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.     

In silico prediction of drug solubility. 3. Free energy of solvation in pure amorphous matter

The solubility of drugs in water is investigated in a series of papers. In this work, we address the process of bringing a drug molecule from the vapor into a pure drug amorphous phase. This step enables us to actually calculate the solubility of amorphous drugs in water. In our general approach, we, on one hand, perform rigorous free energy simulations using a combination of the free energy perturbation and thermodynamic integration methods. On the other hand, we develop an approximate theory containing parameters that are easily accessible from conventional Monte Carlo simulations, thereby reducing the computation time significantly. In the theory for solvation, we assume that DeltaG* = DeltaGcav + ELJ + EC/2, where the free energy of cavity formation, DeltaGcav, in pure drug systems is obtained using a theory for hard-oblate spheroids, and ELJ and EC are the Lennard-Jones and Coulomb interaction energies between the chosen molecule and the others in the fluid. The theoretical predictions for the free energy of solvation in pure amorphous matter are in good agreement with free energy simulation data for 46 different drug molecules. These results together with our previous studies support our theoretical approach. By using our previous data for the free energy of hydration, we compute the total free energy change of bringing a molecule from the amorphous phase into water. We obtain good agreement between the theory and simulations. It should be noted that to obtain accurate results for the total process, high precision data are needed for the individual subprocesses. Finally, for eight different substances, we compare the experimental amorphous and crystalline solubility in water with the results obtained by the proposed theory with reasonable success.     

Cavitation free energy for organic molecules having various sizes and shapes

Cavitation free energy DeltaG(cav), corresponding to the formation of an excluded volume cavity in water, is calculated for a large set of organic molecules employing the thermodynamic integration procedure, which is realized as the original two-step algorithm for growing the interaction potential between the hard cavity wall and the water molecules. A large variety of solute systems is considered. Their characteristic radii change in the range 3-7 A; spherical cavities with radii 3-6 A are also studied. The interaction between water molecules is described by the four-site nonpolarizable TIP4P model. The diversity of the trial molecular set is provided by using a specially formulated nonspherical criterion classifying the cavity shapes according to their deviation from a sphere. Molecular objects were partly taken from the data base NCI Diversity with the aid of this criterion. The so-computed free energies are approximated by the linear volume dependence DeltaG(cav)V = XiV, where V is the cavity volume. This relation works fairly well until the cavity size becomes very large (the effective radius larger than 7 A). The volume dependence valid for solutes of arbitrary shapes can be included in a calculation of the nonpolar free energy component as required in the implicit water model.     

Computation of hydration free energies of organic solutes with an implicit water model

A new approach for computing hydration free energies DeltaG(solv) of organic solutes is formulated and parameterized. The method combines a conventional PCM (polarizable continuum model) computation for the electrostatic component DeltaG(el) of DeltaG(solv) and a specially detailed algorithm for treating the complementary nonelectrostatic contributions (DeltaG(nel)). The novel features include the following: (a) two different cavities are used for treating DeltaG(el) and DeltaG(nel). For the latter case the cavity is larger and based on thermal atomic radii (i.e., slightly reduced van der Waals radii). (b) The cavitation component of DeltaG(nel) is taken to be proportional to the volume of the large cavity. (c) In the treatment of van der Waals interactions, all solute atoms are counted explicitly. The corresponding interaction energies are computed as integrals over the surface of the larger cavity; they are based on Lennard Jones (LJ) type potentials for individual solute atoms. The weighting coefficients of these LJ terms are considered as fitting parameters. Testing this method on a collection of 278 uncharged organic solutes gave satisfactory results. The average error (RMSD) between calculated and experimental free energy values varies between 0.15 and 0.5 kcal/mol for different classes of solutes. The larger deviations found for the case of oxygen compounds are probably due to a poor approximation of H-bonding in terms of LJ potentials. For the seven compounds with poorest fit to experiment, the error exceeds 1.5 kcal/mol; these outlier points were not included in the parameterization procedure. Several possible origins of these errors are discussed.     

