© 2011 Nature America, Inc. All rights reserved. U ligand binding discussions of other problems, such as and are they now introduced being in a fixturepapers in on folding, publication, funnel diagrams have become years, is resolved. Since original the discussion of for many Levinthal paradoxLevinthal a pictorial representation of how the chain (on order the of 10 number of configurations polypeptidethe of totime the find native stategiven is theby random search. For such asearch, the state found can be only by an unbiased probable,state) native thatare the so equally of polypeptide the chain (except native the people, means this that conformations all Taken literally, as it by has been many protein folding is arandom search problem. that appropriate the point of reference for many readers. diagram has created amisconception in Thirumalai introduced by Wolynes, Onuchic and more the become all important. foldwill to its or aggregate has apolypeptidedetermine chain whether aknowledge ofdiseases, factors the that shownbeen source the to be of arange of Given that small (100residues or nature chemical biology representation of the factors governing the protein folding reaction than the funnel diagram. approached hinders folding, rather than guiding it. Diagrams are introduced that provide a less ambiguous is discussed, and it is pointed out that the decrease in the configurational entropy as the native state is to understanding the mechanism of protein folding. The primary role of free energy in protein folding protein folding. To aid in the analysis of the funnel diagram, this Commentary reviews historical Thisapproaches Commentary clarifies the meaning of the funnel diagram, which has been widely cited in papers on Karplus Martin diagram Behind the folding funnel time (say,time 10 This leads to an enormously long folding find one configuration(say, 10 protein) multiplied by required time the to years that fact the it studied has for been many fundamental interest inbiology, inspite of The concept introduced Levinthal is by The foldingfunnel Fig. diagram (see 1 . Moreover, now that misfolding has native state remains aproblem of by proteins which fold to their nderstanding mechanism the 2 , is intended to provide 59 seconds or about 10 4 . Unfortunately, . funnel the 3 , which had, which dominated |VOL 7| JULY 2011 |www.nature.com/naturechemicalbiology 70 for a100-residue − 11 seconds). 52 years).

1),

of computational complexity of paradox the was given language inthe aparadox.indeed ultimate The statement such as proline isomerization), there was factorswith special that slow folding, the of to milliseconds seconds (except incases fewer) generally fold on intimes order the nucleation-condensation mechanism Examples include nucleation-growth the or foldingthe to time range. experimental the has searched to be is restricted to reduce to show how conformational the spacethat phenomenological models were proposed paper, Zwanzig et insightful In an folding. in role a play it can stable, relative to the structures, process. For example, only ifanucleus is energeticbe factors that bias folding the for any of models the to work, there must implicit phenomenological inthe models— The change made explicit aconceptthat is reaction orsmall-molecule protein folding. puzzle model diffusion-collision model the mostthe of widely used is which similar two-dimensionalin the funnel diagram, problem would solved. be and configurational the search (Levinthal) would visited be folding inthe reaction, only alimited number of configurations relatively smoothly toward native the state, in the potential such energy that it decreases tomodel demonstrate that ifthere is abias determinants of any reaction surface is one energy the of fundamentalthe is eminentlyThe new focus reasonable, as of surface of apolypeptide energy chain. the consideration of general the characteristics a to models phenomenological from to protein the folding problem shifted The required energy bias is embodied In late the of focus 1980s,the approaches 9 .

al. 10 used asimplified used

8 and jigsaw- the 5 1 . Many , whether a, whether 6,7 , the reference 2. configurational entropy horizontally. Adapted from The effective energy is plotted vertically and the Figure to that shown inFigure also being used being also picturesque three-dimensional funnels are as nativedecrease the state is approached; horizontal(in the direction) of aprotein direction), and configurational the entropy over solvent interactions vertical (inthe potential energy, implicitly averaged representation of how effective the domains appears inastudy of folding the of PDZ bottle.” Atypical statement of type this a protein is like “funneling wine into a many readers have concluded that folding are appealing images very from which Such decreases. energy funnel diagrams configurational entropy,the decreases as configurations,whichthe determine shape number the because of accessible dimensional diagrams have afunnel-like et toward native the state.” Although Wolynes a funnel that guides proteins appear to resemble shape the of that “Free landscapes energy of many

al.

