heat capacity and compactness of denatured proteins

Ž .Biophysical Chemistry 78 1999 207]217

Heat capacity and compactness of denatured proteins

Themis Lazaridisa,U ,1, Martin Karplusa,b,U

aDepartment of Chemistry and Chemical Biology, Har ard Uni ersity, Cambridge, MA 02138, USAbLaboratoire de Chimie Biophysique, ISIS, Uni ersite Louis Pasteur, 67000 Strasbourg, France´

Received 6 November 1998; received in revised form 2 February 1999; accepted 10 February 1999

Abstract

One of the striking results of protein thermodynamics is that the heat capacity change upon denaturation is largeand positive. This change is generally ascribed to the exposure of non-polar groups to water on denaturation, inanalogy to the large heat capacity change for the transfer of small non-polar molecules from hydrocarbons to water.Calculations of the heat capacity based on the exposed surface area of the completely unfolded denatured state givegood agreement with experimental data. This result is difficult to reconcile with evidence that the heat denaturedstate in the absence of denaturants is reasonably compact. In this work, sample conformations for the denaturedstate of truncated CI2 are obtained by use of an effective energy function for proteins in solution. The energyfunction gives denatured conformations that are compact with radii of gyration that are slightly larger than that ofthe native state. The model is used to estimate the heat capacity, as well as that of the native state, at 300 and 350 Kvia finite enthalpy differences. The calculations show that the heat capacity of denaturation can have large positivecontributions from non-covalent intraprotein interactions because these interactions change more with temperaturein non-native conformations than in the native state. Including this contribution, which has been neglected inempirical surface area models, leads to heat capacities of unfolding for compact denatured states that are consistentwith the experimental heat capacity data. Estimates of the stability curve of CI2 made with the effective energyfunction support the present model. Q 1999 Elsevier Science B.V. All rights reserved.

Keywords: Protein thermodynamics; Solvent effects; Non-bonded interactions; Stability of proteins

U Corresponding authors.1Present address: Department of Chemistry, City College of New York, Convent Ave and 138th St., New York, NY 10031, USA.

0301-4622r99r$ - see front matter Q 1999 Elsevier Science B.V. All rights reserved.Ž .PII: S 0 3 0 1 - 4 6 2 2 9 9 0 0 0 2 2 - 8

( )T. Lazaridis, M. Karplus r Biophysical Chemistry 78 1999 207]217208

1. Introduction

One of the fundamental quantities in proteinthermodynamics is the partial molar heat capacity

Dchange, D C , associated with the unfolding re-N pDaction. The non-zero values of D C lead to theN pŽcurvature of the stability curve free energy of

.unfolding as a function of temperature and arelinked to important phenomena, including hyper-thermostability and cold denaturation. Since

DD C can be measured by calorimetry and hasN pbeen determined for many proteins, to under-stand its origin is of particular interest. Measure-

Dments show D C is typically of the order ofN pw x12]20 calrmol K per residue 1 . The partial

molar heat capacity of native proteins increaseslinearly with temperature whereas the partialmolar heat capacity of the denatured state issomewhat non-linear, tending to level off at high

Dw xtemperatures 2 . As a result, D C varies withN ptemperature, has a maximum near room tempera-ture, and decreases at lower and higher tempera-

w xtures 1 .DIt is generally believed that the positive D CN p

values arise primarily from the exposure of non-w xpolar groups to water 3]8 . The basis for this

conclusion is that the transfer of non-polar groupsfrom the gas phase or non-polar liquids intowater is also accompanied by a large positive heat

w xcapacity change 9 . More recently, it was realizedthat polar groups also make a contribution to

DD C that is smaller and of the opposite sign toN pw xthat of non-polar groups 6,10,11 . It has been

Dfound that D C can be well reproduced as-N pŽsuming a fully unfolded denatured state all