Molecular dynamics simulation of the reaction of hydration of formaldehyde using a potential based on solute-solvent interaction energy components

Molecular dynamics simulations of aqueous solutions at infinite dilution of the reaction of water with formaldehyde, H(2)O + H(2)CO --> H(2)C(OH)(2), were performed using Lennard-Jones 12-6-1 potentials to describe the solute-solvent interactions, and TIP3P to describe the water-water interactions. The Morokuma decomposition scheme of ab initio interaction energies at the SCF level and the dispersion component at MP2 level were used to reproduce the molecular parameters of the solute-water interaction potential. The results show that the functions that use the EX-PL-DIS-ES interaction model to describe the solvation of the reactant and product systems lead to good values of the reaction (DeltaG) and activation (DeltaG(#)) free energy as compared with those from using AMBER-derived parameters, and with the available theoretical and experimental data.     

Soft-core potentials in thermodynamic integration: comparing one- and two-step transformations

Molecular dynamics-based free energy calculations allow the determination of a variety of thermodynamic quantities from computer simulations of small molecules. Thermodynamic integration (TI) calculations can suffer from instabilities during the creation or annihilation of particles. This "singularity" problem can be addressed with "soft-core" potential functions which keep pairwise interaction energies finite for all configurations and provide smooth free energy curves. "One-step" transformations, in which electrostatic and van der Waals forces are simultaneously modified, can be simpler and less expensive than "two-step" transformations in which these properties are changed in separate calculations. Here, we study solvation free energies for molecules of different hydrophobicity using both models. We provide recommended values for the two parameters α(LJ) and β(C) controlling the behavior of the soft-core Lennard-Jones and Coulomb potentials and compare one- and two-step transformations with regard to their suitability for numerical integration. For many types of transformations, the one-step procedure offers a convenient and accurate approach to free energy estimates.     

Computation of the contribution from the cavity effect to protein-ligand binding free energy

We present results of the investigation of the cavity creation/annihilation effect in view of formation of the protein-ligand (PL) complexes. The protein and ligand were considered as rigid structures. The change of the cavity creation/annihilation free energy DeltaG(cav) was calculated for three PL complexes using the thermodynamic integration procedure with the original algorithm for growing the interaction potential between the cavity and the water molecules. The thermodynamic cycle consists of two stages, annihilation of the cavity of the ligand for the unbound state and its creation at the active site of the protein (bound state). It was revealed that for all complexes under investigation, the values of DeltaG(cav) are negative and favorable for binding. The main contribution to DeltaG(cav) appears due to the annihilation of the cavity of the ligand. All computations were made using the parallel version of CAVE code, elaborated in our preceding work.     

Free-energy calculation of structure-H hydrates

A molecular-dynamics (MD) simulation of structure-H hydrates was performed under constant pressure and temperature with 6120 TIP4P water molecules, 900 OPLS-UA methane molecules, and 180 large molecular guest substance (LMGS) molecules. The LMGS molecules were represented in the form of a one-site Lennard-Jones (LJ) model using the LJ parameters sigma and epsilon. In order to clarify the thermodynamic stability of structure-H hydrates, we calculated the free-energy difference, changing on the sigma and epsilon only of the LMGS molecules. In this simulation, stable crystals of structure-H hydrates and a minimum value of DeltaG were obtained at sigma approximately 6.2 A and large values of epsilon. All simulations were performed using the special-purpose computer hardware MDGRAPE-2.     

Lennard-Jones parameters for small diameter carbon nanotubes and water for molecular mechanics simulations from van der Waals density functional calculations

Lennard-Jones (LJ) parameters are derived for classical nonpolarizable force fields for carbon nanotubes (CNTs) and for CNT-water interaction from van der Waals (vdW) enhanced density functional calculations. The new LJ parameters for carbon-carbon interactions are of the same order as those previously used in the literature but differ significantly for CNT-water interactions. This may partially originate from the fact that in addition to pure vdW interactions the polarization and other quantum mechanics effects are embedded into the LJ-potential.     

