Folding of small proteins using a single continuous potentialSeung-Yeon Kim, Julian Lee, and Jooyoung Lee Citation: The Journal of Chemical Physics 120, 8271 (2004); doi: 10.1063/1.1689643 View online: http://dx.doi.org/10.1063/1.1689643 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/120/17?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Multicanonical Monte Carlo simulations of a de novo designed protein with end-to-end β-sheet J. Chem. Phys. 140, 065101 (2014); 10.1063/1.4865127 Consistent free energy landscapes and thermodynamic properties of small proteins based on a single all-atomforce field employing an implicit solvation J. Chem. Phys. 127, 145104 (2007); 10.1063/1.2775450 Assessment of protein folding potentials with an evolutionary method J. Chem. Phys. 125, 014904 (2006); 10.1063/1.2210931 Torsional diffusion Monte Carlo: A method for quantum simulations of proteins J. Chem. Phys. 114, 9725 (2001); 10.1063/1.1368402 Comparison of three Monte Carlo conformational search strategies for a proteinlike homopolymer model: Foldingthermodynamics and identification of low-energy structures J. Chem. Phys. 113, 5065 (2000); 10.1063/1.1289533

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Folding of small proteins using a single continuous potentialSeung-Yeon KimSchool of Computational Sciences, Korea Institute for Advanced Study, 207-43 Cheongryangri-dong,Dongdaemun-gu, Seoul 130-722, Korea

Julian LeeSchool of Computational Sciences, Korea Institute for Advanced Study, 207-43 Cheongryangri-dong,Dongdaemun-gu, Seoul 130-722, Korea and Department of Bioinformatics and Life Sciences,Soongsil University, Seoul 156-743, Korea

Jooyoung Leea)

School of Computational Sciences, Korea Institute for Advanced Study, 207-43 Cheongryangri-dong,Dongdaemun-gu, Seoul 130-722, Korea

~Received 29 October 2003; accepted 2 February 2004!

Extensive Monte Carlofolding simulations for four proteins of various structural classes are carriedout, using asinglecontinuous potential~united-residue force field!. In all cases, collapse occurs ata very early stage, and proteins fold into their nativelike conformations at appropriate temperatures.We also observe that glassy transitions occur at low temperatures. The simulation resultsdemonstrate that the folding mechanism is controlled not only by thermodynamic factors but also bykinetic factors: The way a protein folds into its native structure is also determined by theconvergence point of early folding trajectories, which cannot be obtained by the free energysurface. ©2004 American Institute of Physics.@DOI: 10.1063/1.1689643#

I. INTRODUCTION

Understanding how a protein folds is a long-standingchallenge in modern science. Computer simulations1–14 havebeen carried out to understand the folding mechanism. How-ever, simulation of protein-folding processes with an atomis-tic model is a very difficult task. The difficulties come fromtwo sources:~i! Direct folding simulations using an all-atompotential require astronomical amounts of CPU time, andtypical simulation times are only about a few nanoseconds.~ii ! Atomic pairwise interactions including solvation effectsmay not be accurate enough. An extensive folding simulationhas been carried out for the villin headpiece subdomain~HP-36!, where a 1-ms molecular dynamics simulation with anall-atom potential has been performed, producing only can-didates for folding intermediates.1 For this reason, directfolding simulations have been mainly focused on simplemodels, such as lattice models,2,3 models where only nativeinteractions are included~Go-type models!,4–7 and a modelwith discrete energy terms whose parameters are optimizedseparatelyfor each protein.8 Alternative indirect approacheshave also been proposed including unfoldingsimulations7,9–11 starting from the folded state of a protein.However, it is not obvious whether the folding is the reverseof the unfolding.3,7 Moreover, to the best of our knowledge,no one has yet succeeded in folding more than one proteininto its native state using asingle potential, even with sim-plified models.8

In this work, we propose a method to fold several ofsmall proteins using Monte Carlo dynamics. This method

uses asinglecontinuous potential which includesall ~nativeand non-native! interactions, and yet allows us to carry outfolding simulations starting from non-native conformations.We observe that all proteins fold into their nativelike confor-mations at appropriate temperatures. We find that the foldingmechanism is controlled by both kinetic and thermodynamicfactors: The way a protein folds into its native structure isdetermined not only by the free energy surface, but also bythe convergence point of early folding trajectories relative tothe native state.

