arXiv:physics/9807001v2 [physics.chem-ph] 18 Sep 1998 aes rti odn a etems extensively most in the Thermodynamic process be solution. aqueous self-assembly may macromolecular folding bi- characterized Protein proteins, membrane and folded con- acids, of layers. nucleic groups stability helical in the nonpolar hydropho- base-stacking and to Attractive these bases, significantly between tribute nucleic-acid lipids. interactions of of bic faces tails the amino- hydrocarbon aromatic the chains, and side aliphatic frac- significant groups: acid a nonpolar contain of lipids tion and acids, nucleic teins, iaiedsrpino yrpoi rvn ocsa a as forces driving factor. contributing hydrophobic major of am- quan- description hydrated thorough, titative a of quan- necessitates stability a macromolecules the Nevertheless, phiphilic of understanding assemblies. titative their amphiphilic and of stability molecules full thermodynamic a the in of neglected treatment be cannot water- such dominating bonding, or hydrogen interactions, intramolecular mediated the other and not interactions Clearly, electrostatic if as folding. significant protein in a role plays nonpolar acids, predominantly amino comprising core, hydrophobic a ini ri tissue. over- forma- brain plaque during amyloid in bodies disease-causing tion inclusion in ag- in or non-specific expression proteins the of com- formation and the gregation and complexes; solutions; membranes, macromolecular surfactant of micelles, in of mesophases formation plex fold- the protein ing; solution: aqueous in processes self-assembly Introduction I. studies ept aydcdso eerho yrpoi ef- hydrophobic on research of decades many Despite yrpoi neatospa eta oei many in role central a play interactions Hydrophobic ∗ tblt.Fnly suso dwtig fmlclrylrennoa s approach. nonpolar theory large perturbation molecularly developed of recently contrad “dewetting” a denaturation of p issues core pressure Finally, the hydrophobic whether stability. of the puzzle, analysis penetrates long-standing an water a from that follows suggesting compres proteins gates, hydrophob isothermal of the macroscopic relates denaturation the pro model pressure to IT effects experiments The pressure unfolding folding. and tein protein temperature of study tors kinetics the to and to approach solu funct contributions this nonpolar distribution solvent of spherical and radial molecules, of and methane hydration density between the effects: – approach, hydrophobic (IT) water simplest theory bulk information simples of an the properties using of energy effec estimated bu free hydrophobic is in hydration probability of formation the cavity origins and for molecular fluctuations, probability density the the into between insight relation of damental hydr goal hydrophobic quantify the to with developed is approach theoretical A orsodn uhr alSo 70 hn:(0)665-192 (505) Phone: K710. Stop Mail Author. Corresponding 9 , 10 npoen aesonta h omto of formation the that shown have proteins on † eateto hmclEgneig on okn Univers Hopkins Johns Engineering, Chemical of Department hoeia iiin o lmsNtoa aoaoy Los Laboratory, National Alamos Los Division, Theoretical .Hummer, G. 1 , 2 , 3 , 4 , 5 iplmr uha pro- as such Biopolymers yrpoi ffcso oeua Scale Molecular a on Effects Hydrophobic ∗ .Gre .E acı,M .Paulaitis, E. Garc´ıa, M. E. A. Garde, S. 1 , 2 , 6 , 7 , 8 n structural and L-R98-2758) (LA-UR 1 edvlpddtie oeua xlntosfrteun- the processes. for hy- this, derlying explanations achieve the molecular To detailed of folding. developed protein context we of the model core within drophobic proteins pressure and of temperature denaturation for results contradictory ently change. volume associated the de- of pressure protein the pendences opposite and in diametrically process exhibit proteins transfer denaturation hydrocarbon of the model that sense hydrophobic-core poignantly the out dur- tradict pointed solvent as to Kauzmann, However, by exposed residues become denaturation. nonpolar interior ing gas that protein suggests or the This hydrocarbon in trans- a water. of into from those phase to molecules similar nonpolar are character- ferring denaturation thermodynamic heat the of that istics show studies These rnfretoisfo aoier experiments. hydrocarbon- calorimetry from of entropies convergence transfer temperature observed the eauecnegneo h nrp fpoenunfolding protein tem- of observed entropy the the of convergence with perature coincides convergence entropy oe fpoen,wihwsetbihdo h ba- the on studies. unfolding established core temperature was hydrophobic extensive de- which of the Heat proteins, sis by of explained example. el- model be immediate at can an and naturation is temperatures that pressures elevated fact evated at well-known denature The con- proteins salt composition. as such and properties, centration solution key the or even pressure, temperature changing and variables, of thermodynamic effects elementary most the include Sur- seemingly these explained. and prisingly, be to incomplete, yet have still observations presumably is contradictory phenomena origin key hydrophobic of of understanding our fects, n olo u okhsbe orcnieteeappar- these reconcile to been has work our of goal One .Fx 55 6-43 -al [email protected] E-mail: 665-3493. (505) Fax: 3. fpoen thg rsue.Ti resolves This pressures. high at proteins of ienwisgt notethermodynamics the into insights new vide cstehdohbccr oe fprotein of model hydrophobic-core the icts to n neatoso oeua scale, molecular a on interactions and ation yrpoi oue,hr atce.This particles. hard solutes, hydrophobic t t,Blioe ayad228 USA 21218, Maryland Baltimore, ity, lms e eio855 USA 87545, Mexico New Alamos, oa qiiru fbtn.Applications butane. of equilibrium ional iiiyo ae.Anvlepaainfor explanation novel A water. of sibility e,teptnilo enfre(PMF) force mean of potential the tes, cetoycnegneosre npro- in observed convergence ic-entropy esr tblt fhdohbcaggre- hydrophobic of stability ressure kwtrrsligfo molecular-scale from resulting water lk o.TeI prahrpoue the reproduces approach IT The ion. noprtn xeietlyavailable experimentally incorporating ltsaedsusdi h otx of context the in discussed are olutes 13 s h oe sbsdo h fun- the on based is model The ts. rsuedntrto per ocon- to appears denaturation pressure 14 † , 15 n .R Pratt R. L. and ee ewl ou pcfial on specifically focus will we Here, 6 , 14 7 , 8 That , 11 , 12 and thus provides a thermodynamic foundation for the we find the “best-possible” description of the fluctua- hydrophobic-core model of protein folding.6,7,11,12,16,17,18 tions that satisfies certain experimental constraints. The In addition, we will suggest a molecular mechanism virtues of such an approach are its simplicity, efficiency, for pressure denaturation of proteins19,20,21,22,23,24,25,26 and accuracy for molecule-size solutes. In addition, the and discuss its consequences on folding kinetics and IT model expresses the simple hydrophobic phenomena the characteristics of the ensemble of unfolded protein using properties of bulk water alone. This allows us to structures.15 Our explanation of pressure denaturation relate hydrophobic effects to the peculiar properties of invokes experimental observations regarding the differ- water that distinguishes it from nonpolar solvents. The ences in the structures of heat and pressure-denatured IT model builds on concepts introduced in SPT, specif- proteins, the latter being more compact.23,27,28 We will ically the relation between the probability of finding a also give preliminary results for the effects of salt con- molecule-size cavity in water and the chemical potential centration on hydrophobic hydration, as characterized by of solvation of hard particles. In its simplest form, the IT the Hofmeister series,29 which ranks salts by their abil- model is also related to Ornstein-Zernike integral equa- ity to increase (“salt in”) or decrease (“salt out”) the tion theories and the PC theory.50,52,53,54 solubility of nonpolar molecules in water. The paper is organized as follows: After developing To achieve these goals, we need a model that quanti- the IT model for hard nonpolar solutes and a general- fies hydrophobic hydration and hydrophobic interactions, ization to continuous solute-solvent interactions, we will and the effects of temperature, pressure, and salt concen- discuss practical aspects of the implementation. Limi- tration on them. A number of models of the hydrophobic tations in the macroscopic limit will be analyzed before effect have been developed in the past,1,2,4,30,31,32,33,34 we view the IT model from a historical perspective. We all having their merits and limitations.35 One of the will then present results for the simplest hydrophobic ef- most influential models is scaled-particle theory (SPT)36 fects – solvation of spherical solutes, methane-methane and its extension to hydrophobic hydration.37,38 SPT interaction in water, and contributions to the torsional laid a molecular foundation for surface area mod- equilibrium of butane. The temperature dependence of els of hydrophobic hydration with macroscopic surface hydrophobic hydration and the entropy convergence will tensions.39,40,41,42,43,44,45,46 However, this invocation of be studied subsequently. We will then quantify the ef- macroscopic parameters limits applications of SPT to fects of pressure on hydrophobic interactions, leading to study the subtle changes due to temperature and pres- a model for the pressure denaturation of proteins. The sure, for example, that are closely associated with the effects of salt on hydrophobic hydration will be discussed molecule-scale properties. A thoroughly molecular the- briefly. We will also introduce a recently developed per- ory was developed by Pratt and Chandler (PC),47 based turbation theory model55 based on the energetic loss of on the theory of simple liquids. The PC theory invokes water molecules at the solute-water interface. This model radial correlation functions in the context of Ornstein- allows us to extend our analysis of hydrophobic effects Zernike integral equations with appropriate closures. further to mesoscopic and macroscopic solutes. We will Lazaridis and Paulaitis48 (LP) use an expansion of the conclude with an attempt to give answers to the question conformational entropy in terms of particle distribution how water differs from hydrocarbon liquids as a solvent functions. Both PC and LP theories relate the structural for nonpolar solutes. ordering of water by the nonpolar solute to the hydration thermodynamics through solute-water pair correlations. II. Information theory model of hydrophobic hy- Probably the most powerful approach is the direct dration computer simulation of hydrophobic hydration phenom- ena using molecular dynamics or Monte Carlo methods.49 a. Simple hydrophobic effects: thermodynamics Computer simulations are flexible and, with the advent of dissolving a hard particle in water of ever faster computers, the system size and time scale The simplest hydrophobic solute is a cavity or, equiva- limitations are becoming less of a concern. However, the lently, a hard-core particle which excludes water-oxygen interpretation of computer simulation results necessitates atoms from a volume v of a given shape and molecular a well-founded theoretical framework. Here, and in the size. Cavity formation constitutes the important, first aforementioned theories, computer simulations provide step in solvating any nonpolar solute in water. The sol- critical input that is not readily available from experi- vation thermodynamics of a hard particle is determined ments. by the excess chemical potential µex corresponding to The theoretical approach pursued here describes hy- the free energy of its transfer from an ideal gas into the drophobic hydration and hydrophobic interactions using aqueous environment. Statistical mechanics relates this chemical potentials of the simplest hydrophobic solutes excess chemical potential to the probability p0 of finding in water – “hard” molecules exerting entirely repulsive an empty volume v in water, i.e., a cavity of a given size interactions on water molecules. The solvation chemical and shape,36,56 potentials for hard-core solutes are directly related to the ex presence of cavity volumes of molecular size due to den- µ = −kBT ln p0 . (1) sity fluctuations in bulk water. We quantify these density fluctuations using an approach motivated by informa- To determine the chemical potential µex, we need to 50 51 tion theory (IT). From a maximum-entropy principle, quantify the probability p0 of successfully inserting a

