Chemical Physics Letters 499 (2010) 79–82

Contents lists available at ScienceDirect

Chemical Physics Letters

journal homepage: www.elsevier .com/ locate /cplet t

On the pairwise hydrophobic interaction of fullerene

Giuseppe Graziano ⇑Dipartimento di Scienze Biologiche ed Ambientali, Università del Sannio, Via Port’Arsa 11, 82100 Benevento, Italy

a r t i c l e i n f o a b s t r a c t

Article history:Received 15 August 2010In final form 7 September 2010Available online 15 September 2010

0009-2614/$ - see front matter � 2010 Elsevier B.V. Adoi:10.1016/j.cplett.2010.09.020

⇑ Fax: +39 0824 23013.E-mail address: graziano@unisannio.it

Makowski et al. [9] determined the potential of mean force for the association of two fullerene moleculesin TIP3P water, at room temperature, and found that the water contribution contrasts pairwise hydropho-bic interaction. In the present Letter this result is rationalized by showing that the decrease in wateraccessible surface area upon association causes both a gain in configurational/translational entropy ofwater molecules and a loss of a significant fraction of fullerene–water attractive energetic interactions.The latter term proves to be larger in magnitude than the former because the carbon atom density onfullerene molecular surface is markedly greater than that of normal hydrocarbons.

� 2010 Elsevier B.V. All rights reserved.

1. Introduction

Hydrophobic interaction, HI, is considered to be the major driv-ing force for the folding of globular proteins, the assembly of mi-celles and double-layer membranes, and the formation ofprotein–substrate and protein–protein complexes [1–5]. It is wellknown that the HI magnitude depends not only on the existenceof direct attractions between the interacting molecules, but alsoupon the pivotal role played by water molecules. In referred pro-cesses HI works inextricably tied to other effects, and it is very dif-ficult to single out its molecular mechanism. In order to reach thistarget, computer simulations on the pairwise association of purelynonpolar solutes in reliable water models have proven to be veryuseful [6–8].

Recently, Makowski et al., MCLS [9], determined the potential ofmean force, PMF, of a pair of fullerene molecules, which are verylarge particles, by means of a series of umbrella-sampling molecu-lar dynamics, MD, simulations, using the weighted histogram anal-ysis method, in both the TIP3P water model [10], and in vacuo. ThePMF calculated in water showed the contact minimum, the desolv-ation barrier and the solvent-separated minimum, as expected onthe basis of the results obtained for small hydrocarbons, such asmethane and neopentane [6–8]. The PMF calculated in vacuoshowed the characteristic shape of the Lennard–Jones-like poten-tial. The unexpected result was that the Gibbs energy of the contactminimum configuration of two fullerene molecules (the distancebetween the geometric centres of the interacting moleculesR = 9.7 Å) is markedly smaller in vacuo than in TIP3P water:�38.5 versus �18.0 kJ mol�1, respectively, at 298 K and 1 atm [9].These numbers indicate that pairwise HI of fullerene molecules ismuch more favoured in vacuo than in water: the water contribu-

ll rights reserved.

tion to PMF is positive, opposing association. This result contrastswith that obtained for smaller nonpolar solutes, such as methaneand neopentane [8], and the widely accepted view that water playsthe major role in HI.

The result obtained by MCLS should be considered reliable be-cause they performed very long MD simulations (i.e., 30 windows,each one lasting 10 ns and corresponding to a fixed distance be-tween the fullerene molecules) and used customary force fieldparameters [9–11]. In addition, it is in line with other computersimulation studies: (a) Li and colleagues found that pairwise HIof fullerene molecules is stronger in vacuo than in TIP4P water[12,13]; (b) Hotta and colleagues determined that the water contri-bution contrasts association [14]. On this basis MCLS wrote that‘fullerenes cannot be treated as classical hydrophobic particles’ be-cause the influence of van der Waals interactions is more impor-tant than that of water in determining pairwise HI [9,12]. Theyconcluded that the favourable effect of the decrease in the wateraccessible surface area, WASA [15], upon association is over-whelmed by the unfavourable effect of the water molecules closeto the intersection of the hydration shells of the two fullerene mol-ecules, which have restricted motion (decreased entropy) and can-not form many H-bonds with neighbouring water molecules [9].

