Collapse kinetics and chevron plots from simulationsof denaturant-dependent folding of globular proteinsZhenxing Liua, Govardhan Reddya, Edward P. O’Briena, and D. Thirumalaia,b,1

aBiophysics Program, Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742; and bDepartment of Chemistry,University of Maryland, College Park, MD 20742

Edited* by William A. Eaton, National Institutes of Health-National Institute of Diabetes and Digestive and Kidney Diseases, Bethesda, MD, and approvedMarch 8, 2011 (received for review December 24, 2010)

Quantitative description of how proteins fold under experimentalconditions remains a challenging problem. Experiments often useurea and guanidinium chloride to study folding whereas the nat-ural variable in simulations is temperature. To bridge the gap, weuse themolecular transfer model that combinesmeasured denatur-ant-dependent transfer free energies for the peptide group andamino acid residues, and a coarse-grained Cα-side chain modelfor polypeptide chains to simulate the folding of src SH3 domain.Stability of the native state decreases linearly as ½C� (the concentra-tion of guanidinium chloride) increases with the slope,m, that is inexcellent agreement with experiments. Remarkably, the calculatedfolding rate at ½C�¼0 is only 16-fold larger than the measuredvalue. Most importantly lnkobs (kobs is the sum of folding andunfolding rates) as a function of ½C� has the characteristic V (chev-ron) shape. In every folding trajectory, the times for reaching thenative state, interactions stabilizing all the substructures, andglobal collapse coincide. The value of mf

m (mf is the slope of the fold-ing arm of the chevron plot) is identical to the fraction of buriedsolvent accessible surface area in the structures of the transitionstate ensemble. In the dominant transition state, which does notvary significantly at low ½C�, the core of the protein and certainloops are structured. Besides solving the long-standing problemof computing the chevron plot, our work lays the foundation forincorporating denaturant effects in a physically transparent man-ner either in all-atom or coarse-grained simulations.

kinetic cooperativity ∣ self-organized polymer model ∣ pathway diversity ∣protein denaturation

Understanding how proteins fold in quantitative moleculardetail can provide insights into protein aggregation and dy-

namics of formation of multisubunit complexes. As a result therehave been intense efforts in deciphering the folding mechanismsof proteins using a variety of experiments (1–6), theories (7–13),and simulations (14–16). Despite these advances, a number ofissues such as the denaturant-dependent characteristics of theunfolded states and the link between protein collapse and fold-ing remain unclear (17–21). Single molecule experiments haveprovided clear evidence for folding pathway diversity, and haveshown that polypeptide chains undergo almost continuous col-lapse as the denaturant concentration (½C�) is decreased (1,19–21). These and other experiments (5, 22) raise the need forcomputational models whose predictions can be directly com-pared to experiments, which often use denaturants [guanidiniumchloride (GdmCl) and urea] to initiate folding and unfolding.

Global thermodynamic and kinetic behavior of small proteinsoften exhibit two-state behavior (23), which implies that at allvalues of ½C� only the population of native (N) and the unfolded(U) states are detectable. Experimentally two-state behavior ischaracterized by (i) the free energy of stability, ΔGNUð½C�Þð¼ GNð½C�Þ −GUð½C�ÞÞ, of N with respect to U, varies linearlywith ½C�, and (ii) the logarithm of the relaxation rate kobs, whichis the sum of folding and unfolding rates, is V -shaped (“chevronplot”) when plotted as a function of ½C�. For two-state systemschevron plots show a linear decrease in ln kobs as ½C� increases

(folding arm) followed by an increase (unfolding arm) at high½C�. In previous computational studies, chevron plots were gen-erated by mapping temperature or certain interaction parametersin the force-field to ½C� (16). Although these studies have pro-vided insights into the role of nonnative interactions they fallshort of directly probing folding of proteins in the presence ofdenaturants. Here, we solve the long-standing problem of howto calculate chevron plots, and in the process describe the entirefolding reaction of a small protein in near quantitative detail.

