Monte Carlo simulations for free energies of hydration: Past to present

Abstract

A summary of the development of Monte Carlo statistical mechanics simulations for the computation of free energies of hydration of organic molecules is followed by presentation of results with the latest version of the optimized potentials for liquid simulations–all atom force field and the TIP4P water model. Scaling of the Lennard-Jones interactions between water, oxygen, and carbon atoms by a factor of 1.25 is found to improve the accuracy of free energies of hydration for 50 prototypical organic molecules from a mean unsigned error of 1.0–1.2 to 0.4 kcal/mol.

I. HISTORICAL BACKGROUND

The measurement and prediction of the solubility of gaseous substances in solvents have been fundamental activities with much practical significance in physical chemistry.^1,2 In order to approach the problem for substance A, the equilibrium constant K_eq for the dissolution process in Eq. (1) must be determined. Experimentally, the results for gases are often expressed in terms of Ostwald coefficients or Henry’s law constants,¹ which can be converted to K_eq and

$$\text{A}\left(\text{gas}\right)\rightleftharpoons \text{A}\left(\text{sol}\right),$$ (1)

for example, to the concentration of A in the solution at the measured temperature and pressure. Computationally, one needs to compute the free energy of transfer, ΔG_sol, of A from the gas phase into the solution, and ΔG_sol = −RT ln K_eq. In the usual case, it is assumed that the transfer is for A in the ideal gas state into an infinitely dilute solution. For realistic treatment, it is necessary to perform statistical mechanics for a system consisting of one molecule of A both in the gas phase and in the presence of a large number of solvent molecules under periodic boundary conditions, as introduced by Metropolis et al. in 1953.^3,4 However, it took more than 30 years before reliable calculations of free energies of hydration for molecular systems began to appear from Metropolis Monte Carlo (MC) and molecular dynamics (MD) simulations.

There were significant challenges to progress, including inadequate computing resources, as well as the needs for simulation software, intermolecular potential functions, and proven free-energy methodology. Until the 1980s, calculations had to be run on “mainframe” computers at computer centers that typically had only a few processors; here, the break was the advent of lower-cost “minicomputers,” such as the DEC VAX-11/780, Harris H80, and Gould 32/8750, and then eventually Unix workstations from Sun Microsystems and Silicon Graphics, Inc.⁵ These advances along with the introduction of the Cray-1 and CDC Cyber 205 supercomputers made MC and MD simulations of organic and biomolecular systems in solution broadly viable by the late 1980s.

By the end of the 1970s, simulations for only a few molecular liquids had been reported, notably the earliest MC and MD simulations of water.^6,7 At the time, the author also noted MC studies of water and methane in water by the groups of Clementi and Beveridge.^8,9 As the author was interested in solvent effects on organic reactions and the influence of the solvent on intermolecular interactions, he decided to pursue MC simulations, initially of pure liquids and then solutions of increasing complexity. Consequently, in 1978, he wrote an MC program (MCLiquid) to model simple molecular liquids with rigid monomers, starting with HF and water.^10,11 As in the development of the MCY water model,⁸ these early studies and the first simulation of liquid methanol¹² used intermolecular potential functions that were obtained from fitting to the results of ab initio quantum mechanical calculations on hundreds of geometries for dimers. Currently, this would be termed “machine learning potentials.” In addition to providing detailed structural information on the liquids, it was found here and in work from Lifson et al. that a “12-6-1” description, i.e., Lennard-Jones plus Coulomb terms, was adequate to represent the intermolecular interactions.^12,13 This then led to the development of the TIPS (transferable intermolecular potential functions) models and the original three-site water model.¹⁴

Liquid simulations up to this point were predominantly carried out in the NVE (MD) or NVT (MC) ensembles with the volume fixed at the experimentally observed density. However, in order to compute both heats of vaporization and densities of liquids, the MCLiquid program was enhanced to perform MC simulations in the NPT ensemble.^15,16 With increasing computational resources, it became possible to perform iterative calculations to refine the force-field parameters, which delivered the SPC water model from MD simulations¹⁷ as well as the TIP3P and TIP4P water models from NPT-MC simulations.¹⁸ There was communication on the progress between Peter Kollman, Martin Karplus, and the author, which led to adoption of the TIP3P model in the AMBER and CHARMM programs.^19,20