Effective energy 2

were aware that of occurrence the 1 | Aschematicfolding funneldiagram. 12 ; in the introduction,; inthe it is stated C commentary onfigur

11 . Both the two- the and. Both three- the folding the process ational entr

1. It is aschematic op y 401 © 2011 Nature America, Inc. All rights reserved. commentary information that contribute to dissecting dissecting to contribute that information that provide and structural kinetic folding aided by has been experiments proteins ref. (see 7and references therein). small formost observed behavior folding that barrier resultsenergy two-state inthe two the generallybetween to leads afree- configuration entropy. The delicatebalance contribution of inthe decrease the favors folding, and unfavorable the native state is approached and therefore potential energy, as decreases the which the is diagram.freesum The energy the of rather shown than energy the funnel inthe 402 diagram has continued to proliferate. misconception concerning funnel the have not emphasized point, this and the from paper, seen original their be they such “guiding” is amisconception, as can and b a functionofthefraction ofnative contacts for the27-mer lattice polymer:a Figure “landscape”) of polypeptide the chain (often surface free-energy referred to as a major determinant of folding the rate is the folding process it must that realized be the paradox. Levinthal the To understand the configurations theis essentially origin of folding. The difficulty ofthese finding state is approached, tends actually to slow of available configurations,the as native thatis number inthe true; is, decrease the nativethe state. In opposite the exactly fact, finding in chain polypeptide the aids shape, giveswhich diagram the its funnel-like that inconfiguration decrease the entropy, The developing understanding of protein The essence the misconceptionof is

anddatahightemperature. Adapted withpermission from reference 22. a

c 2 | Free energy Average energy −75 −70 −65 −75 −70 −65 −60 −55 Folding ofalattice polymer. (a–d)The effective energy (a 0 0 0.2 0.2 0.4 0.4 Q 0.6 0.6

0.8 0.8 1,2,7,13 , 1 1 d b of NMR mechanism,the such making use as those calculated asafunction ofthefraction ofnative hydrogen bondsfor adesignedα Figure can be extendedcan from be down milliseconds folding and unfolding that so measurements it has become possible to rapidly trigger not shown becauseofpoorstatistics. reference c −105 −100 −100 atalow temperature (270 K);b −95 −90 −85 −80 −60 −40 0 0 c a

3

| –1

Folding ofadesignedα 1

21; somedatapoints for low andhighQ Average energy (kcal mol ) and protein engineering –1 −150 −140 −130 −120 −110 −100