The explanation suggested by MCLS cannot be considered satis-factory because it is widely recognized that the structural reorga-nization of water–water H-bonds is characterized by almostcomplete enthalpy–entropy compensation [16–20], and so cannotaffect the Gibbs energy balance. In the present Letter, I would liketo show that, by considering the full statistical mechanical expres-sion of the Gibbs energy change associated with pairwise HI [21],and by performing simple physically-based calculations, it is possi-ble to obtain a satisfactory rationalization of the finding that thewater contribution opposes pairwise HI of fullerene molecules.The analysis indicates that the strength of the energetic interac-tions between fullerene carbon atoms and water molecules plays

80 G. Graziano / Chemical Physics Letters 499 (2010) 79–82

the fundamental role because a significant fraction of them is lostupon association.

2. Theory

Bringing two nonpolar solutes, such as two C60 molecules, froma fixed position at infinite separation to a fixed position at contactdistance in water, keeping constant temperature and pressure, iscalled pairwise HI. The associated Gibbs energy change can rigor-ously be splitted in two terms [21]:

DGðHIÞ ¼ EaðC60 � � �C60Þ þ dGðHIÞ ð1Þ

where Ea(C60� � �C60) is the fullerene–fullerene van der Waals inter-action energy in the contact configuration, and does not dependon the presence of the solvent and its nature (i.e., it can be calcu-lated in vacuo or in an ideal gas phase); dG(HI) is the indirect partof the reversible work to carry out the process, and accounts forthe specific features of the solvent in which pairwise HI occurs. Ageneral relationship connects dG(HI) to the Ben–Naim standardhydration Gibbs energy of the C60� � �C60 contact configuration andC60 molecule, respectively [21,22]:

dGðHIÞ ¼ DG�ðC60 � � �C60Þ � 2DG�ðC60Þ ð2Þ

where DG� represents the Gibbs energy change associated with thetransfer of a species from a fixed position in the ideal gas phase to afixed position in water, at constant temperature and pressure.Application of a physical concept and of statistical mechanics allowsthe exact splitting of DG� in two contributions [23,24]:

DG� ¼ DGc þ DGa ð3Þ

where DGc is the reversible work to create at a fixed position inwater a cavity suitable to host the solute molecule, and DGa is thereversible work to turn on the attractive interactions between thesolute molecule inserted in the cavity and all the surrounding watermolecules (the second step is conditional to the first: there is noadditivity of independent contributions). The reorganization ofwater–water H-bonds upon nonpolar solute insertion does not con-tribute to DG� because it is characterized by an almost complete en-thalpy–entropy compensation [16–20,24]. The use of Eq. (3) in thedefinition of dG(HI) leads to:

dGðHIÞ ¼ ½ðDGcðC60 � � �C60Þ � 2DGcðC60Þ� þ ½ðDGaðC60 � � �C60Þ� 2DGaðC60Þ�

¼ dGcðHIÞ þ dGaðHIÞ ð4Þ

Equation (4) shows that a quantitative estimate of the watercontribution to pairwise HI can be obtained from: (a) the calcula-tion of DGc to create in water a cavity suitable to host a coupleof C60 molecules in the contact configuration, and a cavity suitableto host a single C60 molecule; (b) the calculation of DGa to turn onthe attractive interactions between a couple of C60 molecules in thecontact configuration or a single C60 molecule and all the surround-ing water molecules. By means of classic scaled particle theory[25], SPT, calculations, I have shown that: (a) by keeping fixedthe van der Waals cavity volume, the DGc magnitude depends onthe cavity shape, and proves to be proportional to the water acces-sible surface area of the cavity [26], WASAc, that, in turn, is a mea-sure of the solvent-excluded volume due to cavity creation; (b) thevalue of the DGc/WASAc ratio calculated for spherical cavities canbe used, to a good approximation, also for non-spherical cavities[26–29]. Thus dGc(HI) can be calculated from the knowledge ofWASA buried upon association:

dGcðHIÞ ¼ ðDGc=WASAcÞDWASAðassociationÞ ð5Þ

I assume, as a working hypothesis, that also the DGa magnitudeshould scale linearly with WASAc; this assumption is in line withthe results of the detailed analysis performed by Levy and col-leagues [30]. Moreover, I have verified that, in order to achieve aclose agreement with the computer simulation results by MCLS,DWASA(association) has to be multiplied times a factor equal to1.2; this 20% increase should be a consequence of the manner inwhich solute-water van der Waals energetic interactions take place[30]. On this basis one can write:

dGaðHIÞ ¼ ðDGa=WASAcÞ1:2DWASAðassociationÞ ð6Þ

The WASA buried upon association is a negative quantity; theDGc/WASAc ratio is a positive quantity, and so, according to Eq.(5), dGc(HI) provides a negative Gibbs energy change favouringpairwise association; the DGa/WASAc ratio is a negative quantity,and so, according to Eq. (6), dGa(HI) provides a positive Gibbs en-ergy change contrasting pairwise association. The rationale is thatbringing two C60 molecules from a fixed position at infinite dis-tance to a fixed position at contact distance in water causes: (a)a decrease in the solvent-excluded volume, measured by the WASAdecrease upon association, that produces a significant gain of con-figurational/translational entropy of water molecules [26–29]; (b)a decrease in the number of water molecules contacting the twoC60 molecules, that leads to a significant loss of favourable fuller-ene–water energetic interactions [29].

Estimates of DGc have been calculated by means of classic SPT[25], using the experimental water density at 298 K and 1 atm[31], and an effective diameter for water molecules rw = 2.8 Å, sothat the volume packing density nw = 0.383. The selected rw valueis close to the location of the first peak in the oxygen–oxygen radialdistribution function of water [32], and allows a good description,by means of classic SPT, of the cavity size distribution in water[33]. Transient cavities in liquids are produced by molecular-scaledensity fluctuations at equilibrium, and computer simulations al-low direct observation of such cavities in model liquids. A cavityof 10 Å diameter, however, has a very low occurrence probabilityin computer simulations due to the finite size of the system, andthe use of the analytic SPT formula to calculate DGc proves to beadvantageous.

It is straightforward to show that the WASA buried in the con-tact configuration of two spheres can be calculated, analytically, bymeans of the following relationship:

fWASA ¼ 2pð1� cos aÞ=4p ð7Þ

where fWASA is the fraction of total WASA buried in the contact con-figuration, cosa = rsphere/(rsphere + rw), rsphere is the sphere diame-ter, and rw = 2.8 Å, as usually fixed (it is the same value used forSPT calculations). This relationship indicates that, on increasingthe sphere diameter, fWASA decreases markedly; for instance,fWASA = 0.241 for two spheres of 3 Å diameter each, 0.179 for twospheres of 5 Å diameter each, and 0.109 for two spheres of 10 Ådiameter each.

3. Results

According to MCLS results [9], a single fullerene molecule canbe assimilated to a sphere of 9.7 Å diameter, whose WASA amountsto 491 Å2 The reversible work to create in water a cavity suitable tohost a single fullerene molecule, calculated by means of classic SPTat 298 K and 1 atm, is DGc = 124.2 kJ mol�1. A reliable estimate forthe contribution of the energetic interactions between a fullerenemolecule and surrounding water molecules can be obtained inthe following manner. Garde and co-workers calculated that theBen-Naim standard hydration Gibbs energy of fullerene DG� =�54.1 kJ mol�1 at 300 K and 1 atm [34], in the SPC water model

Table 1Values of the hard sphere diameter, of the fraction of buried WASA calculated by means of Eq. (7), of the work to create a cavity suitable to host the solute molecule, of the solute–water interaction energy, of the Lennard–Jones solute–solute interaction energy in vacuo, ELJ, calculated by MCLS [9], of dGc, dGa and DG(HI) calculated according to Eqs. (1)–(6)for the contact configuration of pairwise HI; the numbers in parentheses are the DG(HI) values calculated by MCLS [9]. All the numbers refer to 298 K and 1 atm; see text forfurther details.