The final component, free-energy calculations for molecules in the solution, was slow in developing owing to the computational demands and lack of software. Through the early 1980s, the few free-energy calculations for molecular systems in solution largely used the umbrella sampling method of Patey and Valleau, which had only been introduced in 1975,²¹ and a Lennard-Jones solvent.^22,23 The effect of hydration on the gauche/anti equilibrium for butane was not investigated until 1982.^24,25 Although umbrella sampling was also used to compute the free-energy profile for an S_N2 reaction in water in 1984,²⁶ it is complicated by the need to devise biasing potentials to improve the configurational sampling. A turning point came in 1985 with the finding that Zwanzig’s statistical perturbation theory (SPT)²⁷ could be used to determine, with high precision, the difference in the free energy of hydration for two molecules, ethane and methanol, in TIP4P water.²⁸ The procedure required multiple simulations in “steps” or “windows” using a coupling parameter to interconvert the solutes in the spirit of the thermodynamic integration (TI) method of Kirkwood.²⁹ For the initial SPT calculations, the author had added the necessary code to the MC program that evolved from MCLiquid into the general modeling program BOSS (Biochemical and Organic Simulation System).³⁰ Although the SPT approach met some resistance for the required computer time to run the multiple windows, it soon became widely applied and its name was changed to FEP (free-energy perturbation) theory.^31–33 It may be noted that with only nine citations in 1984 and eight in 1985 to Zwanzig’s paper,²⁷ SPT was not prominent at that time.

Then, to obtain the absolute free energy of solution, ΔG_sol, of a molecule A, it is necessary to compute the difference in free energies for completely removing or “annihilating” A in the gas phase and in the solution. This could be done by converting molecules by FEP or TI to a reference molecule such as methane, which was annihilated. Such calculations for absolute free energies of hydration of small molecules began to appear in the late 1980s,^34–38 and they have evolved to become a regular element of testing in the development of force fields.^39–45

The improvements of force fields for simulations of organic and biomolecular systems have been steady over the past 40 years.^39–49 An important advance was again the introduction of NPT Monte Carlo simulations, which allowed refinement of the non-bonded parameters for diverse molecules through simulations of pure organic liquids.³⁹ Nevertheless, a nagging problem has been that it is difficult to obtain simultaneously excellent results for both the properties of pure organic liquids and for free energies of hydration. For example, with the current version of the OPLS-AA (Optimized Potentials for Liquid Simulations–All Atom) force field, the average errors for densities and heats of vaporization of organic liquids are 0.01 g/cm³ and 0.2 kcal/mol, while the average error for ΔG_hyd is 1.2 kcal/mol.⁴⁴ The problem stems from a systematic overestimation of the hydrophobicity of hydrocarbons. Remedies have been to use a polarizable force field⁴³ or to scale the Lennard-Jones interactions for water oxygen atoms interacting with carbon atoms^45,50–52 or all non-hydrogen atoms.⁵³ In the present work, results are reported for optimization of the latter approach for use with the OPLS-AA force field.