2 ∞ hard-core solute of a given size and shape into equilib- 2 2 2 ′ ′ rium conformations of water, as illustrated in Figure 1. hn i = pnn = hni + ρ dr dr g(|r − r |) , (3c) n=0 v v A virtue of such an approach is that the solvation ther- X Z Z ex modynamics characterized by µ is determined by the where ρ is the number density of bulk water and g(r) properties of pure water, with the solute entering through is the radial distribution function between water-oxygen its molecular volume v. atoms in bulk water, which can be determined from X- ray or neutron scattering measurements or computer sim- ulations. These moment conditions provide constraints on the pn’s, and guarantee that the pn are normalized and have the correct first and second moments. Higher moments would require knowledge of triplet and higher- n=3 order correlation functions which are not generally ac- cessible to experiments, but can be calculated from com- puter simulations.58 The IT approach attempts to provide the “best- n=0 possible” estimate of the probabilities pn under the con- straints of the available information,51 defined as the set {pn} that maximizes an information entropy η subject to the information constraints,

Figure 1. Schematic two-dimensional representation of max η({pn}) . (4) {constraints} the probabilities pn of observing n solvent centers inside a randomly positioned volume v in the bulk fluid. Two In the most general form, we adopt a relative or cross cases are shown: A successful insertion with n = 0 that entropy,59 would contribute to p0, and an insertion with n = 3 ∞ contributing to p3. p η({p })= − p ln n , (5) n n pˆ n=0 n b. Information theory approach to excess chemi- X   cal potentials of solvation wherep ˆn represents an empirically chosen “default model.” The goal is to estimate accurately the probability p0 that a given hard-core solute inserted into water does We consider two natural choices of default models: the not overlap with any of the solvent centers, defined as Gibbs default modelp ˆn ∝ 1/n!, which leads to a Poisson the positions of water-oxygen atoms. The IT approach distribution for a given mean, as would be expected for an does not attempt to model this quantity directly. In- ideal gas, and a flat distribution (ˆpn = 1for n ≤ nmax and pˆn = 0 otherwise), which results in a discrete Gaussian stead, it focuses on the set of probabilities pn of find- ing n water-oxygen atoms inside the observation volume, form of pn with given mean and variance. Empirically, with p being just one of the p . We will attempt to get we find that the latter choice is accurately applicable to 0 n molecule-size cavities.50 accurate estimates of the pn, and p0 in particular, us- ing experimentally available information as constraints 1 on the pn. The moments of the fluctuations in the num- -1 ber of solvent centers within the observation volume v 10 -2 provide such constraints. 10 For a given observation volume v in bulk water, the 10-3 moments of the fluctuations in the particle number n are 10-4 n p determined from the pn as follows: 10-5 10-6 0.18 ∞ 0.22 k k 10-7 0.26 0.30 0.34 0.38 hn i = pnn , (2) -8 n=0 10 X 10-9 0 2 4 6 8 10 12 14 where h· · ·i denotes a thermal average. The zeroth, first, n and second moment can be expressed in terms of exper- imentally accessible quantities: Figure 2. Probabilities pn for observing n water- 50 ∞ oxygen atoms in spherical cavity volumes. Results from Monte Carlo simulations of SPC water57 are shown as h1i = pn = 1 (3a) n=0 symbols. The parabolas are the predictions of the flat X∞ default model of the IT approach. The center-to-center hni = pnn = ρv , (3b) exclusion distance d (in nanometers) is noted next to the n=0 curves. X

3 ex Maximizing the information entropy under the con- the estimated values of p0 and µ for the solute-size straints of eq 3 leads to range considered in Figures 2 and 3.

2 λ0+λ1n+λ2n pn =p ˆne , (6) c. Continuous information theory where λ0, λ1, and λ2 are the Lagrange multipliers chosen A more general statement of the principles underly- to satisfy the moment conditions eq 3. ing the IT approach follows from a generalization to Figure 2 shows pn distributions for spherical observa- continuous interactions between the solvent and the hy- tion volumes v calculated from computer simulations of drophobic solute. Such a generalization can be achieved 57 50 SPC water. The solute exclusion volume is defined by by explicitly considering solvent positions within a mi- the distance d of closest approach of water-oxygen atoms croscopic volume, in addition to the occupancy num- to the center of the sphere. For the range of solute sizes bers considered in the basic IT approach. We de- studied, we find that ln p values are closely parabolic in n fine probability distributions p(j; r1,..., rj ) of observ- n. This would be predicted from the flat default model ing exactly j particles within the observation volume v with n → ∞, as shown in Figure 2. max at positions r1,..., rj in infinitesimal volume elements dr1,...,drj . The p(j; r1,..., rj ) allow us to calculate 14 solvation chemical potentials of a solute with continu- -1 12 ous interactions u(r1, r2,...) between solvent molecules at positions r1, r2,... and a solute at the origin. Within 10 -1.5 the observation volume v, the solute-solvent interactions

T 8

B are treated explicitly. Finite-range interactions provide a /k

ex µex -2 natural choice for v; otherwise, a finite volume v can be

µ 6 /k T (flat) µex B /kBT (Gibbs) chosen with corrections for long-range interactions. 4 I (flat) 62 I (Gibbs) Widom’s formula relates the excess chemical poten- -2.5 Shannon information I µex/k T ex 2 B tial of the solute µ to the normalized p(j; r1,..., rj ),

