Molecular-scale hydrophobic interactions betweenhard-sphere reference solutes are attractiveand endothermicMangesh I. Chaudhari, Sinead A. Holleran, Henry S. Ashbaugh, and Lawrence R. Pratt1

Department of Chemical and Biomolecular Engineering, Tulane University, New Orleans, LA 70118

Edited by Benjamin Widom, Cornell University, Ithaca, NY, and approved November 5, 2013 (received for review July 1, 2013)

The osmotic second virial coefficients, B2, for atomic-sized hardspheres in water are attractive (B2 < 0) and become more attrac-tive with increasing temperature (ΔB2/ΔT < 0) in the temperaturerange 300 K ≤ T ≤ 360 K. Thus, these hydrophobic interactions areattractive and endothermic at moderate temperatures. Hydro-phobic interactions between atomic-sized hard spheres in waterare more attractive than predicted by the available statistical me-chanical theory. These results constitute an initial step towarddetailed molecular theory of additional intermolecular interac-tion features, specifically, attractive interactions associated withhydrophobic solutes.

protein folding | self-assembly | Pratt-Chandler theory

Solvent-mediated, noncovalent interactions within biomolec-ular structures are decisive for their stability and function-

ality over an extended range of conditions (1). Hydrophobicinteractions—a principal category of noncovalent interactions inwater—exhibit strong and characteristic temperature depend-ences (2). Molecular-scale theories of hydrophobic interactionsare judged by their ability to capture those temperature depen-dences. Defensible theories might eventually illuminate a validexplanation and might find broader utility in modeling bio-molecular structure and function.The osmotic second virial coefficient

B2 = limR→∞

−2πZR

0

½ gAAðrÞ− 1�r2dr8<:

9=;≡ lim

R→∞B2½R� [1]

has so far been the only direct experimental check on themolecular theory of hydrophobic interactions between slightlysoluble gases (A) in liquid water (3–6). Here gAA(r) is the usualradial distribution function of AA pairs at infinite dilution. Be-cause of low solubilities for solutes of interest, the necessary experi-ments are challenging. The initial comparisons between the onlyavailable molecular-scale theory, the Pratt–Chandler (PC) theory(7), and directmeasurements ofB2 showed poor agreement (3–5, 8).Explanations for the discrepancy have been suggested (9–11), butthe underlying disagreement has persisted (12).One explanation for this discrepancy focused on the differ-

ences between the actual interactions for accessible experimentalcases and the hard-sphere solute–water interactions natural forthe molecular theory. In this setting, direct high-resolution de-termination of hydrophobic interactions for the hard-spheremodels treated by the theory would be a helpful step, but that hasnot been accomplished so far. The case of atomic-sized hard-sphere solutes has not been treated specifically, mostly becausehard-sphere models are inconvenient in available moleculardynamics simulations.These problems are of basic importance because hydrophobic

interactions are acknowledged as the dominant factor that drivesprotein folding (13, 14). Hydrophobic interactions are also ex-pected to become more favorable with increasing temperature

for physiological temperatures. Hydrophobic interactions canthen be described as favorable for aggregation and endothermicat moderate temperatures. This is a primary conceptual puzzlethat theories of hydrophobic effects should clarify.Summaries of the substantial efforts to clarify these issues

are available (15, 16). As an example, B2 values for the specificcases of Ar and CH4 solutes have been estimated (6, 17) to beunusually small due to substantial cancellation between con-tributions from repulsive and attractive intermolecular forces.In the same context (6), B2 for Kr(aq) has been estimated tobe repulsive (positive). Of course, interactions that are re-pulsive on balance raise the osmotic pressure and are charac-terized by positive values of B2. A broad conclusion is thatattractive and repulsive interactions can play conflicting roles(18) with the consequence that merely realistic simulation ofa case of interest, e.g., CH4, might not provide transparentphysical conclusions. In addition, the integrated quantity B2,particularly the R → ∞ limit, is a subtle target for molecularsimulation calculations.In parallel with simulation efforts, the foundation of the mo-

lecular theory of hydrophobic effects underwent a surprisingrenovation (19, 20). The physical concern for the PC theory (7),the available theory for gAA(r) for hard-sphere solutes, was itsdisregard of orientational correlations between a solute andneighboring water molecules. That orientational structure hadlong been regarded as the essence of the challenge of a statisticalmechanical theory of hydrophobic effects. Of course, the orien-tational structure is implicit in the experimental equation of statewhich is used in the molecular theory (19, 20).