II. COMPUTATIONAL APPROACH

Absolute free energies of hydration were computed with the automated procedure previously described⁴⁴ using BOSS version 5.1.³⁰ The key features are that solute annihilations are carried out using MC/FEP simulations in both the gas phase and in a periodic cube containing 500 TIP4P water molecules¹⁸ in the NPT ensemble at 25 °C and 1 atm. The latest version of the OPLS-AA force field (OPLS/2020) was used to represent the solutes.⁴⁴ The water molecules only translate and rotate, while sampling for all the internal degrees of freedom is also performed for the solutes using a Z-matrix with random changes of bond lengths, bond angles, and dihedral angles. Separate simulations were run to remove the Coulomb and Lennard-Jones interactions using 20 FEP windows in all the cases (Δλ = 0.05). A cutoff distance of 10 Å was applied, such that if any intermolecular heavy atom–heavy atom distance was within the cutoff, then the entire molecule–molecule interaction was included in the potential energy. Interactions were quadratically smoothed to zero over the last 0.5 Å, and standard cutoff corrections were made for the Lennard-Jones interactions neglected beyond the cutoffs. The averaging period for each gas-phase window was 2 × 10⁶ (2 M) configurations. For the aqueous-phase simulations, the charge and Lennard-Jones annihilations used 200 and 500 M configurations per window, respectively. The latter are performed in the creation direction starting from a shrunken state with all the bond lengths at 0.3 Å.⁵⁴ The uncertainties in the computed free-energy results average ± 0.05 kcal/mol for molecules of the present size. Including the equilibration periods, the total number of sampled MC configurations for the aqueous simulations are 2 × 10⁹ and 5 × 10⁹ for the charge and Lennard-Jones annihilations. On a single 3 GHz processor, the required times are ∼1 and 2.5 days. For comparison, the ethane to methanol calculation in 1985 used six windows and 125 TIP4P water molecules with total sampling of 12 M configurations; this required 6 weeks of computing on our only computer, a Harris H80, or the equivalent of about one month on VAX 11/780.^28,31

The OPLS force fields have used the same geometric combining rules for Lennard-Jones parameters as the CFF force field of Lifson et al.,¹³ namely, ε_ij = (ε_iiε_jj)^½ and σ_ij = (σ_iiσ_jj)^½. For the present calculations, scaling the well-depth for the interactions between water oxygen atoms and all carbon atoms, j has been considered such that ε_Oj = κ(ε_OOε_jj)^½. Investigation of alternatives for κ led to the choice of κ = 1.25 as optimal for reducing the average error for the computed free energies of hydration. Best et al. adopted κ = 1.10 with Amber ff03w and TIP4P/2005,⁵³ but they scaled the interactions between water oxygens and all other non-water atoms and considered protein stability, radii of gyration as well as free energies of hydration of side-chain analogs. The results with κ = 1.25 are reported here for OPLS-AA and TIP4P water. Of course, modification of the interactions with water does not affect the results for pure organic liquids.

III. RESULTS AND DISCUSSION

The results for 13 gaseous solutes are first presented in Table I, where experimental Ostwald coefficients have been reported by Wilhelm et al.² The correspondence between Ostwald coefficients and computed free energies of hydration is worth reviewing. An Ostwald coefficient L gives the ratio of the volume of gas dissolved in a volume of solvent, L = V(gas dissolved)/V(solvent). Assuming the gas is ideal and using molar concentrations, the volume of gas dissolved in 1 l of solvent is then L liters and the concentration [A(sol)] = L/V₀ where V₀ is the volume of one mole of ideal gas = RT/P, and the concentration [A(gas)] = 1/V₀. Thus, K_eq for Eq. (1) is the same as L and ΔG_sol = -RT ln L. For example, for methane L = 0.033 95, the solubility S = L/V₀ = L/24.4 = 0.001 39 M, and ΔG_hyd = 2.004 kcal/mol. This result ±0.01 kcal/mol appears in many compendia, which use molar standard states for the gas and solution.^40,55 If alternative standard states are used, corrections are needed.⁵⁶ The MC simulations are performed for a single solute at effectively infinite dilution, so the same concentration units are assumed for the gas and solution, and comparison with the molar ΔG_hyd values is appropriate.

TABLE I.

