Proc. Nad. Acad. Sci. USAVol. 90, pp. 809-813, February 1993Biophysics

Exploring the energy landscape in proteinsJOHN E. STRAUB* AND D. THIRUMALAIt*Department of Chemistry, Boston University, Boston, MA 02215; and tDepartment of Chemistry and Biochemistry, Institute for Physical Science andTechnology, University of Maryland, College Park, MD 20742

Communicated by Peter G. Wolynes, October 15, 1992 (received for review July 10, 1992)

ABSTRACT We present two methods to probe the energylandscape and motions of proteins in the context of moleculardynamics simulations of the helix-forming S-peptide ofRNase Aand the RNase A-3'-UMP enzyme-product complex. The firstmethod uses the generalized ergodic measure tocompute the rateof conformational space sampling. Using the dynamics of non-bonded forces as a means ofprobing the time scale for ergodicityto be obtained, we argue that even in a relatively short time (<10psec) several different conformational substates are sampled. Atlonger times, barriers on the order of a few kcal/mol (1 cal =4.184 J) are involved in the large-scale motion of proteins. Wealso present an approximate method for evaluating the distri-bution of barrier heights g(EB) using the instantaneous normal-mode spectra ofa protein. For the S-peptide, we show thatg(EB)is adequately represented by a Poisson distribution. By com-paring with previous work on other systems, we suggest that thestatistical characteristics of the energy landscape may be a"universal" feature of all proteins.

The dynamics of the internal motions in proteins spanningseveral decades in time is thought to be a direct consequenceof the "complexity" of the underlying energy landscape (1).The wide distribution of time scales for protein motion and itsconsequences has been described in the experiments ofFrauenfelder and coworkers on ligand binding in heme pro-teins (2). Based on the relaxation characteristics over a widetemperature range, they have argued that a single monomericprotein can possess many nearly isoenergetic conformations,which are referred to as conformational substates (1, 2). Theindividual substates are believed to be separated by barriers ofdiffering heights. There appears to be a wide distribution ofbarrier heights ranging from a few hundredths to many kcal/mol (1 cal = 4.184 J). The existence ofthe multivalley structureof the energy landscape together with the notion of a distri-bution of barriers has been used to analyze quantitatively thekinetics of binding of small ligands to myoglobin over a widetemperature range (2, 3). Frauenfelder and coworkers (1, 2)have argued that it is necessary to postulate the existence ofconformational substates (CS) in which the CS have structureon several energy and length scales. This idea explainsnaturally the presence of an array of internal motions inproteins as well as fluctuations on several length scales (1).The existence of many time scales implies that, in general,relaxation functions should exhibit nonexponential kinetics.In addition, the rate constant for rebinding apparently ex-hibits marked deviation from the Arrhenius law; non-Arrhenius temperature dependence is obtained only when therebinding constant is examined over a range oftemperatures.

Molecular dynamics simulations have also provided someevidence for the existence of conformational substates. Fol-lowing the analysis of a 300-psec trajectory generated bysolving Newton's equation for myoglobin, Elber and Karplus(4, 5) have suggested that there are many thermally accessibleminima in the neighborhood of the native structure. Structural

differences between distinct CS corresponded to relativeorientations of the helices, which seem to be initiated byside-chain rearrangements. The measure used by Elber andKarplus is approximate and simple arguments can be used toshow that these authors have overestimated the number ofdistinct minima explored by myoglobin in 300 psec at roomtemperature. More recently, a much more ambitious study ofthe topography ofthe energy landscape in flexible systems hasbeen completed by Czerminski and Elber (6, 7). They wereable to map out the distinct minima in a tetrapeptide and obtainthe location of the transition states separating the minima inthe small peptide molecule. The protocol used by Czerminskiand Elber yields a set of "optimized" reaction pathwaysseparating the minima. Perhaps the most important aspect oftheir study is that the general features (see below) of thedistribution of barrier heights in this small molecule seems tobe quite similar to that postulated for a much larger proteinmolecule-namely, myoglobin (8). Finally, Czerminski andElber showed by using an approximate kinetic model forconformational transitions that as a consequence of the dis-tribution of saddle points there are several relaxation times.The study by Czerminski and Elber, while interesting, iscomputationally intensive and a similar characterization oftheenergy landscape in larger polypeptides will be far moredifficult.Although the studies cited above have gone a long way in

establishing the existence and importance of CS, relativelylittle has been done to elucidate the time scales involved inexploration of the underlying energy landscape. The ultimategoal, of course, is to reveal the relationship between proteinfunction and the associated energy landscape (9). Our goalsin this paper are the following. (i) To devise simple methodsthat can be readily used to assess the time scales on which aprotein samples the available conformational space. We userecently developed techniques for quantitatively measuringthe approximate rate at which proteins search the energylandscape as a function of temperature. In the process, wegain insight into the rate of sampling for the conformationaldegrees offreedom. (ii) To compute the distribution ofbarrierheights in the energy landscape ofproteins. We show that thedistribution of saddle point energies is approximately de-scribed by the Poisson law for energies greater than a certaincut-off value. We wish to stress that our objective is to devisemethods that can be used in conjunction with standardmolecular dynamics simulations to obtain qualitative infor-mation about the dynamics of the energy surface that isexplored at a specified temperature. The complete charac-terization of the energy landscape along the lines attemptedby Czerminski and Elber, while important, may not be asrelevant for finite temperature dynamics.

