Free energy of liquid water from a computer simulation via cell theory

Energetic and Kinetic Competition on the Stability of Pd13 Clusters: Ab Initio Molecular Dynamics Simulations

Material stability is the focus on both experiments and calculations, which includes the energetic stability at the static state and the thermodynamic stability at the kinetic state. To show whether energetics or kinetics dominates on material stability, this study focuses on the Pd13 clusters, because of their observable magnetic moment in experiment. Energetically, the CALYPSO searching method and first-principles calculations find that Pd13(C2) is the ground state at 0 K while the static frequency calculations demonstrate that the icosahedron Pd13(Ih) becomes more favorable on free energy as temperature increases. However, their magnetic moments (8 μB) are not in agreement with the experimental value (<5.2 μB). Kinetically, ab initio molecular dynamics simulations reveal that Pd13(C3v) (6 μB) has supreme isomerization temperature and the other 11 low-lying isomers transform to Pd13(C3v) directly or indirectly, demonstrating that Pd13(C3v) has the maximum probability to be observed in experiment. The magnetic moment difference between experiment (<5.2 μB) and this calculation (6 μB) may be due to the spin multiplicities. Our result suggests that the magnetic moment disparity between theory and experiment (in Pd13 clusters) originates from the kinetic stability.

Energy-entropy multiscale cell correlation method to predict toluene-water log P in the SAMPL9 challenge

The energy-entropy multiscale cell correlation (EE-MCC) method is used to calculate toluene-water log P values of 16 drug molecules in the SAMPL9 physical properties challenge. EE-MCC calculates the free energy, energy and entropy from molecular dynamics (MD) simulations of the water and toluene solutions. Specifically, MCC evaluates entropy by partitioning the system into cells of correlated atoms at multiple length scales and further partitioning the local coordinates into energy wells, yielding vibrational and topographical terms from the energy-well sizes and probabilities. The log P values calculated by EE-MCC using three 200 ns MD simulations have a mean average error of 0.82 and standard error of the mean of 0.97 versus experiment, which is comparable with the best methods entered in SAMPL9. The main contribution to log P is from energy. Less polar drugs have more favourable energies of transfer. The entropy of transfer consists of increased solute vibrational and conformational terms in toluene due to weaker interactions, fewer solute positions in the larger-molecule solvent, reduced water vibrational entropy, negligible change in toluene vibrational entropy, and gains in solvent orientational entropy. The solvent entropy contributions here may be slightly underestimated because software limitations and statistical fluctuations meant that only the first shell could be included while averaged over the whole solution. Nonetheless, such issues will be addressed in future software to offer a general method to calculate entropy directly from MD simulation and to provide molecular understanding or guide system design.

Total free energy analysis of fully hydrated proteins

The total free energy of a hydrated biomolecule and its corresponding decomposition of energy and entropy provides detailed information about regions of thermodynamic stability or instability. The free energies of four hydrated globular proteins with different net charges are calculated from a molecular dynamics simulation, with the energy coming from the system Hamiltonian and entropy using multiscale cell correlation. Water is found to be most stable around anionic residues, intermediate around cationic and polar residues, and least stable near hydrophobic residues, especially when more buried, with stability displaying moderate entropy-enthalpy compensation. Conversely, anionic residues in the proteins are energetically destabilized relative to singly solvated amino acids, while trends for other residues are less clear-cut. Almost all residues lose intraresidue entropy when in the protein, enthalpy changes are negative on average but may be positive or negative, and the resulting overall stability is moderate for some proteins and negligible for others. The free energy of water around single amino acids is found to closely match existing hydrophobicity scales. Regarding the effect of secondary structure, water is slightly more stable around loops, of intermediate stability around β strands and turns, and least stable around helices. An interesting asymmetry observed is that cationic residues stabilize a residue when bonded to its N-terminal side but destabilize it when on the C-terminal side, with a weaker reversed trend for anionic residues.

Spatially Resolved Hydration Thermodynamics in Biomolecular Systems

Water is essential for the structure, dynamics, energetics, and thus the function of biomolecules. It is a formidable challenge to elicit, in microscopic detail, the role of the solvation-related driving forces of biomolecular processes, such as the enthalpy and entropy contributions to the underlying free-energy landscape. In this Perspective, we discuss recent developments and applications of computational methods that provide a spatially resolved map of hydration thermodynamics in biomolecular systems and thus yield atomic-level insights to guide the interpretation of experimental observations. An emphasis is on the challenge of quantifying the hydration entropy, which requires characterization of both the motions of the biomolecules and of the water molecules in their surrounding.

