J. Phys. Chem. B 1997, 101, 4343-4348 4343

Hydrophobic Effect, Water Structure, and Heat Capacity Changes

Kim A. Sharp* and Bhupinder Madan Department of Biochemistry & Biophysics, UniVersity of PennsylVania, 3700 Hamilton Walk, Philadelphia, PennsylVania 19104-6059 ReceiVed: January 16, 1997; In Final Form: April 1, 1997X

hyd The hydration heat capacity (∆Cp ) of nine solutes of varying hydrophobicity was studied using a combination of a random network model equation of water and Monte Carlo simulations. Nonpolar solutes cause a concerted decrease in the average length and angle of the water-water hydrogen bonds in the first hydration shell, while polar and ionic solutes have the opposite effect. This is due to changes in the amounts, relative to bulk water, of two populations of hydrogen bonds: one with shorter and more linear bonds and the other with longer and more bent bonds. Heat capacity changes were calculated from these changes in water structure using a random network model equation of state. The calculated changes account for observed hyd differences in ∆Cp for the various solutes. The simulations provide a unified picture of hydrophobic and hyd polar hydration, and both a structural and thermodynamic explanation of the opposite signs of ∆Cp observed for polar and nonpolar solutes.

Introduction nonpolar groups means that at higher temperatures (T > 80 °C) their insolubility results from unfavorable changes in enthalpy, In recent years analysis of heat capacity changes has become not entropy. Second, the hydration of a majority of polar and of central importance in understanding the thermodynamics of ionic solutes is also accompanied by a decrease in entropy, and protein folding, protein-protein binding, protein-nucleic acid thus by some kind of structuring of water, but in this case binding, and the hydrophobic effect.1-7 There are several hyd 7,16-20 reasons for this. First, the improved sensitivity and accuracy ∆Cp is negative. Thus, analysis of Cp changes has of calorimeters has provided a wealth of heat capacity (Cp) data become essential for a full understanding of polar and nonpolar for these processes. Second, Cp relates the other three major hydration and the hydrophobic effect. hyd thermodynamic quantitiessenthalpy (H), entropy (S), and free In spite of the importance of ∆Cp changes in a wide range energy (G)sto each other of processes involving solutes, proteins, and nucleic acids, basic features remain unexplained at the molecular level. From the 2 2 ∂H ∂S ∂2G 〈∆H〉 data of Privalov and Gill the increase in heat capacity for Cp T T2 (1) nonpolar groups amounts to about 3 cal/(mol K) for each water ) ) )- 2 ) 2 ∂T ∂T ∂T kT in the first hydration shell or 20% of the total heat capacity of (i) (ii) (iii) (iv) a single water molecule. There is no clear picture of how an increase in water structure can give rise both to this remarkably where T is the temperature, k is Boltzmann’s constant, and 〈∆H2〉 hyd large positive ∆Cp for nonpolar solutes and yet result in a is the mean-squared fluctuation in enthalpy. These four hyd negative ∆Cp for polar and ionic groups. Explanations for equivalent definitions of Cp illustrate the diverse thermodynamic hyd the positive ∆C of hydrophobic hydration in terms of the implications for processes that involve changes in Cp. For p induction of more ordered water which is “melted” at increasing example, unfolding of many proteins is accompanied by a large temperatures15,21,22 do not explain the negative ∆Chyd for polar increase in Cp,1,2 implying (eqs 1i and 1ii) that the unfolding p and ionic groups. The latter phenomenon is perhaps the more entropy and enthalpy depend strongly on temperature and puzzling because the entropy of polar and ionic hydration is implying (eq 1iii) that the stability curve (∆Gunfold vs T) will usually negative. Thus, one deduces from eq 1ii that the be an inverted U-shape with the consequent possibility of both ordering of water induced by these solutes increases with heat and cold denaturation. Changes in Cp associated with increasing temperature. This runs counter to the general protein folding and binding come almost entirely from changes hyd expectation that it becomes harder to induce order (decrease in hydration heat capacity (∆Cp ) due to burial (desolvation) entropy) at higher temperatures. of polar and nonpolar groups.3,8,9 Burial of nonpolar groups Monte Carlo or molecular dynamics simulations of solutes also provides the dominant stabilizing interaction (the hydro- in water using standard water and solute potential functions have phobic interaction) in protein folding and binding.10,11 