Molecular simulations of coarse-grained off-lattice models (14,24, 25), are ideally suited to make (semi)quantitative predictionsof the folding reaction. A combination of statistical mechanicalapproach and simulations using simple off-lattice models hasprovided insights into protein folding mechanisms. In particular,these approaches have shown how linear free energy relations forthe seemingly complex folding reactions emerge naturally (26).More importantly, the theoretical ideas rationalize the robustnessof folding in terms of topology alone as illustrated in the simula-tions of structure-based models of small globular proteins (27).Of particular relevance to the present study is the demonstrationthat solvation forces determine the origin of barriers in SH3domains, which helped rationalize the effect of specific mutationson the folding rates (28). Here, we extend the molecular transfermodel (MTM) (29–31) to map in quantitative detail the thermo-dynamics and to a lesser extent kinetics of folding of src SH3 do-main as a function of GdmCl and urea. We show that ΔGNUð½C�Þdecreases linearly as ½C�, with the slopes (m) being in quantitativeagreement with experiments. Simulations of the ½C�-dependentln kobs has the characteristic chevron shape, and the values ofthe slopes of the folding (mf ) and unfolding (mu) arms are in ex-cellent agreement with experiments. There is a single dominanttransition state whose structure shows that the core β-sheets arewell packed and the stiff distal loop is ordered. Contacts involvingthe hydrophobic core and the other secondary structural ele-ments, overall collapse, and formation of the entire native struc-ture form nearly simultaneously at zero and low ½C�. Our worksolves the long-standing problem of describing the entire dena-turant-dependent folding of a protein, including the creation ofthe chevron plot, albeit at a coarse-grained level.

ResultsStructure of src SH3. To establish the utility of MTM we computedseveral experimentally measurable thermodynamic and kineticproperties of the 56-residue src SH3 domain (Fig. 1A) (29) (seeMethods for details). The core of SH3 domains, whose folding hasbeen extensively investigated using experiments (32–35) and si-

Author contributions: Z.L., G.R., E.P.O, and D.T. designed research; Z.L., G.R., and D.T.performed research; D.T. contributed new reagents/analytic tools; Z.L., G.R., and D.T.analyzed data; and Z.L., G.R., and D.T. wrote the paper.

The authors declare no conflict of interest.

*This Direct Submission article had a prearranged editor.1To whom correspondence should be addressed. E-mail: thirum@umd.edu.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1019500108/-/DCSupplemental.

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mulations (28, 36–38), consists of two well-packed three strandedβ-sheets that are connected by RT, n-src, and short distal loops(Fig. 1A).

Denaturant-Dependent Folding Thermodynamics. Given the approx-imations in our model (see Methods and SI Text), it is difficultto get absolute free energy differences between the folded andunfolded states from simulations. To make comparisons withexperiments, we choose a simulation temperature, Ts, at whichthe calculated free energy of stability of the native state (N) withrespect to the unfolded state (U), ΔGNUðTsÞ (GNðTsÞ −GUðTsÞ),and the measured free energy at TEð¼ 295 KÞ ΔGNUðTEÞ coin-cide. The use of ΔGNUðTsÞ ¼ ΔGNUðTEÞ (in water with ½C� ¼ 0)to fix Ts is equivalent to choosing a overall reference energyscale in the simulations. For src SH3, ΔGNUðTE ¼ 295 KÞ ¼−4.1 kcal∕mol at ½C� ¼ 0 (34), which results in Ts ¼ 339 K.Besides the choice of Ts no other parameter is adjusted to fitany other aspect of src SH3 folding. With Ts ¼ 339 K fixed, wecalculated the dependence of fraction of molecules in the nativebasin of attraction (39), fNBAð½C�Þ, on ½C� using Eq. S18. Theagreement between measured and simulated results forfNBAð½C�Þ versus ½C� is excellent (Fig. 1B). The midpoint concen-tration, Cm, obtained using fNBAð½Cm�Þ ¼ 0.5 is ½C� ¼ 2.5 M, alsoagrees with the measured experimental value of 2.6 M (34).

The native state stability with respect to U, ΔGNUð½C�ÞðGNð½C�Þ −GUð½C�ÞÞ, can be calculated using a two-state fit to

fNBAð½C�Þ leading to ΔGNUð½C�Þ ¼ −kBTs lnð fNBAð½C�Þ1−fNBAð½C�ÞÞ. The line-

ar fit (Fig. 1C), ΔGNUð½C�Þ ¼ ΔGNUð½0�Þ þm½C�, yieldsΔGNUð½0�Þ ¼ −3.4 kcal∕mol and m ¼ 1.34 kcal∕mol·M. We alsocomputed ΔGNUð½C�Þ using free energy profiles as a function of χ(Eq. S16), and obtained m ¼ 1.47 kcal∕mol·M. The experimen-tally inferred m value using circular dichroism is 1.5 kcal∕mol·Mwhereas m ¼ 1.6 kcal∕mol·M using fluorescence measurements.The values from our simulation for m ð1.34–1.47Þ kcal∕mol·Mare in excellent agreement with experimental estimates, whichfurther shows that MTM can predict the global thermodynamicproperties accurately.

