The hydration free energies of ions exhibit an approximately quadratic dependence on the ionic charge, as predicted by the Born model. We analyze this behavior using second-order perturbation theory. This provides effective methods to calculating free energies from equilibrium computer simulations. The average and the fluctuation of the electrostatic potential at charge sites appear as the first coefficients in a Taylor expansion of the free energy of charging. Combining the data from different charge states (e.g., charged and uncharged) allows calculation of free-energy profiles as a function of the ionic charge. The first two Taylor coefficients of the free-energy profiles can be computed accurately from equilibrium simulations; but they are affected by a strong system-size dependence. We apply corrections for these finite-size effects by using Ewald lattice summation and adding the self-interactions consistently. An analogous procedure is used for reaction-field electrostatics. Results are presented for a model ion with methane-like Lennard-Jones parameters in SPC water. We find two very closely quadratic regimes with different parameters for positive and negative ions. We also studied the hydration free energy of potassium, calcium, fluoride, chloride, and bromide ions. We find negative ions to be solvated more strongly (as measured by hydration free energies) compared to positive ions of equal size, in agreement with experimental data. We ascribe this preference of negative ions to their strong interactions with water hydrogens, which can penetrate the ionic van der Waals shell without direct energetic penalty in the models used. In addition, we consistently find a positive electrostatic potential at the center of uncharged Lennard-Jones particles in water, which also favors negative ions. Regarding the effects of a finite system size, we show that even using only 16 water molecules it is possible to calculate accurately the hydration free energy of sodium if self-interactions are considered.

A molecular model of poorly understood hydrophobic effects is heuristically developed using the methods of information theory. Because primitive hydrophobic effects can be tied to the probability of observing a molecular-sized cavity in the solvent, the probability distribution of the number of solvent centers in a cavity volume is modeled on the basis of the two moments available from the density and radial distribution of oxygen atoms in liquid water. The modeled distribution then yields the probability that no solvent centers are found in the cavity volume. This model is shown to account quantitatively for the central hydrophobic phenomena of cavity formation and association of inert gas solutes. The connection of information theory to statistical thermodynamics provides a basis for clarification of hydrophobic effects. The simplicity and flexibility of the approach suggest that it should permit applications to conformational equilibria of nonpolar solutes and hydrophobic residues in biopolymers.Hydrophobic interactions are widely believed to be of dominating importance for protein structure, aggregation, and function. However, the molecular theories of hydrophobic interactions (1-10) have not been used so far in molecular studies of protein structure. This is partly because these theories have limitations that are still being clarified (5,(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21) and partly because of their complexity. This paper suggests a new approach to molecular theories of hydrophobic effects and then tests the simplest model to which this suggestion leads. It is argued that the simplicity and flexibility of this approach should eventually permit its application to issues of protein structure in solution.Alternative descriptions of hydrophobic effects that are used are based upon parameterizations of solubility data (22)(23)(24)(25)(26). Those hydrophobicity models have not changed essentially from the concepts of Kauzmann (27) but the solubility data have been parameterized in a variety of ways (28-31). Although solubility models of hydrophobic effects have been useful, molecular-level theories are expected to have wider applicability and to improve our understanding of hydrophobic effects on biomolecular structure. This could be particularly important to recent work that probes protein solution structure in new ways.One example of such work is reversible denaturation experiments. The observed destabilization of folded proteins with decreasing temperature is an evidence of hydrophobic interactions. Cold/heat denaturation of globular proteins (32-34), pressure denaturation (35-39), and the effects of osmotic stress (40)(41)(42)(43)(44)(45)(46)(47)(48)(49) demonstrate that the solvent activity affects the structure. However, parameterizations of hydrophobicity models that reflect the activity of the aqueous medium have not been pursued extensively (50).The adequacy of solubility models is also not obvious in studies of the structures of folding intermediates on renaturation pathways. These stu...