provided much structural and thermodynamic information about The hydrophobic interaction is driven by the low solubility hydration. However, calculation of ∆Chyd directly from such of nonpolar compounds in water due, at room temperature, to p simulations via eq 1 using standard water and solute potential the unfavorable decrease in the entropy (increase in structure) function appears to be impractical without prohibitive size and of the hydrating waters. This has been described in many 23 hyd ways: as “icelike structures”, icebergs, flickering clusters, length computer simulations. Even determination of ∆Cp clathrates, and pentagonal clusters.12-14 It has become clear, by the most robust approach, using eq 1i, requires taking the however, that a description in terms of entropic changes alone difference of a difference in mean enthalpy from at least four s does not explain the distinctive features of the hydrophobic separate simulations between pure water and water plus one 15 hyd solutesand evaluating this difference at two or more temper- effect. First, the large positive ∆Cp for solvation of hyd atures. Direct determination of ∆Cp from simulations is thus * To whom correspondence should be addressed. subject to considerably more statistical uncertainty than the X Abstract published in AdVance ACS Abstracts, May 1, 1997. determination of the hydration free energy or enthalpy. S1089-5647(97)00245-9 CCC: $14.00 © 1997 American Chemical Society 4344 J. Phys. Chem. B, Vol. 101, No. 21, 1997 Sharp and Madan

Figure 1. (a) Definition of H-bond length, d, and angle, θ. (b) Hydrogen bond structure and heat capacity changes. The solid surface shows the rn dependence of the RN model incremental heat capacity ∆Cp (vertical axis) on the mean H-bond length, d, and the rms H-bond angle, θ (horizontal axes). Values of the bond length and angle (and the associated heat capacity change) for the first hydration shell of solutes obtained from Monte Carlo simulations are shown for the nonpolar (2), polar (9), and mixed ( ) H-bond classes. The solutes are water (W), argon (A), ethanol (E), methane (M), K+ (K), Cs+ (C), N-methylacetamide (N), tetramethylammonium× (T), I- (I), and benzene (B). (c) Linear regression analysis of H-bond length vs angle.

hyd We show here how an equation of state for ∆Cp derived with contributions from H-bond bending, stretching, vibrations, from a random hydrogen-bond (H-bond) network model of water a zero point distortion term, and van der Waals interactions, bend strecth vib zpd vdw can be combined with simulations of water around solutes to Cp , Cp , Cp , Cp , and Cp , respectively. These five hyd provide statistically reliable estimates of ∆Cp . Application contributions, expressed as cal/K per mole of H-bonds, are given of this approach to nine solutes of varying hydrophobicity by provides a unified description of hydrophobic, polar and ionic bend -θ2/2 -5.41θ2 hydration that explains at a molecular level how ordering of Cp ) 0.139e + 1.51e (3) water around solutes can result in either an increase or a decrease in Cp. stretch 2 Cp ) 2.10(1 - 3.72x) - 45.88σ (4)

Theory and Methods 6 kFi Fi Fi kFi Fi vib Cp ) 0.9936T∑ coth( ) - ( - ) (5) The random network (RN) model is based on the observation 2 i)1 T T Fi T T that liquid water retains much of the tetrahedral H-bonding [ 2 ] T sinh ( ) network of ice I, albeit in a randomly distorted form due to T thermal fluctuations, and that the essential features of this 2 network can be described in terms of just three structural Czpd ) 0.235e-1.240θ -2.79x (6) parameters: the mean and standard deviation in H-bond length, p d, and s, respectively, and the root-mean-square (rms) H-bond 0.242e-θ 1 1 angle, θ.24,25 The H-bond length and angle used in the RN Cvdw p ) ( 12 - 6) + model are defined in Figure 1a. The RN model provides an θ (1 + x) (1 + x) accurate description of the structure of pure liquid water in terms 6 12 0.558(1 e-θ) (7) - ( 7 - 13) of the oxygen-oxygen radial distribution function, and it (1 + x) (1 + x) reproduces subtle properties of water, such as the bulk dielectric constant over the range 283-573 K, with remarkable where accuracy.26-28 The RN model also provides good equations of state for the free energy, enthalpy and entropy of pure liquid hνi -1.24θ2-2.79x -0.381hνi -1.24θ2-2.79x 26-28 F ) e , kF ) e (8) water over a wide range of temperatures, from which one i 2k i 2000k can derive an equation of state for the heat capacity in terms of the H-bond structural parameters23 where the mean and standard deviation in H-bond length are expressed in dimensionless form as x ) (d - 2.75)/2.75 and σ2 2 2 