Although the calculated value of ΔGNUð½0�Þ is in good agree-ment with the experimental result, it is not as accurate asm-valuebecause of the known difficulties in obtaining accurate results atlow GdmCl concentrations (40). To ensure that MTM can predictΔGNUð½0�Þ accurately we calculated fNBAð½C�Þ for urea (greensquares in Fig. 1B) from which we obtained ΔGNUð½C�Þ (Fig. 1C).The linear fit yields ΔGNUð½0�Þ ¼ −4.2 kcal∕mol, which is inexcellent agreement with measurements. The m value for ureais 0.92 kcal∕mol·M (less than m for GdmCl), which is consistentwith the observation that folding is less cooperative in ureathan in GdmCl (Fig. 1B). The results also confirm that GdmClis more efficient (Cm ¼ 2.5 M) in unfolding proteins than urea(Cm ¼ 4.7 M). Our predictions for equilibrium folding in ureacan be experimentally tested.

Equilibrium Collapse. The dependence of hRgi ¼ h 12N2 ∑ r2iji1∕2,

where rij is the distance between all interaction centers of theprotein and N (¼112 for src SH3) is the total number of beads,on ½C� shows compaction of the denatured state ensemble (DSE)structures of SH3 domain below Cm (Fig. 2A). By following thepopulation of conformations in the unfolded basin of attraction(UBA) and native basin of attraction (NBA) separately, (seeFig. S1 for description of UBA and NBA using χ as the orderparameter) we find that Rg of the NBA (RN

g ) is nearly indepen-

Fig. 1. Thermodynamics of folding. (A) Cartoon representation of src SH3

[Protein Data Bank (PDB) code 1SRL]. Sequence and the corresponding loca-tion of secondary structures are given below. The numbering starts at posi-tion 9. In ref. 34 there is an additional aspartic acid residue at the C terminal.In addition, the sequence in ref. 34 has arginine at the 52nd position whereasour sequence has glutamine. (B) Fraction of molecules in the native state as afunction of denaturant concentration. The folded-unfolded transition regionis assessed using full width at half-maximum of dfNBAð½C�Þ

d½C� , which in experimentsand simulations for GdmCl is 1.42 M and 1.71 M respectively. Red (green) arethe simulation results for GdmCl (urea) and the black dots are experimentalresults. (C) The dependence of ΔGNUð½C�Þ on ½C� for GdmCl (Left) and urea(Right). In (B) and (C) the denaturant concentration is in molarity.

Fig. 2. Equilibrium collapse. (A) Average hRgi (black circles) as a function ofGdmCl concentration. Red and green corresponds to hRgi for folded and un-foled structures, respectively. Blue symbol is constructed using RN

g ð¼ 11.50 ÅÞ,RUg ð¼ 24.04 ÅÞ at ½C� ¼ 8.0 M, and fNBAð½C�Þ (see text for details). The Inset

shows PðRgÞ at ½C� ¼ 1.0 M for DSE. (B) Distribution PðRgÞ of radius of gyrationRg for various concentrations of GdmCl. The Inset shows PðRgÞ for ½C� ¼ 0 M(black), ½C� ¼ 2.5 M (navy), and ½C� ¼ 5.0 M (orange) corresponding to theextended conformations (Rg > 15 Å).

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dent of ½C� (Fig. 2A). In contrast, the size of the structures in theUBA decrease continuously (41) and is about 17%more compactunder folding condition than at high ½C�ð> CmÞ (Fig. 2A). Thevariations in the size of the UBA, hRUBA

g ð½C�Þi, as ½C� is de-creased, is reflected in the ½C�-dependent changes in hRgi (blackcircles in Fig. 2A). If the collapse of the DSE is neglected thenhRUBA

g ð½C�Þi ≈ RUg ¼ 24.04 Å, the value at ½C� ¼ 8.0 M (Fig. 2A).