Dw x.residues fully exposed to solvent 2 if D C isN ptaken to be proportional to the change in theexposed surface area on unfolding. In such calcu-lations the proportionality constants appropriatefor the amino acids are obtained from small model

w xcompound transfer experiments 6,11]13 .Such a fully unfolded thermally induced dena-

tured state is difficult to reconcile with otherexperimental evidence showing that the dena-tured state in the absence of denaturants is rathercompact on the average and includes populationsranging from highly compact to rather unfolded

w xconfigurations of the polypeptide chain 14,15 .

w xAlready in the work of Tanford 16 , it was shownthat the heat denatured state is not as unfoldedas the chemically denatured state. Recent solu-

w xtion X-ray scattering 17 and CD and FTIRw x17]19 experiments on heat denatured ribonu-clease A showed that this state is compact andcontains significant amounts of ‘residual struc-ture’, although it is not native-like because it doesnot offer significant protection from hydrogen

w xexchange 20 . Residual structure has also beenfound in the heat denatured states of other pro-

w xteins 21,22 . Experiments indicating that the de-natured state is compact have been summarized

w xby Shortle 23 . Theoretical work on simplifiedw xprotein models 24,25 and molecular dynamics

Ž . w xMD simulations in explicit solvent 26,27 alsosuggest a relatively compact denatured state.Residual structure and compactness implies thata large fraction of the protein residues are inter-acting with each other and are, at least in part,sequestered from solvent, although the fluctua-tions in the structures are such that little hydro-gen exchange protection is present.

The experiments and simulation results on thecompact nature of the denatured state appearinconsistent with the standard interpretation of

DD C which assumes a fully unfolded state. ToN presolve this question we generalize a recently

Ž .developed model EEF1 that approximates theŽ .effective energy potential of mean force of pro-w xteins in solution 28,29 for the calculation of the

partial molar heat capacity of polypeptides andproteins and show how to use this model to

Devaluate D C . The native state is obtained fromN pthe X-ray structure and the denatured state isgenerated by MD simulations. We use the sameapproach to estimate the stability curve, DD G ofNunfolding, for both compact and extended modelsfor the denatured state. The system we use toillustrate the approach is the 64-residue truncatedversion of the small protein CI2, for which experi-mental data and unfolding simulations are avail-able.

2. Theory

The heat capacity of unfolding at constantpressure, as determined by calorimetric experi-

( )T. Lazaridis, M. Karplus r Biophysical Chemistry 78 1999 207]217 209

w xments 30 , is the difference in partial molar heatŽ .capacity between the native N and denatured

Ž .D states and is equal to the temperature deriva-Žtive of the enthalpy of unfolding the correspond-

.ing difference in partial molar enthalpy ; that is,

DD hND ŽD . ŽN . Ž .D C sC yC s 1N p p p ž / T P

where the overbar denotes a partial molarŽproperty of the protein the increment in that

property when a mole of protein is added to the.solution at constant temperature and pressure .

For simplicity we omit the constant pressure sub-script on the derivatives in what follows. Forclassical, non-polarizable Hamiltonians, such asthose used in most molecular mechanics energy

w xfunctions 31,32 , and for sufficiently dilute solu-tions so that the protein molecules do not inter-act with each other, the partial molar enthalpy ofthe N or D state can be divided into two con-tributions: the partial molar enthalpy in an ideal

Ž .gas ig state at the same temperature and den-Žsity, and the excess partial molar enthalpy or

solvation enthalpy, the enthalpy change upontransfer of the protein from the gas phase to

.solution at constant T and P :

ŽK . i g ,ŽK . e x ,ŽK . Ž .h sh qh , KsN or D 2

The first term in the RHS contains the con-tribution from kinetic energy and from the intra-

Žmolecular potential energy the PvsRT terme xcan be included with h to bring it closer to

w x.Ben-Naim’s standard solvation enthalpy 33 . Thepartial molar heat capacity can be written

i g ,ŽK . e x ,ŽK . h hŽK . Ž .C s q , KsN or D 3p T T

and the heat capacity of unfolding

D i g D e x ,ŽK .D h D hN ND Ž .D C s q 4N p T T

Theoretical calculation of the gas phase con-Ž .tribution in Eq. 3 can in principle be made with

classical mechanics. However, many components

of the intramolecular energy in proteins, such asbond stretching terms and to a smaller extentbond angle bending terms, are associated withhigh frequency vibrations, and their contributionsto the heat capacity at room temperature are

w xgreatly overestimated by classical mechanics 34 .For example, the contribution of a typical protein

