into how a putative hydride interacts with solvent but do not change the ability to predict the relative reactivity of two species using relative hydricity scales.

Single-Ion Thermodynamics from First Principles: Calculation of the Absolute Hydration Free Energy and Single-Electrode Potential of Aqueous Li+ Using ab Initio Quantum Mechanical/Molecular Mechanical Molecular Dynamics Simulations

A recently proposed thermodynamic integration (TI) approach formulated in the framework of quantum mechanical/molecular mechanical molecular dynamics (QM/MM MD) simulations is applied to study the structure, dynamics, and absolute intrinsic hydration free energy Δs GM+,wat◦ of the Li+ ion at a correlated ab initio level of theory. Based on the results, standard values (298.15 K, ideal gas at 1 bar, ideal solute at 1 molal) for the absolute intrinsic hydration free energy [Formula: see text] of the proton, the surface electric potential jump χwat◦ upon entering bulk water, and the absolute single-electrode potential [Formula: see text] of the reference hydrogen electrode are calculated to be -1099.9 ± 4.2 kJ·mol-1, 0.13 ± 0.08 V, and 4.28 ± 0.04 V, respectively, in excellent agreement with the standard values recommended by Hünenberger and Reif on the basis of an extensive evaluation of the available experimental data (-1100 ± 5 kJ·mol-1, 0.13 ± 0.10 V, and 4.28 ± 0.13 V). The simulation results for Li+ are also compared to those for Na+ and K+ from a previous study in terms of relative hydration free energies ΔΔs GM+,wat◦ and relative electrode potentials [Formula: see text]. The calculated values are found to agree extremely well with the experimental differences in standard conventional hydration free energies ΔΔs GM+,wat• and redox potentials [Formula: see text]. The level of agreement between simulation and experiment, which is quantitative within error bars, underlines the substantial accuracy improvement achieved by applying a highly demanding QM/MM approach at the resolution-of-identity second-order Møller-Plesset perturbation (RIMP2) level over calculations relying on purely molecular mechanical or density functional theory (DFT) descriptions. A detailed analysis of the structural and dynamical properties of the Li+ hydrate indicates that a correct description of the solvation structure and dynamics is achieved as well at this level of theory. Consideration of the QM/MM potential-energy components also shows that the partitioning into QM and MM zones does not induce any significant energetic artifact for the system considered.

Re-examining the tetraphenyl-arsonium/tetraphenyl-borate (TATB) hypothesis for single-ion solvation free energies

Attempts to establish an absolute single-ion hydration free energy scale have followed multiple strategies. Two central themes consist of (1) employing bulk pair thermodynamic data and an underlying interfacial-potential-free model to partition the hydration free energy into individual contributions [Marcus, Latimer, and tetraphenyl-arsonium/tetraphenyl-borate (TATB) methods] or (2) utilizing bulk thermodynamic and cluster data to estimate the free energy to insert a proton into water, including in principle an interfacial potential contribution [the cluster pair approximation (CPA)]. While the results for the hydration free energy of the proton agree remarkably well between the three approaches in the first category, the value differs from the CPA result by roughly +10 kcal/mol, implying a value for the effective electrochemical surface potential of water of -0.4 V. This paper provides a computational re-analysis of the TATB method for single-ion free energies using quasichemical theory. A previous study indicated a significant discrepancy between the free energies of hydration for the TA cation and the TB anion. We show that the main contribution to this large computed difference is an electrostatic artifact arising from modeling interactions in periodic boundaries. No attempt is made here to develop more accurate models for the local ion/solvent interactions that may lead to further small free energy differences between the TA and TB ions, but the results clarify the primary importance of interfacial potential effects for analysis of the various free energy scales. Results are also presented, related to the TATB assumption in the organic solvents dimethyl sulfoxide and 1,2-dichloroethane.

Absolute proton hydration free energy, surface potential of water, and redox potential of the hydrogen electrode from first principles: QM/MM MD free-energy simulations of sodium and potassium hydration.