Application of cell models to the melting and sublimation lines of the Lennard-Jones and related potential systems

Harmonic cell models (HCMs) are shown to predict the melting line of the Lennard-Jones (LJ) but not the sublimation line. In addition, even for the melting line, the HCMs are found to be physically unrealistic for inverse power potential systems near the hard-sphere limit, and for the Weeks-Chandler-Andersen system at extremely low temperatures. Despite this, the HCM accurately predicts the LJ mean-square displacement (MSD) from molecular-dynamics (MD) simulations along both lines after simple scaling corrections, to include the effects of anharmonicity and correlated dynamics of the atoms, are applied. Single caged atom molecular dynamics and Monte Carlo simulations provide further quantitative characterization of these additional effects, which go beyond harmonicity. The melting indicator and a modification of the cell model in a similar form are shown to be approximately constant along the melting line, which indicates an isomorph. The less well studied LJ sublimation line is shown not to be an isomorph, yet it still can be represented analytically very accurately by the relationship k_{B}T=aρ^{4}+bρ^{2}, where a and b are constants (k_{B} is Boltzmann's constant, T is the temperature, and ρ is the number density). This relationship has been found previously for the melting line, but the two constants have opposite signs for the sublimation and melting lines. This simple formula is also predicted using a nonharmonic static lattice expression for the pressure. The probability distribution function of the melting factor indicates departures from harmonic or Gaussian behavior in the lower wing. Nevertheless, the mean melting factor is shown to follow a simple MSD Debye-Waller factor dependence along both the melting and sublimation lines. This work combining HCM and MD simulations provides a comparison of the melting and sublimation lines of the LJ system, which could provide the foundations for a more unified statistical mechanical description of these two solid boundary lines.

Energy-entropy method using multiscale cell correlation to calculate binding free energies in the SAMPL8 host-guest challenge

Free energy drives a wide range of molecular processes such as solvation, binding, chemical reactions and conformational change. Given the central importance of binding, a wide range of methods exist to calculate it, whether based on scoring functions, machine-learning, classical or electronic structure methods, alchemy, or explicit evaluation of energy and entropy. Here we present a new energy-entropy (EE) method to calculate the host-guest binding free energy directly from molecular dynamics (MD) simulation. Entropy is evaluated using Multiscale Cell Correlation (MCC) which uses force and torque covariance and contacts at two different length scales. The method is tested on a series of seven host-guest complexes in the SAMPL8 (Statistical Assessment of the Modeling of Proteins and Ligands) "Drugs of Abuse" Blind Challenge. The EE-MCC binding free energies are found to agree with experiment with an average error of 0.9 kcal mol-1. MCC makes clear the origin of the entropy changes, showing that the large loss of positional, orientational, and to a lesser extent conformational entropy of each binding guest is compensated for by a gain in orientational entropy of water released to bulk, combined with smaller decreases in vibrational entropy of the host, guest and contacting water.

Spectrally Resolved Estimation of Water Entropy in the Active Site of Human Carbonic Anhydrase II

A major challenge in understanding ligand binding to biomacromolecules lies in dissecting the underlying thermodynamic driving forces at the atomic level. Quantifying the contributions of water molecules is often especially demanding, although they can play important roles in biomolecular recognition and binding processes. One example is human carbonic anhydrase II, whose active site harbors a conserved network of structural water molecules that are essential for enzymatic catalysis. Inhibitor binding disrupts this water network and changes the hydrogen-bonding patterns in the active site. Here, we use atomistic molecular dynamics simulations to compute the absolute entropy of the individual water molecules confined in the active site of hCAII using a spectrally resolved estimation (SRE) approach. The entropy decrease of water molecules that remain in the active site upon binding of a dorzolamide inhibitor is caused by changes in hydrogen bonding and stiffening of the hydrogen-bonding network. Overall, this entropy decrease is overcompensated by the gain due to the release of three water molecules from the active site upon inhibitor binding. The spectral density calculations enable the assignment of the changes to certain vibrational modes. In addition, the range of applicability of the SRE approximation is systematically explored by exploiting the gradually changing degree of immobilization of water molecules as a function of the distance to a phospholipid bilayer surface, which defines an "entropy ruler". These results demonstrate the applicability of SRE to biomolecular solvation, and we expect it to become a useful method for entropy calculations in biomolecular systems.

Energy-entropy prediction of octanol-water logP of SAMPL7 N-acyl sulfonamide bioisosters

Partition coefficients quantify a molecule's distribution between two immiscible liquid phases. While there are many methods to compute them, there is not yet a method based on the free energy of each system in terms of energy and entropy, where entropy depends on the probability distribution of all quantum states of the system. Here we test a method in this class called Energy Entropy Multiscale Cell Correlation (EE-MCC) for the calculation of octanol-water logP values for 22 N-acyl sulfonamides in the SAMPL7 Physical Properties Challenge (Statistical Assessment of the Modelling of Proteins and Ligands). EE-MCC logP values have a mean error of 1.8 logP units versus experiment and a standard error of the mean of 1.0 logP units for three separate calculations. These errors are primarily due to getting sufficiently converged energies to give accurate differences of large numbers, particularly for the large-molecule solvent octanol. However, this is also an issue for entropy, and approximations in the force field and MCC theory also contribute to the error. Unique to MCC is that it explains the entropy contributions over all the degrees of freedom of all molecules in the system. A gain in orientational entropy of water is the main favourable entropic contribution, supported by small gains in solute vibrational and orientational entropy but offset by unfavourable changes in the orientational entropy of octanol, the vibrational entropy of both solvents, and the positional and conformational entropy of the solute.