Crn(d,s,θ) ) Cbend(d,s,θ) + Cstretch(d,s,θ) + Cvib(d,s,θ) + ) [(s/2.75) + x ], respectively, h is Planck’s constant, and νi p p p p -1 zpd vdw ) 555, 660, 840, 60, 230, and 315 cm for i ) 1-6, Cp (d,s,θ) + Cp (d,s,θ) (2) respectively, are the reference frequencies of the three transla- Hydration Heat Capacity Simulations J. Phys. Chem. B, Vol. 101, No. 21, 1997 4345 tional and three librational intermolecular vibrational modes of previously,23 and eq 10 used to evaluate Cp changes caused by water. These equations were derived as described previously,23 the solute-induced water distortions. Nine solutes encompassing but their presentation has been simplified somewhat by com- a diverse group of properties were examined, including four bining the different numerical constants and expressing the ions (Cs+,K+,I-, and tetramethylammonium (TMA+)), two contributions per mole of H-bonds rather than per mole of water purely nonpolar solutes (argon and methane), an aromatic solute molecules. They are based on the Henn and Kauzmann25 form (benzene), and two solutes containing polar and nonpolar groups of the RN model which incorporates the constant pressure (ethanol and N-methylacetamide). condition under which experiments and simulations are per- Simulations were carried out with 216 water molecules, using formed. the TIP4P potential29 which reproduces the structure and key In the RN model for pure water d, s, and θ are obtained self- thermodynamic properties of bulk water well, and is widely used consistently at any given temperature by minimizing the total in explicit water simulations. The solute and its interaction with free energy with respect to these H-bond structural parameters.25 water were modeled using the OPLS potential function which, The presence of a solute, however, will perturb the H-bonding in conjunction with the TIP4P water potential, provides accurate 30 structure of the surrounding water. If do, so, and θo are the hydration free energies for a wide range of solutes. Sampling average H-bond structural parameters for water in the absence of water configurations was carried out using a Metropolis of solute, i.e., for bulk water, then the change in heat capacity Monte Carlo scheme implemented in the program BOSS31 under from an H-bond with solute-perturbed values of d, s, and θ is constant temperature and pressure conditions (T ) 298 K, P ) given by 1 atm). The systems were equilibrated for 50 106 Monte Carlo steps, and then configurations sampled every× 1000 steps rn rn rn 6 ∆Cp (d,s,θ) ) Cp (d,s,θ) - Cp (d0,s0,θ0) (9) from 10 consecutive runs of 10 10 steps to compute the desired quantities. Solute flexibility× was included for multiatom rn The function ∆Cp (d,s,θ) defined by eq 9 describes the solutes. The solute-water radial distribution function was incremental heat capacity per H-bond as a function of solute obtained from the simulations and used to determine which induced H-bond distortions. The dependence of the incremental hydration shell each water belonged to. Thus, waters lying heat capacity function on the two most sensitive RN variablessthe within the first peak of this distribution function were classified mean length, d, and the rms angle, θsis shown in Figure 1b. as first shell, and so on. Each water was also classified as polar An increase in H-bond length and rms angle decreases the heat or nonpolar hydrating based on whether the nearest solute atom capacity contribution, while decreases in length and angle have or group was polar or nonpolar respectively. All pairs of the opposite effect. The heat capacity also depends to a lesser hydrogen-bonded waters were identified, using a distance cutoff extent on s, decreasing as this variable increases. (taken as 3.4 Å, the boundary of the first peak in the O-O Using eq 9, the net Cp change produced by a solute can be radial distribution function of pure water), the hydrogen bond obtained by summing the contributions from all of the hydrogen type was classified as polar, nonpolar or mixed according to bonds it perturbs: the type of both waters and the hydrogen bond length and angle were evaluated. Mean values of d, s and θ, and their probability hyd rn ∆Cp ) ∑Ni∆Cp (di,si,θi) (10) distributions were accumulated over the course of the simulation. i Standard deviations were obtained from the variation in batch means between different Monte Carlo runs. Convergence was rn where ∆Cp (di,si,θi) is the contribution to heat capacity change checked by monitoring the batch averages of energy and arising from a group of Ni perturbed H-bonds with average hydrogen-bond structural parameters. The mean structural parameters