Solvation of hydrophobes in water and simple liquids

The solvation of nonpolar molecules in water and that in simple liquids are compared and contrasted. First, solvation thermodynamics is reviewed in a way that focuses on how the enthalpy and entropy of solvation depend on the choice of microscopic volume change v in the solvation process--including special choices v being zero (fixed-volume condition) and v being the partial molecular volume of a solute molecule (fixed-pressure condition)--and how the solvation quantities are related with temperature derivatives of the solvation free energy. Second, the solvation free energy and the solvation enthalpy of a Lennard-Jones (LJ) atom in model water are calculated in the parameter space representing the solute size and the strength of the solute-solvent interaction, and the results are compared with those for an LJ atom in the LJ solvent. The solvation diagrams showing domains of different types of solvation in the parameter space are obtained both for the constant-volume condition and for the constant-pressure condition. Similarities between water and the simple liquid are found when the constant-volume solvation is considered while a significant difference manifests itself in the fixed-pressure solvation. The domain of solvation of hydrophobic character in the parameter space is large in the constant-volume solvation both for water and for the simple liquid. When switched to the constant-pressure condition accompanying a microscopic volume change, the hydrophobic domain remains large in water but it becomes significantly small in the simple liquid. The contrasting results are due to the smallness of the thermal pressure coefficient of water at low temperatures.     

Extending the simple weighted density approximation for a hard-sphere fluid to a Lennard-Jones fluid II. Application

A simple weighted density approximation (SWDA) was extended to nonuniform Lennard-Jones fluids by following the spirit of a partitioned density function theory [S. Zhou, Phys. Rev. E 68 (2003) 061201] and mapping the hard-core part onto an effective hard-sphere fluid whose higher order terms beyond the second order of the functional perturbation expansion are treated by the SWDA. The resultant DFT formalism performs well for Lennard-Jones fluids under the influence of diverse external fields. With the present DFT formalism, we investigate in detail the structure and adsorption properties of a low-density LJ gas in a spherical cavity with a wall consisting of hard-sphere or LJ particles. It was found that when the cavity wall exerts an attractive external potential on the LJ particles in the cavity, the excess adsorption decreases as the temperature increases, while when the cavity wall exerts a hard repulsive external potential on the LJ particles in the cavity, the excess adsorption increases as the temperature increases.     

Box size effects are negligible for solvation free energies of neutral solutes

Hydration free energy calculations in explicit solvent have become an integral part of binding free energy calculations and a valuable test of force fields. Most of these simulations follow the conventional norm of keeping edge length of the periodic solvent box larger than twice the Lennard-Jones (LJ) cutoff distance, with the rationale that this should be sufficient to keep the interactions between copies of the solute to a minimum. However, for charged solutes, hydration free energies can exhibit substantial box size-dependence even at typical box sizes. Here, we examine whether similar size-dependence affects hydration of neutral molecules. Thus, we focused on two strongly polar molecules with large dipole moments, where any size-dependence should be most pronounced, and determined how their hydration free energies vary as a function of simulation box size. In addition to testing a variety of simulation box sizes, we also tested two LJ cut-off distances, 0.65 and 1.0 nm. We show from these simulations that the calculated hydration free energy is independent of the box-size as well as the LJ cut-off distance, suggesting that typical hydration free energy calculations of neutral compounds indeed need not be particularly concerned with finite-size effects as long as standard good practices are followed.     

Fundamental measure density functional theory studies on the freezing of binary hard-sphere and Lennard-Jones mixtures

Free energies and correlation functions of liquid and solid hard-sphere (HS) mixtures are calculated using the fundamental measure density functional theory. Using the thermodynamic perturbation theory the free energies of solid and liquid Lennard-Jones (LJ) mixtures are obtained from correlation functions of HS systems within a single theoretical approach. The resulting azeotrope- and spindle-type solid-liquid phase diagrams of HS and LJ binary mixtures are in good agreement with the corresponding ones from computer simulations.     

The free energy of the metastable supersaturated vapor via restricted ensemble simulations. II. Effects of constraints and comparison with molecular dynamics simulations

Extensive restricted canonical ensemble Monte Carlo simulations [D. S. Corti and P. Debenedetti, Chem. Eng. Sci. 49, 2717 (1994)] were performed. Pressure, excess chemical potential, and excess free energy with respect to ideal gas data were obtained at different densities of the supersaturated Lennard-Jones (LJ) vapor at reduced temperatures from 0.7 to 1.0. Among different constraints imposed on the system studied, the one with the local minimum of the excess free energy was taken to be the approximated equilibrium state of the metastable LJ vapor. Also, a comparison of our results with molecular dynamic simulations [A. Linhart et al., J. Chem. Phys. 122, 144506 (2005)] was made.     

