Proc. Natl. Acad. Sci. USAVol. 88, pp. 10880-10884, December 1991Biophysics

Contribution of the hydrophobic effect to protein stability: Analysisbased on simulations of the Ile-96 -- Ala mutation in barnaseMARTINE PREVOST*, SHOSHANA J. WODAK*, BRUCE TIDORtt, AND MARTIN KARPLUSt*Unite de Conformation des Macromolecules Biologiques, Universitd Libre de Bruxelles, Avenue Paul Hdger-CP160, B-1050 Brussels, Belgium; andtDepartment of Chemistry, Harvard University, Cambridge, MA 02138

Contributed by Martin Karplus, April 4, 1991

ABSTRACT Molecular dynamics simulations have beenused to compute the difference in the unfolding free energybetween wild-type barnase and the mutant in which Ile-96 isreplaced by alanine. The simulations yield results (-3.42 and-5.21 kcal/mol) that compare favorably with experimentalvalues (-3.3 and -4.0 kcal/mol). The major contributions tothe free energy difference arise from bonding terms involvingdegrees of freedom of the mutated side chain and from non-bonded interactions of that side chain with its environment inthe folded protein. By comparison with simulations of anextended peptide in the absence of solvent, used as a referencestate, hydration effects are shown to play a minor role in theoverall free energy balance for the Ile -* Ala transformation.The implications of these results for our understanding of thehydrophobic effect and its contribution to protein stability arediscussed.

Hydrophobic interactions are believed to make a majorcontribution to stabilizing the native structure of proteins inan aqueous environment (1-4). According to the classicalpicture (1) and more recent theoretical analyses (4), thehydrophobic effect leads to structures in which many, but notall, of the nonpolar side chains are packed together in theprotein interior where they avoid contact with water. Mea-surements of solvation effects have shown that the hydro-phobic effect is largely entropic in origin at room temperature(5). The unfavorable entropy of solvation is ascribed to theentropy decrease of water surrounding the nonpolar groups(6, 7); this is in accord with simulation studies ofhydrophobicsolutes (8, 9) and peptides in water (10).Although the general role of the hydrophobic effect in

protein folding is understood, a more detailed quantitativedescription of its contributions to protein stability is essen-tial. Site-directed mutagenesis experiments combined withthermal and spectroscopic stability measurements are nowbeing used to dissect the contributions of individual aminoacids (11). Substitutions of buried or partly buried nonpolarresidues in several proteins (12-15) have shown that thechange in thermodynamic stability between the wild type andmutant can be related to the free energies of transfer (16-18)and/or the accessible surface areas (19) of the individualsubstituted residues. Although there is a good overall corre-lation when a wide range of substitutions are considered (12),the variation seen among the hydrophobic aliphatic sidechains does not show any simple relation with transfer freeenergies. Moreover, the interpretation is somewhat confusedby the use of transfer free energy values from differentsolvents or from the gas phase to water for analyzing theexperimental data (12, 14). There are also more basic ques-tions concerning the effect of hydrophobic solvation ofindividual nonpolar groups and its relationship to their bulkproperties in the protein interior (20). Whether the protein

interior corresponds to a nonpolar or a slightly polar liquid(e.g., alcohol-like) is not clear (17, 21). Because such ques-tions are difficult to approach experimentally (11), theoreticalanalyses based on a detailed microscopic description areneeded to provide fuller understanding. Methods for com-puting free energy changes by use of molecular dynamics (22)and Monte Carlo (23, 24) techniques are particularly wellsuited for this purpose. Although such calculations are stillnot routine (25, 26), they are of considerable utility becausethey evaluate thermodynamic quantities that can be com-pared directly with experiment. More important, a recentformulation of the free energy simulation method (27) makesit possible to decompose the computed values into contribu-tions from different parts of the system (protein residuesversus solvent), as well as individual energy-such as non-bonded interactions (van der Waals, electrostatic) and vibra-tional-terms.The present study describes the application of the free

energy simulation method to the substitution of isoleucine byalanine at position 96 in barnase, an extracellular ribonucle-ase from Bacillus amyloliquefaciens. This 110-residue en-zyme is of particular interest because it is being used as aparadigm for studying protein stability and folding (13, 14).The crystal structure is known to 2.0-A resolution (28). Theenzyme is a one-domain protein that undergoes reversiblethermal and urea-induced denaturation closely approximat-ing a two-state equilibrium (29). The protein contains signif-icant secondary structure, including a }-sheet in whichposition 96 occupied by isoleucine is situated (see Fig. 1).This residue is fully buried and participates in a closelypacked hydrophobic interface with an a-helix.We report the results and analysis of simulations of the

"alchemical" transformations (27) of Ile-96 into alanine in thenative solvated protein and in an extended conformation usedas a model for the denatured state in water. For comparisonthe same transformation with the extended model was sim-ulated in the absence of solvent. The implications of theresults for our understanding of the hydrophobic effect andits contribution to protein stability are outlined.