Žbond to the heat capacity frequency ¨ f 1240y1 .cm , if viewed as an independent harmonic

oscillator, is approximately 0.1 R at room temper-ature, whereas the classical harmonic oscillatorgives R from the derivative of the potential andkinetic energy. On the other hand, the non-bonded interactions are associated with muchlower frequencies and can be treated classically.For example, a typical Lennard]Jones interactionŽ y1¨ f 36 cm , from a Taylor expansion of the

.potential around the minimum gives ;0.998 R,essentially the classical limit. The alternative toclassical mechanics is to perform a normal modeanalysis, calculate the frequencies of all normalmodes, and use the equations of a quantum me-chanical harmonic oscillator to compute the ther-

w xmodynamic properties 35,36 . However, this ap-proach would not allow us to study the effects oflarge conformational fluctuations that are not de-scribed by the harmonic approximation and toseparate out the contribution of non-bonded in-teractions.

In this work we follow the classical approachand make use of molecular dynamics but neglectthe kinetic energy contribution. We also do notinclude the contribution of the non-polar hydro-gens, which are not treated explicitly inCHARMM 19 and EEF1. This approach overesti-mates the contribution of stiff degrees of freedomand underestimates the contribution of soft de-grees of freedom. The reasonable values for thepartial molar heat capacities obtained by this

Ž .method see Section 4 must, therefore, resultfrom a cancellation between the two types oferrors. However, disregarding the fortuitous can-cellation, it is the ‘difference’ in partial molarheat capacity between N and D which is ofprimary interest here. This is expected to bereliable in the classical limit because the kineticenergy contribution does not change and thequantum mechanical effects involving stiff bonds


are expected to be the same in N and D. In whatfollows we write

i g ,ŽK . int r a ,ŽK . h E Ž .f , KsN or D 5 T T

where Eint r a is the intramolecular potential en-Ž .ergy, realizing that Eq. 5 is most accurate for

the difference between N and D; i.e. for theD Ž .calculation of D C as in Eq. 1 .N p

The various classical intramolecular terms inŽthe energy function, such as the covalent bond

.stretching and bending, etc. and non-covalentŽ .van der Waals and electrostatic interactions, canbe assumed to be approximately independent oftemperature, so that Eint r a varies with tempera-ture primarily through the change in the distribu-tion of conformations:

Eint r a ,ŽK .Ž .s0, KsN or D 6ž / ŽK . T Ž .p q

where q is a set of variables describing the con-Ž .ŽK .formation of the protein and p q is the con-

formational distribution of the protein in the stateŽ .K. The solvation term in Eq. 4 has two contribu-

tions, one from the change in solvation enthalpyŽ .with temperature at constant p q , and another

from the change in the distribution of conforma-tions with temperature. Thus, the partial molarheat capacity of state K can be written:

Ž .Kint r a ,ŽK . Ž .dE p qŽK .C s dqHp Ž .K TŽ .d p q

e x ,ŽK . hq ž / ŽK . T Ž .p q

Ž .Ke x ,ŽK . Ž .d h p qq dq, KsN or DH Ž .K TŽ .d p qŽ .7

Ž .where the derivatives with respect to p q arew xfunctional derivatives 37 .

