On the mechanism of hydrophobic association of nanoscopic solutes

The hydration behavior of two planar nanoscopic hydrophobic solutes in liquid water at normal temperature and pressure is investigated by calculating the potential of mean force between them at constant pressure as a function of the solute-solvent interaction potential. The importance of the effect of weak attractive interactions between the solute atoms and the solvent on the hydration behavior is clearly demonstrated. We focus on the underlying mechanism behind the contrasting results obtained in various recent experimental and computational studies on water near hydrophobic solutes. The length scale where crossover from a solvent separated state to the contact pair state occurs is shown to depend on the solute sizes as well as on details of the solute-solvent interaction. We find the mechanism for attractive mean forces between the plates is very different depending on the nature of the solute-solvent interaction which has implications for the mechanism of the hydrophobic effect for biomolecules.     

Water properties and potential of mean force for hydrophobic interactions of methane and nanoscopic pockets studied by computer simulations

We consider model systems consisting of a methane molecule and hemispherical pockets of subnanometer radii whose walls are made of hydrophobic material. The potential of mean force for process of translocation of the methane molecule from bulk water into the pockets' interior is obtained, based on an explicit solvent molecular dynamics simulations. Accompanying changes in water density around the interacting objects and spatial distribution of solvent's potential energy are analyzed, allowing for interpretation of details of hydrophobic interactions in relation to hydrophobic hydration properties. Applicability of surface area-based models of hydrophobic effect for systems of interest is also investigated. A total work for the translocation process is not dependent on pocket's size, indicating that pocket desolvation has little contribution to free energy changes, which is consistent with the observation that solvent density is significantly reduced inside "unperturbed" pockets. Substantial solvent effects are shown to have a longer range than in case of a well investigated methane pair. A desolvation barrier is present in a smaller pocket system but disappears in the larger one, suggesting that a form of a "hydrophobic collapse" is observed.     

Enthalpy-entropy contributions to salt and osmolyte effects on molecular-scale hydrophobic hydration and interactions

Salts and additives can significantly affect the strength of water-mediated interactions in solution. We present results from molecular dynamics simulations focused on the thermodynamics of hydrophobic hydration, association, and the folding-unfolding of a hydrophobic polymer in water and in aqueous solutions of NaCl and of an osmolyte trimethylamine oxide (TMAO). It is known that addition of NaCl makes the hydration of hydrophobic solutes unfavorable and, correspondingly, strengthens their association at the pair as well as the many-body level (Ghosh, T.; Kalra, A.; Garde, S. J. Phys. Chem. B 2005, 109, 642), whereas the osmolyte TMAO has an almost negligible effect on the hydrophobic hydration and association (Athawale, M. V.; Dordick, J. S.; Garde, S. Biophys. J. 2005, 89, 858). Whether these effects are enthalpic or entropic in origin is not fully known. Here we perform temperature-dependent simulations to resolve the free energy into entropy and enthalpy contributions. We find that in TMAO solutions, there is an almost precise entropy-enthalpy compensation leading to the negligible effect of TMAO on hydrophobic phenomena. In contrast, in NaCl solutions, changes in enthalpy dominate, making the salt-induced strengthening of hydrophobic interactions enthalpic in origin. The resolution of total enthalpy into solute-solvent and solvent-solvent terms further shows that enthalpy changes originate primarily from solvent-solvent energy terms. Our results are consistent with experimental data on the hydration of small hydrophobic solutes by Ben-Naim and Yaacobi (Ben-Naim, A.; Yaacobi, M. J. Phys. Chem. 1974, 78, 170). In combination with recent work by Zangi, Hagen, and Berne (Zangi, R.; Hagen, M.; Berne, B. J. J. Am. Chem. Soc. 2007, 129, 4678) and the experimental data on surface tensions of salt solutions by Matubayasi et al. (Matubayasi, N.; Matsuo, H.; Yamamoto, K.; Yamaguchi, S.; Matuzawa, A. J. Colloid Interface Sci. 1999, 209, 398), our results highlight interesting length scale dependences of salt effects on hydrophobic phenomena. Although NaCl strengthens hydrophobic interactions at both small and large length scales, that effect is enthalpy-dominated at small length scales and entropy-dominated for large solutes and interfaces. Our results have implications for understanding of additive effects on water-mediated interactions, as well as on biocompatibility of osmolyte molecules in aqueous solutions.     