METHODThe free energy difference, AG, between two states A and Brepresenting, respectively, wild-type and mutant proteins, iscalculated by "computer alchemy", in which one amino acidis transformed into another (27). This operation is achievedby using a hybrid potential function V(rN, A) = (1 - A)VA(rN)+ AVB(rN) where A is a coupling parameter varied from 0 to1, VA(rN) and VB(rN) are the empirical potentials describingthe wild-type and mutant protein, respectively. The symbol

Abbreviations:'EF, exponential formula; TI, thermodynamic integra-tion.tPresent address: Whitehead Institute for Biomedical Research,Nine Cambridge Center, Cambridge, MA 02142.

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summation of contributions involving the interactions of themutated side chain (27, 32); e.g., one can separate thecontributions of the solvent and the protein or can determinethe contributions of different types of terms in the energyfunction. In the present study, which concerns the mutationIle-96 -> Ala, it is convenient to write AG as a sum of fourterms, AG = AGc, + AGnb. + AGci + AGnbi, each of whicharises from specific contributions to AV. The first two"side-chain" terms represent contributions that dependsolely on atoms of the mutated side chain. The other twoterms represent "interaction" terms of the side chain with itsenvironment. The potential energy contributions included inthe four terms and the specific atoms involved are illustratedin Fig. 2.

Previous studies have shown that the method described isadequate for treating charged to nonpolar [e.g., Asp -3 Ala(27)] and charged to charged [e.g., Argo- His (32)] mutations.In the present application, which involves a nonpolar-to-nonpolar mutation, a somewhat more elaborate procedurewas used. Six intermediate states were defined along apathway from A (wild type) to B (mutant). The states weregenerated by successively modifying van der Waals param-eters and bond lengths of the relevant side chains in the

FIG. 1. Environment of Ile-% in the crystal structure of barnase.A ribbon tracing of the backbone of barnase is shown in yellow, andthe residues included in the molecular dynamics simulation zone areshown in magenta, except for Ile-96 shown in red; sequence num-bering is as follows: 7-12, 14, 15, 63, 71, 88-91, 93-98, 107-110.Residues subjected to Langevin dynamics (displayed in bluegreen)are as follows: 6, 9, 13, 16, 18, 20, 64, 69-70, 72-74, 76, 87, 92, 99,106-107. Twenty-one water molecules included in the simulationsare displayed in light blue. The unfolded system (data not shown)includes residues 94-99 in an extended conformation and 145 watermolecules.

rN represents atomic coordinates of the system composed ofparts of the protein and solvating waters.The free energy difference AG between the two states A

(wild type) and B (mutant) can be obtained from either oftwoformally exact expressions. The first is called the "exponen-tial formula" (EF) (30) and has the form

V(r N, A,+1) - V(r N, A,)AG = -kBT In (e kBT )

AVAAi= -kT>EIn (e kBTe A.' [1]

where AV = VB(rN) - VA(rN), AAi = Ai+1 - Ai, kB theBoltzmann constant and T the absolute temperature; theangle brackets represent an ensemble average obtained withthe potential V(rNA,), to represent the system. An equivalentformulation, called thermodynamic integration (TI), yieldsthe following expression (31):

AG = F (aV/aA)A dA - (AV)A.k AA, [2]Jo

where the canonical average (dV/aA)A is equal to (AV)A whenthe hybrid potential energy function V(rN, Ai) is linear in A.An important advantage of Eq. 2, relative to Eq. 1, is that

the total free energy change can be represented as an exact

FIG. 2. Terms included in different free energy components.Different contributions to these terms are depicted by arrows. (a andb) Terms that depend only on the isoleucine side chain and involveexclusively atoms Cp, Cyl, Cv2, and C81. (a) Contributions to AG_,that comprise covalent terms consisting of bond stretching and anglebending. The entire contribution comes from Ile-96 because the C;atom of alanine has no self-energy component in the potential owingto the fact that aliphatic groups are treated as extended atoms (33).(b) Contributions to AGnb,, the intra-side-chain nonbonded term,which are limited to van der Waals interaction between C81 and Cy2atoms. (c and d) Contributions concerning interactions of atomsbelonging to the isoleucine side chain with atoms of the remainder ofprotein and/or solvent. (c) Terms included in AGcj; these comprisecovalent components-bond stretching and valence angle bend-ing-as well as torsional terms. (d) Nonbonded terms that contributeto AGnbi, which are limited to van der Waals interaction. Equivalentdrawings could be made for the alanine side chain with the exclusionof a and b.