Since the denatured state is a collection ofŽ .basins local minima on the effective energy sur-

face, its heat capacity will contain one contribu-

tion from the protein conformational relaxationwithin each basin and another from the redis-tribution of the protein population among thebasins as the temperature is changed. In this workwe calculate only the first contribution. Estima-tion of the second contribution requires samplingof a large number of denatured basins and calcu-lation of how their statistical weights change with

w xtemperature 38]40 . The native state can beviewed as occupying a single basin on the effec-tive energy hypersurface. Therefore, there is nosummation over basins and no contribution to theheat capacity from redistribution among basins.Reasonable estimates of the thermodynamicproperties of protein native states, including theheat capacity, can be made by use of the har-

w xmonic approximation 35 , although multiple con-formers of sidechains make a small additional

w xcontribution 41,42 .Empirical models for the calculation of the

Ž .heat capacity account for the terms in Eq. 7 indifferent ways. Models which use data for thetransfer of model compounds from non-polar liq-

w xuids to water 8,11 implicitly account for thenon-covalent contributions in the first term andfor the second term; i.e. the interactions of themodel compound in the non-polar liquid are as-sumed to mimic the interactions of similar groupsin the protein interior. Most such models neglectthe contribution of covalent intramolecular inter-

Dactions to D C . Models which use data for theN ptransfer of model compounds from the gas phase

w xto water 1 estimate the second term based onaccessible surface area and the first term by dif-

w xference from the experimental value 1,7,43 ; inthe latter, a covalent contribution is includedimplicitly. Because all models to date use a single,fixed conformation as a description for the dena-

Žtured state a fully extended chain or the surface.area of Gly]X]Gly tripeptides , none of themŽ .accounts for the last term in Eq. 7 , which is due

to changes in the conformational distribution ofthe denatured polypeptide chain.

3. Methods

D Ž .In this paper we estimate D C using Eq. 7N pfor N and D. We perform molecular dynamics


simulations at different temperatures and use thewresulting structures to calculate the enthalpy Eq.

Ž .x1 and estimate the various contributions toDD C by finite differences. The simulations areN p

performed with the implicit solvent model em-w xbodied in EEF1 29 , a recently proposed effective

energy function consisting of the CHARMM polarw xhydrogen energy function 32,44 and an implicit

w xsolvation term 28,29 . EEF1 has been shown togive stable native structures in room temperature

w xMD simulation 29 , reasonable energies for un-w xfolded and misfolded conformations 29 and un-

folding pathways in agreement with explicit waterw x w xsimulations 28 . As pointed out in 29 , the same

implicit solvation model can be used to calculatesolvation enthalpies by substituting solvation en-

w xthalpy group contributions 45 for solvation freeenergy group contributions. The method was ap-plied to estimate the unfolding enthalpy of he-

w xlices 29 .To obtain models for the denatured state, we

started from a fully extended chain and per-Ž .formed three simulations D1, D2, D3 with dif-

ferent initial velocities. The simulations lastedŽ .1.15 ns 100 ps at 600 K and 1050 ps at 300 K ,

during which the chain collapsed to a compactconformation. Starting from the three models ob-tained at the end of these simulations, we did20-ps MD simulations at 280 and 320 K; the last 2ps of the simulations were used for calculatingaverages. For D1 the simulations were extendedto 50 ps and averaged over the last 6 ps; theresults were very similar. The native protein wasalso subjected to 20-ps MD simulations at 280and 320 K; these simulations started from anequilibrated structure of native CI2 after 50 ps of

ŽMD at 300 K. The contributions of internal cova-.lent and non-covalent interactions and of solva-

tion to the enthalpy are readily obtained from thesimulations with the EEF1 model and enthalpyparameters. The group solvation free energies attemperatures other than 300 K are calculatedfrom the experimental estimates of the group

w xsolvation enthalpies and heat capacities 29 .Temperature-independent group heat capacities

w xat 298 K were used in this calculation 7 .The heat capacity was also estimated at 350 K.

The native and the three denatured conforma-tions were first simulated at 350 K for 50 ps. Eachof the resulting structures was simulated at 330and 370 K for 20 ps, as above, and the last 2 pswere used in calculating averages. This calcula-tion accounts for the change in conformationaldistribution within individual basins on the effec-tive energy hypersurface but not for the possibleredistribution of the protein population among

Ž .basins see above . Thus, the calculation is ex-pected to underestimate the heat capacity of thedenatured state.