Molecular dynamics simulation of potassium along the liquid-vapor coexistence curve

The applicability of pair potential functions to liquid alkali metals is questionable. On the one hand, some recent reports in the literature suggest the validity of two-parameter pair-wise additive Lennard-Jones (LJ) potentials for liquid alkali metals. On the other hand, there are some reports suggesting the inaccuracy of pair potential functions for liquid metals. In this work, we have performed extensive molecular dynamics simulations of vapor-liquid phase equilibria in potassium to check the validity of the proposed LJ potentials and to improve their accuracy by changing the LJ exponents and taking into account the temperaturedependencies of the potential parameters. We have calculated the orthobaric liquid and vapor densities of potassium using LJ (12–6), LJ (8.5–4) and LJ (5–4), effective pair potential energy functions. The results show that using an LJ (8.5–4) potential energy function with temperature-independent parameters, ε and σ, is inadequate to account for the vapor-liquid coexistence properties of potassium. Taking into account the temperature-dependencies of the LJ parameters, ε(T) and σ(T), we obtained the densities of coexisting liquid and vapor potassium in a much better agreement with experimental data. Changing the magnitude of repulsive and attractive contributions to the potential energy function shows that a two-parameter LJ (5–4) potential can well reproduce the densities of liquid and vapor potassium. The results show that LJ (5–4) potential with temperature-dependent parameters produces the densities of liquid and vapor potassium more accurately, compared to the results obtained using LJ (12–6) and LJ (8.5–4) potential energy functions.     

On the long-range corrections to computer simulation results for the Lennard-Jones vapor-liquid interface

The long-range corrections (LRCs) to the configurational energy have been taken into consideration in the Monte Carlo simulation of the vapor-liquid interface for a pure Lennard-Jones (LJ) fluid. The simulated bulk densities agree satisfactorily with those obtained from the Gibbs ensemble method, and the simulated surface tension values agree reasonably well with those reported in the literature for a larger number of molecules and a larger cut-off distance. To compare the influence of the potential forms on the simulation results, a truncated LJ potential, and a shifted and truncated LJ potential have been examined. Although the bulk densities and surface tensions calculated for different model fluids are strongly affected by the LRC, the different potentials essentially lead to similar density values and similar surface tension values when the respective calculated values are compared on the basis of a reduced temperature scale.     

Is there a relation between excess volume and miscibility in binary liquid mixtures?

Molecular dynamics computer simulations of various symmetrical Lennard-Jones (LJ) models are used to elucidate how the excess volume in dense binary liquids is related to the microscopic interactions between the particles. Both fully miscible systems and systems with a liquid-liquid phase separation are considered by varying systematically the parameters of the LJ potentials. The phase diagrams with the critical points of the demixing systems are determined by means of Monte Carlo simulations in the semigrandcanonical ensemble. The different LJ models are investigated by computing Bhatia-Thornton structure factors, enthalpy of mixing, and excess volume. For the demixing systems, the LJ models show a positive enthalpy of mixing while it is negative for the systems without miscibility gap. In contrast to that, the excess volume can be negative and positive for both demixing and fully miscible systems. This behavior is explained in terms of the interplay between the repulsive and attractive terms in the LJ potential. Whereas repulsions dominate the packing of particles as reflected by the number-density structure factor, the chemical ordering and thus the concentration structure factor are strongly affected by attractive interactions, leading to the "anomalies" of the excess volume.     

On the cavitation energy of water

Free-energy-perturbation theory from molecular dynamics calculations has been used to obtain the DeltaG of adjoining cavities' formation in water. The DeltaGs for systems with three, five and seven cavities are compared with that of a single cavity of the same volume, and found to be in good agreement. The conditions under which the analytical formulation of the energy of cavity formation proposed by Pierotti holds are discussed. The data for a single cavity have been tabulated and can lend themselves to a simple numerical implementation in standard quantum chemical packages, which can be used when high accuracy for DeltaG(cav) is required.