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direction of the transformation. The overall free energy

difference was computed as the sum of the individual freeenergy differences AGj between these successive states,yielding AG(A -- B) = 7_,1 AGj. Individual AGj values were

computed from Eqs. 1 and 2 with three intermediate valuesof A: 1/6, 3/6, and 5/6. This protocol corresponds to takingdiscrete points along a nonlinear pathway between states Aand B and interpolating linearly between these points. Thisapproach was chosen to improve convergence of the rela-tively small energy difference dominated by van der Waalsterms (25) while preserving the possibility of decomposingthe results into individual contributions.The ensemble averages over configuration space required

for Eqs. 1 and 2 were obtained for both the folded and unfoldedstates, by stochastic boundary molecular dynamics simula-tions at 300 K (34) within an 11-A sphere having a 2-Aboundary region, using the CHARMM program (33). The sim-ulations were done on the folded state starting with thehigh-resolution crystallographic coordinates of barnase (28).The unfolded state was modeled from a heptapeptide com-prising the protein fragment centered on residue 96. Thepeptide was taken to be in an extended conformation.

RESULTSThe simulations yield values of the alchemical free energydifferences for the transformation of Ile -- Ala in the solvated

native protein (AGfA), in the solvated unfolded state(AGI MA), and the unsolvated reference state (AGh MA); seeTable 1. By use of the thermodynamic cycle (Fig. 3), thecorresponding difference in unfolding free energy, AAGf mucan be obtained from the following expression (36): AAGf mu= AGhIPA - AGh PA = AGQ.u - AGf mu. The computed valuefor AAGf mu is -3.42 and -5.21 kcal/mol, when calculated bythe EF and the TI methods, respectively. This result is to becompared with the experimental values of -3.3 and -4.0kcal/mol, obtained from slightly different analyses of thesame experimental data (13, 14). Correspondingly, the sol-vation free energy difference AAG is given as follows:AAGru = AGhIA - AGrhA = AG ,1v - AG,1o, where AGOlv

is the measured solvation free energy corresponding to thetransfer from the gas phase to aqueous solution. The calcu-lated values are -0.25 and -1.15 kcal/mol from the EF andTI methods, respectively; the experimental estimate for thedifference in solvation energies of isoleucine versus alanineis -0.21 kcal/mol (16). Finally, we can calculate the free

AGr-*fI I

protein Ef E

AGf u water AGsoiv

<>A AGu'->A

i A AGf ->u waAer AGsoivprotein Ef - Eu I

E, reference

1->AAGr A

AEr reference

AG, _~

FIG. 3. Thermodynamic cycles used to describe AAG in calcu-lations and experiments. The vertical direction concerns alchemicalprocesses corresponding to the Ile -. Ala transformation; the hori-zontal direction concerns chemical steps ofthe unfolding or solvationreactions. The thermodynamic cycle, on the left, refers to theunfolding process of barnase. The left-hand side concerns thetransformation in the folded protein (Ef). The right-hand side con-cerns the transformation in the unfolded state (E.). Unfolding of thewild-type protein in the presence of isoleucine is shown at top, andunfolding of the alanine-containing mutant is shown at bottom. Thethermodynamic cycle on the right refers to the solvation process ofthe unfolded protein-i.e., transfer from the reference phase (gasphase) to water. On the right side of the cycle, the transformationoccurs in the gas phase (E,). The process of transferring the iso-leucine-containing unfolded protein from the gas phase to water isshown on top. The solvation process for the alanine-containingunfolded protein is shown on bottom. The outer thermodynamiccycle brings the unfolded protein in the reference (gas phase) to thefolded state in solution. The process of transferring the isoleucine-containing unfolded protein from the gas phase to the isoleucine-containing folded state is shown on top. The same process in thepresence of alanine is shown on bottom.