di, si, and θi. The summation in eq 10 over different parameters, d, s, and θ were found to converge satisfactorily sets of H-bonds accounts for the fact that different H-bonds with these size and length simulations, unlike direct estimates may be perturbed to different degrees, depending on their hyd of ∆Cp obtained via differences in enthalpy and from distance from the solute, what hydration shell they are in, and fluctuations in enthalpy.23 the polarity of the different solute groups neighboring the water. hyd hyd The contribution of water to ∆Cp for the ionic solutes is In addition to the H-bond contribution to ∆Cp , potential potentially long-ranged, and to include the effect of water contributions can arise from (i) the fluctuation in the pressure- beyond the first hydration shell, a Born correction was used. volume contribution to the hydration enthalpy, (ii) fluctuations Application of eq 1iii to the Born expression for hydration free in the intrasolute energy, and (iii) fluctuations in the mean energy yields solute-solvent interaction energy, Esw.23 At 1 atm pressure, the absolute value of the PV term even for the largest solute 2 2 2 born q T 1 ∂  2 ∂ studied here is ,0.01 kcal/mol, so fluctuations in this term are Cp ) - (11) 2a ( 2 2 3[∂T] ) negligible. Even for large flexible solutes such as proteins cut  ∂T  contribution ii is small,9 and so for the smaller, more rigid where θ is the charge,  is the dielectric constant of water, and solutes here this term can also be ignored. Contribution iii can a is the cutoff radius for explicit determination of solvent in principle be evaluated directly from simulations by obtaining cut contributions, taken at the outer boundary of the first hydration the variation in mean solute-solvent interaction energy with shell, at 3.5 Å for K+, 4 Å for Cs+, 6.4 Å for TMA+, and 3.5 temperature, but for most of the solutes shown here, this Å for I-. The experimental temperature dependence of the water derivative cannot be distinguished statistically from zero in the bulk dielectric constant was taken from ref 32. simulations. In the one case where there is a statistically hyd significant contribution (for argon23), it is a factor of 2 less than Experimental values of ∆Cp for CsI, KI, and TMAI were the water contribution. In the present study we restricted taken from refs 33 and 34, for argon from ref 2, and for ethanol, ourselves to an analysis of the direct water H-bond contribution methane, N-methylacetamide, and benzene from ref 16. to hydration heat capacity only. Results and Discussion To evaluate d0, s0 and θ0 for pure water and di, si, and θi for water surrounding a solute, Monte Carlo simulations of pure Data on the structure of hydrogen bonds between two first water and a solute in water were carried out as described hydration shell waters, between two second shell waters, 4346 J. Phys. Chem. B, Vol. 101, No. 21, 1997 Sharp and Madan between a first and a second shell water, etc., were collected separately. Statistically significant changes in d and θ were seen only for first shell-first shell H-bonds. The standard deviation in hydrogen-bond length, s, also showed no significant change for any group of H-bonds and thus has a negligible role in heat capacity changes. Significant changes in structure were only seen for mean values of d and θ for water-water H-bonds in the first hydration shell. The results for each solute are shown in Figure 1b. Three classes of H-bonds are distinguished in this figure: (i) “Nonpolar” if they are between waters that each hydrate a nonpolar solute or group; (ii) “polar” if they are between waters that each hydrate a polar (or ionic) solute or group; (iii) “mixed” if one water is hydrating a nonpolar group and the other a polar group (present in ethanol and N- methylacetamide only). Waters hydrating ions or polar groups show an increase in mean H-bond length and rms angle compared to the case of pure water, while waters around nonpolar solutes or groups show decreased mean H-bond lengths and rms angles. H-bonds of the “mixed” type show smaller distortions because of the opposing influence of the polar and nonpolar groups, but in the same direction as the nonpolar solutes. Interestingly, distortions in H-bond length and angle take place in a highly concerted hyd Figure 2. Comparison of experimental and calculated ∆Cp obtained fashion, falling on the same straight line regardless of solute using the data of Figure 1b combined with eq 10. type (Figure 1c, d ) 2.74 Å + 0.067θ, correlation coefficient (r2) ) 0.94). This suggests that one can in fact collapse the TABLE 1: Structural and Vibrational Contributions