r fWASA DGc DGa ELJ dGc dGa DG(HI)Å kJ mol�1 kJ mol�1 kJ mol�1 kJ mol�1 kJ mol�1 kJ mol�1

CH4 3.7 0.215 22.9 �14.5 �1.3 �9.8 7.5 �3.6 (�3.1)C2H6 4.4 0.194 30.2 �22.7 �1.9 �11.7 10.6 �3.0 (�2.9)C(CH3)4 5.8 0.163 49.0 �39.1 �4.0 �16.0 15.4 �4.6 (�4.4)C60 9.7 0.112 124.2 �178.3 �38.5 �27.8 47.9 �18.4 (18.0)

G. Graziano / Chemical Physics Letters 499 (2010) 79–82 81

[35]. This large negative number indicates that the transfer of a sin-gle fullerene molecule from a fixed position in the ideal gas phaseto a fixed position in water is a thermodynamically favoured pro-cess, at odds with the hydration of simple alkane molecules thatshow positive DG� values [36]. By using Eq. (3), DGa(C60–water) =DG� � DGc = �54.1–124.2 = �178.3 kJ mol�1, at 298 K and 1 atm;the latter number indicates that the energetic interactions betweenfullerene carbon atoms and water molecules are so strong to over-whelm the positive DGc term.

In the contact configuration of two spheres of 9.7 Å diametereach, fWASA = 0.112 according to Eq. (7), and so DWASA = �110 Å2;application of Eqs. (5) and (6) leads to: dGc(HI) = �27.8 kJ mol�1

and dGa(HI) = +47.9 kJ mol�1. These two estimates, coupled to theEa(C60� � �C60) value calculated in vacuo by MCLS [9], lead to: DG(HI,R = 9.7 Å) = �38.5 � 27.8 + 47.9 = �18.4 kJ mol�1. The latter num-ber is in close agreement with the depth minimum obtained byMCLS, �18.0 kJ mol�1, in the TIP3P water model [9], and by Liand colleagues, �17.6 kJ mol�1, in the TIP4P water model [12,13].

In order to verify the soundness of the devised approach, I haveused it to calculate DG(HI) for the contact minimum configurationof two neopentane molecules (the distance between the geometriccentres of the interacting molecules R = 5.8 Å [8,9]). A single neo-pentane molecule can be assimilated to a sphere of 5.8 Å diameter,whose WASA amounts to 232 Å2. The reversible work to create inwater a cavity suitable to host a single neopentane molecule, calcu-lated by means of classic SPT at 298 K and 1 atm, is DGc =49.0 kJ mol�1. A reliable estimate for the contribution of the ener-getic interactions between a neopentane molecule and surroundingwater molecules can be obtained adopting a procedure similar inspirit to that used for fullerene. Thus, DGa(neopentane–water) =DG�(exp) � DGc = 9.9 � 49.0 = �39.1 kJ mol�1, at 298 K and 1 atm(the experimental DG� value comes from [36]); this estimate isclose to the numbers calculated in different water models by sev-eral authors [37–39]. In the contact configuration of two spheresof 5.8 Å diameter each, fWASA = 0.163 according to Eq. (7), and soDWASA = �76 Å2; application of Eqs. (5) and (6) leads to:dGc(HI) = �16.0 kJ mol�1 and dGa(HI) = +15.4 kJ mol�1. These twoestimates, coupled to the Ea(neopentane� � �neopentane) value invacuo reported by MCLS [9], lead to: DG(HI, R = 5.8 Å) =�4.0 � 16.0 + 15.4 = �4.6 kJ mol�1. The latter number is in closeagreement with the depth minimum obtained by MCLS,�4.4 kJ mol�1, in the TIP3P water model [9].