Ostwald coefficients and free energies of hydration (kcal mol⁻¹) for gases.^a

		ΔGhyd	ΔGhyd	ΔGhyd
Molecule	Ost. Coeff.b	Exptl.c	OPLS/2020d	OPLS4e
Methane	0.033 95	2.004	2.355	2.0
Ethane	0.045 30	1.833	2.211	2.0
Propane	0.036 61	1.960	2.197	2.1
Cyclopropane	0.280 90	0.753	1.220	1.7
Propene	0.180 90	1.013	1.674	1.2
Propyne	1.671 00	−0.304	0.163	−0.7
Butane	0.029 75	2.082	2.222	2.1
2-Methylpropane	0.019 82	2.323	2.068	2.0
2-Methylpropene	0.139 40	1.167	1.735	1.5
1,3-Butadiene	0.352 30	0.618	0.848	0.7
Neopentane	0.014 58	2.505	2.321	1.9
Ammonia	312.7	−3.404	−2.928	−2.9
CO2	0.828 00	0.112	−0.334
Mean unsigned error			0.374	0.31

^a For transfer from the ideal gas phase to aqueous solution with M standard states at 25 °C and 1 atm.

^b Reference 2.

^c Computed from the Ostwald coefficient.

^d Computed by FEP in TIP4P water; the uncertainties are ca. ±0.05 kcal/mol.

^e Reference 45.

The computed results presented in Table I agree well with the experimental data, giving a mean unsigned error (mue) of 0.374 kcal/mol. The testing was expanded to include the 37 more diverse and larger molecules presented in Table II. The mean unsigned error for these molecules is similar at 0.437 kcal/mol. The mue for the 50 molecules is 0.421 kcal/mol and the rmse is 0.544 kcal/mol; the results are shown in Fig. 1. There are only four molecules with errors greater than 1.0 kcal/mol, specifically, 1.1, 1.1, 1.4, and 1.6 kcal/mol, for 2-ethoxyethanol, benzyl alcohol, tetrahydrofuran, and 1,2-ethanediol, respectively. The simple scaling of the well-depth for the water oxygen–carbon atom Lennard-Jones interactions provides striking improvement over the mue values of 1.0 and 1.2 kcal/mol from the unscaled (κ = 1.0) calculations with OPLS-AA for similar collections of molecules.^44,57 Notably, the mean signed error has been reduced from 1.0 to 1.2 kcal/mol (too hydrophobic) to only 0.10 kcal/mol. Thus, the systematic problem is overcome by this simple deviation from the geometric combining rules. The present results also reflect greater accuracy than the rmse of 1.46 kcal/mol for the 14 neutral, side-chain analogs considered by Best et al. using the scaled Amber ff03w and TIP4P/2005.⁵³ In addition, there are reported results with OPLS4 for 45 of the molecules presented in Tables I and II;⁴⁵ the mue of 0.42 kcal/mol for the 45 molecules with OPLS4 is similar to the present results.

TABLE II.

Additional computed and experimental free energies of hydration.^a

	ΔGhyd	ΔGhyd	ΔGhyd	ΔGhyd	ΔGhyd
Molecule	Exptl.b	OPLS/2020c	Coul.d	L-Je	OPLS4f
Pentane	2.32	2.396	−0.038	2.434	2.0
Isopentane	2.38	2.191	−0.012	2.203	2.1
Hexane	2.48	2.384	−0.076	2.460	2.3
Cyclohexane	1.23	1.273	0.019	1.254	1.3
Octane	2.88	2.325	−0.140	2.465	2.5
2-Methyl-2-butene	1.31	1.680	−0.248	1.928
Cyclohexene	0.37	0.528	−0.392	0.920	0.6
Isoprene	0.68	0.961	−0.745	1.706	0.8
1,5-Hexadiene	1.01	1.288	−0.643	1.931
Benzene	−0.86	−1.032	−1.744	0.712	−1.1
Ethylbenzene	−0.79	−1.698	−2.700	1.002	−1.0
Naphthalene	−2.40	−3.142	−2.891	−0.251	−2.5
Chlorobenzeneg	−1.12	−0.906	−1.459	0.553	−1.3
Cyanobenzene	−4.10	−4.226	−3.945	−0.281	−4.7
Nitrobenzene	−4.12	−3.841	−3.861	0.020	−4.6
Aniline	−5.49	−5.635	−6.268	0.633	−4.7
Anisole	−2.45	−2.111	−2.816	0.705	−2.6
Phenol	−6.61	−6.177	−6.665	0.488	−6.3
Benzyl alcohol	−6.62	−7.706	−8.446	0.740	−5.3
Methanol	−5.10	−4.920	−6.701	1.781	−4.3
Ethanol	−5.00	−5.114	−7.028	1.914	−4.2
Butanol	−4.72	−5.130	−7.186	2.056	−4.7
Hexanol	−4.40	−5.344	−7.462	2.118	−4.4
2-Methyl-2-propanol	−4.47	−4.176	−6.173	1.997	−4.5
Ethanethiolg	−1.14	−0.811	−2.406	1.595	−1.1
Diethyl sulfideg	−1.61	−1.601	−3.396	1.795	−1.0
1,2-Ethanediol	−9.30	−10.879	−12.542	1.663	−9.6
2-Ethoxyethanol	−6.69	−5.625	−7.852	2.227
Diethyl ether	−1.59	−1.157	−3.436	2.279	−0.5
Tetrahydrofuran	−3.47	−2.041	−3.335	1.294	−2.7
1,2-Dimethoxyethane	−4.84	−4.103	−6.436	2.333	−5.3
Ethylamine	−4.51	−3.648	−5.881	2.233	−3.2
Butylamine	−4.30	−4.017	−6.309	2.292	−3.0
Diethylamine	−4.07	−3.507	−5.782	2.275	−2.6
Triethylamine	−3.04	−2.759	−5.056	2.297	−3.3
N-Methylacetamide	−10.00	−9.964	−10.612	0.648	−10.0
Water (TIP4P)	−6.32	−6.190	−8.500	2.310
Mean unsigned error		0.437			0.46