To provide an additional criterion for the valid-ity of the model, we use it to estimate the stabilityof the protein as a function of temperature. The

Žstandard free energy of unfolding the differencein the chemical potential of the denatured and

.native forms is

² : con fDGsD W yTDS

² int r a: ² sl¨ : con f Ž .sD E qD DG yTDS 8

² :where D W is the change in the average effec-² sl¨ :tive energy, D DG is the change in the aver-

age solvation free energy and DScon f is the changein conformational entropy upon unfolding. The

w xvalue of W is obtained from EEF1 29 . For the² int r a: con fcalculation we take D E and DS to be

independent of temperature. This is not strictlytrue, but the two terms compensate each other toa large extent in the free energy of unfoldingŽ .both increase with temperature . Consequently,the T factor in the conformational entropy termand the temperature dependence of the solvationfree energy should yield a qualitatively correctstability curve with the present model. For each

Žstructure of the denatured state D1, D2, D3 and. ² int r a:completely extended, E , we calculate D E

² sl¨ : con fand D DG at 300 K and determine DS byfitting the experimental DG at that temperature;

Ž .the value used 7 kcalrmol is that given byw x ² int r a:Jackson and Fersht 46 . Then, we keep D E

and DScon f constant, as indicated above, and² sl¨ :determine D DG as a function of tempera-

ture from the implicit solvation model.


4. Results and discussion

All three simulations starting from the fullyextended chain resulted in a relatively compactstructure. The energetic and structural character-istics of the three compact denatured conforma-tions, designated D1, D2, and D3, are shown inTable 1 and compared with the native state. Theeffective energies of the three denatured statesare between 38 and 65 kcalrmol higher than thatof the native state. This is a reasonable differ-ence, considering the experimental protein stabil-ity and the estimated change in conformational

w x Ž .entropy 29 . The radius of gyration R of thegdenatured conformations is only 12]18% greaterthan that of the N state; this is in agreement withthe experimental data cited in the introduction.Nevertheless, the root mean square deviationŽ .RMSD of all backbone atoms from the crystalstructure and between the three denatured states

˚is more than 10 A; the RMSD of the native state˚simulation from the crystal structure is 1.46 A.

The ‘hydrophobic collapse’ from the extendedstate found in the MD simulations is accom-panied by formation of a large number of pro-tein]protein hydrogen bonds; the number of hy-drogen bonds in the denatured forms is onlyslightly smaller than that in the native state butvery few are native-like. Moreover, the three de-natured conformations have almost no native

Žcontacts pairs of atoms more than three residues˚apart in sequence that are within 4 A from each

w x.other in the crystal structure 28 . D1 has onenative hydrogen bond between the b4 and b6

Ž .strands 58N]50O within 1.5 times the distancein the crystal structure, D3 has no native con-tacts, and D2 has the 13 CG2-51 CG1 hy-drophobic contact within twice the native dis-tance. Thus, the three structures correspond to areasonable, though limited, sample of the dena-tured ensemble.

From the average calculated enthalpy at 280Ž .and 320 K and Eq. 1 , the partial molar heat

capacity of the native protein is estimated to be2.75 kcalrmol K. Of this, 1.725 kcalrmol K arisesfrom internal interactions and the rest from sol-

Ž .vation see Table 2 . Although the calculation ishighly approximate, this estimate is in the rangeof the experimental results. The value for the full,

Žuntruncated CI2 83 residues instead of 64 con-. w xsidered here at 298 K is 3.18 kcalrmol K 1 . The

internal contribution is quite close to the valuecalculated for BPTI at 300 K from normal mode

w xcalculations 35 .Most of the internal contributions to C , comep

Žfrom the covalent interactions 1.425 out of 1.725.kcalrmol K ; these correspond primarily to the

harmonic terms in the potential function. Thetotal heat capacity arises 52% from covalent con-tributions, 11% from non-bonded contributionsŽ .van der Waals and electrostatic and 37% fromthe solvation term. However, the covalent con-tribution is likely to be overestimated because of

Ž .the classical treatment see Section 2 .