Significance

Hydrophobic interactions and other solvent-mediated non-covalent interactions are decisive for the stability and func-tionality of biomolecular structures over an extended range ofconditions. A molecular theory should explain characteristictemperature dependences and help in modeling biomolecularstructures and functions. So far, there has been just one theoryof hydrophobic interactions, the Pratt–Chandler (PC) theory.The PC theory has not been definitively tested because of lack ofquantitative correspondence between systems convenient fortheory, simulation, and experiment. The present work adaptssimulation results to atomic-scale hard-core models of the PCtheory. Surprisingly, from the perspective of the PC theory,these results show that atomic-scale hydrophobic interactionsbetween hard-sphere solutes are attractive and endothermic,thereby identifying a target for improved theories.

Author contributions: M.I.C., S.A.H., H.S.A., and L.R.P. designed research, performed re-search, analyzed data, and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.1To whom correspondence should be addressed. E-mail: lpratt@tulane.edu.

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The point of view that the water orientational structure isessential was analyzed by separating the hydration entropy ofhydrophobic solutes into translational and orientational con-tributions (21, 22). It was found that the orientational contri-bution to the hydration entropy could be about one-third of thewhole, and thus not at all negligible.At about the same time it was noticed that the whole of the

statistical mechanical theory could be formulated without ex-plicit identification of orientational structure (20, 23–26). Thatorientational structure becomes involved in the theory thusrenovated (19, 20) by inclusion of n-body correlations with n ≥3 (19, 25–27). In the information theory format (28–30), n-bodycorrelations can be added incrementally. When that is donewith the natural Poisson default model, it is found that freeenergy predictions worsen in going from n = 2 to n = 3 (28–30).The n = 2 theory is accidentally accurate in this sense. This un-expected accuracy cannot be explained by the argument that then = 2 theory is consistent with the successful description ofpacking effects in the Percus–Yevick theory for the hard-sphere fluid for the following reason: when the informationtheory is designed to be exact for the case of the hard-spheresolvent, by using the default model numerically determinedfor that case, the free energy predictions worsen again (28).Of course, a sophisticated default model such as the defaultmodel numerically determined for the hard-sphere solventdoes reflect n-body correlations for n ≥ 3. Thus, an n = 2theory is simple and realizable and can be accurate. However,accuracies for specific physical issues are questions forspecific verification.A conclusion of this discussion is that for the assessment of

attractive interactions associated with hydrophobic solutes, areference system, e.g., the hard-sphere solute case, is not wellestablished. The work below determines gAA(r) numericallyexactly for hard-sphere solutes in a computer simulation modelof water and thus takes the initial step toward a detailed mo-lecular theory of additional intermolecular interaction fea-tures. This is accomplished by developing in the next sectiona specialized analysis that can be applied to simulation dataobtained from standardized calculations on sufficiently largespace and time scales.

AnalysisWe seek the cavity distribution function

yAAðrÞ= exp½ βuAAðrÞ� gAAðrÞ [2]

for atomic-sized hard spheres on the basis of the potentialdistribution theorem (or test particle) approach (31, 32)

yAAð1; 2Þ=

De−βΔU

ð2ÞAA

���1; 2E0D

e−βΔUð1ÞA

E0

De−βΔU

ð1ÞA

E0

: [3]

Here 1=kBβ=T. Eq. 3 anticipates evaluation with trial place-ments of hard spheres at specific points, such as the two points(1, 2). yAA(1,2) = yAA(r) will depend only on the magnitude r of thedisplacement between positions 1 and 2. The notation ⟨. . .⟩0 indi-cates the average over the configurations of water without thesolutes present. ΔUð1Þ

A =UðN + 1Þ−UðNÞ−Uð1Þ is the bindingenergy for an inserted A atom and for the hard-sphere caseconsidered here is either zero (no overlap with a water oxygenatom) or positive infinity. Thus, e−βΔU

ð1ÞA is an indicator function for

permissibility of an insertion at a specific point. We will treat thecase that ΔUð2Þ

AA for two trial placements is additive, ΔUð1ÞA +ΔUð1Þ

Afor the AA atoms considered.We rearrange this formula to make the numerical interpre-

tation transparent. Note that the denominator factors of Eq. 3,

being averages of indicator functions, are probabilities. Followingthe primitive understanding of conditional probabilities pðAjBÞ=pðABÞ=pðBÞ; we use one of those denominator probabilities tointroduce the expectation conditional on permissibility of aninitial insertion. Taking the position of that first insertion to bethe origin 0

→, we write

yAAðrÞ=

De−βΔU

ð1ÞA

���r→E0D

e−βΔUð1ÞA

E0

; [4]

where r→ is the position of a trial placement relative to a permis-sible insertion. The average indicated in the numerator is condi-tional on the permissible placement at 0

→, although we do not set

up a further notation for that. Eq. 4 expresses the well-knownzero-separation theorem (33–35)

yAAð0Þ= 1De−βΔU

ð1ÞA

E0

; [5]

because the numerator is 1 under the condition of a permissibleinsertion at 0

→.