Free energy (kcal mol ) 0.2 0.2 , −4 −2 b 0 2 4 ) andfree energy (c 0 0 0.4 0.4 andc 0.2 0.2 Q nature chemical biology at a low temperature; 0.6 0.6 0.4 0.4 -helical peptide.(a–d)The effective energy ( a anddatahightemperature (390K).Adapted withpermission from Q , d 0.8 0.8 ) calculated as 0.6 0.6 7 Also, . Also, 1 1 0.8 0.8 270 K values at the low and high temperature, respectively, are |VOL 7| JULY 2011 |www.nature.com/naturechemicalbiology 1 1 b nanoseconds to microseconds and more recently to how paradox Levinthal the resolved can be for(see, example, ref. 7). propose folding pathways for proteins small data,experimental have to used been in explicit solvent, combined with solvent.the Also, unfolding simulations for example, ref. 17)representations of for example, ref. 16)and explicit (see, atomistic simulations with implicit (see, models thermodynamic parameters include lattice above by focusing and on structural the phenomenological approaches mentioned data. Models that go beyond the experimental supplementthe to used be to a range of theoretical continues methods to awell-defined native state.Consequently, from multiconfiguration the denatured state going in aresampled that conformations the protein—that is, acomplete description of folding trajectories evenfor asimple to determine detailed the experimentally However, it far has so not possible been twothe that is larger than afew kilocalories. separating barrier a is there that and state, states, denatured the state and native the there are only two significantlypopulated to mean, as inchemical reactions exponential course. time This is interpreted reaction of many proteins small follows an folding the that finding the is studies such d −150 −140 −130 −120 −110 −100 The first concrete demonstration of −4 −2 0 2 4 0 0 1,13 , coarse-grained models 0.2 0.2 14 . An important result of 0.4 0.4 , b -helical peptide:a ) andfree energy (c Q 0.6 0.6 0.8 0.8 390 K 1 15 , that and and and and , 1 1 d ) © 2011 Nature America, Inc. All rights reserved. 10 sequences, whereas there are on order the of simulation above described on 27-mer the latticebased Monte Carlo and nature chemical biology favoring native the state interactions neighboring between beads by achain ofon beads acubic lattice with of a27-residue polypeptide represented was provided by Monte Carlo simulations free energy values (Fig. energy free structure approaches native the state. The have amonotonic as 27-mer decrease the configurations is ‘rugged’ notand does resulting surface energy from sampled the synthetic α temperature range have been reported for a in implicit solvent over appropriate the folding of aprotein, all-atom studies peptide showing and for energy energy free the the all-atom simulations or results experimental (two-state) folding kinetics. Thelatter is responsible forthe exponential (Fig. energy native state is approached, whereas free the configurationthe smoothly decreases as contrast, sampled effectivethe the energy of For temperature higher the (Fig. it is region inthe a barrier; of Q configurationaladded entropy introduces lower temperature (Fig. progress variable for system. the At the of native contacts, Q temperature, as of afunction number the temperature and Figure freeeffectiveand energy energy at alow shows lattice the results model for the enough (10 of denatured configurations largeis folding number the evenwhen occurred 18). The simulations demonstratedthat than funnel the picture and are less subject to misinterpretations information about folding the mechanism distance from native the state provide more showing as of afunction energy the free the about 10 et conceptualthe framework of Zwanzig the simulationthe time. In accordance with random search tothe find native state in denatured configurations to visit only an infinitesimalfraction the of and solves paradox Levinthal the by having time, folds peptide the to native the state simulations; during average the folding accord obtained with those from lattice the potential showed function energy results in hairpin β forms astable three-stranded double- of folding the of a20-residue that peptide Analysis of moleculardynamics simulations

al.model 16 Although there are no corresponding One-dimensional representations 3illustrate point; this are they configurationsthe denatured in state. 7 -sheet represented by an all-atom steps were required to fold the -helix and are shown inFigure 18 10

2d) has asignificantbarrier. ) to make it impossible fora impossible it make to ) , it was shown that only , which is, which asatisfactory 1,7,13,20

|VOL 7| JULY 2011 |www.nature.com/naturechemicalbiology

13 2c) show that the

), the effective2a), the 2b,d 19 (see also ref. also (see . 13 . Figures . Figure at ahigh

=

2b), in

0.8.