energy change in going from the gas phase to the foldedprotein by use of the relation, AAGf = AGhA - AGIhPA.The resulting values are + 3.17 and +4.06 kcal/mol from theEF and TI methods, respectively. No direct measurement ofthis quantity is available from experiment, although it can beobtained by difference from the other experimental resultsgiven in Table 1; the estimated values of +3.1 and +3.8kcal/mol correspond to the two different measured values ofAAGf .u. Thus, the simulation and experiment agree that theessential effect of the mutation is the difference in the

Table 1. Computed free energy changes (in kcal/mol) for Ile -* Ala mutation in barnase/aGf--A Y-OG A YOI-A &v-+ Avl-vuA~~hA ~AGhA AGhA fV~ AU

Contribution* protein water reference AAGfou AAGrb.u AAGrf protein water

cs -3.28 -4.73 -3.18 -1.45 -1.55 -0.10 -1.88 -1.63ci -1.40 -3.09 -5.07 -1.69 1.98 3.67 -1.80 -1.60nbs -0.97 -0.79 -0.24 0.18 -0.55 -0.71 -1.12 -0.51nbi 2.56 0.31 1.35 -2.25 -1.04 1.21 7.7 8.0TI (total) -3.09 -8.3 -7.15 -5.21 -1.15 4.06 2.90 4.26EF -3.39 -6.81 -6.56 -3.42 -0.25 3.17EXP -3.3t; -4.0t -0.21§AGf, AGu, and AGr are free energies for alchemical transformation Ile -. Ala, in folded state (protein), unfolded state (water), and unfolded

state in gas phase (reference), respectively. A&AGf, taG,., and vAGr f are as explained in text. EXP, experimental results. Different freeenergy contributions considered (see text and Fig. 2) are listed in column 1. The next to last column lists values of AV and its componentsevaluated using ensemble averages computed for the folded protein at states A, (close to isoleucine) and A,, (close to alanine) by the formula:AV"A = (VAia exp(-AVAA)/kT)A, /(exp(-AVAk)/kT)A, - (VIje exp(-AVAA)/kT)Ak/(exp(-AVAA)/kT)A,, where Vile and VAa are theappropriate contributions to the potential energy of the side chain, and ,Ak = 0.17. A corresponding expression is used for the unfolded state(last column). All calculations were done with the CHARMM program (33) using parameters and described procedures (32). Calculations of AGfor a given A consist of an equilibration simulation of 5 ps followed by an averaging period of 10 ps. All computations were done on a Cyber205 computer at the former John von Neuman Center.*For definitions, see text and Fig. 2.tData from ref. 14.tData from ref. 13.§Data from ref. 16.

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stability of the folded state rather than the differential sol-vation of isoleucine and alanine in the unfolded state.

In view of the sensitivity of free energy values to theconvergence of simulations (37), it is important to assess thestatistical precision of the results. The computations havebeen performed with two formally equivalent procedures (theEF and TI methods) that would give identical results if fullconvergence had been achieved. The discrepancy betweenthe free energy differences for isoleucine and alanine ob-tained by these methods is 1.8 kcal/mol for denaturation and0.9 kcal/mol for solvation. The two calculations show veryconsistent behavior, as the largest difference between indi-vidual AGj values obtained with the EF and TI proceduresdoes not exceed 0.6 kcal/mol. Computations based on thestandard deviation of individual AGj free energy valuesobtained by the TI procedure estimate the precision of theoverall free energy difference to be -1 kcal/mol.To obtain insight into the origin of the free energy changes,

the thermodynamic integration procedure given in Eq. 2 wasused. The calculated free energies for the folded protein, theunfolded protein in solution, and the reference state aredecomposed into four distinct contributions listed in Table 1;for definitions, see Fig. 2.Most of the computed free energy difference of unfolding,

AAGf..,, arises from three terms: the nonbonded interactionof the mutated side chain with the rest of the system (AAGnbi= -2.25 kcal/mol), the covalent interactions ofthe side chainwith the rest ofthe system (AAGci = -1.69 kcal/mol), and theintra-side-chain covalent term (AAGCS = -1.45 kcal/mol).The contribution from nonbonded intra-side-chain interac-tions is negligible for this case (AAGnbs = 0.18 kcal/mol).The computed contributions of the nonbonded interaction

term, AGnbi, which in this case is entirely due to van derWaals interactions, are unfavorable for alanine relative toisoleucine in both the folded and unfolded states (columns 2and 3 in Table 1), but the unfavorable effect in the unfoldedstate is marginal, leading to an overall negative contributionto AAGnbi.The free energy changes from side-chain covalent terms