Structure and dynamics of an aqueous solution containing poly-(acrylic acid) and non-ionic surfactant octaethylene glycol n-decyl ether (C10E8) aggregates and their complexes investigated by molecular dynamics simulations

A detailed molecular dynamics simulation study of the self-assembly, intermolecular structure and thermodynamic behavior of an aqueous solution of non-ionic surfactant octa ethylene glycol n-decyl ether (C₁₀E₈) in the presence of a non-ionic polar polymer poly(acrylic acid) PAA is presented. The aggregation number N_agg and concentration of surfactant C_s in the simulation systems were varied in the range 0.01-0.32 M and 5 < N_agg < 101 (dilute to concentrated) with a dilute polymer concentration (C_p = 0.01 M). Lamellar aggregates of non-ionic surfactant in bulk aqueous solution are shown by molecular level computations for the first time. Spherical micellar aggregates and lamellar aggregates are formed at low and high N_agg, respectively. The transition from the spherical micelle phase to the lamellar phase in a binary solution is captured for the first time. A conformational transition from coiled to extended PAA chains adsorbed on the surfactant aggregate occurs at a particular value of N_agg, commensurate with the transition from spherical micelle aggregates to anisotropic lamellar aggregates. Formation of the surfactant aggregate in binary and ternary solutions and the polymer-surfactant complex in a ternary solution is enthalpically favored. Adsorption of PAA on the surfactant aggregate surface is driven by hydrogen bonds (HBs) between carboxylic acid groups of PAA and ethylene oxide groups of C₁₀E₈. A significant number of HBs occur between polar oxygens of C₁₀E₈ and hydroxyl oxygens of PAA. The results are in agreement with the limited available experimental data on this system.

Reentrant melting and multiple occupancy crystals of bounded potentials: Simple theory and direct observation by molecular dynamics simulations

Aspects of the phase coexistence behavior of the generalized exponential model (GEM-m) and bounded versions of inverse power potentials based on theory and molecular dynamics (MD) simulation data are reported. The GEM-m potential is ϕ(r)=exp(-r^{m}), where r is the pair separation and m is an adjustable exponent. A simple analytic formula for the fluid-solid envelope of the Gaussian core model which takes account of the known low- and high-density limiting forms is proposed and shown to represent the simulation data well. The bounded inverse power (BIP) potential is ϕ(r)=1/(a^{q}+r^{q})^{n/q}, where a, n, and q are positive constants. The BIP potential multiple occupancy crystal or cluster crystals are predicted to form when q>2 and a>0, for n>3, which compares with the corresponding GEM-m condition of m>2. Reentrant melting should occur for the BIP potential when q≤2 and a>0. MD simulations in which the system was gradually compressed at constant temperature using the BIP potential produced cluster states in the parameter domain expected but it was not possible to establish conclusively whether a multiply occupied crystal or a cluster fluid had formed owing to assembly structural fluctuations. The random phase approximation reproduces very well the BIP MD energy per particle without any discontinuities at the phase boundaries. The Lindemann melting rule is shown analytically to give a more rapidly decaying reentrant melting curve boundary than the so-called melting indicator (MI) empirical melting criterion which has also been investigated in this study. The MI model gives a better match to the high-density phase boundary for small m and q values.

Probing the temperature profile across a liquid-vapor interface upon phase change

Understanding the temperature profile across a liquid-vapor interface in the presence of phase change is essential for the accurate prediction of evaporation, boiling, and condensation. It has been shown experimentally, from non-equilibrium thermodynamics and using molecular dynamics simulations, the existence of an inverted temperature profile across an evaporating liquid-vapor interface, where the vapor-side interface temperature observes the lowest value and the vapor temperature increases away from the interface, opposite to the direction of heat flow. It is worth noting, however, that an inverted temperature profile is not always the case from other experiments and simulations. In this study, we apply non-equilibrium molecular dynamics simulations to systematically study the temperature profile across a liquid-vapor interface during phase change under various heat fluxes in a two-interface setting consisting of both an evaporating and a condensing interface. The calculated vapor temperature shows different characteristics inside the Knudsen layer and in the bulk vapor. In addition, both the direction and magnitude of the vapor temperature gradient, as well as the temperature jump at the liquid-vapor interface, are functions of the applied heat flux. The interfacial entropy generation rate calculated from the vibrational density of state of the interfacial liquid and vapor molecules shows a positive production during evaporation, and the results qualitatively agree with the predictions from non-equilibrium thermodynamics.

Entropy of Proteins Using Multiscale Cell Correlation

A new multiscale method is presented to calculate the entropy of proteins from molecular dynamics simulations. Termed Multiscale Cell Correlation (MCC), the method decomposes the protein into sets of rigid-body units based on their covalent-bond connectivity at three levels of hierarchy: molecule, residue, and united atom. It evaluates the vibrational and topographical entropy from forces, torques, and dihedrals at each level, taking into account correlations between sets of constituent units that together make up a larger unit at the coarser length scale. MCC gives entropies in close agreement with normal-mode analysis and smaller than those using quasiharmonic analysis as well as providing much faster convergence. Moreover, MCC provides an insightful decomposition of entropy at each length scale and for each type of amino acid according to their solvent exposure and whether they are terminal residues. While the residue entropy depends weakly on solvent exposure, there is greater variation in entropy components for larger, more polar amino acids, which have increased conformational entropy but reduced vibrational entropy with greater solvent exposure.

