In silico prediction of drug solubility: 2. Free energy of solvation in pure melts

The solubility of drugs in water is investigated in a series of papers and in the current work. The free energy of solvation, DeltaG*(vl), of a drug molecule in its pure drug melt at 673.15 K (400 degrees C) has been obtained for 46 drug molecules using the free energy perturbation method. The simulations were performed in two steps where first the Coulomb and then the Lennard-Jones interactions were scaled down from full to no interaction. The results have been interpreted using a theory assuming that DeltaG*(vl) = DeltaG(cav) + E(LJ) + E(C)/2 where the free energy of cavity formation, DeltaG(cav), in these pure drug systems was obtained using hard body theories, and E(LJ) and E(C) are the Lennard-Jones and Coulomb interaction energies, respectively, of one molecule with the other ones. Since the main parameter in hard body theories is the volume fraction, an equation of state approach was used to estimate the molecular volume. Promising results were obtained using a theory for hard oblates, in which the oblate axial ratio was calculated from the molecular surface area and volume obtained from simulations. The Coulomb term, E(C)/2, is half of the Coulomb energy in accord with linear response, which showed good agreement with our simulation results. In comparison with our previous results on free energy of hydration, the Coulomb interactions in pure drug systems are weaker, and the van der Waals interactions play a more important role.     

Application of a two-length-scale field theory to the solvation of neutral and charged molecules

We develop a continuous self-consistent theory of solute-water interactions that allows determination of the hydrophobic layer around a solute molecule of any geometry, with an explicit account of solvent structure described by its correlation function. We compute the mean solvent density profile n(r) surrounding the solute molecule as well as its solvation free energy deltaG. We compare the two-length-scale field theory to the numerical data of Monte Carlo simulations found in the literature for spherical molecules and discuss the possibility of self-consistent adjustment of the free parameters of the theory. In the framework of this approach, we compute the solvation free energies of alkane molecules and the free energy of interaction of two spheres of radius R separated by the distance D. We describe the general setting of the self-consistent account of electrostatic interactions in the framework of our model where the water is considered not as a continuous medium but as a gas of dipoles. We analyze the limiting cases where the proposed theory coincides with the electrostatics of a continuous medium.     

Solubility of water in a benzene-cyclohexane mixture

The solubility of a water molecule in a binary mixture of nonpolar cyclohexane and quadrupolar benzene is studied with the ab initio method. A novel self-consistent reaction field theory that properly accounts for benzene quadrupole moments in the continuum solvent framework is used to describe the solvation effects of the solvent mixture. The free energy of transfer from pure cyclohexane to the mixture solvent is obtained with the neglect of nonelectrostatic contributions. A reasonable agreement with experiments indicates that the theoretical method presented here provides a promising approach to electronic structure calculations in quadrupolar solvents and their mixtures with nonpolar solvents.     

Evaluating the Free Energies of Solvation and Electronic Structures of Lithium‐Ion Battery Electrolytes

Adaptive biasing force molecular dynamics simulations and density functional theory calculations were performed to understand the interaction of Li⁺ with pure carbonates and ethylene carbonate (EC)‐based binary mixtures. The most favorable Li carbonate cluster configurations obtained from molecular dynamics simulations were subjected to detailed structural and thermochemistry calculations on the basis of the M06‐2X/6‐311++G(d,p) level of theory. We report the ranking of these electrolytes on the basis of the free energies of Li‐ion solvation in carbonates and EC‐based mixtures. A strong local tetrahedral order involving four carbonates around the Li⁺ was seen in the first solvation shell. Thermochemistry calculations revealed that the enthalpy of solvation and the Gibbs free energy of solvation of the Li⁺ ion with carbonates are negative and suggested the ion–carbonate complexation process to be exothermic and spontaneous. Natural bond orbital analysis indicated that Li⁺ interacts with the lone pairs of electrons on the carbonyl oxygen atom in the primary solvation sphere. These interactions lead to an increase in the carbonyl (C=O) bond lengths, as evidenced by a redshift in the vibrational frequencies [ν(C=O)] and a decrease in the electron density values at the C=O bond critical points in the primary solvation sphere. Quantum theory of atoms in molecules, localized molecular orbital energy decomposition analysis (LMO‐EDA), and noncovalent interaction plots revealed the electrostatic nature of the Li⁺ ion interactions with the carbonyl oxygen atoms in these complexes. On the basis of LMO‐EDA, the strongest attractive interaction in these complexes was found to be the electrostatic interaction followed by polarization, dispersion, and exchange interactions. Overall, our calculations predicted EC and a binary mixture of EC/dimethyl carbonate to be appropriate electrolytes for Li‐ion batteries, which complies with experiments and other theoretical results.     