0 -3 ex −βµ −βu(r1,r2,...) 1 2 3 4 5 6 7 8 9 e = e number of moments i=Dj E −βu(r1,...,rj ) Figure 3. Effect of increasing the number of mo- = dri p(j; r1,..., rj )e , (8) j "i=1 v # ments used in the IT calculations. Shown as a function X Y Z of the number of moments is the predicted excess chem- −1 ical potential of a methane-size hard sphere with exclu- where β = kBT . sion radius d =0.33 nm for solvation in SPC/E water60 Calculating p(j; r1,..., rj ) directly would require (left-hand scale). Also shown for reference is the actual knowledge of higher-order correlation functions. Instead, chemical potential (dot-dashed line). The Shannon in- our goal is to infer the p(j; r1,..., rj ) from available in- formation I (right-hand scale) illustrates that the third formation, such as the one- and two-particle densities in and higher moments do not result in an appreciable gain the observation volume v, expressed as constraint func- in information. tionals: ρ(r)= hδ(r − r )i (9a) Figure 3 illustrates the effect of including higher mo- α α ments in the IT model. Results are shown both for the X i=j Gibbs and the flat default models. We find that the j = dr δ(r − r )p(j; r ,..., r ) , prediction of µex is greatly improved when the second 1 i 1 1 j "i=1 v # moment (i.e., the variance of the particle number) is Xj≥1   Y Z 2 (2) ′ ′ included in addition to the mean. However, inclusion ρ g (r, r )/2= h δ(r − rα)δ(r − rγ )i (9b) of higher moments initially makes the prediction worse. α>γ X Only when seven or more moments are used is the predic- i=j j tion as accurate as the two-moment model. Also shown in = dr δ(r − r )δ(r − r )p(j; r ,..., r ) . 2 i 1 2 1 j Figure 3 is the calculated Shannon information I({pn}), "i=1 v # Xj≥2   Y Z ∞ ′ I({p })= p ln p . (7) where r and r are positions inside the observation vol- n n n ume v. For a homogeneous fluid, the density will be uni- n=0 X form, i.e., ρ(r) will be independent of position; and the Including the second moment results in a large gain of two-particle density distribution will be the radial distri- information, whereas the gains from including additional bution function of the homogeneous system, g(2)(r, r′)= higher moments are small: moments of order three and g(|r − r′|). higher are, in this respect, “un-informative,” as measured In analogy to eq 5, we define an information entropy η by the gain in Shannon information I({pn}) (see also for continuous solvent positions and discrete occupancy Ref. 61). Those higher moments do not significantly alter numbers,63

4 i=j be non-zero. Similar procedures apply to non-uniform j η ≡− dri p(j; r1,..., rj ) ln[v j!p(j; r1,..., rj )]. systems such as the solvation structure near a fixed so- j "i=1 v # X Y Z lute, but the positioning of the observation volume is then (10) relevant.

j The observation volume may be subdivided into sev- We include the factor of v in the logarithm of eq 10 for eral strata. This introduces cross moments in the general dimensional consistency. Maximizing this entropy func- case and permits conceptually interesting possibilities for tional under the constraints of eq 9 results in an expres- modeling the probabilities. As an example, consider the sion for the probability distribution p(j; r1,..., rj ), mean density in a volume element external to a core re- j gion conditional on the requirement that the core region −1 j (1) be empty. With such considerations, the IT modeling −β ln[v j!p(j; r1,..., rj )] = ω (rk) naturally produces a prediction of the hydration struc- Xk=1 j ture near a hydrophobic solute without special geomet- 1 + βω(2)(r , r ) . (11) rical limitations. Such possibilities have scarcely been 2 k l studied so far, however. For the calculation of the re- k,l=1 X quired moment information, stratification is no particular (1) (2) problem but it may affect the computational efficiency. The Lagrange multipliers ω (rk) and ω (rk, rl) are Note, that a stratified representation can also be derived chosen such that the p(j; r1,..., rj ) satisfy the constraint functionals eq 9. Incorporating additional n-particle cor- from coarse-graining the continuous IT of section II c. relation information (n > 2) is straightforward. In a Next, we will illustrate several methods to calculate practical implementation, one can subdivide the obser- the second moment of the particle-number fluctuations. vation volume into a finite number of volume elements. Then we will describe numerical methods for the entropy The constraint functionals, eqs 9, then reduce to a finite maximization. number of constraints. Calculation of particle-number variances. The second- Note that the probability distributions p(j; r1,..., rj ) moment constraint eq 3c requires the calculation of a have a Boltzmann-Gibbs structure, i.e., they are propor- double integral over the volume v involving the distance- (1) tional to an exponential of effective interactions ω (rk) dependent pair correlation function g(r). Such integrals (2) and ω (rk, rl) in kBT units, divided by j!. Summation can be evaluated using Monte Carlo techniques. The ′ and integration of the p(j; r1,..., rj ) results in the fa- average of the pair correlation function, hg(|r − r |)i can miliar form of a grand-canonical partition function for a be determined by placing two points r and r′ randomly microscopic volume v embedded in a large bath of sol- inside the volume v according to a uniform probability vent molecules. The effective interactions entering the density. This average is then multiplied by the square of Boltzmann-Gibbs factors in this grand-canonical parti- the volume v to give an estimate of the integral in eq 3c. (1) tion function are the Lagrange multipliers ω (rk) and As suggested above, the variance and higher moments (2) ω (rk, rl). These effective interactions are chosen to of particle-number fluctuations can be determined di- satisfy available information constraints, rather than be- rectly from molecular simulations. Unlike p0, the mean ing derived directly from integrating out bath degrees of and variance can be calculated accurately from insertion freedom. also for large solutes. Objects with volume v (shape ω(1)(r) is expected to be a spatial constant approxi- and size) are inserted with random positions and ori- mately the negative chemical potential of the solvent.55 entations into water configurations taken from an equi- Similarly, ω(2)(r, r′) will be approximately an interatomic librium molecular dynamics or Monte Carlo simulation. pair potential, except in a small region near the surface The mean and variance of the particle number n is then of the subsystem. determined directly. Volumes of certain shapes are amenable to analytical 64 d. Some practical aspects of the information the- evaluation. For spherical volumes v of radius d, Hill de- ory model rived a transformation of the six-dimensional integral to one-dimensional form. Such transformations can be ac- Here, we provide details on the practical implementa- complished by determining the pair-distance distribution tion the IT model. Before discussing specific details, we P (R) well known from small-angle scattering theory,65 note that the required moments of particle-number fluc- tuations can be obtained in situ from simulation data. P (R)= dr dr′δ(R − |r − r′|) , (12) This idea is most direct for a uniform liquid. The ob- v v servation volume v is planted as a stencil in the simula- Z Z tion volume and the required moments are extracted as where δ(x) is Dirac’s delta function. This leads to the averages over a set of simulation configurations. For a following expression for the second moment: uniform liquid the positioning of the observation volume is irrelevant and many positions can be used simultane- hn2i = hni + ρ2 dR P (R)g(R) . (13) ously. It is worth noting further that since the occupation numbers can only be a finite number of non-negative inte- Z n gers, only a finite number of binomial moments h k i will We can also use the three-dimensional analog to eq 12,
5 P (R)= dr dr′δ[R − (r − r′)] . (14) e. Some pitfalls of the simplest models v v Z Z It is a helpful, heuristic view that the IT approach Fourier transformation relates P (R) to the form factor studies a model grand partition function associated with S(k) of the volume v, a molecular scale volume and involving effective interac- tions. Because this may be an unusual setting for the consequent statistical thermodynamic calculations, con- ˜ ik·r 2 P (k)= drP (r)e = |S(k)| , (15) ventional results are not guaranteed. Here we note some Z of the pitfalls that have been encountered for the sim- where plest models. These discussions emphasize that the ori- entation for development of this approach is to discover ik·r models that work. S(k)= dre . (16) Non-separability of chemical potentials of distant sites v Z within a Gaussian model. Consider two spheres of exclu- The Fourier transform eq 15 can be inverted as sion radius λ separated by a distance large compared to λ. The excess chemical potential of the two spheres should be ex ex 1 k r two times that of the individual spheres, µ (1, 2)=2µ . P (R)= dkP˜(k)e−i · . (17) (2π)3 The mean and variance of the particle-number fluctua- Z tions are also additive. For the combined volume, we will 2 2 2 As a generalization of Hill’s result for a single sphere,64 have m2 = 2m1 and σ2 = 2σ1 , where mi and σi are we find for a collection of N non-overlapping spheres of the means and variances of the particle-number distribu- radius d: tions of an individual sphere (i = 1) and the two spheres (i = 2). Following the flat default model, we approximate 14 4 R R3 p0 by a Gaussian form, P (R)=4πR2 Nπd3 − + θ(2d − R) 3 d 12d3 2 2    e−mi /2σi N−1 N 3/2 3 p0 ≈ . (20) 4 y − |R − r | 2 + πd3 (1 − x ) − ij ij (18) (2πσi ) 3 ij 3r Rd i=1 j=i+1 ij X X   We then find that thep corresponding chemical potential 5/2 5 yij − |R − rij | of an infinitely-distant pair of spheres is not exactly that + θ[R − (r − 2d)]θ(r +2d − R) , 60r Rd3 ij ij of two individual spheres, ij  