Solubility Advantage of Amorphous Ketoprofen. Thermodynamic and Kinetic Aspects by Molecular Dynamics and Free Energy Approaches

Thermodynamic and kinetic aspects of crystalline (c-KTP) and amorphous (a-KTP) ketoprofen dissolution in water have been investigated by molecular dynamics simulation focusing on free energy properties. Absolute free energies of all relevant species and phases have been determined by thermodynamic integration on a novel path, first connecting the harmonic to the anharmonic system Hamiltonian at low T and then extending the result to the temperature of interest. The free energy required to transfer one ketoprofen molecule from the crystal to the solution is in fair agreement with the experimental value. The absolute free energy of the amorphous form is 19.58 kJ/mol higher than for the crystal, greatly enhancing the ketoprofen concentration in water, although as a metastable species in supersaturated solution. The kinetics of the dissolution process has been analyzed by computing the free energy profile along a reaction coordinate bringing one ketoprofen molecule from the crystal or amorphous phase to the solvated state. This computation confirms that, compared to the crystal form, the dissolution rate is nearly 7 orders of magnitude faster for the amorphous form, providing one further advantage to the latter in terms of bioavailability. The problem of drug solubility, of great practical importance, is used here as a test bed for a refined method to compute absolute free energies, which could be of great interest in biophysics and drug discovery in particular.

Role of Water Hydrogen Bonding on Transport of Small Molecules inside Hydrophobic Channels

We conducted a systematic analysis of water networking inside smooth hyperboloid hydrophobic structures (cylindrical, barrel, and hourglass shapes) to elucidate the role of water hydrogen bonding on the transport of small hydrophobic molecules (ligands). Through a series of molecular dynamics simulations, we established that a hydrogen-bonded network forming along the centerline resulted in a water exclusion zone adjacent to the walls. The size of the exclusion zone is a function of the geometry and the nonbonded interaction strength, defining the effective hydrophobicity of the structure. Exclusion of water molecules from this zone results in lower apparent viscosity, leading to acceleration of ligand transport up to 7 times faster than that measured in the bulk. Transport of ligands into and out of the hydrophobic structures was shown to be controlled by a single water molecule that capped the narrow regions in the structure. This mechanism provides physical insights into the behavior and role of water in the bottleneck regions of hydrophobic enzyme channels. These findings were then used in a sister publication [ Escalante , D. E. , Comput. Struct. Biotechnol. J. 2019 17 757 760 ] to develop a model that can accurately predict the transport of ligands along nanochannels of broad-substrate specificity enzymes.

Entropy of Simulated Liquids Using Multiscale Cell Correlation

Accurately calculating the entropy of liquids is an important goal, given that many processes take place in the liquid phase. Of almost equal importance is understanding the values obtained. However, there are few methods that can calculate the entropy of such systems, and fewer still to make sense of the values obtained. We present our multiscale cell correlation (MCC) method to calculate the entropy of liquids from molecular dynamics simulations. The method uses forces and torques at the molecule and united-atom levels and probability distributions of molecular coordinations and conformations. The main differences with previous work are the consistent treatment of the mean-field cell approximation to the approriate degrees of freedom, the separation of the force and torque covariance matrices, and the inclusion of conformation correlation for molecules with multiple dihedrals. MCC is applied to a broader set of 56 important industrial liquids modeled using the Generalized AMBER Force Field (GAFF) and Optimized Potentials for Liquid Simulations (OPLS) force fields with 1.14*CM1A charges. Unsigned errors versus experimental entropies are 8.7 J K - 1 mol - 1 for GAFF and 9.8 J K - 1 mol - 1 for OPLS. This is significantly better than the 2-Phase Thermodynamics method for the subset of molecules in common, which is the only other method that has been applied to such systems. MCC makes clear why the entropy has the value it does by providing a decomposition in terms of translational and rotational vibrational entropy and topographical entropy at the molecular and united-atom levels.

A simple heuristic approach to estimate the thermochemistry of condensed-phase molecules based on the polarizable continuum model

A simple model based on a quantum chemical approach with polarizable continuum models (PCMs) to provide reasonable translational and rotational entropies for liquid phase molecules was developed.

Binding Thermodynamics and Kinetics Calculations Using Chemical Host and Guest: A Comprehensive Picture of Molecular Recognition