Ionic solvation studied by image-charge reaction field method

In a preceding paper [J. Chem. Phys. 131, 154103 (2009)], we introduced a new, hybrid explicit/implicit method to treat electrostatic interactions in computer simulations, and tested its performance for liquid water. In this paper, we report further tests of this method, termed the image-charge solvation model (ICSM), in simulations of ions solvated in water. We find that our model can faithfully reproduce known solvation properties of sodium and chloride ions. The charging free energy of a single sodium ion is in excellent agreement with the estimates by other electrostatics methods, while offering much lower finite-size errors. Similarly, the potentials of mean force computed for Na-Cl, Na-Na, and Cl-Cl pairs closely reproduce those reported previously. Collectively, our results demonstrate the superior accuracy of the proposed ICSM method for simulations of mixed media.     

Solubility of naproxen in several organic solvents at different temperatures

By using the van’t Hoff and Gibbs equations the thermodynamic functions Gibbs free energy, enthalpy, and entropy of solution, were evaluated from solubility data of naproxen (NAP) determined at several temperatures in octanol, isopropyl myristate, chloroform, and cyclohexane, as pure solvents. The water-saturated organic solvents also were studied except cyclohexane. The excess free energy and the activity coefficients of the solutes, and the mixing and solvation thermodynamic quantities were also determined. The NAP solubilities were higher in chloroform and octanol with respect to those obtained in cyclohexane. In addition, by using literature values for NAP aqueous solubility, the thermodynamic functions relative to transfer of this drug from water to organic solvents were also estimated.     

Redox entropy of plastocyanin: developing a microscopic view of mesoscopic polar solvation

We report applications of analytical formalisms and molecular dynamics (MD) simulations to the calculation of redox entropy of plastocyanin metalloprotein in aqueous solution. The goal of our analysis is to establish critical components of the theory required to describe polar solvation at the mesoscopic scale. The analytical techniques include a microscopic formalism based on structure factors of the solvent dipolar orientations and density and continuum dielectric theories. The microscopic theory employs the atomistic structure of the protein with force-field atomic charges and solvent structure factors obtained from separate MD simulations of the homogeneous solvent. The MD simulations provide linear response solvation free energies and reorganization energies of electron transfer in the temperature range of 280-310 K. We found that continuum models universally underestimate solvation entropies, and a more favorable agreement is reported between the microscopic calculations and MD simulations. The analysis of simulations also suggests that difficulties of extending standard formalisms to protein solvation are related to the inhomogeneous structure of the solvation shell at the protein-water interface combining islands of highly structured water around ionized residues along with partial dewetting of hydrophobic patches. Quantitative theories of electrostatic protein hydration need to incorporate realistic density profile of water at the protein-water interface.     

In silico prediction of drug solubility: 4. Will simple potentials suffice?

In view of the extreme importance of reliable computational prediction of aqueous drug solubility, we have established a Monte Carlo simulation procedure which appears, in principle, to yield reliable solubilities even for complex drug molecules. A theory based on judicious application of linear response and mean field approximations has been found to reproduce the computationally demanding free energy determinations by simulation while at the same time offering mechanistic insight. The focus here is on the suitability of the model of both drug and solvent, i.e., the force fields. The optimized potentials for liquid simulations all atom (OPLS‐AA) force field, either intact or combined with partial charges determined either by semiempirical AM1/CM1A calculations or taken from the condensed‐phase optimized molecular potentials for atomistic simulation studies (COMPASS) force field has been used. The results illustrate the crucial role of the force field in determining drug solubilities. The errors in interaction energies obtained by the simple force fields tested here are still found to be too large for our purpose but if a component of this error is systematic and readily removed by empirical adjustment the results are significantly improved. In fact, consistent use of the OPLS‐AA Lennard‐Jones force field parameters with partial charges from the COMPASS force field will in this way produce good predictions of amorphous drug solubility within 1 day on a standard desktop PC. This is shown here by the results of extensive new simulations for a total of 47 drug molecules which were also improved by increasing the water box in the hydration simulations from 500 to 2000 water molecules. © 2008 Wiley Periodicals, Inc. J Comput Chem, 2009     

Solution Thermodynamics and Preferential Solvation of Meloxicam in Propylene Glycol + Water Mixtures