Potential of mean force between hydrophobic solutes in the Jagla model of water and implications for cold denaturation of proteins

Using the Jagla model potential we calculate the potential of mean force (PMF) between hard sphere solutes immersed in a liquid displaying water-like properties. Consistent estimates of the PMF are obtained by (a) umbrella sampling, (b) calculating the work done by the mean force acting on the hard spheres as a function of their separation, and (c) determining the position dependent chemical potential after calculating the void space in the liquid. We calculate the PMF for an isobar along which cold denaturation of a model protein has previously been reported. We find that the PMF at contact varies non-monotonically, which is consistent with the observed cold denaturation. The Henry constant also varies non-monotonically with temperature. We find, on the other hand, that a second (solvent separated) minimum of the PMF becomes deeper as temperature decreases. We calculate the solvent-solvent pair correlation functions for solvents near the solute and in the bulk, and show that, as temperature decreases, the two pair correlation functions become indistinguishable, suggesting that the perturbation of solvent structure by the solute diminishes as temperature decreases. The solvent-solute pair correlation function at contact grows as the temperature decreases. We calculate the cavity correlation function and show the development of a solvent-separated peak upon decrease of temperature. These observations together suggest that cold denaturation occurs when the solvent penetrates between hydrophobic solutes in configurations with favorable free energy. Our results thus suggest that cold denatured proteins are structured and that cold denaturation arises from strong solvent-solute interactions, rather than from entropic considerations as in heat denaturation.     

Orientational dynamics of water trapped between two nanoscopic hydrophobic solutes: A molecular dynamics simulation study

We investigate thoroughly the effect of confinement and solute topology on the orientational dynamics of water molecule in the interplate region between two nanoscopic hydrophobic paraffinlike plates. Results are obtained from molecular dynamics simulations of aqueous solutions of paraffinlike plates in the isothermal-isobaric ensemble. An analysis of survival time auto correlation function shows that the residence time of the water molecule in the confined region between two model nanoscopic hydrophobic plates depends on solute surface topology (intermolecular distance within the paraffinlike plate). As expected, the extent of confinement also changes the residence time of water molecules considerably. Orientational dynamics was analyzed along three different directions, viz., dipole moment, HH, and perpendicular to molecular plane vectors. It has been demonstrated that the rotational dynamics of the confined water does not follow the Debye rotational diffusion model, and surface topology of the solute plate and the extent of confinement have considerable effect on the rotational dynamics of the confined water molecules.     

Enthalpy-entropy compensation in the effects of urea on hydrophobic interactions

By comparison of neopentane pair potentials of mean force (PMFs) in room temperature water and 6.9 molar aqueous urea, it was recently shown that urea molecules affect the PMF minima in an unexpected way (Lee, M.-E.; van der Vegt, N. F. A. J. Am. Chem. Soc. 2006, 128, 4948). While the first PMF minimum in urea solution has an identical shape and depth to those of the corresponding minimum in water, the second minimum in urea solution is broader, deeper, and shifted out to a slightly larger distance. Here, we present a study of the enthalpic and entropic contributions to these PMFs. Its significance for understanding the driving forces responsible for thermodynamically favorable neopentane contact and solvent-separated distances in urea solution is discussed. We propose that the solute-solvent entropy and solute-solvent enthalpy changes should be analyzed for obtaining an unambiguous molecular-scale picture. In urea solution, enthalpy-entropy compensation effects associated with structural solvent reorganization processes are large, causing changes of the system's enthalpy and entropy with hydrophobic pair separation to be very different from the solute-solvent enthalpy and entropy changes. The entropies are discussed in terms of the molecular-scale solvent reorganization processes.     

Binless estimation of the potential of mean force

For a system in thermal equilibrium, described by classical statistical mechanics, we derive an unbiased estimator for the marginal probability distribution of a coordinate of interest, rho( x). This result provides a "binless" method for estimating the potential of mean force, Phi = -beta (-1) ln rho, eliminating the need to construct histograms or perform numerical thermodynamic integration. In our method, the distribution that we seek to compute is expressed as the sum of a reference distribution, rho 0(x)essentially an initial guess or estimate of rho( x)and a correction term. While the method is valid for arbitrary rho 0, we speculate that an accurate choice of the reference distribution improves the convergence of the method. Using a model molecule, simulated both in vacuum and in solvent, we validate our proposed approach and compare its performance with the histogram and thermodynamic integration methods. We also discuss and validate an extension in which our approach is used in combination with a biasing force, meant to improve uniform sampling of the coordinate of interest.     