II. METHODS

We study the folding dynamics of proteins using aunited-residue15–17 ~UNRES! force field where a protein isrepresented by a sequence ofa-carbon (Ca) atoms linked byvirtual bonds with attached united sidechains~SCs! andunited peptide groups located in the middle between the con-secutive Ca’s ~Fig. 1!. All the virtual bond lengths are fixed:the Ca – Ca distance is taken as 3.8 Å, and Ca – SC distancesare given for each amino acid type. The energy of a protein isgiven by

E5Udis1(i , j

@Uel-loc(4) ~ i , j !1Uss~ i , j !#1(

iÞ jUsp~ i , j !

1 (i , j 21

Upp~ i , j !1(i

@Ub~ i !1U t~ i !1U r~ i !#.

Here,Udis denotes the energy term which forces two cysteineresidues to form a disulfide bridge. The four-body interactionterm Uel-loc

(4) results from the cumulant expansion of the re-a!Author to whom correspondence should be addressed. Electronic mail:jlee@kias.re.kr

JOURNAL OF CHEMICAL PHYSICS VOLUME 120, NUMBER 17 1 MAY 2004

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stricted free energy of the protein.Uss( i , j ) represents themean free energy of the hydrophobic~hydrophilic! interac-tion between the sidechains of residuesi and j , which isexpressed by Lennard-Jones potential,Usp( i , j ) correspondsto the excluded-volume interaction between the sidechain ofresiduei and the peptide group of residuej , and the potentialUpp( i , j ) accounts for the electrostatic interaction betweenthe peptide groups of residuesi and j . The termsUb( i ),U t( i ), andU r( i ) denote the short-range interactions, corre-sponding to the energies of virtual angle bending, virtualdihedral angle torsions, and sidechain rotamers, respectively.

The parameters of the UNRES force field weresimulta-neouslyoptimized18,19 for four proteins of betanova11,20 ~20residues, three-strandedb-sheet!, 1fsd~Ref. 21! ~28 residues,oneb-hairpin, and onea-helix!, HP-36~Refs. 1 and 12! ~36residues, three-helix bundle!, and fragment B of staphylococ-cal protein A~Refs. 5, 7, 9, 10, 13, and 22! ~46 residues,three-helix bundle!. The low-lying local-energy minima forthese proteins were found by the conformational space an-nealing method.23,24The parameters were modified in such away that the nativelike conformations are energetically morefavored than the others. The global minimum-energy confor-mations found using the optimized force field are of the root-mean-square deviation~RMSD! values 1.5 Å, 1.7 Å, 1.7 Å,and 1.9 Å from theexperimentalstructures for betanova,1fsd, HP-36, and protein A, respectively. For betanova, the‘‘native’’ structure is shown to be only marginally stable andis in equilibrium with less ordered structures.25 For this rea-son, it might be more appropriate to call this structure as the‘‘predominant’’ one. In this paper, we choose to use the term‘‘native’’ to refer to ‘‘predominant’’ for betanova, to be con-sistent with the other three proteins. After the parameter op-timization, one set of the parameters is obtained for fourproteins. The optimized parameters are not overfitted to the

four proteins, but are transferable to other proteins to someextent.19 In Ref. 19 the procedures to obtain the optimizedparameters used in this work are described in detail.

In the UNRES force field there are two backbone anglesand two sidechain angles~see Fig. 1! per residue~nosidechains for glycines!. The values of these angles are per-turbed one at a time, typically about 15°, and the backboneangles are chosen three times more frequently than thesidechain angles. The perturbed conformation is accepted ac-cording to the change in the potential energy, following theMetropolis rule. Since only small angle changes are allowedone at a time, the resulting Monte Carlo dynamics can beviewed as equivalent to the real dynamics.2,8,14 At a fixedtemperature, at least ten independent simulations startingfrom various non-native states of a protein were performedup to 109 Monte Carlo steps~MCS! for each run, which wecall long-time simulations. Hundreds of long-time simula-tions were conducted for a protein.