The equilibrium solubilities of the analgesic drug meloxicam (MEL) in propylene glycol + water mixtures were determined at several temperatures from 293.15 to 313.15 K. The Gibbs energy, enthalpy, and entropy of solution and of mixing were obtained from these solubility data. The solubility was maximal in neat propylene glycol and very low in pure water at all temperatures studied. A nonlinear plot of Δ_soln H° versus Δ_soln G° gave a negative slope from pure water up to 0.80 mass fraction of propylene glycol and a positive slope above this composition up to neat propylene glycol, at the mean temperature 303.15 K. Accordingly, the driving mechanism for MEL solubility in the water-rich mixtures was the entropy, probably due to water-structure loss around nonpolar moieties of the drug, while for the propylene glycol-rich mixtures it was the enthalpy, probably due to better solvation of the drug. The preferential solvation of MEL by the components of the solvent was estimated by means of the inverse Kirkwood-Buff integral method, showing rather small preferential solvation of MEL by propylene glycol at all compositions.     

In silico prediction of drug solubility: 1. Free energy of hydration

As a first step in the computational prediction of drug solubility the free energy of hydration, DeltaG*(vw) in TIP4P water has been computed for a data set of 48 drug molecules using the free energy of perturbation method and the optimized potential for liquid simulations all-atom force field. The simulations were performed in two steps, where first the Coulomb and then the Lennard-Jones interactions between the solute and the water molecules were scaled down from full to zero strength to provide physical understanding and simpler predictive models. The results have been interpreted using a theory assuming DeltaG*(vw) = A(MS)gamma + E(LJ) + E(C)/2 where A(MS) is the molecular surface area, gamma is the water-vapor surface tension, and E(LJ) and E(C) are the solute-water Lennard-Jones and Coulomb interaction energies, respectively. It was found that by a proper definition of the molecular surface area our results as well as several results from the literature were found to be in quantitative agreement using the macroscopic surface tension of TIP4P water. This is in contrast to the surface tension for water around a spherical cavity that previously has been shown to be dependent on the size of the cavity up to a radius of approximately 1 nm. The step of scaling down the electrostatic interaction can be represented by linear response theory.     

Coupled semiempirical quantum mechanics and molecular mechanics (QM/MM) calculations on the aqueous solvation free energies of ionized molecules

The aqueous solvation free energies of ionized molecules were computed using a coupled quantum mechanical and molecular mechanical (QM/MM) model based on the AM1, MNDO, and PM3 semiempirical molecular orbital methods for the solute molecule and the TIP3P molecular mechanics model for liquid water. The present work is an extension of our model for neutral solutes where we assumed that the total free energy is the sum of components derived from the electrostatic/polarization terms in the Hamiltonian plus an empirical “nonpolar” term. The electrostatic/polarization contributions to the solvation free energies were computed using molecular dynamics (MD) simulation and thermodynamic integration techniques, while the nonpolar contributions were taken from the literature. The contribution to the electrostatic/polarization component of the free energy due to nonbonded interactions outside the cutoff radii used in the MD simulations was approximated by a Born solvation term. The experimental free energies were reproduced satisfactorily using variational parameters from the vdW terms as in the original model, in addition to a parameter from the one-electron integral terms. The new one-electron parameter was required to account for the short-range effects of overlapping atomic charge densities. The radial distribution functions obtained from the MD simulations showed the expected H-bonded structures between the ionized solute molecule and solvent molecules. We also obtained satisfactory results by neglecting both the empirical nonpolar term and the electronic polarization of the solute, i.e., by implementing a nonpolarization model. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 1028–1038, 1999     

Accurate estimation of solvation free energy using polynomial fitting techniques

This report details an approach to improve the accuracy of free energy difference estimates using thermodynamic integration data (slope of the free energy with respect to the switching variable λ) and its application to calculating solvation free energy. The central idea is to utilize polynomial fitting schemes to approximate the thermodynamic integration data to improve the accuracy of the free energy difference estimates. Previously, we introduced the use of polynomial regression technique to fit thermodynamic integration data (Shyu and Ytreberg, J Comput Chem, 2009, 30, 2297). In this report we introduce polynomial and spline interpolation techniques. Two systems with analytically solvable relative free energies are used to test the accuracy of the interpolation approach. We also use both interpolation and regression methods to determine a small molecule solvation free energy. Our simulations show that, using such polynomial techniques and nonequidistant λ values, the solvation free energy can be estimated with high accuracy without using soft‐core scaling and separate simulations for Lennard‐Jones and partial charges. The results from our study suggest that these polynomial techniques, especially with use of nonequidistant λ values, improve the accuracy for ΔF estimates without demanding additional simulations. We also provide general guidelines for use of polynomial fitting to estimate free energy. To allow researchers to immediately utilize these methods, free software and documentation is provided via http://www.phys.uidaho.edu/ytreberg/software . © 2010 Wiley Periodicals, Inc. J Comput Chem, 2010     