Hydrophobic interactions between methane and a nanoscopic pocket: three dimensional distribution of potential of mean force revealed by computer simulations

We consider a model system of methane molecule and a hemispherical, hydrophobic pocket of an 8 A radius, remaining together in aqueous environment. A spatial map of potential of mean force acting on methane molecule due to presence of pocket is constructed, based on a series of explicit solvent molecular dynamics simulations. A relation between free energy variations associated with methane translocations and accompanying changes in solvent density distribution is analyzed. A funnel-like area where free energy is diminished with respect to bulk is present over the pocket entrance and extends up to 9 A toward the bulk solvent. In order to get into the pocket, however, methane has to cross a free energy barrier, which is more prominent around the circumferential part of pocket entrance, while achieving bulklike free energy values at the very center. As a methane molecule crosses this barrier, the pocket gets completely dehydrated, which leads to "hydrophobic collapse," manifested by a sharp decrease in free energy. We find that the observed free energy changes are closely related to interactions between the methane hydration shell and the surrounding solvent. Results presented here are a continuation of our previous studies of methane-pocket systems.     

A comparison of methods to compute the potential of mean force.

Most processes occurring in a system are determined by the relative free energy between two or more states because the free energy is a measure of the probability of finding the system in a given state. When the two states of interest are connected by a pathway, usually called reaction coordinate, along which the free-energy profile is determined, this profile or potential of mean force (PMF) will also yield the relative free energy of the two states. Twelve different methods to compute a PMF are reviewed and compared, with regard to their precision, for a system consisting of a pair of methane molecules in aqueous solution. We analyze all combinations of the type of sampling (unbiased, umbrella-biased or constraint-biased), how to compute free energies (from density of states or force averaging) and the type of coordinate system (internal or Cartesian) used for the PMF degree of freedom. The method of choice is constraint-bias simulation combined with force averaging for either an internal or a Cartesian PMF degree of freedom.     

Orientation-dependent potential of mean force for protein folding

We present a solvent-implicit minimalistic model potential among the amino acid residues of proteins, obtained by using the known native structures [deposited in the Protein Data Bank (PDB)]. In this model, the amino acid side chains are represented by a single ellipsoidal site, defined by the group of atoms about the center of mass of the side chain. These ellipsoidal sites interact with other sites through an orientation-dependent interaction potential which we construct in the following fashion. First, the site-site potential of mean force (PMF) between heavy atoms is calculated [following F. Melo and E. Feytsman, J. Mol. Biol. 267, 207 (1997)] from statistics of their distance separation obtained from crystal structures. These site-site potentials are then used to calculate the distance and the orientation-dependent potential between side chains of all the amino acid residues (AAR). The distance and orientation dependencies show several interesting results. For example, we find that the PMF between two hydrophobic AARs, such as phenylalanine, is strongly attractive at short distances (after the obvious repulsive region at very short separation) and is characterized by a deep minimum, for specific orientations. For the interaction between two hydrophilic AARs, such a deep minimum is absent and in addition, the potential interestingly reveals the combined effect of polar (charge) and hydrophobic interactions among some of these AARs. The effectiveness of our potential has been tested by calculating the Z-scores for a large set of proteins. The calculated Z-scores show high negative values for most of them, signifying the success of the potential to identify the native structure from among a large number of its decoy states.     