The distribution PðRgÞ of Rg shows that as ½C� increases src SH3

samples extended conformations with Rg exceeding 30 Å(Fig. 2B). Comparison of PðRgÞ of the DSE below Cm (Fig. 2A,Inset) and above Cm (the DSE region at high [C] in Fig. 2B, Inset)shows that compact conformations (Rg < 15 Å) in DSE aresampled only under folding conditions. Compaction of DSEbelow Cm implies that the dynamics of unfolded states mustbe different under folding conditions relative to high ½C� as shownrecently for protein L (42).

Folding Kinetics and the Chevron Plot.We calculated the ½C�-depen-dent folding (unfolding) rates from folding (unfolding) trajec-tories, which were generated from Brownian dynamics usingthe effective energy function, HPðfrig;½C�Þ (Eq. 3 and SI Textfor details). From sixty (fifty for ½C� ¼ 0) folding trajectoriesthe fraction of unfolded molecules at time t, is computed usingPuðtÞ ¼ 1 − ∫ t

0PfpðsÞds, where PfpðsÞ is the distribution of firstpassage times. For some of the ½C� values, we fit PuðtÞ ≈ e−tkf ½C�under folding conditions (½C� < Cm) from which kf ð½C�Þ can beextracted. Similarly, a single exponential fit for some values of½C� > Cm yields kuð½C�Þ. At high (low) ½C�, we can approximatekobs ¼ kf ð½C�Þ þ kuð½C�Þ as kuð½C�Þ (kf ð½C�Þ). For ½C� ¼ 2.5 M and3.0 M (≈Cm) the folding and unfolding rates are comparable,kuð½C�Þ ≈ kf ð½C�Þ. At these concentrations we performed bothfolding and unfolding simulations and by fitting the results forPuðtÞ using exponential function we extracted kuð½C�Þ andkf ð½C�Þ (see SI Text for details). We obtained kuð½0�Þ by globallyfitting the relaxation rate, kobs ¼ kf ð½C�Þ þ kuð½C�Þ, using ln kobs ¼ln½kf ð½0�Þe−mf ½C�∕RT þ kuð½0�Þemu½C�∕RT �, where mf (mu) is the slopeof the folding (unfolding) arm with ln kf ¼ ln kf ð0Þ −mf ½C�∕RTand ln ku ¼ ln kuð0Þ þmu½C�∕RT.

The dependence of PuðtÞ at ½C� ¼ 0.0 M, 1.0 M, and 5.0 M as afunction of t is given in Fig. 3A. For ½C� ¼ 5.0 M, PuðtÞ representsthe fraction of conformations in the NBA. Clearly, kf ð½C�Þ de-creases as ½C� increases, and kuð½C�Þ increases as ½C� increases.Most importantly, plot of ln kobs as a function of ½C� over the con-centration range (0 M ≤ ½C� ≤ 6.5 M) of GdmCl shows a classicchevron shape (Fig. 3B). For ½C� ¼ 0 ∼ 2.0 M, ku ≪ kf , so thatkobs ≈ kf and similarly for ½C� above 3.5 M, kobs ≈ ku. In thetransition region, kf ≈ ku, which requires generation of a largenumber of trajectories, kobsð½C�Þ converges slowly resulting inlarger errors. Comparison of the simulation and experimentalresults (filled black circles in Fig. 3B) allows us to draw two majorconclusions. (i) The slopes (from the folding and unfolding arms)of the simulated chevron plot are surprisingly similar to theexperimental values. (ii) Within error bars in simulations andexperiments, we do not find any deviation from linearity in thechevron plot. Taken together, our simulation results for kobs½C�semiquantitatively capture all of the experimental features, whichis remarkable given the simplicity of the MTM.

Consistency Between Thermodynamics and Kinetics. From the slopeof the folding arm (simulation results in Fig. 3B), we obtainmf ¼ 0.95 kcal∕mol·M and mu ¼ 0.60 kcal∕mol·M from theunfolding arm. The corresponding experimental values are mE

f ¼0.99 kcal∕mol·M and mE

u ¼ 0.45 kcal∕mol·M, which shows verygood agreement between experiments and simulations althoughmu is somewhat higher than mE

u . For a two-state description,thermodynamics and kinetics simulations should be consistent,

and hence we expect m ≈mf þmu. From the simulated chevronplot, we obtain m ≈ 1.55 kcal∕mol·M, which is in excellent agree-ment with m ð1.34–1.47 kcal∕mol·MÞ obtained from equilibriumcalculations (Fig. 1C). The midpoint Cm at which kf ≈ ku is Cm ≈2.2 M (Fig. 3B), which nearly coincides with Cm ¼ 2.5 M calcu-lated using fNBA½Cm� ¼ 0.5 (Fig. 1B). Thus, both thermodynamicand kinetic results obtained using MTM simulations are in har-mony, and justifies the two-state description for src SH3 folding.