Solvation free energies of molecules. The most stable anionic tautomers of uracil

Anionic states of nucleic acid bases are suspected to play a role in the radiation damage processes of DNA. Our recent studies suggested that the excess electron attachment to the nucleic acid bases can stabilize some rare tautomers, i.e. imine-enamine tautomers and other tautomers with a proton being transferred from nitrogen sites to carbon sites (with respect to the canonical tautomer). So far, these new anionic tautomers have been characterized by the gas-phase electronic structure calculations and photoelectron spectroscopy experiments. In the current contribution we explore the effect of water solvation on the stability of the new anionic tautomers of uracil. The accurate free energies of solvation are calculated in a two step approach. The major contribution was calculated using the classical free-energy perturbation adiabatic-charging approach, where it is assumed that the solvated molecule has the charge distribution given by the polarizable continuum model. In the second step the free energy of solvation is refined by taking into account the real, average solvent charge distribution. This is done using our accelerated QM/MM simulations, where the QM energy of the solute is calculated in the mean potential averaged over many MD steps. We found that in water solution three of the recently identified anionic tautomers are 6.5-3.6 kcal mol(-1) more stable than the anion of the canonical tautomer.     

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.     

Electronic Structure Insights into the Solvation of Magnesium Ions with Cyclic and Acyclic Carbonates

A computational framework to rank the solvation behavior of Mg²⁺ in carbonates by using molecular dynamics simulations and density functional theory is reported. Based on the binding energies and enthalpies of solvation calculated at the M06‐2X/6‐311++G(d,p) level of theory and the free energies of solvation from ABF‐MD simulations, we find that ethylene carbonate (EC) and the ethylene carbonate:propylene carbonate (EC:PC) binary mixture are the best carbonate solvents for interacting with Mg²⁺. Natural bond orbital and quantum theory of atoms in molecules analyses support the thermochemistry calculations with the highest values of charge transfer, perturbative stabilization energies, electron densities, and Wiberg bond indices being observed in the Mg²⁺(EC) and Mg²⁺(EC:PC) complexes. The plots of the noncovalent interactions indicate that those responsible for the formation of Mg²⁺ carbonate complexes are strong‐to‐weak attractive interactions, depending on the regions that are interacting. Finally, density of state calculations indicate that the interactions between Mg²⁺ and the carbonate solvents affects the HOMO and LUMO states of all carbonate solvents and moves them to more negative energy values.     

Incorporating the excluded solvent volume and surface charges for computing solvation free energy

Gauss's law or Poisson's equation is conventionally used to calculate solvation free energy. However, the near‐solute dielectric polarization from Gauss's law or Poisson's equation differs from that obtained from molecular dynamics (MD) simulations. To mimic the near‐solute dielectric polarization from MD simulations, the first‐shell water was treated as two layers of surface charges, the densities of which are proportional to the electric field at the solvent molecule that is modeled as a hard sphere. The intermediate water was treated as a bulk solvent. An equation describing the solvation free energy of ions using this solvent scheme was derived using the TIP3P water model. © 2013 Wiley Periodicals, Inc.     

Diabatic free energy curves and coordination fluctuations for the aqueous Ag+Ag2+ redox couple: a biased Born-Oppenheimer molecular dynamics investigation

Biased Born-Oppenheimer molecular dynamics simulations are performed to compute redox potential and free energy curves for the redox half reaction Ag(+)-->Ag(2+)+e(-) in aqueous solution. The potential energy surfaces of reactant and product state are linearly coupled and the system transferred from the reduced state to the oxidized state by variation of the coupling parameter from 0 to 1. The redox potential is obtained by thermodynamic integration of the average ionization energy of Ag(+). Diabatic free energy curves of reduced (R) and oxidized (O) states are obtained to good statistical accuracy by reweighting and combining the set of biased distributions of the ionization energy. The diabatic free energy curves of Ag(+) and Ag(2+) are parabolic over a wide range of the reaction coordinate in agreement with the linear response assumption that underlies Marcus theory. However, we observe deviations from parabolic behavior in the equilibrium region of Ag(+) and find different values for the reorganization free energy of R (1.4 eV) and O (0.9 eV). The computed reorganization free energy of Ag(2+) is in good agreement with the experimental estimate of 0.9-1.2 eV obtained from photoelectron spectroscopy. As suggested by our calculations, the moderate deviation from linear response behavior found for Ag(+) is likely related to the highly fluxional solvation shell of this ion, which exhibits water exchange reactions on the picosecond time scale of the present molecular dynamics simulation.