Potential of mean force of hydrophobic association: dependence on solute size

The potentials of mean force (PMFs) were determined for systems involving formation of nonpolar dimers composed of methane, ethane, propane, isobutane, and neopentane, respectively, in water, using the TIP3P water model, and in vacuo. A series of umbrella-sampling molecular dynamics simulations with the AMBER force field was carried out for each pair in either water or in vacuo. The PMFs were calculated by using the weighted histogram analysis method (WHAM). The shape of the PMFs for dimers of all five nonpolar molecules is characteristic of hydrophobic interactions with contact and solvent-separated minima and desolvation maxima. The positions of all these minima and maxima change with the size of the nonpolar molecule, that is, for larger molecules they shift toward larger distances. The PMF of the neopentane dimer is similar to those of other small nonpolar molecules studied in this work, and hence the neopentane dimer is too small to be treated as a nanoscale hydrophobic object. The solvent contribution to the PMF was also computed by subtracting the PMF determined in vacuo from the PMF in explicit solvent. The molecular surface area model correctly describes the solvent contribution to the PMF together with the changes of the height and positions of the desolvation barrier for all dimers investigated. The water molecules in the first solvation sphere of the dimer are more ordered compared to bulk water, with their dipole moments pointing away from the surface of the dimer. The average number of hydrogen bonds per water molecule in this first hydration shell is smaller compared to that in bulk water, which can be explained by coordination of water molecules to the hydrocarbon surface. In the second hydration shell, the average number of hydrogen bonds is greater compared to bulk water, which can be explained by increased ordering of water from the first hydration shell; the net effect is more efficient hydrogen bonding between the water molecules in the first and second hydration shells.     

Hydrophobicity within the three-dimensional Mercedes-Benz model: potential of mean force

We use the three-dimensional Mercedes-Benz model for water and Monte Carlo simulations to study the structure and thermodynamics of the hydrophobic interaction. Radial distribution functions are used to classify different cases of the interaction, namely, contact configurations, solvent separated configurations, and desolvation configurations. The temperature dependence of these cases is shown to be in qualitative agreement with atomistic models of water. In particular, while the energy for the formation of contact configurations is favored by entropy, its strengthening with increasing temperature is accounted for by enthalpy. This is consistent with our simulated heat capacity. An important feature of the model is that it can be used to account for well-converged thermodynamics quantities, e.g., the heat capacity of transfer. Microscopic mechanisms for the temperature dependence of the hydrophobic interaction are discussed at the molecular level based on the conceptual simplicity of the model.     

Dynamics of water trapped between hydrophobic solutes

We describe the model dynamical behavior of the solvent between two nanoscopic hydrophobic solutes. The dynamics of the vicinal water in various sized traps is found to be significantly different from bulk behavior. We consider the dynamics at normal temperature and pressure at three intersolute distances corresponding to the three solvent separated minima in the free energy profile between the solutes with attractions. These three states correspond to one, two, and three intervening layers of water molecules. Results are obtained from a molecular dynamics simulation at constant temperature and pressure (NPT) ensemble. Translational diffusion of water molecules trapped between the two solutes has been analyzed from the velocity correlation function as well as from the mean square displacement of the water molecules. The rotational behavior has been analyzed through the reorientational dynamics of the dipole moment vector of the water molecule by calculating both first and second rank dipole-dipole correlation functions. Both the translational and reorientational mobilities of water are found to be much slower at the smaller separation and increases as the separation between solutes becomes larger. The occupation time distribution functions calculated from the trajectories also show that the relaxation is much slower for the smallest intersolute separation as compared to other wider separations. The sublinear trend in mean square displacement and the stretched exponential decay of the relaxation of dipolar correlation and occupation distribution function indicate that the dynamical behavior of water in the confined region between two large hydrophobic solutes departs from usual Brownian behavior. This behavior is reminiscent of the behavior of water in the vicinity of protein surface clefts or trapped between two domains of a protein.     

Temperature dependence of three-body hydrophobic interactions: potential of mean force, enthalpy, entropy, heat capacity, and nonadditivity

Temperature-dependent three-body hydrophobic interactions are investigated by extensive constant-pressure simulations of methane-like nonpolar solutes in TIP4P model water at six temperatures. A multiple-body hydrophobic interaction is considered to be (i) additive, (ii) cooperative, or (iii) anti-cooperative if its potential of mean force (PMF) is (i) equal to, (ii) smaller than, or (iii) larger than the corresponding pairwise sum of two-methane PMFs. We found that three-methane hydrophobic interactions at the desolvation barrier are anti-cooperative at low to intermediate T, and vary from essentially additive to slightly cooperative at high T. Interactions at the contact minimum are slightly anti-cooperative over a wider temperature range. Enthalpy, entropy, and heat capacity are estimated from the computed PMFs. Contrary to the common expectation that burial of solvent-accessible nonpolar surface area always leads to a decrease in heat capacity, the present results show that the change in heat capacity upon three-methane association is significantly positive at the desolvation barrier and slightly positive at the contact minimum. This suggests that the heat capacity signature of a hydrophobic polymer need not vary uniformly nor monotonically with conformational compactness. Ramifications for protein folding are discussed.     