The absolute intrinsic hydration free energy GH+,wat◦ of the proton, the surface electric potential jump χwat◦ upon entering bulk water, and the absolute redox potential VH+,wat◦ of the reference hydrogen electrode are cornerstone quantities for formulating single-ion thermodynamics on absolute scales. They can be easily calculated from each other but remain fundamentally elusive, i.e., they cannot be determined experimentally without invoking some extra-thermodynamic assumption (ETA). The Born model provides a natural framework to formulate such an assumption (Born ETA), as it automatically factors out the contribution of crossing the water surface from the hydration free energy. However, this model describes the short-range solvation inaccurately and relies on the choice of arbitrary ion-size parameters. In the present study, both shortcomings are alleviated by performing first-principle calculations of the hydration free energies of the sodium (Na+) and potassium (K+) ions. The calculations rely on thermodynamic integration based on quantum-mechanical molecular-mechanical (QM/MM) molecular dynamics (MD) simulations involving the ion and 2000 water molecules. The ion and its first hydration shell are described using a correlated ab initio method, namely resolution-of-identity second-order Møller-Plesset perturbation (RIMP2). The next hydration shells are described using the extended simple point charge water model (SPC/E). The hydration free energy is first calculated at the MM level and subsequently increased by a quantization term accounting for the transformation to a QM/MM description. It is also corrected for finite-size, approximate-electrostatics, and potential-summation errors, as well as standard-state definition. These computationally intensive simulations provide accurate first-principle estimates for GH+,wat◦, χwat◦, and VH+,wat◦, reported with statistical errors based on a confidence interval of 99%. The values obtained from the independent Na+ and K+ simulations are in excellent agreement. In particular, the difference between the two hydration free energies, which is not an elusive quantity, is 73.9 ± 5.4 kJ mol-1 (K+ minus Na+), to be compared with the experimental value of 71.7 ± 2.8 kJ mol-1. The calculated values of GH+,wat◦, χwat◦, and VH+,wat◦ (-1096.7 ± 6.1 kJ mol-1, 0.10 ± 0.10 V, and 4.32 ± 0.06 V, respectively, averaging over the two ions) are also in remarkable agreement with the values recommended by Reif and Hünenberger based on a thorough analysis of the experimental literature (-1100 ± 5 kJ mol-1, 0.13 ± 0.10 V, and 4.28 ± 0.13 V, respectively). The QM/MM MD simulations are also shown to provide an accurate description of the hydration structure, dynamics, and energetics.

Solvation of the Guanidinium Ion in Pure Aqueous Environments: A Theoretical Study from an “Ab Initio”-Based Polarizable Force Field

We report simulation results regarding the hydration process of the guanidinium cation in water droplets and in bulk liquid water, at a low concentration of 0.03 M, performed using a polarizable approach to model both water/water and ion/water interactions. In line with earlier theoretical studies, our simulations show a preferential orientation of guanidinium at water-vacuum interfaces, i.e., a parallel orientation of the guanidinium plane to the aqueous surface. In an apparent contradiction with earlier simulation studies, we show also that guanidinium has a stronger propensity for the cores of aqueous systems than the ammonium cation. However, our bulk simulation conditions correspond to weaker cation concentrations than in earlier studies, by 2 orders of magnitude, and that the same simulations performed using a standard nonpolarizable force field leads to the same conclusion. From droplet data, we extrapolate the guanidinium single hydration enthalpy value to be -82.9 ± 2.2 kcal mol-1. That is about half as large as the sole experimental estimate reported to date, about -144 kcal mol-1. Our result yields a guanidinium absolute bulk hydration free energy at ambiant conditions to be -78.4 ± 2.6 kcal mol-1, a value smaller by 3 kcal mol-1 compared to ammonium. The relatively large magnitude of our guanidinium hydration free energy estimate suggests the Gdm+ protein denaturing properties to result from a competition between the cation hydration effects and the cation/protein interactions, a competition that can be modulated by weak differences in the protein or in the cation chemical environment.

Unraveling halide hydration: A high dilution approach.