2a,c 2

3

and all-atom models dependence ofdependence folding the rate inlattice deviations from Arrhenius the temperature confirmed that the fact by there are striking the entropy in the protein folding reaction is protein?) is not known. proteins (for example, is Fig. are more complex. What happens in actual folding and energy free-energy the diagrams three-stranded β of asomewhat more complex peptide, the turn is formed. In corresponding studies to occur very early when the first helical is notbarrier clearly visible, though it is likely temperature the helical state is not stable; a a stable minimum at Q Figure plots (Fig. bonds folded inthe state. The effective energy of fraction number the of hydrogen progress coordinate Q as andof afunction energy energy the free and high-temperature results for effective the lattice model, the Figure energy (Fig. energy native state as inFigure smoothly the decreases effectiveto energy latticethe higher-temperature the model; result is somewhat than for less rugged engendered infuture discussions of protein misconceptions that funnelthe diagram has eliminate will Commentary this the for example, reference 24.Ihope that studies ofuseful protein folding; see, on depend specific the protein. to a‘low’ and ‘high’ temperature is likely to is increased constant that decreases however, there is aturnover rate inthe Arrhenius formula. At temperatures, higher at low temperatures, inaccord with the rate as of afunction temperature increases at temperature high arethey ‘soft matter’ (entropy dominated) temperature (enthalpy dominated), whereas viewpoint, proteins are ‘hard matter’ at low nativethe one. Thus,from the physicist’s configurationsthat searchedbe must to find ofbecause larger the number of accessible (unfavorable) entropic contribution increasing bythe dominated is and activation remains energy free positive becomes negative (Fig. increases, apparent the activation energy activation energy. As temperature the potential that surface to leads an apparent rate is dominated by of ruggedness the the At low temperature (Fig. of behavior this is evident from Figure Fig. experimentally have observed been (for Ferrara details, see et The important role the enthalpyof and The funnelThe concept has stimulated

18.2 inref. 7

2a,b

3a,b , except that low-temperature the 22

3c, ; corresponding results ) are in similar to those d -sheet mentioned above ), at low temperature there is ). Apossible explanation 23 21 . What corresponds , here in defined terms (see Fig. (see

as temperature the =

2b. As to free the 2b), whereas the presents both low-

0.8, whereas at high

al. ), the folding folding the 2a), 21

valid for1 valid any ). As with the

4a,b 7 (see (see ). The

2. 19 13,22 ,

funnel diagram. folding the in present nor experiment from there are hiddencomplexities not evident folding reaction. The reference showsthat descriptionmeaningful of protein the and references therein) provide amore transitionthe temperature) ref. (see 25 and unfolding at equilibrium (near folding repeatedly peptides of simulations on analysis the of moleculardynamics Figures folding. Diagrams shown like those here in temperature for thelattice polymerinFigure ( e-mail: [email protected] Strasbourg, Strasbourg, France. d’IngénierieSupramoléculaires, Université de de Chimie Biophysique, Institut de Science et Laboratoireandthe in USA Massachusetts, ChemicalBiology, Harvard University, Cambridge, & ofChemistry Martinin theDepartment Karplus is and 22. for CI2. Adapted with permission from references reference in Figure temperature for thedesignedα ( adapted with permission from reference 2. Figure a b ) Calculated rate constant as a function of ) Calculated rate constant as a function of a b c ln(k)

ln(k/T) −18.5 −17.5 −16.5 −0.5 −2.5 −1.5

−19 −18 −17 −16 4 | log(rate (1/ns)) −2 −1 −4 −3 −2 −1

0.4 3. Adapted withpermission from

0 1 2 3 2–4, supplemented by figuresbased 21. (c Arrhenius plots for folding reaction. 2.8 ) Measured refolding rate constant 2

0.6 2.5 1,000/T(K) 3 1/T 1/T commentary

-helical peptide 3 u 0.8 3.2 3.5 × 10

2; −3 3.4 1 403 4

◼ 7 © 2011 Nature America, Inc. All rights reserved. commentary 404 6. 5. 4. 3. 2. 1. References 8. 7. 9.

Wetlaufer, D.B. Proc. Natl. Acad. USA Sci. K. Jr. Grand, &Le 435–508(Birkhäuser, S.,eds.), Boston, 1994). Protein Folding Problem and Tertiary Structure Prediction. (Merz, protein structure prediction, and paradox. Levinthal the inThe Ngo, J.T., Marks, J. &Karplus, M.Computational complexity, 1181–1190 (1999). Tsai, C.J., Kumor, S.,Ma, B.Y. &Nussinov, Protein R. . 8 Sci Münck, 22(University E.,eds.) of Illinois Press, Urbana, 1969). House, Monticello, Illinois (Debrunner, P., Tsibris, J.C.M. & in Biological Systems, Proceedings of aMeeting held at Allerton C.HowLevinthal, to fold graciously. inMossbauer Spectroscopy 1619–1620 (1995). Wolynes, P.G., Onuchic, J.N. &Thirumalai, D. Science Engl Dobson, C.M.,Sali, A.&Karplus, M.Angew Karplus, M.&Weaver, D.L. Nature Enzyme Catalysis and Protein Folding (WHFreeman, 1999). Fersht, A.Structure and Mechanism in Protein Science: Aguide to 4028–4030 (1985). Harrison, Proc. S.C.&Durbin, R. Natl. Acad. USA Sci.