to the length and angle distortions into a single structural distortion Hydration Heat Capacity coordinate, with the hydrophobic groups at one end of the % of total spectrum, and small ions at the other end. The H-bond length solute bending stretching vibrational of 2.74 ( 0.02 Å corresponding to an angle of θ ) 0° K+ -64 -59 23 (representing a hypothetical nonpolar solute of very large Cs+ -53 -78 31 structuring power) is very close to the value of 2.75 Å seen in TMA+ 41 82 -23 low-temperature (T <-175 K) ice I where also θ 0°,35 Ar 69 47 -16 ≈ reinforcing the idea that temperature and solutes act in a similar CH4 60 54 -14 way as perturbants of water structure. In fact, the concept of a benzene 54 63 -17 “structural temperature” to describe hydration waters is often used in experimental studies.36 polar solutes, the relative magnitudes of structural and vibra- tional contributions are similar but are of opposite sign. This Figure 1b also shows that each set of d, θ values obtained should be contrasted with the relative contributions to the total from the simulations locates that group of H-bonds at particular heat capacity of pure water in the RN model, where only 33% point on the incremental heat capacity surface, ∆Crn(d,s,θ), p comes from the structural part and 66% from the vibrational with the pure water point (d ) 2.95 Å, θ ) 29.4°) lying on the part. The zero-point and van der Waals terms make a negligible ∆Cp ) 0 contour. From these values and eq 10 the net contribution to ∆Chyd for all the solutes examined here. ∆Chyd for each solute was calculated. For the ionic groups, p p The results in Figure 1 are consistent with the standard view the values for the neutral iodide salt were obtained by summing that water in some way becomes more “icelike” around nonpolar the cation and anion contributions. This allows for comparison groups and less “ice-like” around polar groups. A more detailed with experimental measurements which can directly access hyd analysis of the structural changes reveals a more complex ∆Cp of neutral salts. Comparison of experimental and picture. The H-bond angle probability distribution for the ionic calculated hydration heat capacity changes (Figure 2) shows a hyd solutes and polar groups shows that a large population have very good correlation (linear regression ∆Cp (calc) ) 0.47∆ angles greater than 30o, with a prominent peak around 50°- hyd 2 Cp + 0.33 cal/(mol K), r ) 0.98). Figures 1b and 2 show 60°, while a smaller population has a more linear geometry with that the combination of random network model and water a peak centered around 12° (Figure 3A). This distribution is a simulations captures the most striking feature of hydration heat consequence of the strong electrostatic solute-solvent interac- capacities: the opposite behavior of polar and nonpolar groups. tion which tends to align the water dipoles along the solute- hyd The switch from negative to positive ∆Cp as one goes from water axis (inset, Figure 3A), so that many, but not all, H-bonds salts containing small inorganic ions such as Cs+ and K+ to within the first shell are substantially bent compared to bulk larger hydrophobic ions such as TMA+ 17 is also seen in the water. simulations. For nonpolar solutes, the first hydration shell geometry is Eisenberg and Kauzmann have identified two major contribu- best understood in comparison to pure water. The H-bond angle tions to the heat capacity of water: structural and vibrational.35 probability distribution for pure water shows a large peak again The relative importance of these two contributions is difficult centered around 12° (Figure 3B). There is, however, a small to obtain from experiments, but estimates can be obtained from but significant second population centered at about 50°. Pure these simulations using eqs 2-8. For nonpolar solutes, large water is known to have a coordination number of 4.4-4.8.35 positive contributions to the hydration heat capacity come from Examination of water configurations from the computer simula- the structural (bending and stretching) terms, with a smaller, tions shows that the low angle peak arises from the majority of negative contribution from the vibrational term (Table 1). For water molecules that are typically coordinated in a quasi- Hydration Heat Capacity Simulations J. Phys. Chem. B, Vol. 101, No. 21, 1997 4347

Figure 4. Heat capacity as a function of the difference in energy levels, ∆E, in the two-state model, eq 12. The energy difference and relative heat capacity corresponding to water in bulk solvent and around hydrophobic and polar solutes are indicated schematically in the plot.