^a For transfer from the ideal gas phase to an aqueous solution with M standard states at 25 °C and 1 atm.

^b Reference 40.

^c Computed by FEP; uncertainties are ca. ±0.05 kcal/mol.

^d Free-energy change for introducing the Coulombic interactions.

^e Free-energy change for introducing the Lennard-Jones interactions.

^f Reference 45.

^g Includes the X-sites on chlorine and sulfur.

FIG. 1.

Comparison of experimental and computed free energies of hydration for the 50 solutes in kcal/mol. The line is the best linear fit and has r² = 0.9761.

Looking at the results in more detail, the treatment of alkanes is good with the computed values for methane through octane and isomers of 2.1–2.4 kcal/mol consistent with the observed narrow range of 1.8–2.9 kcal/mol. Thus, the enthalpy–entropy balance with increasing size appears to be well reproduced.⁵⁷ The computed results for the unsaturated hydrocarbons, propene, propyne, and 1,3-butadiene are all a little high by 0.2–0.5 kcal/mol (Table I), which could likely be remedied by a small increase in charge separation for unsaturated C–H bonds.⁴⁴ For the n-alcohols presented in Table II, the computed results for methanol and ethanol are within 0.2 kcal/mol of the experimental values; however, there is a drift toward too favorable ΔG_hyd on progressing to butanol and hexanol. In addition, the computed results for the nine aromatic compounds, benzene through phenol, presented in Table II show errors of less than 0.5 kcal/mol, except for ethylbenzene and naphthalene for which the errors are 0.9 and 0.7 kcal/mol. The origin of the larger error, 1.1 kcal/mol, for benzyl alcohol is unclear as the combined error for its components, benzene and methanol, is small.

There is also a larger error for tetrahydrofuran (THF), 1.4 kcal/mol. Similarly, the other ethers (anisole, ethyl ether, and 1,2-dimethoxyethane) are found to be too hydrophobic by 0.3–0.7 kcal/mol. As suggested previously, it is expected that the description of ether oxygen atoms by the force field would benefit from the inclusion of lone-pair or “X” sites as in ST2 and TIP5P water.^7,58 In their absence, ether oxygen atoms are likely too shielded by the adjacent alkyl groups. With OPLS4, ethyl ether and THF are also too hydrophobic; the results are better for 1,2-dimethoxyethane. Obtaining accurate results for both small cyclic and acyclic ethers with the CHARMM Drude force field was also challenging but benefited from using both lone-pair sites on oxygen and atom-pair-specific Lennard-Jones parameters through NBFIX terms.⁵⁹

It should be noted that both OPLS/2020 and OPLS4 use “X sites” to improve the electrostatics for chlorine, bromine, and iodine on aromatic rings and for sulfur atoms.^60,61 For chlorobenzene, ethane thiol, and diethyl sulfide, the computed free energies of hydration are improved by 0.2, 0.6, and 1.1 kcal/mol, respectively, by addition of the X-sites with OPLS/2020. In each case, there is greater hydrophilicity with the X sites, with maintenance of accurate descriptions of pure liquids.