To estimate the ratio in Eq. 4, we exploit many (nt) trial place-ments into the system volume V for each of nc configurations (Fig.1). Those trial points will have the density nt/V and be statisticallyuniform. Out of the nt trial points, a smaller number ns(c) arepermissible placements, and we estimate the denominator of Eq. 4with

PcnsðcÞ=ðncntÞ= ns=nt.

We expect ðnt − 1ÞΔV=V of those trial placements to land ina volume ΔV which is a thin shell of radius r > 0 surroundinga permissible insertion. Let us denote by Δnsðr; cÞ the number ofpermissible placements obtained in the shell for configuration c.We estimate the numerator of Eq. 4 as

Fig. 1. A configuration of 5 × 103 water molecules together with thespherical inclusions identified by nt = 2 × 105 trial placements of a hard spherewith distance of closest approach to an oxygen atom of 0.31 nm. This sizecorresponds approximately to an Ar solute for which the van der Waals lengthparameter σA is about 0.34 nm (57), thus adopting 0.31 nm − 0.17 nm = 0.14nm as a van der Waals contact radius of the water oxygen atom. Hard-spheresolutes of this size have about maximal cavity oxygen contact density (19).

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PcΔnsðr; cÞ

ðnt − 1ÞncΔV=V=

ΔnsðrÞðnt − 1ÞΔV=V

: [6]

Combining these results, we have the estimate�nsV

�yAAðrÞΔV =ΔnsðrÞ; [7]

when nt →∞. This is the same formula as if the permissible inser-tions were actual particles, although that is not the case here.The formula in Eq. 7 is operationally consistent with the zero-

separation theorem in Eq. 5 according to the following argu-ment. Consider a small volume element surrounding the posi-tion r→ = 0

→that is known to be a permissible placement. We

expect that all trial placements in this region should be permis-sible, so Δnsð0→Þ≈ ntΔV=V on the right side of Eq. 7. Therefore,yAAð0Þ≈ nt=ns, which is the operational content of Eq. 5.

Results and DiscussionUsing standard methods (detailed below), this approach wasimplemented for the case that the hard-sphere distance of closestapproach to an oxygen atom was 0.31 nm, corresponding ap-proximately to the case of an Ar solute. Larger solutes wouldmake the present calculations prohibitively difficult. The resultsfor ln yAA(r) (Fig. 2) operationally satisfy the zero-separationtheorem and show strong hydrophobic attraction at short dis-tances and solvent-separated hydrophobic attraction qualita-tively consistent with the PC theory. The ln yAA(r) values for thevarious temperatures cross for r ∼ 0.3 nm. At larger distances,hydrophobic interactions are more stabilizing at higher temper-atures judged from the distributions.The radial distribution function gAAðrÞ= yAAðrÞ for r ≥ 2 ×

0.17 nm = 0.34 nm (and zero otherwise; Fig. 3) scrutinizesthese properties more closely. The contact values, gAA(r = 0.34nm), determined here are more than two times larger than thepredictions of the PC theory. The contact values are higher forhigher temperatures, indicating stronger hydrophobic contactattractions at higher temperatures and agreeing with theresults of Mancera et al. and some of the preceding work

discussed therein (15). The contact values of the PC theoryalso increase with increasing T, but those increases are small(7), and the PC contact values are sufficiently different fromthe numerical values found here that the small increases arenot noteworthy.It is commonplace for simulation calculations (36–44) to de-

termine the temperature dependences of the AA potentials ofmean force wAAðrÞ≡ − kBT   ln  gAAðrÞ. That recent work is broadlysupportive of the conclusions here and is discussed further below.However, available wAA(r) results address our present problemsonly indirectly because they have not obtained B2 for the purposesof isolating aspects of interactions of different physical type; thatis, they have not obtained B2 for a simple reference system thatmight be the basis of further development.The B2 integrals (Fig. 4) provide the solution thermodynamic

assessment of these distribution functions. The B2 values are neg-ative for T = 300 K and become more negative at higher tem-peratures. The biggest negative contribution is associated withcontact hydrophobic attractions. Solvent-separated hydrophobicattractions near R ∼ 0.7 nm are distinct but smaller than contacthydrophobic interactions.The significance of the solvent-separated hydrophobic in-