As a final test, I have calculated DG(HI) for the contact mini-mum configuration of two methane molecules. A single methanemolecule can be assimilated to a sphere of 3.7 Å diameter, whoseWASA amounts to 133 Å2. The reversible work to create in watera cavity suitable to host a single methane molecule, calculated bymeans of classic SPT at 298 K and 1 atm, is DGc = 22.9 kJ mol�1

[27]. The energetic interactions between a methane molecule andsurrounding water molecules lead to DGa(methane–water) = -14.5 kJ mol�1, at 298 K and 1 atm [27]. In the contact configurationof two spheres of 3.7 Å diameter each, fWASA = 0.215 according toEq. (7), and so DWASA = �57 Å2; application of Eqs. (5) and (6)

leads to: dGc(HI) = �9.8 kJ mol�1 and dGa(HI) = +7.5 kJ mol�1. Thesetwo estimates, coupled to the Ea(methane� � �methane) value invacuo reported by MCLS [9], lead to: DG(HI, R = 3.7 Å) =�1.3 � 9.8 + 7.5 = �3.6 kJ mol�1. The latter number is in agreementwith the depth minimum obtained by MCLS, �3.1 kJ mol�1, in theTIP3P water model [9]. All the relevant numbers for the pairwise HIof methane, ethane, neopentane and fullerene, treated as simplespheres, are collected in Table 1. It is evident that the present ap-proach works well also in the case of ethane, even though this mol-ecule is absolutely not spherical in shape.

Methane and neopentane differ from fullerene for the strengthof the energetic interactions with water molecules: the carbonatoms of fullerene are described as aromatic [9,34], and so theirvan der Waals interactions with water are stronger than those ofalkanes [40]. In fact, at room temperature and 1 atm, DG� is largenegative for fullerene, but positive for methane and neopentane[34,36]. The association of two fullerene molecules causes a gainin configurational/translational entropy of water molecules dueto the decrease in the solvent-excluded volume [i.e., the dGc(HI)term], and a loss of a significant fraction of attractive interactionsbetween fullerene carbon atoms and water molecules [i.e., thedGa(HI) term]. The latter term overwhelms in magnitude the for-mer and the water contribution opposes pairwise HI of fullerenemolecules. In the case of methane and neopentane, the energeticinteractions between nonpolar solute and surrounding water mol-ecules are not so strong to render the dGa(HI) term larger in mag-nitude than the dGc(HI) one.

4. Discussion

Fullerene is an interesting solute because, with a diameter ofabout 10 Å, is a real nanoscale molecule and the finding that itspairwise HI is dominated by direct van der Waals attractions be-cause the water contribution contrasts association [9,12–14], mer-its a careful analysis. The statistical mechanical Eqs. (1)–(4)indicate that water affects pairwise HI in two different ways: a gainin configurational/translational entropy of water molecules for thedecrease in the solvent-excluded volume upon association, and aloss of solute–water energetic attractions for the burial of molecu-lar surface upon association. It has been shown that the first con-tribution scales linearly with WASA [26–29], and I have assumedin the present work that a linear dependence on WASA holds alsofor the second contribution. The obtained results seem to lend sup-port to this assumption.

No term accounting for the structural reorganization ofwater–water H-bonds upon association is present in the DG(HI)expressions of the Theory section. This is because such a processis characterized by an almost complete enthalpy–entropy compen-sation, as demonstrated by several authors using differentapproaches [16–20]. In particular, Lee’s analysis indicates thatthe physical origin of this enthalpy–entropy compensation is therelative weakness of solute–water attractions in comparison to

82 G. Graziano / Chemical Physics Letters 499 (2010) 79–82

the strength of water–water H-bonds [18,40]. Thus, the emphasison the structural features of water–water H-bonds in the hydrationshell of the contact configuration is not really useful to gain insighton the driving forces of pairwise HI [9,12–14]. In addition, notwith-standing a very careful analysis of MD simulations, Levitt and co-workers did not find any clear evidence of ordered orientationsof water molecules in the hydration shell of a single fullerene mol-ecule [41].