Comparison of simulation and experiments shows thatalthough MTM simulations reproduce the chevron shape well,the dependence of ln kobs on ½C� does not quantitatively agreewith experiments. In particular, kf ð½0�Þ from simulations is925.9 s−1 whereas the extrapolated value to ½C� ¼ 0 from experi-ment is 56.7 s−1. Thus, the folding rate from MTM simulations isabout 16 times faster that the experimental rate. Deviations fromexperiments increase to about 40 at C ∼ Cm. The differencesbetween kuð½C�Þ obtained using simulations and experimentsare larger. For example, kuð½C� ¼ 5.0 MÞ differs by a factor ofapproximately 150 from experiments (Fig. 3B). The unfoldingrate kuð½0�Þ (the value in water) from our simulations is kuð½0�Þ≈7.60 s−1, whereas the extrapolated rate from experiment is0.1 s−1. The larger discrepancy in kuð½0�Þ between simulationsand experiments is due to the substantial error in the simulations,which occurs because relatively small number of unfolding eventsare observed at low ½C�. The agreement with experiments couldbe improved further by taking into account the changes in thedenaturant-dependent changes in viscosity. Nevertheless, fromthe calculated kf ð½0�Þ and kuð½0�Þ values, we infer thatΔGNUð½0�Þ≈−kBTs ln

kf ð½0�Þkuð½0�Þ ¼ −3.2 kcal∕mol, which compares favorably with

estimates from thermodynamic simulations. The coincidence ofm value calculated from the linear dependence of ΔGNUð½C�Þwith mf þmu obtained from chevron plot, and the similaritybetween ΔGNUð½0�Þ and the estimate using kf ð½0�Þ and kuð½0�Þallows us to conclude that a two-state model is accurate for fold-ing of src SH3. Although such a conclusion has been reachedusing experiments it is gratifying that we can capture the global

Fig. 3. Folding kinetics. (A) Fraction of molecules that have not folded(½C� ¼ 0.0 M, and ½C� ¼ 1.0 M) or unfolded (½C� ¼ 5.0 M) as a function oftime. The lines are exponential fits to the data. (B) Comparison of chevronplots obtained from simulations and experiment. The scale for the experi-mental plot for ln kobs is on the left, and for the simulations is on the right.The slopes of the calculated folding (mf ) and unfolding (mu) are0.95 kcal∕mol·M and 0.60 kcal∕mol·M, respectively, which are in good agree-ment with experiment.

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aspects of folding thermodynamics and kinetics using simulationsbased on MTM.

Collapse Kinetics, Folding, and Unfolding Pathways. The kinetics ofcollapse of src SH3 domain at ½C� ¼ 0 and ½C� ¼ 1.0 M shows(Fig. 4A) that hRgðtÞi decays with a single rate constant, kcð½C�Þ,the rate of collapse. The values of kcð½C�Þ from the data inFig. 4A are similar to kf ð½C�Þ, which shows compaction andfolding occur nearly simultaneously. The inset shows PðRg; tÞat t ¼ 0, t ≈ τc ≈ ðkcð½C� ¼ 1.0 MÞÞ−1 ≈ 4.2 ms, and t ¼ 34 ms(the first passage time for the slowest trajectory), respectively.Even after collapse (t ≈ τc) the protein samples, with small prob-ability (middle inset), conformations with Rg > 12 Å (RN

g ¼11.5 Å). Thus, the ensemble of kinetically collapsed conforma-tions is a mixture of folded and specific (native-like) and nonspe-cific compact structures (43). Although at an ensemble levelkcð½C�Þ ≈ kf ð½C�Þ, examination of the dynamics of acquisition ofstructure (χðtÞ as a function of t in Fig. S2) shows heterogeneity,which is masked in the ensemble averages. The link between com-paction and acquisition of secondary structure, measured interms of fraction (f ss) of secondary structure element, shows thatthese two processes also occur simultaneously (Fig. 4B) in srcSH3. Due to the cooperative nature of folding there is very littledependence of these processes on ½C�.