population, that decreases θ and thus increases Cp. In this analysis, one might say that nonpolar groups do not so much increase the ordering of water as decrease the disordering of water. The division between the two H-bond classes occurs around θ ) 35° (Figure 3A,B), which we note is similar to the angle used to distinguish broken and made H-bonds in recent neutron scattering studies of water38 and in a computer simula- tion of water H-bond dynamics.39 The overall picture that emerges from our structural analysis is one of two populations of H-bond angles, with nonpolar groups decreasing the population with more distorted geometry and polar groups increasing their population. The same trend is also seen in the H-bond length probability distribution (not shown), but it is less pronounced. In fact, a two-state model, which is the simplest thermodynamic model with a nonzero heat capacity, may be usefully employed to interpret the simulation results. For a system with only two energy levels, separated by a gap ∆E, application of eq 1iv gives

Figure 3. H-bond angle probability distribution functions for pure 2 2 -∆E/kT + ∆E ∆E 1 e water (solid line) and the first hydration shell of (A) Cs (b) and the Cp ) [p1p2] ) [ ] (12) 2 2 -∆E/kT -∆E/kT hydroxyl group of ethanol (0). (B) Methane (2) and the methyl group kT kT 1 + e 2 + e of ethanol (+). where p1 and p2 are the probabilities of the system being in the tetrahedral fashion around another water molecule (inset, Figure lower and higher energy states, respectively. For there to be a 3B), while the second peak arises from an occasional extra or significant heat capacity there must be a sizable fluctuation in mismatch water37 that, due to thermal fluctuations, approaches energy, so the two states must be of different energy and yet within H-bonding distance. This water is constrained by the significantly populated. Thus if the energy gap is very small, other waters, however, to approach from the direction of the the heat capacity will be small, as the ∆E2/kT term is small. “face of the tetrahedron” where it must make a bent H-bond Conversely, if the energy gap is very large, then the probability with the central water at an average angle, it turns out close to of populating the upper energy level, p2, is very small, and again half the tetrahedral angle ( 109°/2). A nonpolar solute, which the fluctuation, and thus the heat capacity, is small. Increasing can interact only weakly and≈ nondirectionally with water through ∆E at first will increase Cp as the first factor in eq 12 increases, van der Waals interactions, must nevertheless coexist with the but further increases in ∆E will decrease Cp because p2 considerable amount of structure that persists in liquid water. decreases. The two-state heat capacity function plotted in Figure It thus tends to occupy the same position with respect to the 4 illustrates this behavior. The results of our random network/ quasi-tetrahedral lattice that the more weakly interacting water explicit water simulation may thus be interpreted in terms of a would, in effect competing for this mismatch site (inset, Figure two-state model as follows. Nonpolar solutes weakly perturb 3B). It is the reduction of the high angle population, not a the surrounding water, creating a greater disparity in energy of decrease in the average angle of the more linear H-bond different water microstates, but not so great that the waters 4348 J. Phys. Chem. B, Vol. 101, No. 21, 1997 Sharp and Madan cannot fluctuate between these states. The predominant effect explanation for several important features of small solute is to increase the energy difference term, resulting in a positive hydration