A fundamental point is that, as calculated by Garde and co-workers, for the hydration of fullerene DG� = �54.1 kJ mol�1 at300 K and 1 atm [34]. Clearly, this is not an experimental datum(it is very difficult to try to perform such measurements for fuller-ene), but it is worth noting that other authors, using a different cal-culation procedure, obtained that DG� is large negative for thehydration of fullerene at room temperature [42]. These negativeDG� estimates are in line with the experimental finding that, forthe hydration of the aromatic hydrocarbons benzene and toluene,DG� = �3.6 and �3.7 kJ mol�1, respectively, at 298 K and 1 atm[40]; in fact, fullerene was considered a three-dimensional ana-logue of benzene [41]. Graziano and Lee clarified that the latternegative values are due to the strength of van der Waals interac-tions between the aromatic ring and surrounding water molecules;the weak benzene–water H-bonds should not play a role due to analmost complete enthalpy–entropy compensation [24,40]. The sit-uation should be very similar for fullerene. Both MCLS [9] andGarde and co-workers [34] considered all the carbon atoms of ful-lerene as aromatic carbon atoms of the AMBER force-field [11],their charge was set to zero and their interactions with water wereof the Lennard–Jones type. Since there are no charges on the fuller-ene molecule, there is no possibility to form the weak H-bondswith water molecules, and the DGa(C60–water) magnitude is solelydetermined by van der Waals interactions. The latter prove to be sostrong to overwhelm the DGc contribution because the density ofcarbon atoms on the fullerene surface is very large [12,13] (i.e.,each carbon atom is an interaction centre): 0.20 carbon atomsper Å2 of molecular surface in the case of fullerene versus 0.05 car-bon atoms per Å2 of molecular surface in the case of neopentane.

A final general consideration is in order. If the decrease in thesolvent-excluded volume (i.e., WASA decrease) were the majordriving force of HI, pairwise association of two spheres wouldnot be a good model because the WASA fraction buried in the con-tact configuration, fWASA, is at most around 0.25 and decreases onincreasing the sphere diameter, as emphasized by Eq. (7). ThesefWASA values are markedly smaller than those characterizing otherprocesses dominated by HI: (a) in the case of two large disks, about50% of total WASA is buried upon association [43]; (b) in the caseof globular proteins, about 60% of total WASA is buried upon fold-ing [44,45]. This point is important also to achieve a good balancebetween the Ea(solute� � �solute) term, that is a negative quantity,and the dGa(HI) term, that is a positive quantity; if these two ener-getic terms would balance each other, the HI driving force wouldbe the dGc(HI) term. The latter entropic term is mainly determinedby geometric features of interacting molecules; fixed the geometry,introduction of suitably selected interaction centres can beexploited to strengthen pairwise HI.

MCLS determined the PMF for the association of two fullerenemolecules in TIP3P water, at room temperature and atmospheric

pressure, and found that the water contribution contrasts pairwiseHI [9], in line with previous computer simulation results [12–14].This result is rationalized by showing that the decrease in WASAupon association causes both a gain in configurational/transla-tional entropy of water molecules and a loss of a significant frac-tion of fullerene–water attractive energetic interactions. Thelatter term proves to be larger in magnitude than the former be-cause the density of carbon atoms on the fullerene molecular sur-face is markedly greater than that of normal hydrocarbons.

Acknowledgements

The article is dedicated to professor Attilio Immirzi in the occa-sion of his 72th birthday.

References

[1] W. Kauzmann, Adv. Protein Chem. 14 (1959) 1.[2] C. Tanford, The Hydrophobic Effect: Formation of Micelles and Biological

Membranes, second edn., Wiley, New York, 1980.[3] W. Blokzijl, J.B.F.N. Engberts, Angew. Chem. Int. Ed. Engl. 32 (1993) 1545.[4] D. Chandler, Nature 437 (2005) 640.[5] P. Ball, Chem. Rev. 108 (2008) 74.[6] S. Shimizu, H.S. Chan, J. Chem. Phys. 113 (2000) 4683.[7] T. Ghosh, A. Kalra, S. Garde, J. Phys. Chem. B 109 (2005) 642.[8] E. Sobolewski, M. Makowski, C. Czaplewski, A. Liwo, S. Oldziej, H.A. Scheraga, J.