ex ex kBT 2 where rij is the distance of the centers of spheres i and µ (1, 2) − 2µ ≈− ln(πσ1 ) . (21) 2 2 2 2 2 2 j, xij = (R + rij − 4d )/(2Rrij ), and yij = R + rij − 2Rrij xij . Using this method, analytical results can also This non-additivity is a direct consequence of the Gaus- be found for rotation ellipsoids with axes a = b and c. sian model and can be simply repaired by insisting ac- Maximizing the information entropy. Maximization curately on zero probability for negative occupancies. In of the information entropy eq 5 is easily performed us- fact, the definition of our problem here requires that the ing Lagrange multipliers for the constraints. For con- occupancies be non-negative integers. straints on mean and variance, this leads to the expres- Additivity can be recovered by dividing the two sphere sion eq 6 for pn. Lagrange multipliers λ0, λ1, and λ2 volume into two strata: one for each sphere or, when the are then calculated such that the corresponding pn sat- spheres overlap, the individual strata can be defined by isfy the moment conditions eq 3. This corresponds to utilizing the plane of intersection of the spherical surfaces solving three non-linear equations with three unknowns as a bounding surface. Suppose again that the spheres which can be accomplished, for instance, using a Newton- are far apart and that the probability of occupancy of Raphson method. Alternatively, one can use standard one sphere is independent of the occupancy of the other. minimization packages, minimizing the squared differ- The cross correlation between the numbers n1 and n2 of ences of the left and right-hand sides of the moment solvent centers in the two non-overlapping strata is then constraint equations. This is consistent with the idea given by of minimizing the “thermodynamic potential” 2 hn1n2i = ρ dr1 dr2g(|r1 − r2|) kmax kmax j Zv1 Zv2 f(λ1,...,λkmax ) ≡ ln 1+ pˆj exp − λk 2  k  = ρ v1v2 = hn1ihn2i , (22) j=1 ! X kX=1   kmax  corresponding to uncorrelated particle-number fluctua- j + λk , (19) tions. An expanded IT model uses the joint probabilities k p(n ,n ) of finding n particles in v and n particles in Xk=1   1 2 1 1 2 v2. Including the constraint eq 22 derived from stratifi- here expressed in terms of the binomial moment infor- cation in that expanded IT model gives the correct re- mation and for a general default model. sult that the hydration free energy of the two spheres is

6 the sum of the hydration free energies of the individual An indication of the large-solute effects can already spheres. be seen in Figure 2. For solutes of size d = 0.3 nm The non-additivity in the simple model is caused by and larger, we find that p1 is depressed relative to the the possibility of having a zero occupancy of the volume flat default model. This means that it is relatively less v with a negative number −n of particles in sub-volume likely to find an isolated water molecule in a cavity than v1 that is compensated by a positive number n of parti- would be predicted from that simple model. Based on cles in sub-volume v2. By imposing explicitly that each the foregoing discussion, for solutes of increasing size we of the sub-volumes has zero occupancy when the total expect that p0 will be become larger than predicted from occupancy is zero, the non-additivity is eliminated. This the flat default model. argument illustrates that the restriction of the probabil- ities pn to non-negative integers can be significant. This f. Historical perspective example also emphasizes that arguments well justified about the center of a distribution can lead to fundamen- Scaled particle theory ideas. The IT approach can be tal errors when applied to the wings of a distribution. viewed in the context of preceding studies of the insertion 66 Macroscopic limit. For the flat default model, we can probability p0. Mayer and Montroll and later Reiss et 36 calculate the limit of large solutes. The variance in the al. expressed p0 in terms of higher-order correlation particle number is then given by the isothermal compress- functions of the bulk fluid, ibility χT , ∞ m 2 2 (−1) m hn i − hni p0 =1+ dr1 dr2 ··· drmρ = ρk Tχ for v → ∞ . (23) m! B T m=1 v v v hni X Z Z Z ×g(m)(r , r ,..., r ) , (26) For the flat default model and a spherical solute with 1 2 m exclusion radius λ, we find that the chemical potential where ρmg(m) is the m-body joint density for solvent grows with the volume of the solute, centers (here, water-oxygen atoms). These are standard ∂µ 2πλ2 combinatorial results, more frequently seen in forms such ≈ , (24) as67 ∂λ χT ∞ where we used the Gaussian approximation eq 20. The n p =1+ (−1)m , (27) flat default model predicts for the water density ρG(∞) 0 m m=1 at contact with a hard wall: X  

2 −1 ∂µ −1 and G(∞) = lim (4πλ ρkBT ) ≈ (2ρkBTχT ) , (25) λ→∞ ∂λ j + m n p = (−1)m , (28) which for water under standard conditions results in j j j + m m=0 G(∞) ≈ 8. The contact value for a flat wall, however, is X     well known to be G(∞) = p/ρkBT , where p is the pres- where n is the number of solvent centers within v. The sure. If we use bulk density and pressure of water under first of these can be derived directly by noting that −4 n n standard conditions, we obtain p/ρkBT ≈ 7.4 × 10 ; for p0 = h(−1+1) i where (−1+1) is an indicator func- 38 −5 saturated water vapor, we find p/ρkBT ≈ 2 × 10 . tion that is one for n = 0 and zero otherwise. These sums Clearly, the contact value at a flat wall, G(∞), would be truncate sharply if only a finite maximum number of par- grossly overestimated if the flat default model were used ticles can be present in the observation volume. They are in this macroscopic limit. of practical value for small volumes where the maximum In the limit of large solutes we expect the multipha- number of solvent centers that can possibly occupy that 38 sic nature of water to become important. Stillinger volume is small. For larger volumes, large terms of alter- pointed out that the solvation of hard spheres much nating sign appear in the expansion eq 27 that make it larger than a water molecule can be quantified by con- difficult to use estimated values for the factorial moments sidering that a molecularly thick vapor layer surrounds in this linear fashion. the solute. For large solutes, the formation of a cor- Equating the excess chemical potential µex to the respondingly large water-vapor bubble with subsequent quasi-static work of creating a cavity results in an ex- insertion into that bubble will cost less free energy than pression for p0 in terms of the contact value G(λ) of the inserting directly into the liquid phase. The simplest IT solvent density at the surface of the exclusion volume v. model uses a flat or Gibbs default model and does not For a spherical exclusion volume of radius λ we have:36,68 account for this possibility, as density fluctuations are d of either Gaussian or Poisson-like character, respectively. ex 2 More elaborate default models should however be able to µ = −kBT ln p0 =4πρkBT G(λ)λ dλ . (29) 0 account for the possibility of vaporizing a certain small Z volume. But clearly, defining such models would require This result is formally exact for a hard-sphere solute and 2 knowledge about the free energy of microscopic vapor- establishes that 4πρkBT G(λ)λ is the compressive force bubble formation which is a task as formidable as the exerted by the solvent on a hard spherical solute of diam- calculation of solvation chemical potentials. eter λ. If the solvent is considered to contain a hard core