Kinetic cooperativity is most vividly illustrated in Fig. 4C for½C� ¼ 1.0 M and Fig. 4D for ½C� ¼ 6.0 M. In Fig. 4c, we plot thefirst passage times for reaching the native state at ½C� ¼ 1.0 Mand for establishment of interactions involving various structuralelements for the sixty folding trajectories. All the first passagetimes coincide (all measures of structure content form at thesame time), which shows that in the final stages (after crossingthe free energy barrier) the entire structure is consolidated simul-taneously. Similarly, unfolding at ½C� ¼ 6.0 Mas measured by lossof structures of the whole protein or various fragments (β12, β23,β34, and β45) also occur simultaneously (Fig. 4D). The black

symbols in Fig. 4 C and D shows the correlation between the firstpassage times for establishing interactions between β1 − β2 andβ2 − β3. The quantitative picture that emerges is that unfoldingand folding in src SH3 occurs in a single step in a highly coopera-tive manner although there is pathway diversity in the dynamics ofindividual folding trajectories (Fig. S2).

Structures of the Transition State Ensemble (TSE). We first deter-mined PðδÞ, the probability of forming native contacts as a func-tion of the progress variable δ ¼ t∕τ1i, where τ1i is the first passagetime for the ith trajectory. The values of δTS where dP∕dδ starts toincrease rapidly is identified with the transition state (TS) region,and the conformations in this neighborhood are grouped to ob-tain the TSE (44). We determined the structures of the TSE usinga procedure that is similar to the progress variable clusteringmethod (45, 46), which uses a neural network algorithm to classifyTSE structures without using surrogate reaction coordinates.

The global characteristics of the TSE is experimentally de-scribed using the Tanford βf ¼ mf∕m (or βu ¼ 1 − βf ). From theexperimental chevron plot βEf ¼ 0.69 whereas our simulationsgive βSf ¼ 0.61. It is generally assumed that βf is related to theburied solvent accessible surface area (SASA) at the TS. Forthe TSE obtained in our simulations, we calculated the ½C�-de-pendent distribution PðΔRÞ (see Fig. 5A for ½C� ¼ 1.0 M), whereΔR ¼ ðΔU − ΔTSEÞ∕ðΔU − ΔNÞ and ΔU , ΔTSE, ΔN are the SASAvalues in the DSE (½C� ¼ 6.0 M), TSE, and the NBA (½C� ¼1.0 M), respectively. We found that the average hΔRi ¼ 0.68. The

Fig. 4. Folding and unfolding pathways. (A) Collapse kinetics monitored bythe time-dependent average hRgðtÞi as a function of t. Collapse (expansion)decreases (increases) in hRgðtÞi can be fit using a single exponential function.The Inset shows PðRg; tÞ at t ¼ 0, t ≈ τc ≈ ðkcð½C� ¼ 1.0 MÞÞ−1 ≈ 4.2 ms, andt ¼ 34 ms, the first passage time for the slowest trajectory, respectively.(B) Projection of kinetic folding trajectories in terms of fraction of secondarystructure formation, f ss, and hRgi. (C) First passage time for formation of var-ious interactions, indicated in the figure, from the sixty folding trajectories at½C� ¼ 1.0 M. The solid black symbols show the correlation between τβ12i andτβ23i , which are first passage times for formation of interactions β1 − β2 andβ2 − β3, respectively. (D) Same as (C) except the results are for unfoldingat ½C� ¼ 6.0 M. Similarity between (C) and (D) establishes the reversibilitybetween folding and unfolding.

Fig. 5. Transition state ensemble structures. (A) Distribution PðΔRÞ, of theΔR ¼ ðΔU − ΔTSEÞ∕ðΔU − ΔNÞ, which is the fraction of buried solvent accessiblesurface area relative to the unfoled structures. The average values of ΔN andΔU are 5270 Å2 and 8189 Å2, respectively. The average hΔRi ¼ 0.68, whichcoincides with Tanford β parameter. (B) Contact map of the native stateensemble (Upper Left) and the one for the TSE structures (Lower Right).The scale on the right gives the probablity of contact formation. The numberof native contacts in the folded state is 507. (C) Superposition of a fewstructures of the dominant TSE cluster (Left) shows formation of the corewith disorder mostly in the RT loop (see text). The subdominant cluster haswell-formed β1, β2, and β3 hydrophobic core.

