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.