7 itself, there is a maximum number of solvent particles ideal gas with water density. Reasonable and customary nmax that can occupy a certain volume v. This results values are used for the van der Waals radii of the solvent in useful, exact expressions for the case of nmax = 1 and interaction sites in each case. Figure 4 shows that the 2. For larger solutes, the expected macroscopic solvation most probable cavity radii are quite small in all those behavior37,38 cases as is expected for liquids. Also, the difference in the most probable cavity size between liquid n-hexane ex 3 2 µ ∼ p(4πλ /3)+ γ(4πλ ) , (30) and liquid water is not large, and smaller than the cor- responding difference between liquid water and an ideal is used as a basis for extrapolation. Here p is the pressure gas with water density. The fact that the differences and γ is the liquid-vapor surface tension. In the inter- seen here are slight is, presumably, defined by the fact mediate molecular region, the approximate asymptotic that the most basic units considered in n-hexane are the form is connected smoothly to the exact curve for small 37,38 methyl and methylene groups. These are not so different solutes. in size from a water molecule. In contrast, simple equa- Computer simulation studies. The cavity formation tion of state models might treat the n-hexane molecule as probability p0 lends itself to direct computer simula- a sphere of substantially larger size. It can also be noted tion studies by test-particle insertion. Spherical vol- that on a packing fraction basis, typical organic liquids umes v in water were studied extensively using com- are denser than liquid water.70 puter simulations.69,70,71,72,73,74 Similar insertion meth- ods were also used to study polymer solubilities.75 The One feature in Figure 4 that does distinguish water simulation studies of Pohorille and Pratt69,70,71 clarified is the breadth of the distributions. The distribution of a number of interesting speculations on hydrophobicity the nearest solvent site neighboring an arbitrary position and provided the first discriminating tests of theoretical in the liquid is narrower in liquid water. The widths of models for G(λ). these distributions are independent of the van der Waals radii assigned to the solvent sites. The interpretation is 1.0 that, compared to other liquids, water is “stiffer” when water • larger cavities must be opened. )

-1 random • ) (Å

S n-hexane • 0.5 2.0 water (R+R ) λ nn

p 1.0 G( n-hexane

0.0 0.0 -1 0 1 0 1 2 3 R (Å) λ (Å) Figure 4. Distributions pnn of the distance R + RS from a random point to the nearest neighboring site in Figure 5. Contact densities G(λ) of water and n- water, n-hexane, and an ideal gas with water density.71 hexane at the surface of hard-sphere solutes.69 G(λ) gives R is the van der Waals radius of the solvent. For each of S the compressive force of the solvent on a hard spherical the three fluids, the solvent van der Waals volumes are solute. represented by a superposition of spherical sites of one type only. The ideal-gas curve is the Hertz distribution 2 3 4πρ(R + RS) exp[−4πρ(R + RS) /3] with ρ the molec- This comparative stiffness can be analyzed further by ular (or oxygen) density of liquid water at 301 K and determining the contact functions G(λ) for water and hy- atmospheric pressure. The van der Waals radius of wa- drocarbon liquids and comparing the compressive force ter oxygen is RS =1.35 A;˚ that of united-atom carbons that such solvents exert on a hard spherical solute. As in n-hexane is RS =1.85 A.˚ shown in Figure 5, water exerts a higher compressive force on such a solute than typical hydrocarbon sol- One hypothesis that has sparked interest in recent vents. It is thus an accurate view on molecular-scale hy- years is the idea that the low solubility of inert gases drophobic effects that water “squeezes-out” hydrophobic 78,79 in liquid water is due to the small size of the water solutes. molecule and, consequently, the “interstitial” cavities will Field theory models. A perspective related to the be smaller in water than in coexisting organic liquids of IT approach pursued here has been adopted before by common interest.76,77 Since the present work deals di- Chandler.52 Equilibrium fluctuations of the water density rectly with such cavities, it is straightforward to test that are approximated by a Gaussian model. The partition idea. Figure 4 shows distributions71 of the distances from functions are then calculated for the two systems with an arbitrary point to the nearest interaction site for three and without a cavity of volume v present, using a field- cases of interest: liquid n-hexane, liquid water, and an theoretic method. Chandler then shows that this con-

8 tinuous Gaussian fluctuation model leads to the Pratt- b. Methane-methane potential of mean force Chandler integral equation theory.47 For sufficiently large (PMF) volumes, a Gaussian field model is expected to reproduce accurately the p if the continuous probability densities n We now focus on the simplest model of hy- of particle numbers are mapped to discrete values of n drophobic interactions exemplified by the free en- and the negative values of n are disregarded. ergy of bringing together two methane-size cavities in water.47,49,80,81,82,83,84,85 This free energy profile de- 40 simulation fines a PMF. We obtain the cavity contribution to the 35 information theory methane-methane PMF by calculating the excess chem- ex 30 ical potential, µ (r), of a cavity volume described by )

-1 25 two spheres each of radius d and separated by a dis- ex 20 tance r. The PMF is then defined as W (r) = µ (r) − ex

(kJ mol lims→∞ µ (s). This definition guarantees that W (r) → 15 ex

µ 0 for r → ∞, avoiding the problem of non-additivity dis- 10 cussed above for non-stratified volumes. 5 0 Figure 7 shows a comparison of the cavity PMF cal- 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 culated using IT with that obtained from explicit simu- R (nm) lations by Smith and Haymet.81 The flat default model has been used here and throughout this paper, unless Figure 6. Excess chemical potential of hard-sphere otherwise noted. Also, an exclusion radius of d = 0.33 solutes in SPC water57 as a function of the exclusion nm has been used for methane-size cavities correspond- radius d. The symbols are simulation results, compared ing to a distance where the methane-water oxygen radial with the IT prediction using the flat default model (solid distribution function reaches a value of 1.0 in commonly line).50 used models.86,87 The direct interaction between the two methane molecules has been subtracted from the simula- III. Results for elementary hydrophobic effects tion PMF to get approximately the cavity contribution to the methane-methane PMF. a. Hydration of simple spherical solutes 2 The probabilities pn of observing exactly n oxygen cen- ters inside spherical volumes, shown in Figure 2, are re- trans 1.5

produced accurately using IT with a flat default model. ) The corresponding excess chemical potentials of hydra- -1 ex tion of those solutes, µ = −kBT ln p0, are shown in Fig- 1

50 ex ) (kJ mol cis gauche ure 6. As expected, µ increases with increasing cavity φ W( radius. The agreement between IT predictions and com- 0.5 puter simulation results is excellent over the whole range information theory simulation d ≤ 0.36 nm accessible to direct simulation calculations 0 of p0. 0 π/4 π/2 3π/4 π φ (rad) 2 1 Figure 8. Water contribution to the torsional PMF 50 88 0 of butane from IT and explicit computer simulation )

-1 (dashed line). -1 -2 -3 In agreement with the simulation data, the PMF calcu- W(r) (kJ mol lated using IT shows features characteristic of hydropho- -4 information theory bic interactions. A free energy minimum is observed cor- -5 simulation responding to overlapping hard spheres. That deep min- 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 imum is separated by a barrier from a second minimum r (nm) at a distance of about 0.7 nm. The second minimum corresponds to solvent-separated pairs of methane-size Figure 7. PMF between two methane-size cavities in cavities. In addition, we observe a third shallow min- SPC water,57 comparing explicit computer simulations81 imum which was also seen in computer simulations by (dashed line) with the IT prediction50 (solid line). Note L¨udemann et al.85 The equilibrium between the solvent- that the simulation curve is based on continuous solute- separated and contact minimum as a function of pressure solvent interactions, subtracting the methane-methane provides insights into pressure denaturation of proteins, interaction potential from the methane-methane PMF. as described in section V.

9 c. Solvent contributions to the torsional equilib- with the corresponding water densities are used as input rium of butane in the IT model to calculate chemical potentials of hydra- tion for hard solutes. Solute sizes of d =0.28, 0.31, 0.33, As a final example of a simple hydrophobic effect, Fig- and 0.345 nm were considered, corresponding approxi- ure 8 shows the cavity contribution to the torsional PMF mately to neon, argon, methane, and xenon, respectively. for n-butane in water. IT calculations are compared with Figure 9 shows the chemical potentials for each of the 88 explicit computer simulations by Beglov and Roux. We spherical solutes as a function of temperature T , for T be- find that the the cavity PMF favors the compact cis tween 275 and 625 K. Also shown are the results of direct (φ = 0) structure over the extended trans structure by simulation calculations using test-particle insertion to- about 1.8 kJ/mol. The gauche structure (φ = π/3) is gether with representative statistical errors. We find ex- favored over trans by about 0.7 kJ/mol. Those differ- cellent agreement between simulation and theory for the ences and the overall cavity PMF for the torsional iso- chemical potentials over the whole temperature range. merization of butane are in excellent agreement with the The chemical potentials are approximately parabolic ex- simulation data. cept at the highest temperatures, and the curves for dif-

40 ferent solute sizes are shifted vertically with the maxi- Xe mum at about 400 K. 35 Me 30 100

Ar 25 50 (kJ/mol) ex

µ 20 Ne (J/mol/K) 0 sat T) ∂

15 / Ne ex −50

∂ µ Ar 10 −( Me 300 350 400 450 500 550 600 T (K) −100 Xe

300 350 400 450 500 550 Figure 9. Chemical potential of hard-sphere so- T(K) lutes with exclusion radii corresponding to neon, argon, methane, and xenon as a function of temperature along Figure 10. Hydration entropy of hard-sphere so- the saturation curve of water.14 The solid line is the IT lutes with exclusion radii corresponding to neon, argon, prediction. The symbols are computer simulation data methane, and krypton as a function of temperature along 14 from insertion. Both simulation and theory data are the saturation curve of water. Shown is the entropy de- based on SPC water57 data. fined in eq 31.