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close correspondence between βSf (or βEf ) and hΔRi affirms thestandard interpretation of the Tanford parameter. The smallfluctuation, σ2 ¼ ðhΔ2

Ri − hΔRi2Þ∕hΔRi2 ¼ 0.1, shows that theTSE in src SH3 is conformationally restricted.

From the lower half of Fig. 5B, which shows the contact mapobtained from the TSE structures, it is clear that relative to thenative state (top half) there are variations in the extent to whichthe structure is ordered in the transition region. The core of theprotein involving interactions in strands β1, β2 and to a lesser ex-tent β3 form with substantial probability in the TSE. The distalloop is structured in the TS as are residues involving the divergingturn. It is likely that the necessity to form the distal loop due to itsintrinsic stiffness forces the core of the protein to be ordered inthe TSE (37, 47). Residues from the n-src loop that are part ofthe hydrophobic core form with great probability in the TSE.However, the RT-loop, with large conformational fluctuations inthe transition state, is disordered in the TSE (structures on theleft in Fig. 5C).

We classified the TSE structures according to the similarity tothe native state for ½C� ¼ 1.0 M (Fig. 5C). In the dominant tran-sition state (found in approximately 72% of the folding trajec-tories) the core of the structures comprised of β1, β2, and β3 ispacked orthogonally as in the native state (structures in Fig. 5C,Left). The subdominant member of the TSE (approximately12%) also has a well-formed central β-sheet but are otherwisenot well structured (conformation of Fig. 5C, Right). The calcu-lated average structure of the dominant TSE conformations issimilar to that inferred from experimental ϕ-value analysis.The common features of the TSE is that the central β1, β2, β3strands are well structured as are residues in the distal loop.The present and previous simulations using entirely differentmodels and methods of determining the TSE structures, are alsoin very good agreement (36, 38). Both the structures, which areconsistent with experiments, show that contacts involving β1 − β2sheet and the RT loop forms with low probability. Althoughsimulations predict a single dominant transition state structure,diversity (assessed by probability distribution of contact forma-tion or PðΔRÞ) in the TSE structures, which is hard to glean fromexperiments, points to the inherent plasticity of the TSE, even inSH3 domains in which there is predominantly a single transi-tion state.

DiscussionCollapse Transition. Based on Flory theory we expect that radiusof gyration must scale as RU

g ≈ aDNν (ν ≈ 0.6, aD ≈ 2 Å)(48) athigh ½C�. Similarly, in the folded state RN

g ≈ 3N1∕3Å (49). Thepredicted theoretical values (RN

g ≈ 11.3 Å and RUg ≈ 22.4 Å)

are in very good agreement with simulation results. Using thevalues for RU

g (≈24.04 Å at ½C� ¼ 8.0 M in Fig. 2A) and RNg

(≈11.50 Å) a theoretical estimate hRTg i ¼ fNBAð½C�ÞRN

g þ ð1−fNBAð½C�ÞÞRU

g can be made. However, Fig. 2A shows that hRTg i

does not agree with hRgi. Because the changes in hRUBAg ð½C�Þi

in small single domain proteins are not large (20–30%), whichpredominantly occur below Cm where the population of mole-cules in the UBA is small, it is in general difficult to detect themusing standard small angle X-ray scattering (SAXS) methods.However, a recent study using SAXS has reported continuousreduction in hRUBA

g ð½C�Þi as ½C� is lowered in dihydrofolate reduc-tase (50), a protein with approximately 150 amino acid residues.Single molecule experiments also show unambiguous featuresof collapse for protein L and cold shock protein.

In the folding of src SH3 protein, collapse process and acquisi-tion of native structure occur nearly simultaneously (Fig. 4). Onthe time scale of collapse we find that only about 50% of thesecondary structures are fully formed (Fig. 4B), which impliesthat there is a dynamic coexistence between folded and unfolded

states. In contrast, in the folding of Cytochrome c, monitored bySAXS and CD, it has been shown that collapse is preceded byacquisition of secondary structure (51, 52). We suggest that withdecreased cooperativity there could be a separation in the timescale between these two distinct processes (53).