Understanding the fine balance between changes of entropy and enthalpy and the competition between a guest and water molecules in molecular binding is crucial in fundamental studies and practical applications. Experiments provide measurements. However, illustrating the binding/unbinding processes gives a complete picture of molecular recognition not directly available from experiments, and computational methods bridge the gaps. Here, we investigated guest association/dissociation with β-cyclodextrin (β-CD) by using microsecond-time-scale molecular dynamics (MD) simulations, postanalysis and numerical calculations. We computed association and dissociation rate constants, enthalpy, and solvent and solute entropy of binding. All the computed values of kon, koff, ΔH, ΔS, and ΔG using GAFF-CD and q4MD-CD force fields for β-CD could be compared with experimental data directly and agreed reasonably with experiment findings. In addition, our study further interprets experiments. Both force fields resulted in similar computed ΔG from independently computed kinetics rates, ΔG = -RT ln(kon·C0/koff), and thermodynamics properties, ΔG = ΔH - TΔS. The water entropy calculations show that the entropy gain of desolvating water molecules are a major driving force, and both force fields have the same strength of nonpolar attractions between solutes and β-CD as well. Water molecules play a crucial role in guest binding to β-CD. However, collective water/β-CD motions could contribute to different computed kon and ΔH values by different force fields, mainly because the parameters of β-CD provide different motions of β-CD, hydrogen-bond networks of water molecules in the cavity of free β-CD, and strength of desolvation penalty. As a result, q4MD-CD suggests that guest binding is mostly driven by enthalpy, while GAFF-CD shows that gaining entropy is the major driving force of binding. The study deepens our understanding of ligand-receptor recognition and suggests strategies for force field parametrization for accurately modeling molecular systems.

Assessment of Hydration Thermodynamics at Protein Interfaces with Grid Cell Theory

Molecular dynamics simulations have been analyzed with the Grid Cell Theory (GCT) method to spatially resolve the binding enthalpies and entropies of water molecules at the interface of 17 structurally diverse proteins. Correlations between computed energetics and structural descriptors have been sought to facilitate the development of simple models of protein hydration. Little correlation was found between GCT-computed binding enthalpies and continuum electrostatics calculations. A simple count of contacts with functional groups in charged amino acids correlates well with enhanced water stabilization, but the stability of water near hydrophobic and polar residues depends markedly on its coordination environment. The positions of X-ray-resolved water molecules correlate with computed high-density hydration sites, but many unresolved waters are significantly stabilized at the protein surfaces. A defining characteristic of ligand-binding pockets compared to nonbinding pockets was a greater solvent-accessible volume, but average water thermodynamic properties were not distinctive from other interfacial regions. Interfacial water molecules are frequently stabilized by enthalpy and destabilized entropy with respect to bulk, but counter-examples occasionally occur. Overall detailed inspection of the local coordinating environment appears necessary to gauge the thermodynamic stability of water in protein structures.

Macromolecular Entropy Can Be Accurately Computed from Force

A method is presented to evaluate a molecule's entropy from the atomic forces calculated in a molecular dynamics simulation. Specifically, diagonalization of the mass-weighted force covariance matrix produces eigenvalues which in the harmonic approximation can be related to vibrational frequencies. The harmonic oscillator entropies of each vibrational mode may be summed to give the total entropy. The results for a series of hydrocarbons, dialanine and a β hairpin are found to agree much better with values derived from thermodynamic integration than results calculated using quasiharmonic analysis. Forces are found to follow a harmonic distribution more closely than coordinate displacements and better capture the underlying potential energy surface. The method's accuracy, simplicity, and computational similarity to quasiharmonic analysis, requiring as input force trajectories instead of coordinate trajectories, makes it readily applicable to a wide range of problems.

Prediction of Water Binding to Protein Hydration Sites with a Discrete, Semiexplicit Solvent Model

Buried water molecules are ubiquitous in protein structures and are found at the interface of most protein-ligand complexes. Determining their distribution and thermodynamic effect is a challenging yet important task, of great of practical value for the modeling of biomolecular structures and their interactions. In this study, we present a novel method aimed at the prediction of buried water molecules in protein structures and estimation of their binding free energies. It is based on a semiexplicit, discrete solvation model, which we previously introduced in the context of small molecule hydration. The method is applicable to all macromolecular structures described by a standard all-atom force field, and predicts complete solvent distribution within a single run with modest computational cost. We demonstrate that it indicates positions of buried hydration sites, including those filled by more than one water molecule, and accurately differentiates them from sterically accessible to water but void regions. The obtained estimates of water binding free energies are in fair agreement with reference results determined with the double decoupling method.

Evaluation of water displacement energetics in protein binding sites with grid cell theory

The grid cell theory method was used to elucidate perturbations in water network energetics in a range of protein–ligand complexes.

Current and emerging opportunities for molecular simulations in structure-based drug design

An overview of the current capabilities and limitations of molecular simulation of biomolecular complexes in the context of computer-aided drug design is provided. Steady improvements in computer hardware coupled with more refined representations of energetics are leading to a new appreciation of the driving forces of molecular recognition. Molecular simulations are poised to more frequently guide the interpretation of biophysical measurements of biomolecular complexes. Ligand design strategies emerge from detailed analyses of computed structural ensembles. The feasibility of routine applications to ligand optimization problems hinges upon successful extensive large scale validation studies and the development of protocols to intelligently automate computations.

Prediction of Small Molecule Hydration Thermodynamics with Grid Cell Theory

An efficient methodology has been developed to quantify water energetics by analysis of explicit solvent molecular simulations of organic and biomolecular systems. The approach, grid cell theory (GCT), relies on a discretization of the cell theory methodology on a three-dimensional grid to spatially resolve the density, enthalpy, and entropy of water molecules in the vicinity of solute(s) of interest. Entropies of hydration are found to converge more efficiently than enthalpies of hydration. GCT predictions of free energies of hydration on a data set of small molecules are strongly correlated with thermodynamic integration predictions. Agreement with the experiment is comparable for both approaches. A key advantage of GCT is its ability to provide from a single simulation insightful graphical analyses of spatially resolved components of the enthalpies and entropies of hydration.