We obtain a solvation entropy by taking the derivative IV. Temperature dependence of hydrophobic hy- of the chemical potential along the saturation curve, dration ∂µex S = − . (31) a. Solvation chemical potentials sat ∂T  sat The hydrophobic effect is often associated with charac- Additional equation-of-state contributions to the stan- 11,89 teristic temperature dependences. One of the more dard solvation entropy are negligible, estimated to be less astonishing observations is that the entropies of trans- than 1 and 10 J/(mol K) for temperatures T < 450 and fer of nonpolar molecules from gas phase or a nonpolar T < 550 K, respectively. Figure 10 shows the temper- solvent into water converge at a temperature of about ature dependence of the entropy Ssat for the different 400 K to approximately zero entropy change. Similar solutes. The entropies are large and negative at room behavior was also seen in microcalorimetry experiments temperature for all the solutes and decrease in magni- 6,7,8 on unfolding of several globular proteins. This anal- tude with increasing temperature. The temperature de- ogy to the transfer data supported the hydrophobic-core pendence of entropies is approximately linear with slopes model of protein folding: during unfolding predominantly increasing with the increasing solute size. The resulting nonpolar residues are transferred from a mostly nonpolar heat capacity, protein core into an aqueous environment. We have used the IT model to clarify the molecular ori- ∂Ssat C = T , (32) gin and the quantitative details of this striking entropy- sat ∂T sat convergence behavior.14 Computer simulations were car-   ried out to calculate the water-oxygen radial distribution is large and positive (approximately 40 cal mol−1 K−1 for function g(r) at several temperatures along the experi- a methane-size cavity). Moreover, the entropies converge mental saturation curve of water. The g(r)’s together at about 400 K to approximately zero entropy, although

10 at closer inspection the temperature range of the conver- b. Origin of entropy convergence in hydrophobic gence region is several 10 K and the entropy is not exactly hydration and protein folding zero at convergence. To understand the origin of this ap- proximate convergence we determine in the following the To quantify the different factors contributing to the factors entering into the IT model. entropy convergence,14 we use the approximate Gaussian representation eq 20 for p0 with mean m = hni = ρv and variance σ2 = hn2i − hni2. This results in an estimate of µex ≈ Tρ2(T)x(v) the chemical potential,

ex 2 2 2 2 µ ≈ Tρ {kBv /2σ } + T {kB ln(2πσ )/2} . (33)

T entropy The second term is smaller than the first, and depends only logarithmically on the solute size. The solvation chemical potential may therefore be lowered by lower- ing the product of temperature and density Tρ2(T ), or 2 µex ≈ Tρ2(T)x(v)+Ty(v) by increasing the particle-number fluctuations σ . The temperature dependence of Tρ2(T ) along the saturation curve is non-monotonic. On the other hand, we find that the variance σ2(T, v) changes only little over the range of 14 T temperatures in Figure 9. Accordingly, we can approx- entropy imate the chemical potential as

µex ≈ Tρ2(T )x(v)+ Ty(v) , (34)

µex = -kBT ln p0 2 2 where the functions x(v) = kBv /2σ and y(v) = 2 kB ln(2πσ )/2 depend only on the solute volume, not on the temperature. The contributions of the various terms to the entropy convergence are illustrated schematically T entropy in Figure 11. The term Ty(v) in eq 33 is generally small. If we neglect Ty(v), the solvation entropies eq 31 con- verge exactly at zero entropy for solutes of different size, and the temperature of convergence corresponds to the maximum of Tρ2(T ) along the saturation curve. When Figure 11. Schematic representation of the different the second term Ty(v) is included, the entropy at conver- factors resulting in approximate convergence of solvation gence is different from zero, but the convergence is still entropies. exact. However, both x(v) and y(v) show weak temper- ature dependences. When those are considered as well, the resulting entropy convergence is no longer exact but occurs over a temperature range that is several 10 K wide 10-3 benzene for the solutes considered here. n-heptane carbon-tetrachloride Our analysis shows that the main factors leading to water an entropy convergence are derived from the proper- ties of pure water. They are: (1) the non-monotonic

) 2

-1 behavior of Tρ (T ) along the saturation curve and (2) the weak temperature dependence of the particle-number (bar

T 2

χ fluctuations σ (T, v) for solute excluded volumes. The macroscopic analog of σ2 is the isothermal compressibil- 10-4 ity χT . For macroscopic volumes v, eq 23 relates the 2 2 two, σ = ρ vkBTχT . Figure 12 compares the isother- mal compressibility of water and several organic solvents 280 300 320 340 360 380 400 420 up to their critical point as a function of temperature. T (K) Indeed, we find that water shows far weaker tempera- ture variations of χT than the organic solvents, which Figure 12. Comparison of the isothermal compressibil- can be seen as the origin of the entropy convergence. In 2 2 ity of water and nonpolar solvents as a function of tem- the macroscopic limit, the variance σ = ρ vkBTχT is 2 perature along their respective saturation curves. Data a product of Tρ (T ) with a maximum at about 440 K are taken from Ref. 90. and χT with a minimum at about 320 K. This results in only small variations of σ2 in the range of temperatures between 273 and 420 K.

11 5 To quantify this inverted “liquid-hydrocarbon model,”

4 -0.6 we focus on the stability of hydrophobic aggregates formed in water as a function of pressure. Aggregate 3 -1

W (kJ/mol) formation of spherical, methane-size solutes serves as an ∆ 2 -1.4 idealized model of the stability of protein hydrophobic 0 7000 cores. The free energy of forming such aggregates can be 1 p (kbar)

W (kJ/mol) expressed in terms of two-body and higher order PMF’s 0 between the methane-size solutes in water. In the follow-