(AGcs) are large and favor alanine in both the folded andunfolded states. This is expected because the covalent mo-tional contributions are always positive (destabilizing), andonly isoleucine has covalent interactions within the sidechain. To interpret their origin, an estimate was made of theexpected internal free energy with a classical harmonic modelfor 1-butane, taken as a model for an isoleucine side chain;the value for the single atom representing the alanine sidechain is zero. It yields a value of -3.66 kcal/mol, most ofwhich comes from the vibrational enthalpy of the six classicalinternal degrees of freedom. This value compares favorablywith the simulation results for the sum of AGs and AGnbS inthe gas-phase reference state (-3.42 kcal/mol), indicatingthat the intrinsic vibrational properties of the isoleucine sidechain are unperturbed in the gas-phase-extended heptapep-tide. In both the folded and solvated unfolded states thevalues are somewhat more negative. The origin of thesedifferences, which make significant contributions to AAGf..u(see Table 1) is not clear.The free energy contributions from the covalent interaction

term (AGj1) are favorable to alanine in the unfolded and foldedstates. However, the favorable effect is smaller in the foldedstate, leading to a negative contribution to the wild-type/mutant free energy difference. The destabilizing effect ofisoleucine in the unfolded state may be due to strain intro-duced by the surrounding water molecules, whose H bondswith other water molecules and the neighboring main chainare adjusted to accommodate the larger nonpolar group. ThatAGi computed in the unsolvated reference state is verydestabilizing for isoleucine is somewhat surprising. The dom-inant contribution comes from bond-angle distortions that

may result from the fact that, in the absence of solvent, atomsof the isoleucine side chain interact exclusively with theheptapeptide, particularly with those in the main chain at-oms.

It is of interest to compare the above results with thedecomposition of the average potential energy at the endpoints of the Ile -. Ala transformation pathway, listed in thelast two columns of Table 1. Although the signs for the AVfand A1Vu values are the same as the corresponding AG values,the magnitudes are very different. In particular the "nbi"term is =8 kcal/mol for both folded and unfolded states. Thisresult agrees well with the computed interaction energy oftheisoleucine side chain (C3 excluded) with the rest of theprotein in the crystal conformation (=7 kcal/mol). That thecomputed free energy changes (+2.56 and +0.31 for foldedand unfolded states, respectively; see Table 1) are muchsmaller than these values suggests that other effects areinvolved. Because the present simulations are not preciseenough to evaluate the entropic and energetic contributionsto the free energy, other aspects of the simulations have beenanalyzed in an attempt to gain further insight. In doing so, thestructure of the alanine mutant has been assumed to differlittle from wild type. This assumption is in agreement withevidence from NMR measurements (14) and with the com-putations, which indicate that rms deviations between wild-type and alanine mutant structures are <1 A when all atomsare considered. Atomic volume calculations done by usingcoordinates from end points of the simulation pathway (38)suggest that the packing around residue 96 is looser in thealanine-containing mutant than in the wild type. This shouldlead to favorable entropic contributions in the mutant pro-tein. To examine this conjecture, rms atomic fluctuationsfrom several independent 30-ps vacuum molecular dynamicstrajectories of the wild-type and the mutant proteins werecompared. Although these were found to be similar onaverage, fluctuations of main-chain and C3 atoms of residue96 were about twice as large in the alanine mutant, suggestingthat the latter is indeed more mobile, in agreement with astabilizing entropic contribution. A corresponding analysiswas not made for the unfolded structure. The standardpicture of the hydrophobic effect would suggest an increasein the entropy of the unfolded state of alanine relative toisoleucine, as increased water structure is expected in thepresence of the larger hydrophobic isoleucine side chain.

DISCUSSIONIn what follows, we address two questions: the first concernsthe analysis by Kellis et al. (13, 14) of several apolar side-chain mutations in barnase, and the second deals with themore general question of the role of the hydrophobic effect inprotein stability.