heat capacity behavior, it provides a unified picture heat capacity of hydration. In constrast, polar and small ionic of polar and nonpolar solvation, and it provides new insights solutes strongly perturb the surrounding water. Although this into the phenomenon of hydrophobicity. This approach should creates a still larger disparity in energy different water states, open up opportunities for further study of hydration heat the hydrating waters are so strongly associated with the solute capacities of small solutes and larger molecules such as proteins. that the waters cannot fluctuate so easily between these states. This can result in a net decrease in heat capacity due to a large Acknowledgment. We thank Bill Jorgensen, Erin Duffy, and decrease in the probability term. This description of the effect Dan Severance for advice on using BOSS, Paul Axelson and of nonpolar and polar solutes is illustrated schematically in the William De Grado for a critical reading of the manuscript, and plot of the two-state heat capacity function, Figure 4. Of course, Hugh Gallagher for help with Figure 1. Financial support is the random network/explicit water model describes a more acknowledged from NIH (GM54105) and the E. R. Johnson complex and realistic potential surface with many more than Research Foundation. two states, but the qualitative effects of solute pertubation are References and Notes made clear from the two-state analogy. The idea of rapid exchange between somewhat energetically (1) Baldwin, R. Proc. Natl. Acad. Sci. U.S.A. 1986, 83, 8069-8072. (2) Privalov, P. L.; Gill, S. J. AdV. Protein Chem. 1988, 39, 191-234. different water states, perhaps represented by the two popula- (3) Makhatadze, G.; Privalov, P. J. Mol. Biol. 1990, 213, 385-391. tions of linear and bent hydrogen bonds seen in Figure 3, relates (4) Spolar, R.; Livingstone, J.; Record, M. T. Biochemistry 1992, 31, back to the idea of flickering clusters40,41 and is consistent with 3947-55. (5) Spolar, R. S.; Record, M. T. Science 1994, 263, 777-84. a recent computer simulation of hydrogen-bond kinetics in pure (6) Murphy, K. P.; Privalov, P. L.; Gill, S. J. Science 1990, 247, 559- water39 where rate constants for making and breaking hydrogen 561. bonds between neighboring waters by rotational motions were (7) Murphy, K.; Freire, E. AdV. Protein Chem. 1992, 43, 313-361. determined to be k 1ps-1 and k 0.7 ps-1, respectively, (8) Sturtevant, J. Proc. Natl. Acad. Sci. U.S.A. 1977, 74, 2236-40. f ) b ) (9) Velicelebi, G.; Sturtevant, J. M. Biochemistry 1979, 18, 1180-6. which corresponds to a small effective energy of formation, (10) Kauzmann, W. AdV. Protein Chem. 1959, 14,1-63. given by ∆E )-kT ln (kf/kb), of -0.21 kcal/mol. (11) Dill, K. A. Biochemistry 1990, 29, 7133. In summary, the use of the random network model now makes (12) Frank, H. S.; Evans, M. W. J. Chem. Phys. 1945, 13, 507. (13) Nemethy, G.; Scheraga, H. J. Chem. Phys. 1962, 36, 3382-3400. it feasible to obtain statistically reliable estimates of hydration (14) Tanford, C. H. The Hydrophobic Effect; John Wiley and Sons: New heat capacities for both polar and nonpolar solutes from explicit York, 1980. water simulations and to interpret the results directly in terms (15) Kauzmann, W. Nature 1987, 325, 763-4. (16) Cabani, S.; Gianni, P.; Mollica, V.; Lepori, L. J. Solution Chem. of changes in water structure. It should be emphasized that 1981, 10, 563-595. although the explicit water simulations described here require (17) Marcus, Y. Biophys. Chem. 1994, 51, 111-128. considerably amounts of computer time, they represent the only (18) Murphy, K.; Gill, S. Thermochim. 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