teractions has been much discussed following the ground-breaking work of Pangali et al. (45, 46). Those simulationstreated Lennard–Jones (LJ) model solutes similar to Kr or Xesolutes. Simulation results (with LJ attractions) were comparedwith PC theory (not treating LJ attractions although modi-fied for continuous repulsive interactions). Additionally, thosesimulations determined wAA(r) only to within an additive con-stant, so comparisons proceeded after matching the two re-sults at their minimum values, a convenient choice that hasbeen often followed (47, 48). This comparison can give theimpression that the nonmatched solvent-separated hydropho-bic interactions are unusually variable or significant. If, forthe purposes of comparison, the wAA(r) values were matched atsolvent-separated radii, then the Pangali et al. results showstronger contact hydrophobic attractions than does the PCtheory, qualitatively in agreement with the present work. Recentmolecular dynamics simulations for Xe(aq) or CH4 (aq) pairs(42–44) agree qualitatively with the present results and thussupport this conclusion.

Fig. 2. Cavity distribution functions for hard-sphere solutes in liquid waterat pressure P = 1 atm and four different temperatures. The spheres have vander Waals radii of 0.17 nm and distance of closest approach to a water ox-ygen atom of 0.31 nm. The dots on the left vertical axis are values calculatedindependently on the basis of the zero-separation theorem.

Fig. 3. Radial distribution functions for hard-sphere solutes in liquid waterat P = 1 atm and four different temperatures. The spheres have van derWaals radius of 0.17 nm and distance of closest approach to a water oxygenatom of 0.31 nm. The prediction of the information theory model (25, 26)at T = 300 K is shown by the star and the gray dashed curve. The contactvalue obtained matches the Pratt–Chandler theory numerical result (7) andso is labeled PC

IT .

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Inclusion of longer-ranged attractive interactions, i.e., Londondispersion interactions, can change these B2 values and trendsdepending on the balance of solute–solute and solute–waterinteractions (11, 16, 18). The localmolecularfield theory (49, 50) isone promising suggestion for how to proceed with inclusion oflonger-ranged interactions, has commonalities with earlier in-tuitive proposals (11, 16), and deserves further development.New results that manipulated the balance of attractive

Lennard–Jones interactions appeared recently (17). Those sim-ulations studied the case that solute–solute interactions werepurely WCA-style repulsive interactions, although solute–waterattractive Lennard–Jones interactions were retained. Notsurprisingly, B2 values for those model solutes were morepositive (repulsive) than for the hard-sphere solutes studiedhere. Still, the temperature dependences were similar, moreattractive at higher T. B2 values for methane solutes were thenestimated on the basis of approximate restoration of solute–solute attractive interactions.The temperature dependence observed here is sometimes re-

ferred to as an inverse temperature dependence. This behaviormight be counterintuitive for the following reason: hydrophobicassociation is typically rationalized as clumping of inert solutioninclusions due to specific structuring of their hydration shells(51). That specific structuring might be more significant at lowertemperatures, so perhaps the hydrophobic association should bestronger at lower temperatures, perhaps more important yet insupercooled water. The fact that the ln yAA(r) values for thevarious temperatures cross in the core region (Fig. 2) alreadysuggests that this rationalization is not simple. Whether the ra-tionalized aggregation behavior is obtained depends on the de-gree that the spherical inclusions can overlap one another and isnot a trivial reflection of solubility behavior.Hydration shell structuring surely is an important factor in

hydrophobic interactions. What this argument does not addressis the distinctive equation of state of liquid water. Some well-recognized peculiarities occur at higher-than-physiological tem-peratures; for example, the compressibility minimum occurs at46 °C, under these low-pressure conditions. The eventual sta-tistical mechanical explanation (52) of the similarly counterin-tuitive entropy convergence hydrophobic phenomenon (at T ∼130 °C) depended first on proper involvement of the actual

equation of state of liquid water (19). Indeed, the calculationshere also exploit that specific equation of state.

ConclusionsHydrophobic interactions between atomic-sized hard spheresin water are more attractive than predicted by the availablemolecular theory, the PC theory (7, 12, 20), as judged by os-motic second virial coefficients B2 obtained here. The B2 valuesfor atomic-sized hard spheres in water are attractive (B2 < 0)and become more attractive with increasing temperature(ΔB2/ΔT < 0) in the temperature range 300 K ≤ T ≤ 360 K.Thus, molecular-scale hydrophobic interactions are attractiveand endothermic at moderate temperatures. These resultsconstitute the initial step toward detailed molecular theory ofadditional intermolecular interaction features, specifically, at-tractive solute–water interactions. Definitive assessment of thetemperature dependences studied here has not been availablebefore and is necessary for broader theory founded at the samebasic level.