Assessing the accuracy of inhomogeneous fluid solvation theory in predicting hydration free energies of simple solutes

Accurate prediction of hydration free energies is a key objective of any free energy method that is applied to modeling and understanding interactions in the aqueous phase. Inhomogeneous fluid solvation theory (IFST) is a statistical mechanical method for calculating solvation free energies by quantifying the effect of a solute acting as a perturbation to bulk water. IFST has found wide application in understanding hydration phenomena in biological systems, but quantitative applications have not been comprehensively assessed. In this study, we report the hydration free energies of six simple solutes calculated using IFST and independently using free energy perturbation (FEP). This facilitates a validation of IFST that is independent of the accuracy of the force field. The results demonstrate that IFST shows good agreement with FEP, with an R² coefficient of determination of 0.99 and a mean unsigned difference of 0.7 kcal/mol. However, sampling is a major issue that plagues IFST calculations and the results suggest that a histogram method may require prohibitively long simulations to achieve convergence of the entropies, for bin sizes which effectively capture the underlying probability distributions. Results also highlight the sensitivity of IFST to the reference interaction energy of a water molecule in bulk, with a difference of 0.01 kcal/mol changing the predicted hydration free energies by approximately 2.4 kcal/mol for the systems studied here. One of the major advantages of IFST over perturbation methods such as FEP is that the systems are spatially decomposed to consider the contribution of specific regions to the total solvation free energies. Visualizing these contributions can yield detailed insights into solvation thermodynamics. An insight from this work is the identification and explanation of regions with unfavorable free energy density relative to bulk water. These regions contribute unfavorably to the hydration free energy. Further work is necessary before IFST can be extended to yield accurate predictions of binding free energies, but the work presented here demonstrates its potential.

Entropy from state probabilities: hydration entropy of cations

Entropy is an important energetic quantity determining the progression of chemical processes. We propose a new approach to obtain hydration entropy directly from probability density functions in state space. We demonstrate the validity of our approach for a series of cations in aqueous solution. Extensive validation of simulation results was performed. Our approach does not make prior assumptions about the shape of the potential energy landscape and is capable of calculating accurate hydration entropy values. Sampling times in the low nanosecond range are sufficient for the investigated ionic systems. Although the presented strategy is at the moment limited to systems for which a scalar order parameter can be derived, this is not a principal limitation of the method. The strategy presented is applicable to any chemical system where sufficient sampling of conformational space is accessible, for example, by computer simulations.

Why the solvation water around proteins is more dense than bulk water

The main aim of this work is to propose a rational explanation of the commonly observed phenomenon of increasing water density within solvation shell of proteins. We have observed that the geometry of the water-water hydrogen bond network within solvation layer differs from the one in bulk water, and it is the result of interactions of water molecules with protein surface. Altered geometry of the network reflects changes in the structure of solvation water. Our explanation of the observed changes is based on model proposed by Tanaka (Tanaka, H. J. Chem. Phys. 2000, 112, 799). According to this model, in liquid water exist some special structures formed by water molecules thanks to their unique ability to create the branched network of hydrogen bonds. These structures have two characteristic features: a low potential energy of internal interactions and a large specific volume. We provide some evidence for the supposition that deformation of the geometry of the water-water hydrogen bond network is responsible for destabilization of these structures and therefore for increased local density of water. Our model is constructed on the basis of the analysis of solvation water of some specific protein, the motor head of kinesin. Subsequently, we used it for description of solvation of purely hydrophobic surface. It has been found that in this case an unoccupied space between the hydrophobic surface and neighboring solvation layer exists. It has been found that thickness of this region depends on local geometry of the water-protein interface, and it is a result of maintaining a balance between water-surface interactions and water-water interactions. In our opinion, existence of this space region is one of the main factors that differentiates the hydrophobic hydration from hydration of the native form of kinesin. Its existence also explains why the density is greater for solvation water around the native form of the protein than in the vicinity of the hydrophobic surface.

Correlations in liquid water for the TIP3P-Ewald, TIP4P-2005, TIP5P-Ewald, and SWM4-NDP models

Water is one of the simplest molecules in existence, but also one of the most important in biological and engineered systems. However, understanding the structure and dynamics of liquid water remains a major scientific challenge. Molecular dynamics simulations of liquid water were performed using the water models TIP3P-Ewald, TIP4P-2005, TIP5P-Ewald, and SWM4-NDP to calculate the radial distribution functions (RDFs), the relative angular distributions, and the excess enthalpies, entropies, and free energies. In addition, lower-order approximations to the entropy were considered, identifying the fourth-order approximation as an excellent estimate of the full entropy. The second-order and third-order approximations are ~20% larger and smaller than the true entropy, respectively. All four models perform very well in predicting the radial distribution functions, with the TIP5P-Ewald model providing the best match to the experimental data. The models also perform well in predicting the excess entropy, enthalpy, and free energy of liquid water. The TIP4P-2005 and SWM4-NDP models are more accurate than the TIP3P-Ewald and TIP5P-Ewald models in this respect. However, the relative angular distribution functions of the four water models reveal notable differences. The TIP5P-Ewald model demonstrates an increased preference for water molecules to act both as tetrahedral hydrogen bond donors and acceptors, whereas the SWM4-NDP model demonstrates an increased preference for water molecules to act as planar hydrogen bond acceptors. These differences are not uncovered by analysis of the RDFs or the commonly employed tetrahedral order parameter. However, they are expected to be very important when considering water molecules around solutes and are thus a key consideration in modelling solvent entropy.