-1 ing, we will study the pair and three-body interactions of methane-like solutes in water which control the stability -2 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 and formation of larger aggregates. r (nm) b. Effect of pressure on hydrophobic interactions Figure 13. Pressure dependence of the PMF between two methane-like solutes in water. The PMF’s were Figure 13 shows the PMF’s W (r) between two calculated from information theory using a flat default methane-like particles in water for pressures up to about model.15 A methane-methane Lennard-Jones interaction 7 kilobar (700 MPa) calculated using IT.15 The PMF’s was added. SPC water57 g(r)’s were used. The arrows exhibit two minima, a contact minimum at about 0.4 indicate changes with increasing pressures from −160 to nm distance and a solvent-separated minimum at a dis- 7250 bar. tance of about 0.7 nm. The two minima are separated by the desolvation barrier. The PMF’s are normalized V. Pressure denaturation of proteins at the solvent-separated minimum to illustrate that the contact minimum loses importance with increasing pres- a. Denaturation by water transfer into the pro- sure. The free energy of the contact minimum increases tein interior relative to the solvent-separated minimum by about 0.9 kJ/mol when the pressure is increased from 1 bar to 7 Proteins can be denatured by pressures of typically kilobar. In addition, the desolvation barrier between the a few kilobar.19,20,21,22,23,24,25,26 This pressure-induced two minima also increases with increasing pressure. Sim- unfolding has been a long-standing puzzle in our under- ilar pressure destabilization of contact configurations was standing of protein stability. The “liquid-hydrocarbon observed for three methane molecules in contact.15 model” of protein unfolding11 explains unfolding as a The effect of pressure on pair PMF’s between two transfer of predominantly hydrophobic residues from the methane molecules has also been studied by computer protein interior into the aqueous solvent. Indeed, this simulation.92 These simulation calculations showed a model accurately describes the thermodynamics of tem- pressure destabilization of contact pairs that is com- perature denaturation, in particular, the convergence of parable in magnitude to the IT calculations. In addi- the entropies of unfolding. However, as Kauzmann13 tion, simulation calculations also showed the formation pointed out, such a description fails to explain the pres- of methane aggregates in water at low pressures that dis- sure denaturation of proteins. In particular, the volume solved at high pressures.93 changes associated with transferring hydrocarbons from The pressure destabilization of hydrophobic contact a nonpolar phase into water exhibit behavior exactly op- configurations can be understood from the following sim- posite to that upon pressure unfolding as a function of plified model: At low pressures, the interstitial space be- pressure. The volume change of hydrocarbon transfer is tween two large nonpolar solutes is energetically unfavor- negative at low pressure and positive at high pressure;91 able for water molecules.55,94 As the pressure increases, the volume change of protein unfolding is positive at low the space between the nonpolar solutes is more likely pressure and negative at high pressure.13 to be occupied by water, increasing the importance of To explain the pressure denaturation of proteins, we solvent-separated configurations relative to contact con- invoke additional information about the structural prop- figurations. A more detailed analysis indeed shows that erties of pressure-denatured proteins. Nuclear mag- the destabilizing effect of pressure on hydrophobic con- netic resonance (NMR) measurements reveal consider- tacts increases with increasing size of the two nonpolar ably more structural organization in pressure-denatured solutes.15 proteins compared to that in temperature-denatured 27 21,22,23,24 proteins. These and other experiments indi- c. Stability of proteins cate that the pressure-denatured proteins are relatively more compact than temperature-denatured proteins. Ac- We have shown that the stability of hydrophobic aggre- cordingly, we depart from the commonly used model ex- gates decreases with increasing pressure as a result of wa- plaining protein unfolding thermodynamics as a transfer ter penetration. When applied to the hydrophobic-core of nonpolar residues from the protein interior into so- model of proteins, we predict that pressure denatures lution. Instead, we describe pressure denaturation as a proteins by swelling.15 A swelling mechanism has also transfer of water molecules from the aqueous phase into been proposed for urea-induced protein denaturation.95 the hydrophobic core of a protein. Under sufficiently high pressures water molecules will

12 ‡ ‡ intercalate into the hydrophobic protein interior. The a function of pressure. Thus, ∆Wf and ∆Wu are the resulting structures of pressure-denatured proteins are barriers for forming and breaking hydrophobic contact predicted to be more ordered and more compact than configurations, corresponding to “folding” and “unfold- ‡ those of temperature-denatured proteins. Recent small- ing” reactions, respectively. We find that ∆Wf and 28 angle scattering experiments indeed showed that the ‡ ∆Wu both increase approximately linearly with increas- pressure-denatured staphylococcal nuclease protein has ‡ ing pressure. Activation volumes defined as ∆v = a considerably smaller radius of gyration of R ≈ 3.5 f/u g ‡ ‡ nm compared to the temperature-denatured protein with ∂∆Wf/u/∂p are both positive, ∆vf = 3.8 ml/mol and ‡ Rg ≈ 4.6 nm. ∆vu = 1.6 ml/mol, for folding and unfolding reactions, respectively. Increasing pressure is thus expected to slow d. Formation of clathrate hydrates down both the “folding” and the “unfolding” reactions. Indeed, experimental studies of the pressure-dependent This physical picture of pressure denaturation of pro- folding and unfolding kinetics of the protein staphylococ- teins is relevant also for the formation of clathrate hy- cal nuclease showed a reduction of both rates with in- drates, crystalline compounds forming at elevated pres- creasing pressure.97 The experimentally observed activa- sures from non-polar molecules and water. As solid de- tion volumes for folding and unfolding of staphylococcal posits in gas pipelines, clathrate hydrates cause problems ‡ ‡ nuclease are ∆Vf = 92 ml/mol and ∆Vu = 20 ml/mol, in natural gas transmission. Methane-water clathrate hy- respectively. From the ratio of overall experimental ac- drates, for instance, form at pressures above about 44 tivation volumes and theoretical activation volumes cal- MPa at a temperature of 298 K, with a stoichiometry of culated per hydrophobic contact, we can estimate the about 6-7 water molecules per methane and a methane- number of hydrophobic contacts broken in the folding methane nearest-neighbor distance of about 0.62-0.74 96 transition state. For staphylococcal nuclease we estimate nm. At lower pressures, the two fluid phases sepa- that number to be between 10 and 25, in good agreement rate. Formation of the clathrate structure is consistent with the predictions of energy landscape theory.15,98 with stabilizing the solvent-separated minimum in the methane-methane PMF relative to the contact minimum, 50 as predicted from the analysis of pressure effects on hy- 45 drophobic interactions. In addition, the present theory predicts that larger hydrocarbons require lower pressure 40 for clathrate formation,15 in agreement with the experi- 35 mental observations that led to the gas-gravity method 30 of phase determination.96 25 ( kJ/mol )

ex 20

5 µ 15 4.5 ∆ ³ 10 vu = 1.6 mL/mol 4 5 3.5 0

(kJ/mol) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 ³ 3 W

∆ d ( nm ) 2.5 ∆ ³ vf = 3.8 mL/mol 2 Figure 15. Excess chemical potential of hard-sphere 1.5 solutes in aqueous NaCl solution as a function of solute 0 2000 4000 6000 p (bar) size, given by the exclusion radius with water, and salt concentration. Figure 14. Pressure dependence of the desolvation bar- rier between the contact and solvent-separated minimum VI. Salt effects on hydrophobic hydration in the methane-methane PMF of Figure 13. The activa- tion free energy for forming (open squares) and breaking Addition of salts to water generally reduces the solu- hydrophobic contacts (filled squares) is shown as a func- bility of nonpolar solutes. The Hofmeister series29 ranks tion of pressure. salts, in part, according to this reduction of the solubility of nonpolar compounds. Here, we do not attempt to pro- e. Pressure effects on protein folding kinetics vide a complete description of salt effects on hydrophobic hydration. Instead, we present some preliminary results We noted before that increasing pressure results in for the solubility of small hard-sphere solutes in a solu- a higher desolvation barrier between the contact con- tion of sodium-chloride (NaCl) in water. In particular, figuration and the solvent-separated minimum. Fig- we compare results for NaCl concentrations of 1, 3, and ‡ ‡ ure 14 shows the barrier heights ∆Wf and ∆Wu from the 5 mol/l as well as pure water. The three salt solutions solvent-separated and contact minimum, respectively, as were studied using molecular dynamics simulations ex-

13 tending over 1 nanosecond each. For simulation details, the solvation chemical potential is largely determined by see Ref. 99. the free energy of forming a vapor bubble of appropriate Figure 15 shows the excess chemical potential of spher- size. The free energy of forming large bubbles can be ical solutes as a function of their exclusion radius d with approximated by the product of surface area and liquid- water. The solute-solvent interactions are modeled as vapor surface tension, since the contribution from the 38 those of hard spheres with radii Rw = 0.132 nm of wa- pressure-volume work is small for water. Accordingly, + ter, RNa = 0.085 nm of Na , and RCl = 0.18 nm of we expect that the solvation chemical potential grows Cl−. The exclusion radii with water, Na+, and Cl− are with the surface area for large solutes. Stillinger38 gave d = R + Rw, R + RNa, and R + RCl, respectively, where a justification of this based on the water structure near R is the solute hard-sphere radius. We find that adding a large hard solute. The contact-value theorem gives, in salt indeed increases µex and thus decreases the solubil- the limit of a flat hard wall, a water density at contact of ity for a solute of a given size. For a methane-size solute about 7.4 × 10−4 times the bulk density of water. Still- (d = 0.33 nm), the excess chemical potential increases inger argued that this “dewetting” implies the presence of by about 10 kJ/mol from about 34 to 44 kJ/mol when a molecularly thick vapor layer near a hard wall, and as a increasing the NaCl concentration from 0 to 5 mol/l. result, area laws with surface tension coefficients similar To describe the salt dependence using the IT approach, to that of the liquid-vapor surface tension of water. we extend our formulation to solvent mixtures. For In an attempt to bridge the gap between microscopic ionic solutes, the strong interactions between water and and macroscopic solvation behavior, we have recently ions give rise to the formation of compact hydrated ions developed a quantitative description of the solvation of which result in non-trivial density fluctuations near ions. nonpolar solutes as a function of their size.55 The wa- Therefore, instead of trying to model the full distribu- ter structure near spherical55 and non-spherical Lennard- tion of particle number fluctuations p(nW,nNa,nCl), we Jones solutes100,101 was found to be largely insensitive to focus on a more tractable subset, which contains the the details of the solute-water interactions except for the probability of the event we are ultimately interested in, solute size, motivating the use of perturbation theory. p(nW = 0,nNa = 0,nCl = 0). Further, we factorize this For spherical solutes, we find that the radial distribution probability into an ionic part and a conditional water function of water around the solute can be described ac- part, curately by