Materials and MethodsThe GROningenMAchine for Chemical Simulations (GROMACS) (53) package,with the Extended Simple Point Charge (SPC/E) (54) model, was used tosimulate liquid water. The Parinello–Rahman barostat was used to establishthe pressure at 1 atm, and the Nose–Hoover thermostat maintained thetemperature. Bonds involving hydrogen atoms were constrained by theLINear Contraint Solver (LINCS) algorithm. Conventional periodic boundaryconditions and particle mesh Ewald, with a real-space cutoff at 1 nm, wereused to treat long-range interactions. Simulation cells containing 5 × 103

randomly placed water molecules were created using packing optimizationfor molecular dynamics simulations (PACKMOL) (55) to match the experi-mental density approximately.

After 1 × 104 steps of energy minimization and 2 ns of density equili-bration, trajectories of 20 ns (sampled 1/ps) were obtained at each tem-perature. Each simulation frame was analyzed for cavities based on nt = 2 ×105 trial placements with a distance of closest approach to an oxygen atomof 0.31 nm. Successful placements can be considered as hard spheres of ra-dius 0.17 nm corresponding approximately to an Ar atom. The cavity analysisrequired an order-of-magnitude more computational effort than the gen-eration of the molecular dynamics trajectories.

Considering the integrand of Eq. 1, gAA(r) ∼ 1 in the thermodynamic limit.For fixed particle numbers in a finite system, that subtracted value is less than1, and the correction is O(V−1) (56), vanishing in the thermodynamic limit. Inthe present work, we do not have fixed numbers of A particles.

The information theory calculation (25, 26) (Fig. 3) was based uponthe Poisson default model and the occupancy information indicatedin Fig. 5, both features slightly different from the initial such calcula-tions (20, 25, 26). The reason for these changes was the view that they

Fig. 4. Running integral B2[R] (Eq. 1) for assessment of the R→∞ value. Thedotted curve for R ≤ 0.34 nm is the positive contribution from the hard coreof gAA(r) that is common to all here. The suggested R → ∞ values are de-cidedly negative (attractive) and become more attractive at higher tem-peratures. The biggest negative contribution is associated with contacthydrophobic attractions. Solvent-separated hydrophobic attractions nearR ∼ 0.7 nm are distinct but smaller. The predictions of the PC theory for B2

in these circumstances are repulsive (positive).

Fig. 5. Occupancy data used in the information theory modeling shownin Fig. 3.

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would be preferable for analysis of the dissociation limit (r → ∞). Thenumerical distinctions are not decisive, but we record the details forcompleteness.

Referring to Fig. 5, for each rwe exploit the information ⟨nA⟩, ⟨nA(nA − 1)/2⟩, and ⟨nA nB⟩. These data induce the probability model

pðnA,nBÞ∝ expf−FðnA,nB; ζ1,ζ2,ζ3Þg=nA!nB!, [8]

with

FðnA,nB; ζ1,ζ2,ζ3Þ= ζ1½nA +nB�+   ζ2½nAðnA − 1Þ=2+nBðnB − 1Þ=2�+   ζ3nAnB,

[9]

nA and nB being nonnegative integers and ζj being Lagrange multipliers.Those Lagrange multipliers are obtained by optimizing

lnX

nA,nB≥0expf−FðnA,nB; ζ1,ζ2,ζ3Þg=nA!nB!+ ÆFæ,

as usual. With the optimized ζj, the hydration free energy of the AA diatomis obtained from the normalizing denominator

βμðexÞ = − ln  pð0,0Þ= lnP

nA,nB≥0expf−FðnA,nB; ζ1,ζ2,ζ3Þg=nA!nB! : [10]

Considering the distinction of this information from the initial applications(20, 25, 26), notice that

nðn− 1Þ=nAðnA − 1Þ+nBðnB − 1Þ+ 2nAnB, [11]

where n = nA + nB. Therefore, optimizations that yielded ζ2 = ζ3 wouldconform to use of the reduced information on the left of Eq. 11. Thatoutcome was not seen in these applications. We do expect and observethat ζ3 → 0 when r → ∞. Still, the implied gross free energies and thevariation of the free energies with r are similar. The zero-separationtheorem is reasonably satisfied; the observed discrepancy in ln yAA(0) isby about 0.1 on a magnitude of 18. The r → ∞ level of the gross freeenergies was computed directly, however, rather than relying on thezero-separation theorem.

ACKNOWLEDGMENTS. The financial support of the Gulf of Mexico ResearchInitiative (Consortium for Ocean Leadership Grant SA 12-05/GoMRI-002) isgratefully acknowledged.

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