The hydration properties of halide aqua ions have been investigated combining classical Molecular Dynamics (MD) with Extended X-ray Absorption Fine Structure (EXAFS) spectroscopy. Three halide-water interaction potentials recently developed [M. M. Reif and P. H. Hünenberger, J. Chem. Phys. 134, 144104 (2011)], along with three plausible choices for the value of the absolute hydration free energy of the proton (ΔG [minus sign in circle symbol]hyd[H+]), have been checked for their capability to properly describe the structural properties of halide aqueous solutions, by comparing the MD structural results with EXAFS experimental data. A very good agreement between theory and experiment has been obtained with one parameter set, namely LE, thus strengthening preliminary evidences for a ΔG [minus sign in circle symbol]hyd[H] value of -1100 kJ mol(-1) [M. M. Reif and P. H. Hünenberger, J. Chem. Phys. 134, 144104 (2011)]. The Cl(-), Br(-), and I(-) ions have been found to form an unstructured and disordered first hydration shell in aqueous solution, with a broad distribution of instantaneous coordination numbers. Conversely, the F(-) ion shows more ordered and defined first solvation shell, with only two statistically relevant coordination geometries (six and sevenfold complexes). Our thorough investigation on the effect of halide ions on the microscopic structure of water highlights that the perturbation induced by the Cl(-), Br(-), and I(-) ions does not extend beyond the ion first hydration shell, and the structure of water in the F(-) second shell is also substantially unaffected by the ion.

Prediction of hydration free energies for the SAMPL4 diverse set of compounds using molecular dynamics simulations with the OPLS-AA force field

All-atom molecular dynamics computer simulations were used to blindly predict the hydration free energies of a range of small molecules as part of the SAMPL4 challenge. Compounds were parametrized on the basis of the OPLS-AA force field using three different protocols for deriving partial charges: (1) using existing OPLS-AA atom types and charges with minor adjustments of partial charges on equivalent connecting atoms and derivation of new parameters for a number of distinct chemical groups (N-alkyl imidazole, nitrate) that were not present in the published force field; (2) calculation of quantum mechanical charges via geometry optimization, followed by electrostatic potential (ESP) fitting, using Jaguar at the LMP2/cc-pVTZ(-F) level; and (3) via geometry optimization and CHelpG charges (Gaussian09at the HF/6-31G* level), followed by two-stage RESP fitting. The absolute hydration free energy was computed by an established protocol including alchemical free energy perturbation with thermodynamic integration. The use of standard OPLS-AA charges (protocol 1) with a number of newly parametrized charges and the use of histidine derived parameters for imidazole yielded an overall root mean square deviation of the prediction from the experimental data of 1.75kcal/mol. The precision of our results appears to be mainly limited by relatively poor reproducibility of the Lennard-Jones contribution towards the solvation free energy, for which we observed large variability that could be traced to a strong dependence on the initial system conditions.

Fast and accurate generation of ab initio quality atomic charges using nonparametric statistical regression

We introduce a class of partial atomic charge assignment method that provides ab initio quality description of the electrostatics of bioorganic molecules. The method uses a set of models that neither have a fixed functional form nor require a fixed set of parameters, and therefore are capable of capturing the complexities of the charge distribution in great detail. Random Forest regression is used to build separate charge models for elements H, C, N, O, F, S, and Cl, using training data consisting of partial charges along with a description of their surrounding chemical environments; training set charges are generated by fitting to the b3lyp/6-31G* electrostatic potential (ESP) and are subsequently refined to improve consistency and transferability of the charge assignments. Using a set of 210 neutral, small organic molecules, the absolute hydration free energy calculated using these charges in conjunction with Generalized Born solvation model shows a low mean unsigned error, close to 1 kcal/mol, from the experimental data. Using another large and independent test set of chemically diverse organic molecules, the method is shown to accurately reproduce charge-dependent observables--ESP and dipole moment--from ab initio calculations. The method presented here automatically provides an estimate of potential errors in the charge assignment, enabling systematic improvement of these models using additional data. This work has implications not only for the future development of charge models but also in developing methods to describe many other chemical properties that require accurate representation of the electronic structure of the system.