A microscopic theory is developed which can describe many of the structural and thermodynamic properties of infinitely dilute solutions of apolar solutes in liquid water. The theory is based on an integral equation for the pair correlation functions associated with spherical apolar species dissolved in water. It requires as input the experimentally determined oxygen–oxygen correlation function for pure liquid water. The theory is tested by computing thermodynamic properties for aqueous solutions of apolar solute species. The predictions of both the Henry’s Law constant and the entropy of solution are in good agreement with experiment. The calculation of the latter quantity is essentially independent of any adjustable parameters. It is shown how the correlation functions we have calculated can be used to predict the solubility of more complicated, aspherical, and nonrigid solutes in liquid water. For the more complex molecules it is convenient to study the difference between the excess chemical potential of the molecule and the chemical potentials of its separated components (Ben-Naim’s measure of the hydrophobic interaction strength δA(HI)M). A generalized formula for δA(HI)M is presented which reduces to Ben-Naim’s result in the special case of a rigid solute. The calculations of δA(HI)M for normal alkanes through n-decane are independent of unknown and adjustable mean field parameters. For ethane, the theoretical results for δA(HI)M are in good agreement with experimental data. To treat longer alkanes multipoint correlation functions are required. It is shown that the standard superposition approximation for the multipoint correlation functions predicts qualitatively incorrect results. A correction to the superposition approximation is developed. Its use yields theoretical values for δA(HI)M which agree well with experiment. We also calculate free energies of transfer of n-alkane solutes from a hydrocarbon solvent to liquid water. Here, too, our results are in good agreement with experiment. After testing the theory against experiment, several experimentally inaccessible aspects of the hydrophobic effect are discussed. Analysis of pair correlation functions between spherical apolar species indicates that the hydrophobic interaction becomes more attractive as the solute size is increased at constant temperature, or as the temperature is decreased. The solute–solute correlations in liquid water are compared with those in a nonassociated (hard-spherelike) solvent. Plots are presented which indicate that the insertion of an apolar species increases the intermolecular structure of water in the vicinity of the apolar solute. The effects of solvent environments on the conformations of small chain molecules are discussed. It is shown that hydrocarbon solvents as well as water tend to reduce the spatial extension of the chain molecules from what is found in the gas phase. However, calculations which take cognizance of the detailed structure of the solvent liquids reveal that the average conformational structure of some n-alkanes may be more compact in some nonassociated solvents than in water.

Proteins can be denatured by pressures of a few hundred MPa. This finding apparently contradicts the most widely used model of protein stability, where the formation of a hydrophobic core drives protein folding. The pressure denaturation puzzle is resolved by focusing on the pressure-dependent transfer of water into the protein interior, in contrast to the transfer of nonpolar residues into water, the approach commonly taken in models of protein unfolding. Pressure denaturation of proteins can then be explained by the pressure destabilization of hydrophobic aggregates by using an information theory model of hydrophobic interactions. Pressure-denatured proteins, unlike heat-denatured proteins, retain a compact structure with water molecules penetrating their core. Activation volumes for hydrophobic contributions to protein folding and unfolding kinetics are positive. Clathrate hydrates are predicted to form by virtually the same mechanism that drives pressure denaturation of proteins.A decade ago, Walter Kauzmann (1) challenged the commonly held view that a hydrophobic core stabilizes globular proteins, by poignantly remarking that the ''liquid-hydrocarbon model (2) fails almost completely when one attempts to extend it to the effects of pressure on protein folding.'' Although a variety of forces stabilize folded proteins (3-6), the formation of a hydrophobic core is thought to play a dominant role. This view is supported by the temperature dependence of hydrophobic contributions to protein unfolding showing remarkable similarities to the transfer of hydrocarbons from a nonpolar phase into water, notably a convergence of the entropy of transfer (2,7,8). However, Kauzmann (1) pointed out that the pressure dependence of protein unfolding is at odds with the hydrophobic-core model: The volume change ⌬V upon unfolding is positive at low pressures but negative at pressures of about 100-200 MPa. The transfer of hydrocarbons into water shows exactly the opposite behavior, with ⌬V being negative at low pressures and positive at high pressures.Evidently, pressure unfolding of a protein (9-16) does not correspond to the transfer of a nonpolar molecule from a nonpolar environment into aqueous solution. Unlike heatdenatured proteins, the ensemble of pressure-denatured proteins retains elements of structural organization (13, 17). Consequently, an understanding of the thermodynamics of pressure denaturation might focus on the free energy of water transfer into the hydrophobic core of the protein (18) rather than transfer of nonpolar solutes into water. Our conceptual framework for pressure denaturation is as follows: the protein interior is largely composed of efficiently packed residues, more likely hydrophobic than those at the surface (19). Increasing hydrostatic pressure then forces water molecules into the protein interior, gradually filling cavities, and eventually breaking the protein structure apart.We therefore study the effects of pressure on the association of nonpolar residues in water. We use the informat...

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