The hydrogen bond network structure within the hydration shell around simple osmolytes: urea, tetramethylurea, and trimethylamine-N-oxide, investigated using both a fixed charge and a polarizable water model

Despite numerous experimental and computer simulation studies, a controversy still exists regarding the effect of osmolytes on the structure of surrounding water. There is a question, to what extent some of the contradictory results may arise from differences in potential models used to simulate the system or parameters employed to describe physical properties of the mixture and interpretation of the results. Bearing this in mind, we determine two main aims of this work as follows: description of the water-water hydrogen bond network structure within the solvation layer around solute molecules (urea, trimethylamine-N-oxide, and tetramethylurea), and also comparison of rigid simple point charges (SPC) and polarizable (POL3) models of water. The following quantities have been examined: radial distribution functions of water molecules around the investigated solutes, both local and overall characteristics of the hydrogen bond network structure (using recently elaborated method), along with estimation of the mean energy of a single hydrogen bond, and also the probability distributions which describe the orientation of a single water particle plane relatively to the center of mass of the solute molecule. As an independent method for the evaluation of the degree of changes in local structural ordering, a harmonic approximation has been adopted to estimate the absolute entropy of water. It was found that within the solvation shell of the investigated solutes, the structure of hydrogen bond network changes only slightly comparing to bulk water. Therefore, we conclude that the investigated osmolyte molecules do not disturb significantly the structure of surrounding water. This conclusion was also confirmed by calculations of the absolute entropy of water using a harmonic approximation. In the immediate vicinity of the solutes, we observe that the water-water hydrogen bonds are slightly more stable; they are slightly less distorted and a little shorter than in bulk water. Nevertheless, although this local water structure is more stable and stiffer, our results do not indicate that it is more ordered compared to bulk. Finally, the comparison of both used models of water, the fixed charge and the polarizable, leads to unambiguous conclusion that rigid (SPC) water model may be successfully used in simulations instead of polarizable (POL3), as no significant differences between these two models have been observed.

Topological hydrogen-bond definition to characterize the structure and dynamics of liquid water

A definition that equates a hydrogen bond topologically with a local energy well in the potential energy surface is used to study the structure and dynamics of liquid water. We demonstrate the robustness of this hydrogen-bond definition versus the many other definitions which use fixed, arbitrary parameters, do not account for variable molecular environments, and cannot effectively resolve transition states. Our topology definition unambiguously shows that most water molecules are double acceptors but sizable proportions are single or triple acceptors. Almost all hydrogens are found to take part in hydrogen bonds. Broken hydrogen bonds only form when two molecules try to form two hydrogen bonds between them. The double acceptors have tetrahedral geometry, lower potential energy, entropy, and density, and slower dynamics. The single and triple acceptors have trigonal and trigonal bipyramidal geometry and when considered together have higher density, potential energy, and entropy, faster dynamics, and a tendency to cluster. These calculations use an extended theory for the entropy of liquid water that takes into account the variable number of hydrogen bonds. Hydrogen-bond switching is shown to depend explicitly on the variable number of hydrogen bonds accepted and the presence of interstitial water molecules. Transition state theory indicates that the switching of hydrogen bonds is a mildly activated process, requiring only a moderate distortion of hydrogen bonds. Three main types of switching events are observed depending on whether the donor and acceptor are already sharing a hydrogen bond. The switch may proceed with no intermediate or via a bifurcated-oxygen or cyclic dimer, both of which have a broken hydrogen bond and symmetric and asymmetric forms. Switching is found to be strongly coupled to whole-molecule vibration, particularly for the more mobile single and triple acceptors. Our analysis suggests that even though water is heterogeneous in terms of the number of hydrogen bonds, the coupling between neighbors on various length and time scales brings about greater continuity in its properties.