During simulations the values of RMSD from the nativestructure and the radius of gyration (Rg) were calculatedusing Ca coordinates. The lowest RMSD values from foldingsimulations are 0.78 Å, 1.07 Å, 1.58 Å, and 2.07 Å for bet-anova, 1fsd, HP-36, and protein A, respectively. The frac-tions of the native contacts (Q and r! were also measuredduring simulations, whereQ is calculated from the experi-mental structure. A native contact is defined to exist whentwo Ca’s separated more than two residues in sequence areplaced within 7.0 Å. To definer we first characterize thenative-state conformations by performing simulations start-ing from the experimental structures, at the same tempera-tures where folding simulations were performed. We defineras the fraction of native contacts weighted over their contactprobabilities in the native state simulations.7,10,11Due to thefluctuation of the native conformation, the value ofQ is usu-ally lower than that ofr. The time histories of the typicaltrajectories from the folding simulations of betanova andHP-36 are shown in Fig. 2. The trajectories for 1fsd andprotein A are similar to those in the figure. Distributions ofRMSD, Q, r, andRg are also accumulated during the whole

FIG. 1. United-residue representation of a protein. The interaction sites aresidechain ellipsoids of different sizes~SC! and peptide-bond centers~p!indicated by shaded circles, whereas thea-carbon atoms~small open circles!are introduced to define the backbone-local interaction sites and to assist indefining the geometry. The virtual Ca – Ca bonds have a fixed length of 3.8Å, corresponding to a trans peptide group; the virtual-bond~u! and dihedral~g! angles are variable. Each sidechain is attached to the correspondinga-carbon with a different but fixed bond lengthbi , variable bond anglea i

formed by SCi , and the bisector of the angle defined by Ci 21a , Ci

a , andCi 11

a , and with a variable dihedral angleb i of counterclockwise rotationabout the bisector, starting from the right side of the Ci 21

a , Cia , Ci 11

a frame.

FIG. 2. Typical folding trajectories of betanova atT540 and HP-36 atT570 starting from non-native conformations. Only the first 108 MCS areshown among 109 MCS for clarity. RMSD,Rg , andQ are plotted for every100 MCS in the early part and for every 23105 steps in the subsequent part.The dotted lines represent the values ofRg of the native states.

8272 J. Chem. Phys., Vol. 120, No. 17, 1 May 2004 Kim, Lee, and Lee

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simulations. The distributions of RMSD are shown in Fig. 3and those forr andRg are shown in Fig. 4 as contour plots.

To investigate the early folding trajectories, 200(105 MCS) independent simulations were also performed ata fixed temperature for a protein, which we call short-timesimulations. The averages of early folding trajectories plottedin Fig. 4 were obtained as follows. We divided the initial105 MCS into 19 intervals~ten 103 MCS and subsequentnine 104 MCS), and took the average over conformations ineach interval. These averages were again averaged over 100independent simulations starting from random conforma-tions. The same procedure was applied to 100 independentsimulations starting from a fully extended conformation.

III. RESULTS

The simulation of betanova atT540 (arbitrary units)starting from an unfolded conformation demonstrates thatrapid collapse occurs in about 104 MCS; the value ofRg

decreases, whereas the value ofQ remains below 0.3~Fig.2!. During the next 108 MCS, more compact states appear,and the value ofQ moves up as high as 1.0. There are twovalues of RMSD that are populated@see also Fig. 3~b!#: onecorresponds to less ordered structures (RMSD;5 Å) and theother to nativelike structures (RMSD;2.5 Å). This demon-strates that, although the nativelike structure is the moststable one, thermal fluctuation can temporarily kick the pro-tein out of the native structure.

We observe that collapse occurs at a very early stage(;104 MCS) for all four proteins, but the details of eachfolding process appear to depend on the secondary structurecontents. We now analyze the details of the folding behaviorof each protein. For betanova at low temperatures (T<30),the probability distributions of various quantities such asRMSD depend on initial conformations, showing its glassybehavior @Fig. 3~a!#. At higher temperatures (T>40) thisnonergodic glassy behavior disappears. It should be notedthat nativelike structures are more easily found from thesimulation atT540 @Fig. 3~b!# than from the best of ten runsat T530. When temperature decreases fromT580 to 60, thelocation of the RMSD peak dramatically moves from 8 Å to3 Å. This demonstrates the cooperative folding characteris-tics of betanova. For 40<T<60 betanova folds into its na-tivelike structure. The initial folding trajectories and the dis-tribution of (r,Rg) at T540 are shown in Fig. 4~a! forbetanova. Regardless of its initial conformation~either ran-dom or fully extended!, the average pathways to the foldedconformation initially converge to (r,Rg);(0.35,8.5 Å),and then they move horizontally to the native structure. Thisis consistent with the recent folding scenario for proteinswith b structure.7 The populated states11 around r;0.4– 0.5 andRg;11– 12 Å are not from initial folding tra-jectories, but from the fluctuation of nativelike structures.This kinetic information is difficult to be captured by freeenergy calculations alone.11 Figure 5 shows snapshots of atypical folding trajectory for betanova atT550.