. 37,868–893(1998).

260, 404–406(1976). 70, 697–701(1973). Chem. Int. Edn. Int. Edn. . Chem.

267, 82, , 11. 10. 13. 12. 14. 18. 17. 16. 15. 21. 19. 22. 20.

Dill, K.A.&Chan,Dill, H.S.Nat.Biol Struct. 20–22 (1992). Zwanzig, Szabo, R., B. A.&Bacghi, Proc. Natl. Acad. USA Sci. Sali, A.,Shakhnovich, E.&Karplus, M.Nature (2008). N.et Calosci, Proc. Natl. Acad. USA Sci. Kubelka, J., Henry, Cellmer, E.R., T., Hofrichter, J. &Eaton, W.A. Baldwin, R.L. Nature R.L. Baldwin, Shaw, D.E. et Zagrovic, B. &Pande, V. J. Comput. Chem. 24,1432–1436(2003). (2009). H.,Schug, A.&Onuchic,Lammert, J.Bioinformatics Karplus, M.J. Phys. Chem. B 5000–5010 (2000). Ferrara, P., Apotolakis, J. &Caflisch, A.J. Phys. Chem. B J. Eaton, W.A., Thompson, Chan,P.A., C.-K., Hagen, S.J. & Hofrichter, 1801–1803 (2003). Cavalli, A.,Haberthuer, U., Paci, E.&Caflish, A.Protein. 12, Sci Structure

4 , 1133–1139(1996).

al. Science 330,341–346(2010). al. Proc. Natl. Acad. USA Sci.

369, 183–184(1994). 105, 18655–18662(2008). 104, 11–27(2000). nature chemical biology . 4 , 10–19(1997). , 19241–19246 105, 19241–19246

369, 248–251(1994).

, 881–891 77, 881–891 104, 89, |VOL 7| JULY 2011 |www.nature.com/naturechemicalbiology Corrected print14July2011 after interests. financial competing no declares author The Competing financialinterests Institutes of Health. National US the from grant a by part, in supported, was in reference 1or reference 7.The work done at Harvard found be can many omitted; been have citations original some limitation, space to Owing figures. the with help and manuscript the on comments V. for and Ovchinnikov results and comments on manuscript the and M.Cecchini providing for Qi B. and Dinner A. Caflisch, A. thank I Acknowledgments 23. 25. 24.

112, 8701–8714(2008). Krivov, S.V., Muff,Caflisch, S., A.& Karplus, M.J. Phys. Chem. B Fold. Des . 1 Onuchic, J.N., N.D., Socci, Luthey-Schulten, Z.&Wolynes, P.G. and Mixtures (Gompper, (Wiley, M.,eds.) G.&Schick, 2006). Müller-Plathe, F. Introduction. inSoft Matter Vol. 1, Polymer Melts , 441–450(1996).

© 2011 Nature America, Inc. All rights reserved. correction notice

corrected HTML inthe and PDF versions of article. the been has error The folding. hinders that entropy configurational the in decrease a is there that read should it but folding, hinders In the version of this article initially published, there was an error in the abstract that stated an increase in the configurational entropy Nat. Chem. Biol. 7,401–404(2011);published online 17June 2011;corrected after print 14 July 2011 Karplus Martin Behind thefolding funneldiagram corrigendum Nature chemical biology nature chemical biology