The OPLS/2020 results are good for alkylamines and aniline, while they are too hydrophobic with OPLS4. The need for an X site is not indicated in these cases, although it is beneficial for pyridine-type nitrogen atoms.⁴⁵ Again, the accuracies for free energies of hydration with OPLS-AA and OPLS4 are similar, and they are near the level where improvement is limited by the accuracy of the experimental data.

In Table II, the computed free-energy changes for the introduction of the Coulomb and Lennard-Jones interactions are also separately listed. The latter term includes the unfavorable creation of the cavity in water and the favorable 1/r⁶ van der Waals or dispersion interactions. For alkanes, aliphatic alcohols, and other saturated molecules, the Lennard-Jones term is relatively steady at 1.9–2.5 kcal/mol irrespective of the size of the molecule, indicating a rough cancellation of the cavity and dispersion terms with increasing size. Thus, the variations in ΔG_hyd arise primarily from the differences in the electrostatic interactions between the solutes and water. An interesting point is that the Lennard-Jones component is significantly less unfavorable for the unsaturated and aromatic molecules; it is actually attractive for naphthalene and cyanobenzene. This effect must come from the dispersive term between the water oxygen atoms and unsaturated atoms, including benzenoid carbons being more favorable on average, since water molecules can approach more closely to the π-faces of unsaturated atoms. The electrostatic interactions between water and benzene rings are also favorable by about 2 kcal/mol owing to water-π-type hydrogen bonds.^62,63 An image of the hydration of cyanobenzene is shown in Fig. 2, in which the two water molecules participating in the π-type and conventional hydrogen bonds are highlighted.

FIG. 2.

Rendering from the MC simulation of cyanobenzene in water. Only water molecules within 3.5 Å of the solute are shown. The water molecules in the π-type and conventional hydrogen bonds are enlarged.

Given the small ranges for the Lennard-Jones components of the free energies of hydration, an algorithm for their prediction based on surface areas and atom types could likely be devised. This could potentially remove the need for the most demanding part of the calculations.

IV. CONCLUSION

The computation of free energies of hydration using Metropolis Monte Carlo simulations³ with non-polarizable force fields and explicit representation of solvent molecules has matured over the past 40 years. Attainment of the current state of both high precision and accuracy required persistent improvements in computer resources, software and algorithms, force fields, and free-energy methodology. Our group’s contributions in this regard have included recognition of the power of personal computers,⁶⁴ the BOSS MC program,³⁰ the TIPnP water models,¹⁸ the OPLS force fields,^39,44 and the earliest and numerous FEP calculations for molecular systems.^28,31–33 The advances have been much enabled by the pioneering work of Metropolis et al.³ The present report represents the culmination of the efforts for free energies of hydration, with the final gains in accuracy provided by abandonment of the conventional combining rules for Lennard-Jones interactions between water molecules and carbon atoms.

AUTHOR DECLARATIONS

Conflict of Interest

The author has no conflicts to disclose.

Author Contributions

William L. Jorgensen: Conceptualization (lead); Data curation (lead); Funding acquisition (lead); Methodology (lead); Project administration (lead); Software (lead); Writing – original draft (lead); Writing – review & editing (lead).

DATA AVAILABILITY

The BOSS program and OPLS-AA force-field parameters are available for download from http://zarbi.chem.yale.edu. The software performs Monte Carlo simulations of pure liquids and solutions, QM/MM, and free-energy perturbation (FEP) calculations. It includes sample input for performing the automated calculations of free energies of hydration.