Table 1aCharacteristics of native and denatured conformations

bN D1 D2 D3

cEffective energy y1970 y1931 y1905 y1932Ž . Ž . Ž . Ž . Ž .-vdW, -elec, -solv 453, 1083, 607 411, 1074, 614 399, 1052, 626 389, 1083, 627R at 300 K 11.38 12.91 12.78 13.40gH-bonds 60 53 54 58RMSD from crystal 1.46 13.41 12.96 11.8RMSD from D1 10.85 13.88RMSD from D2 12.11R at 350 K 11.17 12.97 12.64 13.07g

a ˚Energies in kcalrmol, R and RMSD in A; the RMSD refers to the backbone atoms while R includes all atoms.g gbN is the native state after 200 ps of molecular dynamics simulation at 300 K.cAfter 300 steps of energy minimization.


Table 2D aEstimation of D C for CI2 at 300 KN p

co¨ n o n b sl¨² : ² : ² : ² : ² : ² :W Hss H H D H H Css ss p

Tot Int Solv

N 2.75 1.725 1.025280 K y1467 y815 578 y1393 y909 y1724320 K y1361 y746 635 y1381 y868 y1614

D1 3.45 2.675 0.775280 K y1449 y805 592 y1397 y928 y1733320 K y1320 y698 651 y1349 y897 y1595

D2 3.2 1.775 1.425280 K y1423 y762 583 y1345 y948 y1710320 K y1310 y691 652 y1343 y891 y1582

D3 3.175 1.85 1.325280 K y1442 y779 581 y1360 y954 y1733320 K y1331 y705 642 y1347 y901 y1606

UD C NªD1 0.7 0.95 y0.25N p

UD C NªD2 0.45 0.05 0.4N p

UD C NªD3 0.425 0.125 0.3N p

aAverage energies over the last 2 ps of a 20-ps MD simulation at the designated temperature. N is native, D is denatured. Energiesin kcalrmol, heat capacities in kcalrmol K. W is the effective energy, H the intramolecular energy, H co¨ the covalents s s s

Ž . s l¨contribution to H bonds, bond angles, dihedral, and improper dihedral angles , D H the solvation enthalpy, H the totals senthalpy.

The results for the denatured conformationsare also shown in Table 2. D1 has a heat capacity

Žof 3.45 kcalrmol K 2.675 from internal interac-. Žtions , D2 of 3.2 kcalrmol K 1.775 from internal

. Žinteractions , and D3 of 3.175 kcalrmol K 1.85.from internal interactions . Comparing Tables 1

and 2, we see that the solvation contribution isnot simply related to R . The experimental valuegfor denatured untruncated CI2 given by

w xMakhatadze and Privalov 1 at 298 K is 4.04kcalrmol K. This is in reasonable agreement withthe calculated average value of 3.275 kcalrmol Kfor the truncated version of CI2, despite thehighly approximate nature of the calculation.

The estimated heat capacity of unfolding,DD C , varies considerably for the three models ofN p

the denatured state. The average value is 0.525"0.15 kcalrmol K, compared to the experimental

w xvalue of 0.789 kcalrmol K 46 obtained with theuntruncated form of CI2. Although the dis-ordered N-terminal segment of CI2 is expected tocontribute to the absolute heat capacity of thenative and denatured states, its contribution to

DD C may be small. The underestimation ofN pDD C may be due to deficiencies of the hydrationN p

Žmodel for example the group additivity assump-.tion , to the obviously limited sampling of the

denatured state or to the neglect of transitions onŽ .a longer time scale see above . Covalent con-

tributions are approximately equal in the dena-tured and native conformations and thus do notmake a significant contribution to the unfolding

w xheat capacity, as expected 13,39 ; see also Section2. In the case of D1, the solvation contribution isfound to be negative despite the fact that expo-sure of non-polar groups is higher in D. This isdue to the contribution of the third term in Eq.Ž .7 , which is included implicitly in the MD simula-tion at different temperatures but has not beenconsidered in previous analyses. For example, in-crease in temperature may increase the exposureof polar groups in the denatured state and thusdecrease the solvation enthalpy differencebetween D1 and N. Indeed, for D1 the non-polar

˚2surface area increases by 7 A whereas the polar˚2surface area increases by 127 A between the


280-K and 320-K simulations. If the solvationenthalpy at 320 K is calculated from the ensembleof conformations generated at 280 K, the solva-

wtion heat capacity of D1 the second term in theŽ .xRHS of Eq. 7 is calculated to be 1.225 kcalrmol

K instead of the 0.775 that results when thechange in conformational distribution is takeninto account.