The influence of water interfacial potentials on ion hydration in bulk water and near interfaces

The surface potential at the water liquid–vapor interface is discussed in relation to experimental determinations of bulk absolute ion hydration free energies. It is shown that, rather than the surface potential itself, the net electrostatic potential at the center of an uncharged solute can aid both in relating differences between tabulations of hydration free energies and in explaining differing classical and quantum surface potential estimates. Quantum mechanical results for the net potential are consistent with conclusions from previous classical simulations, suggesting a contribution from the net potential that can influence ion density profiles for single ions in water droplets.

Assessing the quality of absolute hydration free energies among CHARMM-compatible ligand parameterization schemes.

Multipurpose atom-typer for CHARMM (MATCH), an atom-typing toolset for molecular mechanics force fields, was recently developed in our laboratory. Here, we assess the ability of MATCH-generated parameters and partial atomic charges to reproduce experimental absolute hydration free energies for a series of 457 small neutral molecules in GBMV2, Generalized Born with a smooth SWitching (GBSW), and fast analytical continuum treatment of solvation (FACTS) implicit solvent models. The quality of hydration free energies associated with small molecule parameters obtained from ParamChem, SwissParam, and Antechamber are compared. Given optimized surface tension coefficients for scaling the surface area term in the nonpolar contribution, these automated parameterization schemes with GBMV2 and GBSW demonstrate reasonable agreement with experimental hydration free energies (average unsigned errors of 0.9-1.5 kcal/mol and R(2) of 0.63-0.87). GBMV2 and GBSW consistently provide slightly more accurate estimates than FACTS, whereas Antechamber parameters yield marginally more accurate estimates than the current generation of MATCH, ParamChem, and SwissParam parameterization strategies. Modeling with MATCH libraries that are derived from different CHARMM topology and parameter files highlights the importance of having sufficient coverage of chemical space within the underlying databases of these automated schemes and the benefit of targeting specific functional groups for parameterization efforts to maximize both the breadth and the depth of the parameterized space.

Absolute Hydration Free Energies of Blocked Amino Acids: Implications for Protein Solvation and Stability

Most proteins perform their function in aqueous solution. The interactions with water determine the stability of proteins and the desolvation costs of ligand binding or membrane insertion. However, because of experimental restrictions, absolute solvation free energies of proteins or amino acids are not available. Instead, solvation free energies are estimated based on side chain analog data. This approach implies that the contributions to free energy differences are additive, and it has often been employed for estimating folding or binding free energies. However, it is not clear how much the additivity assumption affects the reliability of the resulting data. Here, we use molecular dynamics–based free energy simulations to calculate absolute hydration free energies for 15 N-acetyl-methylamide amino acids with neutral side chains. By comparing our results with solvation free energies for side chain analogs, we demonstrate that estimates of solvation free energies of full amino acids based on group-additive methods are systematically too negative and completely overestimate the hydrophobicity of glycine. The largest deviation of additive protocols using side chain analog data was 6.7 kcal/mol; on average, the deviation was 4 kcal/mol. We briefly discuss a simple way to alleviate the errors incurred by using side chain analog data and point out the implications of our findings for the field of biophysics and implicit solvent models. To support our results and conclusions, we calculate relative protein stabilities for selected point mutations, yielding a root-mean-square deviation from experimental results of 0.8 kcal/mol.

Simulation study of ion pairing in concentrated aqueous salt solutions with a polarizable force field

The accuracy of empirical force fields is critical for meaningful molecular dynamics simulations of concentrated ionic solutions. Current models are typically developed on the basis of single ion properties such as the monohydrate energy in the gas phase, or the absolute hydration free energy at infinite dilution. However, the failure of these models to represent accurately the properties of concentrated solutions cannot be excluded. Here, these issues are illustrated for a polarizable potential based on classical Drude oscillators. To model accurately concentrated ionic solutions, the parameters of the potential functions are optimized to reproduce osmotic pressure data. The sodium-chloride potential of mean force in solution calculated from the empirically-adjusted model is consistent with the results from that calculated from ab initio CPMD simulations.