Thermodynamics of liquids: standard molar entropies and heat capacities of common solvents from 2PT molecular dynamics

We validate here the Two-Phase Thermodynamics (2PT) method for calculating the standard molar entropies and heat capacities of common liquids. In 2PT, the thermodynamics of the system is related to the total density of states (DoS), obtained from the Fourier Transform of the velocity autocorrelation function. For liquids this DoS is partitioned into a diffusional component modeled as diffusion of a hard sphere gas plus a solid component for which the DoS(υ) → 0 as υ→ 0 as for a Debye solid. Thermodynamic observables are obtained by integrating the DoS with the appropriate weighting functions. In the 2PT method, two parameters are extracted from the DoS self-consistently to describe diffusional contributions: the fraction of diffusional modes, f, and DoS(0). This allows 2PT to be applied consistently and without re-parameterization to simulations of arbitrary liquids. We find that the absolute entropy of the liquid can be determined accurately from a single short MD trajectory (20 ps) after the system is equilibrated, making it orders of magnitude more efficient than commonly used perturbation and umbrella sampling methods. Here, we present the predicted standard molar entropies for fifteen common solvents evaluated from molecular dynamics simulations using the AMBER, GAFF, OPLS AA/L and Dreiding II forcefields. Overall, we find that all forcefields lead to good agreement with experimental and previous theoretical values for the entropy and very good agreement in the heat capacities. These results validate 2PT as a robust and efficient method for evaluating the thermodynamics of liquid phase systems. Indeed 2PT might provide a practical scheme to improve the intermolecular terms in forcefields by comparing directly to thermodynamic properties.

Comparison of entropic contributions to binding in a "hydrophilic" versus "hydrophobic" ligand-protein interaction

In the present study we characterize the thermodynamics of binding of histamine to recombinant histamine-binding protein (rRaHBP2), a member of the lipocalin family isolated from the brown-ear tick Rhipicephalus appendiculatus. The binding pocket of this protein contains a number of charged residues, consistent with histamine binding, and is thus a typical example of a "hydrophilic" binder. In contrast, a second member of the lipocalin family, the recombinant major urinary protein (rMUP), binds small hydrophobic ligands, with a similar overall entropy of binding in comparison with rRaHBP2. Having extensively studied ligand binding thermodynamics for rMUP previously, the data we obtained in the present study for HBP enables a comparison of the driving forces for binding between these classically distinct binding processes in terms of entropic contributions from ligand, protein, and solvent. In the case of rRaHBP2, we find favorable entropic contributions to binding from desolvation of the ligand; however, the overall entropy of binding is unfavorable due to a dominant unfavorable contribution arising from the loss of ligand degrees of freedom, together with the sequestration of solvent water molecules into the binding pocket in the complex. This contrasts with binding in rMUP where desolvation of the protein binding pocket makes a minor contribution to the overall entropy of binding given that the pocket is substantially desolvated prior to binding.

Relationship between structure, entropy, and diffusivity in water and water-like liquids

Anomalous behavior of the excess entropy (S(e)) and the associated scaling relationship with diffusivity are compared in liquids with very different underlying interactions but similar water-like anomalies: water (SPC/E and TIP3P models), tetrahedral ionic melts (SiO(2) and BeF(2)), and a fluid with core-softened, two-scale ramp (2SRP) interactions. We demonstrate the presence of an excess entropy anomaly in the two water models. Using length and energy scales appropriate for onset of anomalous behavior, we show the density range of the excess entropy anomaly to be much narrower in water than in ionic melts or the 2SRP fluid. While the reduced diffusivities (D*) conform to the excess-entropy-scaling relation, D* = A exp(alphaS(e)) for all the systems (Rosenfeld, Y. Phys. Rev. A 1977, 15, 2545), the exponential scaling parameter, alpha, shows a small isochore dependence in the case of water. Replacing S(e) by pair correlation-based approximants accentuates the isochore dependence of the diffusivity scaling. Isochores with similar diffusivity-scaling parameters are shown to have the temperature dependence of the corresponding entropic contribution. The relationship between diffusivity, excess entropy, and pair correlation approximants to the excess entropy are very similar in all the tetrahedral liquids.

Solvation theory to provide a molecular interpretation of the hydrophobic entropy loss of noble-gas hydration

An equation for the chemical potential of a dilute aqueous solution of noble gases is derived in terms of energies, force and torque magnitudes, and solute and water coordination numbers, quantities which are all measured from an equilibrium molecular dynamics simulation. Also derived are equations for the Gibbs free energy, enthalpy and entropy of hydration for the Henry's law process, the Ostwald process, and a third proposed process going from an arbitrary concentration in the gas phase to the equivalent mole fraction in aqueous solution which has simpler expressions for the enthalpy and entropy changes. Good agreement with experimental hydration free energies is obtained in the TIP4P and SPC/E water models although the solute's force field appears to affect the enthalpies and entropies obtained. In contrast to other methods, the approach gives a complete breakdown of the entropy for every degree of freedom and makes possible a direct structural interpretation of the well-known entropy loss accompanying the hydrophobic hydration of small non-polar molecules under ambient conditions. The noble-gas solutes experience only a small reduction in their vibrational entropy, with larger solutes experiencing a greater loss. The vibrational and librational entropy components of water actually increase but only marginally, negating any idea of water confinement. The term that contributes the most to the hydrophobic entropy loss is found to be water's orientational term which quantifies the number of orientational minima per water molecule and how many ways the whole hydrogen-bond network can form. These findings help resolve contradictory deductions from experiments that water structure around non-polar solutes is similar to bulk water in some ways but different in others. That the entropy loss lies in water's rotational entropy contrasts with other claims that it largely lies in water's translational entropy, but this apparent discrepancy arises because of different coordinate definitions and reference frames used to define the entropy terms.