Folding Kinetics. The most important achievement of our study isthe simulation of the chevron plot. The difference between thecalculated and measured kobs could arise from neglect of nonna-tive interactions and possibly from not using explicit models forwater and GdmCl. Although one can appeal to extensive studiesusing lattice and offlattice models (44), which show that to a largeextent the folding mechanisms for foldable sequences are notgreatly altered by including nonnative interactions, their role hasto be quantitatively assessed. Nonnative interactions are likelynot relevant either at very high or very low ½C� values but do playan important role in affecting the thermodynamics in the transi-tion region (Fig. 1B). Despite these reservations the remarkableagreement between simulations and experiments shows that thetheory outlined here can yield fundamental insights into proteinfolding using higher resolution models.

Concluding RemarksWe have produced the most comprehensive picture of denatur-ant-dependent folding of src SH3 domain using simulations thatinclude denaturant effects in conjunction with self-organizedpolymer-side chain (SOP-SC) model for the polypeptide chain.Results for folding thermodynamics are in quantitative agree-ment with experiments. Somewhat surprisingly it is found thatnot only do the calculations reproduce the chevron plot but thepredicted values of the ½C�-dependent folding rates are in reason-able agreement with measurements. The nearly ½C�-independentstructure of the dominant transition state structure is broadlyconsistent with the structural interpretation based on Φ-valueanalysis and simulations (at ½C� ¼ 0) carried out using entirelydifferent model and methods. Our work shows that the Tanfordβ-parameter measures the fraction of buried SASA in the TSE.The theory underlying the MTM is general and can be adoptedin conjunction with all-atom representation of proteins, thusmaking it feasible to study denaturant and osmolyte effects usingsimulations.

MethodsSOP-Sidechain Model. Simulations were carried out using the SOP model (54)each residue is represented by two interaction centers, one that is located atthe Cα position and the other is at the center of mass of the side chain. Nativestate is stabilized by backbone–backbone, side chain–side chain, and back-bone–side chain interactions. For the side chain–side chain interactions weuse the Betancourt–Thirumalai statistical potential (55). The effective energy,obtained by integrating over solvent (water) degrees of freedom, whichdescribes the intrapeptide interactions of the polypeptide chain with coor-dinates frig, is

EPðfrigÞ ¼ VFENE þ VNATLJ þ VNEI þ VNN

LJ : [1]

Detailed descriptions of the terms in Eq. 1 and the parameters of theforcefield are in Tables S1 and S2. We use low friction Langevin simulationsto obtain thermodynamic properties (56). Folding trajectories are generatedusing a Brownian dynamics algorithm (57) using a friction coefficient thatcorresponds to water viscosity (see SI Text for details).

Molecular Transfer Model. The free energy of transferring a protein confor-mation described by frig fromwater (½C� ¼ 0) to aqueous denaturant solution(½C� ≠ 0) is approximated (58, 59) as

ΔGðfrig;½C�Þ ¼ ∑k

δgðk;½C�ÞαkðfrigÞ∕αGly-k-Gly; [2]

where the sum is over backbone and side chain, δgðk;½C�Þ ¼ mk ½C� þ bk (seeTable S3 formk and bk values, and Table S4 for αGly-k-Gly values) is the transferfree energy of group k (60, 61), αkðfrigÞ is the SASA of k, and αGly-k-Gly is the

Liu et al. PNAS ∣ May 10, 2011 ∣ vol. 108 ∣ no. 19 ∣ 7791

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SASA of the kth group in the tripeptide Gly-k-Gly. Thus, the effective freeenergy function for a protein at ½C� ≠ 0 is

HPðfrig;½C�Þ ¼ EPðfrigÞ þ ΔGðfrig;½C�Þ: [3]

We used the procedure in (29), which is accurate provided exhaustivesampling using EPðfrigÞ is performed, to obtain thermodynamic properties.We use the full effective free energy function (Eq. 3) to generate folding and

unfolding trajectories at ½C� ≠ 0 (see SI Text for details). We compute αkðfrigÞand ∂αk ðfrigÞ

∂ ~riusing the procedure in ref. 62. Note that MTM does not assume

whether the protein folds by a two-state or multistate mechanism. If EPðfrigÞcaptures the nature of folding the MTM can accurately describe the effectof denaturants.

ACKNOWLEDGMENTS. This work is supported by the National Science Founda-tion (NSF CHE 09-14033).

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