The compactness of the denatured conforma-tions does not change substantially between 300

Ž .and 350 K Table 1 . The reason is that, althoughconformational entropy tends to make them moreexpanded as the temperature increases, the solva-tion free energy of both polar and non-polargroups increases with temperature and tends to

Žmake the protein more compact so that the.groups are less exposed as the temperature in-

creases. The heat capacity results at 350 K areshown in Table 3. The heat capacity of the nativestate increases from 2.75 at 300 K to 3.2 kcalrmolK at 350 K. The value at 350 K given by

w xMakhatadze and Privalov 1 is 3.47 kcalrmol K.Both the internal and solvation terms increase

with temperature, although the increase in thesolvation term is larger.

The heat capacities of the three denaturedconformations behave differently with tempera-ture. For D1, the heat capacity at 350 K is lowerthan that at 300 K, but for D2 and D3 it is higherthan at 300 K. The difference in heat capacitybetween N and D is smaller than at 300 K. This

w xis in qualitative agreement with experiment 1 .Experimental dissection of the thermodynamic

quantities of unfolding into internal and solvationcontributions would be very useful but is verydifficult. Recently, an approach was developed forthe estimation of a contribution to the proteinconfigurational entropy change on denaturationbased on the mobility of backbone NH vectors as

w xdetermined from NMR 47,48 . From the temper-ature dependence of this entropy contribution,

w xheat capacity information could be extracted 49 .It was found that the entropy associated withbackbone NH bond vectors increases with tem-perature more for the unfolded than for nativeproteins. This means that there is significant con-

Table 3D aEstimation of D C for CI2 at 350 KN p

co¨ s l¨² : ² : ² : ² : ² :W H H D H H Css ss p

Tot Int Solv

N 3.2 1.8 1.4330 K y1331 y723 653 y861 y1584370 K y1221 y651 725 y805 y1456

D1 3.275 2.525 0.75330 K y1297 y692 670 y870 y1562370 K y1181 y591 723 y840 y1431

D2 3.4 1.5 1.9330 K y1282 y669 659 y885 y1554370 K y1177 y609 733 y809 y1418

D3 3.5 2.65 0.85330 K y1293 y673 655 y889 y1562370 K y1166 y567 719 y855 y1422

UD C NªD1 0.075 0.725 y0.65N p

UD C NªD2 0.2 y0.3 0.5N p

UD C NªD3 0.3 0.85 y0.55N p

aAverage energies over the last 2 ps of a 20-ps MD simulation at the designated temperature. N is native, D is denatured. Energiesin kcalrmol, heat capacities in kcalrmol K. W is the effective energy, H the intramolecular energy, H co¨ the covalents s s s

Ž . s l¨contribution to H bonds, bond angles, dihedral, and improper dihedral angles , D H the solvation enthalpy, H the totals senthalpy.


tribution to the heat capacity of unfolding fromintraprotein interactions, in accord with the re-sults obtained here.

Calculated stability curves based on the com-pact models and the extended chain are shown inFig. 1. The stability curve obtained from the ex-

w xperimental thermodynamic data 46 is shown inthe same figure. It exhibits a maximum at approx-imately 270 K. The calculated temperature de-pendence of the stability exhibits the correct qual-itative features, such as a maximum around roomtemperature, although the results do not agreequantitatively with experiment, as expected forthe model calculations done here. The calcula-tions based on the compact models give changesin DG with temperature comparable to experi-ment, whereas the calculation based on the ex-tended chain gives variations in DG that aresignificantly too large, because the calculated val-

Ž .ues of all terms in the RHS of Eq. 8 are toolarge. This supports the conclusion that compactconformations are more reasonable models forthe thermally denatured state.