FIG. 3. The probability distributions of RMSD.~a! The distributions of ten independent long-time simulations for betanova, starting from random confor-mations atT530. The glassy behavior is apparent.~b! RMSD distributions of betanova at various temperatures. As temperature decreases, the value ofprominent RMSD dramatically moves from 9.0 Å to 2.5 Å.~c! The distributions for 1fsd. The value of the most probable RMSD drops rapidly from 9.0 Åto 3.1 Å. ~d! The distributions for HP-36. The double-peak structure appears forT<100, representing the cooperative~first-order-like! two-state transition. In~b!, ~c!, and~d! the probability distribution at a temperature for the given protein was obtained by accumulating the results from ten independent long-timesimulations.

8273J. Chem. Phys., Vol. 120, No. 17, 1 May 2004 Folding of small proteins

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FIG. 4. The initial folding trajectories~from 200 independent short-time simulations at a fixed temperature for a protein! and the contour plots of thepopulation distributions~from 10 independent long-time simulations at a fixed temperature for a protein! as a function ofr andRg at appropriate temperatures.The triangles represent the averages of 100 early folding trajectories starting from random conformations. The squares are from 100 early folding trajectoriesstarting from a fully extended conformation. The color scale of the contour plots indicates the exponentx of the population 10x15 at given values ofr andRg . ~a! Betanova atT540. ~b! 1fsd atT570. ~c! HP-36 atT570. ~d! Protein A atT580.

FIG. 5. Snapshots of a typical folding trajectory for betanova atT550. In the figures the left side is the N-terminal part.~a! Initial random conformation:RMSD510.0 Å, Q50.08, andRg512.2 Å. ~b! Conformation at 104 MCS: RMSD57.2 Å, Q50.21, andRg510.7 Å. This conformation corresponds to aninitially collapsed state where theb-hairpin structure is formed only in the N-terminal residues.~c! Conformation at 23104 MCS: RMSD55.3 Å, Q50.33, andRg58.8 Å. This conformation has the same topology as the native state, but it is less ordered. In particular, it corresponds to the convergencepoint of the early folding trajectories in (Q,Rg) plane.~d! Conformation at 73105 MCS: RMSD51.4 Å, Q50.75, andRg57.7 Å. The conformation isalmost identical to the native state, and is one of the most populated conformations in Fig. 4~a!.

8274 J. Chem. Phys., Vol. 120, No. 17, 1 May 2004 Kim, Lee, and Lee

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For 1fsd the distributions of various quantities for tenindependent runs (109 MCS each! show glassy behavior forT<50. Again, the nonergodic glassy behavior disappears athigher temperatures (T>70). Figure 3~c! shows the RMSDdistributions at various temperatures. The strength of the co-operativity for 1fsd is not as strong as that of betanova.Again after initial collapse to (r,Rg);(0.3,9 Å), the aver-age trajectories move horizontally@Fig. 4~b!#, although less

prominently so compared to the case of betanova. Figure 6shows snapshots of a typical folding trajectory for 1fsd atT570.

For HP-36, the distributions of various quantities for tenindependent runs (109 MCS each! again show glassy behav-ior at T560. At higher temperatures we have two RMSDpeaks@Fig. 3~d!#. At T590 the peak near the native structurebegins to dominate, and as temperature decreases it becomes

FIG. 6. Snapshots of a typical folding trajectory for 1fsd atT570. In ~a! the upper left side is the N-terminal part. In~b!, ~c!, and ~d! the left side is theN-terminal part.~a! Initial random conformation: RMSD59.8 Å, Q50.03, andRg514.4 Å. ~b! Conformation at 43104 MCS: RMSD55.9 Å, Q50.18,andRg59.5 Å. This conformation corresponds to the convergence point of the early folding trajectories in (Q,Rg) plane. Theb-hairpin is formed but thea-helix is not yet initiated.~c! Conformation at 106 MCS: RMSD53.7 Å, Q50.50, andRg58.4 Å. This conformation has thea-helix formed and itstopology is that of the native state. It is one of the most populated conformations in Fig. 4~b!. ~d! Conformation at 23106 MCS: RMSD51.9 Å, Q50.68, andRg58.0 Å. This conformation is almost identical to the native state.