Iso-energy cutoff for the calculation of interionic potential of mean force in water

An iso-energy cutoff scheme is introduced for the calculation of the potential of mean force between two ions in water. The cutoff criterion is based on the optimal interaction of the water dipole with the ion pair, for which analytical expressions are derived. Formulas are also derived to characterize the solvent reorganization contribution to the potential of mean force. Treatment of the contributions from waters outside the cutoff is also discussed. © 1994 John Wiley & Sons, Inc.     

Hydration of hydrophobic solutes treated by the fundamental measure approach

We have developed a method to calculate the hydration of hydrophobic solutes by the fundamental measure theory. This method allows us to carry out calculations of the density profile and the hydration energy for hydrophobic molecules. An additional benefit of the method is the possibility to calculate interaction forces between solvated nanoparticles. Based on the designed method, we calculate hydration of spherical solutes of various sizes from one angstrom up to several nanometers. We have applied methods to evaluate the free energies, the enthalpies, and the entropies of hydrated rare gases and hydrocarbons. The obtained results are in agreement with available experimental data and simulations.     

Understanding the nitrate coordination to Eu3+ ions in solution by potential of mean force calculations

Coordination of nitrate anions with lanthanoid cations (Ln(3+)) in water, methanol and octanol-1 has been studied by means of molecular dynamics simulations with explicit polarization. Potential of mean force (PMF) profiles have been calculated for a mono-complex of lanthanoid nitrate (Ln(NO(3))(2+)) in these solvents using umbrella-sampling molecular dynamics. In pure water, no difference in the nitrato coordination to lanthanoids (Nd(3+), Eu(3+) and Dy(3+)) is observed, i.e. the nitrate anion prefers the monodentate coordination, which promotes the salt dissociation. Then, the influence of the nature of the solvating molecules on the nitrato coordination to Eu(3+) has been investigated. PMF profiles point out that both monodenate and bidentate coordinations are stable in neat methanol, while in neat octanol, only the bidentate one is. MD simulations of Eu(NO(3))(3) in water-octanol mixtures with different concentrations of water have been then performed and confirm the importance of the water molecules' presence on the nitrate ion's coordination mode.     

Group contributions to the partial molar volume of ionic organic solutes in aqueous solution

Partial molar volumes at infinite dilution in water at 25°C for more than 400 organic electrolytes (carboxylic acid and amine salts, sulfates, sulfonates, selected polyelectrolyes, aminoacids and derivatives) are described through a simple additivity scheme already adopted for non-electrolytes. Twenty-five charged groups are assigned a contribution and the partial molar volumes of more than 150 monofunctional organic ions are reproduced with a standard deviation of 0.8 cm³-mol^–1. The different volumetric behavior of hydrophobic and hydrophilic centers, either charged or uncharged, is discussed. Deviations from additivity for mono-and polyfunctional ions are analyzed in terms of (i) extension of the hydration cosphere of different polar centres; (ii) intramolecular interactions and their dependence on the nature, number and mutual separation of interacting groups.     

Group contributions to the thermodynamic properties of non-ionic organic solutes in dilute aqueous solution

The thermodynamic properties G _h ^o,H _h ^o, and C _p,h ^oassociated with the transfer of non-ionic organic compounds from gas to dilute aqueous solution and the limiting partial molar properties C _p ^o ,2 and V₂ ² of these compounds in water are described through a simple scheme of group contributions. A distinction is made between groups made only of carbon and hydrogen, and functional groups i.e. groups containing at least one atom different from carbon and hydrogen. Each group is assigned a contribution, for each property, through a least squares procedure which utilizes only molecules containing at most one functional group. Finally, for compounds containing more than one functional group, correction parameters are evaluated as the differences between the experimental values and those calculated by means of the group contributions. The different behavior of hydrophilic compared with hydrophobic groups is discussed for the various properties. A rationale for the correction parameters, i.e. for the effects of the interactions among hydrophilic groups on the thermodynamic properties, is attempted.