The maximum in DG near room temperatureresults from competition between the two tem-perature dependent terms in the present model:there is the conformational entropy term,TDScon f, which increasingly favors the D state as

Ž .Fig. 1. Stability curves for CI2: experimental thick solid line ,Ž . Ž .based on D1 thin solid line , based on D2 thin dashed line ,Ž .based on D3 dotted line , and based on the extended chain

Ž .thick dashed line . The curves are fitted to the experimentalŽ .value at 300 K see text .

Fig. 2. Free energy of unfolding as a function of temperatureŽ . Žsolid line , the conformational entropy contribution dotted

. Ž .line , and the solvation free energy contribution dashed line .These are based on the D1 model; the results for the otherdenatured conformations D2 and D3 are similar.

T increases, and the solvation term which in-Žcreasingly favors the D state as T decreases Fig.

. con f2 . Possible increases in DS with temperature,which are neglected in the present calculations,would increase the curvature of the calculatedstability curves and bring them in closer agree-

Ž .ment with experiment. In terms of Eq. 8 , denat-uration is due to conformational entropy at high

Ž ² sl¨ :.temperature and solvation free energy D DG² sl¨ :at low temperatures. In D DG , the solvation

free energy becomes less positive for non-polargroups and more negative for polar groups astemperature decreases. This makes exposure ofnon-polar groups less unfavorable and exposureof polar groups more favorable so that, as sug-

w xgested by Makhatadze and Privalov 50 , bothpolar and non-polar groups contribute to colddenaturation.

5. Conclusions

A method for estimating the heat capacity ofproteins has been presented. The basis of thecalculation is a model for the solvation free en-ergy, enthalpy and heat capacity whose parame-ters were obtained from experimental data on thesolvation of small molecules. This method differsfrom purely empirical approaches in that it does


not assume a fixed conformation for the dena-tured state but uses averages over conformationsobtained from molecular dynamics simulations.Also, it includes the contribution from changes inthe conformational distribution of the proteinwith temperature in the calculation of the heatcapacity; this is neglected in other approaches.Although the calculation of the absolute partialmolar heat capacities is highly approximate dueto the use of classical mechanics, the results areof the correct order of magnitude. The heat ca-pacity of unfolding should be less affected by theclassical approximation.

The exposure of non-polar groups to watermakes a large positive contribution to the heatcapacity of unfolding in the present model, aswell as in more empirical analyses. However, theresults presented here suggest that a significantcontribution to the heat capacity of denaturationcomes from protein]protein non-bonded interac-tions. The denatured state is compact but morelabile than the native state so that temperaturecan break interactions in D more easily than inN. This means that the enthalpy of D increaseswith temperature more than that of N and con-

Dtributes significantly to D C . The same effectN phas been found to be important in lattice modelsfor proteins that use temperature-independent

w xinteraction parameters 51 . The results demon-strate that relatively compact denatured statesare consistent with the calorimetric data. Thesmaller contribution of hydrophobic group expo-sure to the heat capacity in this model for thedenatured state is compensated by the contribu-tion of non-covalent protein interactions.

The picture that emerges from the presentstudy is considerably more complex than thatderived from simple models that assume that allof the denaturation heat capacity arises fromexposure of polar and non-polar surface area. Itis reasonable that some correlation should existbetween denaturation heat capacity and amountof buried non-polar surface area. However, thisdoes not necessarily mean that all of this area willbe fully exposed in the denatured state and sug-gests that the success of heat capacity calculationswith simple surface accessibility models, which

use a fully extended denatured state, may befortuitous, in part.

The results presented here pertain to the ther-mally denatured state in the absence of denatu-rants and agree with experimental and simulationestimates concerning its compact nature. In solu-tions of urea or Gdn HCl, the solvation freeenergy of both polar and non-polar groups is

w xmore favorable 52 and a more expanded dena-tured state is expected.

Acknowledgements

This work was supported by a grant from theNational Science Foundation and a BurroughsWellcome PMMB postdoctoral fellowship to T.L.The authors are grateful to Benoit Roux for manyinsightful comments.

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