FIG. 7. Snapshots of a typical folding trajectory for protein A atT585. In the figures the lower side is the C-terminal part.~a! Initial random conformation:RMSD514.3 Å, Q50.21, andRg516.9 Å. ~b! Conformation at 1.33105 MCS: RMSD57.2 Å, Q50.28, andRg511.3 Å. This conformation correspondsto the convergence point of the early folding trajectories in (Q,Rg) plane. The helix III is already formed.~c! Conformation at 1.0133107 MCS: RMSD510.0 Å, Q50.43, andRg510.7 Å. The RMSD value is larger than that of the conformation~b!. It should be noted that the helix III is stably formed.~d!Conformation at 5.2133107 MCS: RMSD52.6 Å, Q50.56, andRg59.9 Å. The conformation has the same topology as the native state.

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stronger. This double-peak feature demonstrates the coopera-tive two-state transition. The conformations centered at thehigher value of RMSD come from a variety of collapsedstates. The conformations from the other peak are native like.When we examine them, the helix I~residues 4–8! is stablyformed,12 while the others are fluctuating. The initial foldingtrajectories @Fig. 4~c!# converge to (r,Rg);(0.2,11 Å).These collapsed structures fold into nativelike structures(r,Rg);(0.7,10 Å). Compared to the case of betanova and1fsd, the average folding trajectories are more diagonal.

The overall folding characteristics of protein A are simi-lar to those of HP-36. The initial folding trajectories@Fig.4~d!# converge to (r,Rg);(0.25,12 Å). These collapsedstates fold into nativelike structures in a diagonal fashionsimilar to the case of HP-36. When we examine the native-like conformations with 3 Å<RMSD<4 Å, the helix III~residues 42–55! is most stably formed. This is in agreementwith recent investigations.9,13,22We also observe that a first-order-like collapse transition5 @from (Q,Rg);(0.15,18 Å) to~0.15, 12 Å!# occurs nearT5120. Figure 7 shows snapshotsof a typical folding trajectory for protein A atT585.

IV. DISCUSSION

By using a single continuous potential, we could observefolding processes of four small proteins in realistic settings.In all cases, rapid collapse is followed by subsequent foldingprocess that takes place in a longer time scale. We also ob-serve that glassy transitions occur at low temperatures. Thefolding mechanism suggested in this study is as follows.There are two aspects of folding dynamics:~i! nonequilib-rium kinetic properties and~ii ! equilibrium thermodynamicproperties~Fig. 8!. The nonequilibrium kinetic properties,relevant to the early folding trajectories~fast process!, can beexamined only by direct folding simulations. The free energysurface, an equilibrium thermodynamic property, dictates theway an initially collapsed state completes its folding~slowprocess!. The way a protein folds into its native structure—

i.e., either horizontally or diagonally in the (r,Rg) plane—isdetermined by the position~for example, C or C8 in Fig. 8!of (r,Rg) where early folding trajectories converge, relativeto the native state. It appears that slow folding process ofa-proteins occurs in a diagonal fashion compared to that ofproteins containingb-strands.7

In conclusion, we successfully carried out direct foldingsimulations of more than one protein using asinglecontinu-ous potential. In this study, we simultaneously optimized18,19

the parameters of the UNRES force field for four small pro-teins and used this force field for the study of the foldingmechanism of these proteins. The results provide new in-sights into the folding mechanism.

ACKNOWLEDGMENTS

This work was supported by Grants No. R01-2003-000-11595-0~J. L.! and No. R01-2003-000-10199-0~J. L.! fromthe Basic Research Program of the Korea Science & Engi-neering Foundation.

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FIG. 8. A schematic of the folding trajectories. The colored contour repre-sents the free energy surface, which is an equilibrium property. Even forproteins with identical free energy landscape, the early folding trajectories~dashed lines! may converge into different points~C or C8). The solid linesrepresent the later part of the folding trajectories dictated by the free energylandscape. The location~C or C8) of the convergence point of the earlyfolding trajectories for a protein is determined by its kinetic properties. Thisinformation can be obtained only by directfolding simulations.

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