p(nW =0,nNa =0,nCl =0)= p(nNa =0,nCl = 0) gsw(r) = exp[−βurep(r) − βω(r)+ C(r)] , (36) ×p(nW =0|nNa =0,nCl = 0) , (35) where p(nW = 0|nNa = 0,nCl = 0) is the conditional where urep(r) is the repulsive solute-water interac- probability of having no water-oxygen atoms in the wa- tion from, e.g., Weeks-Chandler-Anderson separation;102 ter exclusion volume, given that there are no ions in C(r) is a renormalized attractive interaction; and ω(r) the respective ion exclusion volumes. Such a factor- represents the free energy of a test water molecule that ization combined with accurate simulation data pro- does not interact with the solute as a function of the dis- vides insights into salt effects on hydrophobic solubili- tance r from the solute,55 thus containing the non-trivial ties at a molecular level. The contributions to the so- solvent contributions. Interestingly, ω(r) is directly re- (1) lute chemical potential, −kBT ln p(nNa = 0,nCl = 0) lated to the one-body Lagrange multiplier ω (r) of the and −kBT ln p(nW = 0|nNa = 0,nCl = 0) both increase continuous IT eq 11. We find that ω(r) is dominated by a with increasing NaCl concentration. The increase in the contribution ω0(r−r0) that is identical for various solutes salt contribution is primarily due to an increase in the except for a simple radial translation with the solute size mean of the p(nNa,nCl) distribution. Interestingly, the r0. The remainder ∆ω(r − r0), however, changes non- mean of p(nW|nNa = 0,nCl = 0) is approximately inde- trivially with the solute size, acting as a cavity-expulsion pendent of salt concentration. It is the decrease in the potential for the test water molecule. variance of this conditional water distribution with in- The cavity-expulsion potential, ∆ω(r − r0), quantifies creasing NaCl concentration that leads to a decrease in primarily the loss of energetic interactions of a test wa- p(nW =0|nNa =0,nCl = 0). The direct salt effect of the ter molecule as it crosses the solute-water interface from reduction in free volume due to overlap with salt ions and the water phase towards the center of the solute. For the indirect salt effect of changing the water structure ac- small solutes, ∆ω(r − r0) is negligible. As the solute count for roughly one half each of the total increase in size increases, this loss of interactions becomes impor- ex solute chemical potentials. tant, reaching −µw , the negative excess chemical poten- tial of water, in the limit of large solutes. For solutes VII. New directions: Cavity expulsion and area of diameters more than about 0.5 nm the cavity expul- laws sion results in a slight depression in the water density at the solute-water interface. We find that this “weak We mentioned in section IIe that for macroscopic so- dewetting” with increasing solute size is sufficient to give lutes physical effects intrude that are not captured in the an approximate surface-area dependence of the solvation simplest IT models. This is a consequence of the mul- chemical potential, which would otherwise be dominated tiphasic nature of water. In the limit of large solutes, by solute volume terms.55

14 VIII. Concluding remarks cial account of the known details of hydrophobic solva- tion patterning. These theories yield satisfactory predic- How is water different from hydrocarbon liquids as a tions for the simplest hydrophobic effects in part because solvent for nonpolar solutes? This is the foremost ques- they incorporate information about the equation of state tion asked of theories of hydrophobic effects and several and the oxygen-atom pair correlations of water. A next of the points above constitute relevant parts of the an- stage of basic molecular theory of hydrophobic effects swer. It is appropriate therefore to ask this question more will surely address the hydration structure in more detail specifically and to organize features of the answer that while at the same time preserving equation of state infor- are provided by the IT models considered here. One ba- mation such as the low, temperature insensitive values of sic aspect of the answer is straightforward and on that the isothermal compressibility that have been identified score we do not contribute anything further: water is dif- as particularly important. ferent because of the mechanical potential energies of the IT models of the solvation thermodynamics can eas- interactions among water molecules, including hydrogen ily be generalized to various aqueous and non-aqueous bonding, orientational specificity, and cooperativity. solvents and mixtures, combined with other approaches, In the context of solvation thermodynamics, the ques- or extended to “unusual” models of water, such as a re- tion how water differs from other solvents solicits infor- cently proposed two dimensional model of water106 or an mation about a particular structural pattern of the sol- isotropic water model without directional hydrogen-bond vent that might be principally responsible for hydropho- interactions.83 We have successfully adapted the method bic hydration free energies. For example, clathrate-like to study the solubility of small molecules in polymeric structures have been invoked to explain hydrophobic hy- fluids.107,108 Crooks and Chandler109 compared solva- dration phenomena.103 The answer given by the IT model tion chemical potentials of hard spheres of varying sizes is that water differs from common hydrocarbon solvents, in hard-sphere fluids to simulation data, finding that a in the first place, because the distribution of pairs of ex- Gibbs prior gives a good description of the density fluctu- clusion spheres, oxygen-oxygen sites in water, is distinc- ations and the solvation thermodynamics. Applications tive and has a distinctive temperature dependence. In of the IT method to study phase equilibria have been the second place, the isothermal compressibility of water, suggested.110 Lum, Chandler, and Weeks111 have com- that can be obtained by integration of the oxygen-oxygen bined field-theoretic methods52 with the IT approach to distribution, is insensitive to temperature compared to describe the solvation thermodynamics and structure of the same property of organic solvents. Since the mini- mesoscopic and macroscopic spherical solutes in water mum in the temperature dependence of the isothermal which is tightly associated with “dewetting” of the so- compressibility is very distinctive, this answer is remark- lute surface.38,55 able and remarkably simple. The development of IT models is an ongoing effort tar- From our analysis of pressure effects on hydrophobic geting several directions. A better description for molec- interactions and protein stability, we can deduce fur- ularly large solutes can be expected from improved, phys- ther peculiarities of water. Increasing the pressure in ically motivated default models that incorporate the mul- water shifts the balance from attractive and directional tiphasic nature of water to account for the free energy hydrogen-bonding forces to repulsive and isotropic pack- of forming microscopic vapor bubbles.111 IT models can ing forces between water molecules.104 This results in easily be generalized to study hydrophobic interactions “energetic frustration” of water molecules in the water in inhomogeneous environments, for instance, to describe phase,105 which in turn reduces the relative cost of incor- ligand binding to proteins. Section II c gives an outline porating water molecules into a hydrophobic aggregate. of a continuous IT, which establishes connections to the At least two factors make water special in this context: concepts of cavity expulsion and dewetting of nonpolar its large negative excess chemical potential prevents wa- surfaces.55 The volume stratification discussed in section ter from penetrating hydrophobic aggregates under nor- II d leads naturally to predictions of the hydration struc- mal conditions,55 thus driving the formation of protein ture. hydrophobic cores. Under sufficiently high pressure, how- In conclusion, we believe that the basic IT model pro- ever, the relatively small water molecules occupy pre- vides a simple theoretical framework to study many hy- viously empty cavities, exploiting imperfections in the drophobic phenomena, as it already led to a new under- packing of the protein interior. Note that the small size standing of the temperature dependence of hydrophobic of water molecules here is not used as an argument that hydration,14 and resulted in a new description of the pres- “interstitial” cavities are smaller in water than in typical sure denaturation of proteins.15 organic liquids,76,77 as discussed in section IIf, but that water molecules can occupy small cavities in the protein interior. This incorporation of water molecules is favored ACKNOWLEDGMENTS further by the versatility of water molecules which can form favorable hydrogen-bonding interactions in many L.R.P. thanks Andrew Pohorille for extended collab- different environments, particularly in the interior of a orations on the problems of hydrophobic effects. This protein which is never entirely nonpolar. work was in part supported by the U.S. Department of These arguments also explain in part the success of Energy through the Los Alamos National Laboratory simple theories, such as SPT and PC, that take no spe- LDRD-CD grant for an “Integrated Structural Biology

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