Phys. Chem. B 111 (2007) 10765.[9] M. Makowski, C. Czaplewski, A. Liwo, H.A. Scheraga, J. Phys. Chem. B 114

(2010) 993.[10] W.L. Jorgensen, J. Chandrasekhar, J.D. Madura, R.W. Impey, M.L. Klein, J. Chem.

Phys. 79 (1983) 926.[11] W.D. Cornell et al., J. Am. Chem. Soc. 117 (1995) 5179.[12] L.W. Li, D. Bedrov, G.D. Smith, Phys. Rev. E 71 (2005) 011502.[13] L.W. Li, D. Bedrov, G.D. Smith, J. Chem. Phys. 123 (2005) 204504.[14] T. Hotta, A. Kimura, M. Sasai, J. Phys. Chem. B 109 (2005) 18600.[15] B. Lee, F.M. Richards, J. Mol. Biol. 55 (1971) 379.[16] R. Lumry, E. Battistel, C. Jolicoeur, Faraday Symp. Chem. Soc. 17 (1982) 93.[17] H.A. Yu, M. Karplus, J. Chem. Phys. 89 (1988) 2366.[18] B. Lee, Biophys. Chem. 51 (1994) 271.[19] H. Qian, J.J. Hopfield, J. Chem. Phys. 105 (1996) 9292.[20] D. Ben-Amotz, R. Underwood, Acc. Chem. Res. 41 (2008) 957.[21] A. Ben-Naim, Hydrophobic Interactions, Plenum Press, New York, 1980.[22] A. Ben-Naim, Solvation Thermodynamics, Plenum Press, New York, 1987.[23] B. Lee, Biopolymers 31 (1991) 993.[24] G. Graziano, Chem. Phys. Lett. 429 (2006) 114.[25] H. Reiss, Adv. Chem. Phys. 9 (1966) 1.[26] G. Graziano, J. Phys. Chem. B 113 (2009) 11232.[27] G. Graziano, Chem. Phys. Lett. 483 (2009) 67.[28] G. Graziano, Biopolymers 91 (2009) 1182.[29] G. Graziano, Chem. Phys. Lett. 491 (2010) 54.[30] R.M. Levy, L.Y. Zhang, E. Gallicchio, A.K. Felts, J. Am. Chem. Soc. 125 (2003)

9523.[31] G.S. Kell, J. Chem. Eng. Data 20 (1975) 97.[32] G. Hura, T. Head-Gordon, Chem. Rev. 102 (2002) 2651.[33] G. Graziano, Chem. Phys. Lett. 396 (2004) 226.[34] M. Athawale, S.N. Jamadagni, S. Garde, J. Chem. Phys. 131 (2009) 115102.[35] H.J.C. Berendsen, W.F. van Gunsteren, J.P.M. Postma, J. Hermans, in: B. Pullman

(Ed.), Intermolecular Forces, Reidel, Dordrecht, 1981, p. 331.[36] G. Graziano, Can. J. Chem. 80 (2002) 401.[37] W.L. Jorgensen, J. Gao, C. Ravimohan, J. Phys. Chem. 89 (1985) 3470.[38] D. Trzesniak, N.F.A. van der Vegt, W.F. van Gunsteren, Phys. Chem. Chem. Phys.

6 (2004) 697.[39] M. Almlof, J. Carlsson, J. Aqvist, J. Chem. Theory Comput. 3 (2007) 2162.[40] G. Graziano, B. Lee, J. Phys. Chem. B 105 (2001) 10367.[41] D.R. Weiss, T.M. Raschke, M. Levitt, J. Phys. Chem. B 112 (2008) 2981.[42] A. Muthukrishnan, M.V. Sangaranarayanan, Chem. Phys. 331 (2007) 200.[43] R. Zangi, B.J. Berne, J. Phys. Chem. B 112 (2008) 8634.[44] S. Miller, J. Janin, A.M. Lesk, C. Chothia, J. Mol. Biol. 196 (1987) 641.[45] V.Z. Spassov, A.D. Karshikoff, R. Ladenstein, Protein Sci. 4 (1995) 1516.