Kellis et al. (13) used a thermodynamic cycle very similarto Fig. 3 and considered both alchemical processes (Ile -+ Alatransformations) and chemical processes (the unfolding re-action in the presence of isoleucine or alanine). They as-sumed that AGci, AG,,, and AGnb, have the same value in thefolded and unfolded state and so do not contribute to AAGfu..The present analysis shows that the nonbonded side-chaincontribution, AAGnbb, is indeed small for this case but thatboth of the other neglected terms are significant. In consid-ering the covalent term, Kellis et al. (14) focused on the effectdue to the change in the number of atoms. Clearly, anycontribution due simply to the imbalance of the chemicalequation must cancel out when the chemical processes areconsidered or when a reference state is introduced for thealchemical process. However, this does not mean that thereis no covalent contribution to AAGf.but The side-chain term,AGc, consists only of contributions from isoleucine, but theyare different in the folded and unfolded states, which repre-

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sent significantly different environments and so are a non-negligible part of AAGf,. The total covalent contributionconsisting of the sum of AAGCS and AAGci is -3.14 kcal/mol.It does represent a sizable fraction of the total value ofAAGfM (-5.21 kcal/mol). The quantitative values obtainedin the present simulation may, of course, be in error, but theconceptual conclusion is unlikely to be wrong.Most of the remaining contribution to AAGf M, arises from

the free energy term, AAGnbi, corresponding to nonbondedvan der Waals interactions of the side chain with its envi-ronment. This is the term emphasized by Kellis et al. (13),who concluded furthermore that the nonbonded contributionfrom the alchemical transformation in the folded protein isthe dominant term, whereas that from the alchemical trans-formation in the unfolded state is negligible. The calculationsalso show the importance of the nonbonded terms in thefolded state. They suggest, however, that only about one halfof &AGf .u arises from nonbonded interactions and that theremainder is due to the difference in covalent terms betweenthe folded and unfolded states.

Studies of amino acid substitutions in several proteins(12-15) show that the difference in stability of the wild-typeand mutant proteins is roughly proportional to the free energyof transfer of the individual substituted residues from organicsolvent to water. For Ile -. Ala mutations, the difference inthe transfer free energies ranges from 1.5 to 3.11 kcal/mol,measured for octanol and cyclohexane, respectively (18, 21).These values are of the same order as the computed orexperimental unfolding free energy differences. The differ-ence in the gas phase to water solvation free energies ofisoleucine versus alanine is significantly smaller (16).To understand the origins of these observations it is useful

to decompose the transfer process from water to the proteininterior into two separate solvation processes: (i) transfer ofthe side chain from the gas phase to the protein interior and(ii) transfer of the side chain from the gas phase to water(columns 6 and 7 of Table 1). The total free energy differencebetween isoleucine and alanine in the gas phase to proteintransfer (AAGf) is significantly larger than the correspond-ing difference in solvation free energies (AAG,,,). Thissuggests that although hydration effects provide importantcontributions to the free energy of transfer of the individualside chains from water to the protein interior, they make onlya minor contribution to the difference in stabilization freeenergy between the isoleucine and alanine side chains. Asalready mentioned, important contributions to this free en-ergy balance are provided by nonbonded van der Waals termsin the folded protein. Because the difference in experimentalsolvation free energies of isoleucine versus alanine from thegas phase to cyclohexane [2.9 kcal-mol (21)] is also larger thanthat from gas phase to water (0.21 kcal/mol), it is likely thatnonbonded van der Waals terms are important in the transferof isoleucine versus alanine to homogeneous organic sol-vents. For other types of mutations, particularly those in-volving polar or charged groups, the difference in watersolvation would probably be larger than that for solvation inan organic phase. These considerations suggest that transferfree energies of amino acids and the various hydrophobicityscales derived from them may include contributions, otherthan pure hydration effects. From the present results, therecan be contributions from changes in nonbonded interactionsof the side chains with the organic phase and from changes incovalent side-chain terms. In particular, the suggestion thatinternal degrees of freedom of the solute also contribute towhat is commonly described as hydrophobicity is an impor-tant result of the analysis.

The present theoretical study thus provides valuable newinsights into the origin of the hydrophobic effect and its rolein the thermodynamics of protein folding. In the future,improvements ofthe force field and longer simulations shouldlead to fuller convergence of the ensemble averages andpermit decomposition of the free energy into the enthalpicand entropic terms (23). With the available methods, inter-actions within the environment, either the protein (other thanthe mutated side chain) or the surrounding solvent, contrib-ute to the calculated free energy difference only through theBoltzmann weighting factor inherent in the molecular dy-namics trajectories (39). It would be useful to evaluate theeffect of these contributions, particularly for the presentproblem because the solvent and protein matrix are invokedin most descriptions of the role of the hydrophobic effect onprotein stability.

M.P. and S.J.W. acknowledge support from the European Com-munities Biotechnology Action Program (Contract 0319-B) and thankthe Fonds National de la Recherche Scientifique for a grant to oneof us (M.P.). M.K. and B.T. acknowledge partial support from theNational Science Foundation, the National Institutes of Health, andthe Department of Energy.

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