METHODSThe analysis presented in this paper is based on finite durationmolecular dynamics (MD) simulations in vacuum. We carriedout calculations on the S peptide of bovine pancreatic RNase

Abbreviation: CS, conformational substate(s); MD, molecular dy-namics.

809

The publication costs of this article were defrayed in part by page chargepayment. This article must therefore be hereby marked "advertisement"in accordance with 18 U.S.C. §1734 solely to indicate this fact.

810 Biophysics: Straub and Thirumalai

A and the RNase A-3'-UMP enzyme-inhibitor complex. Theprotein parameters are based on the version 19 CHARMMparameter set. The inhibitor and histidine parameters weredeveloped separately and will be described elsewhere (J.E.S.,C. Lim, and M. Karplus, unpublished data). The dynamicswere performed at constant energy using a modified version 20of the CHARMM simulation program (10).As emphasized in the Introduction, the generic feature of

the complex energy landscape that dictates protein dynamicsis the multivalley structure with a rather wide spectrum ofbarrier heights. A question of interest is to obtain the timescales for the transition between different conformationalsubstates that are being sampled in a given observation time.Let us assume that there are two distinct CS labeled a and ,(separated by an effective free energy of activation AFR.Conformations belonging to a and (3 would mix only if in theprocess of time evolution the molecule makes a transitionfrom a to (. Such an activated transition requires times on theorder of Toexp(AF* /kBT), where T0 is typically on the orderofa vibrational period. Therefore, to assess whether there aredistinct minima separated by barriers, it is necessary tofollow the time development oftwo trajectories starting fromindependent initial conditions. Let

1 rtfaN(t) = - fdsFN(s) [1]

t

be the time average of the force due to the nonbondedpotential experienced by the ith atom. Superscript a indicatesthat the starting conformation maps onto CS a. Similarly, letfiN(t) be the time averaged nonbonded force whose initialconfiguration maps onto CS (. Since the initial conformationshave to be independent, sufficient care must be used togenerate them in any numerical computation. The forcemetric dFN(t) is defined as

i NdFN(t) = - IIfN(t) -fPN(t)II [21N i=1 2

The properties of dFN(t) are easy to discuss. If the initialconformations mix on the time scale of the simulations, thendFN(t) should vanish for long times. On the other hand, if theinitial conformations belong to distinct CS then dFN(t) -O 0only for times longer than roexp(AFa/kBT), which can bevery long even for moderate values ofAF I/kBT. IfdFN(t) forlong times does not vanish, then it follows that the twotrajectories explore distinct conformational substates. Themost important aspect of the force metric is that in cases inwhich there is a bottleneck between the initial conformationsa and X that the protein overcomes in the process of simu-lation, the time scale for such a transition can be obtainedfrom the scaling relation (11, 12)

dFN(t)IdFN(O) - 1DFN(T)t. [3]

The parameter DFN (analogous to the maximal Lyapunovexponent) gives the temperature-dependent rate at which theconformational space is being explored. Estimates of barrierheights separating two conformational substates may be ob-tained from Eq. 3. Since DFN gives roughly the rate forsampling conformations belonging to a distinct CS, it fol-lows that AFRO kBTln(DFNTO). According to the CS picture,there is a wide range ofbarriers in proteins and, consequently,AFts obtained from this relation should be viewed as theaverage free energy barrier between the two CS a and ,(. Thisinterpretation assumes that the distinct CS are dilutely con-nected. It should be noted that other quantities, such as theenergy ofthe ith residue, local density, etc., may also be usedto monitor the rate of sampling of conformational space.

The temperature-dependent parameter DFN gives an esti-mate of the average barrier height separating the various CSthat are sampled during a MD simulation. It is, however,desirable to obtain a direct expression for the distribution ofbarrier heights connecting the various CS. The distribution ofbarrier heights can in principle be obtained by using the forcemetric. To do this, one has to compute DFN(T) for a largenumber of pairs of initial conformations. This is not onlycumbersome but it also does not offer any physical insight.Here we use an approximate method for computing g(EB)from the analysis of the temperature dependence of thefraction of unstable instantaneous normal modes. Typically,normal mode calculations are carried out at zero temperatureby expanding the potential function about a mechanicallystable energy minimum. Originally in the study of glassesRahman et al. (13) and more recently in the context ofmelting(14) and the liquid to glass transition (15-17), a number ofstudies have suggested that the fluidity of the system may infact be associated with the fraction of unstable modes. Theunstable modes, characterized by imaginary frequencies, areassociated with motion over regions of the potential surfacewith negative curvature-namely, the saddle points. Here weshow that by using a few simple approximations g(EB) maybe directly estimated from a molecular dynamics trajectory.Our method is easily used for any protein for which thepotential function is known. A more accurate determinationof g(EB) can be made by extending the time scale of thesimulation. However, we will show that the robustness oftheCS picture becomes apparent even within our limited simu-lation time scales.We use the following method to obtain g(EB). Ifthe potential

energy function is known, then the equilibrium fraction ofunstable modes can be computed as a function of temperature.For example, we can divide the configuration integral at a fixedtemperature into regions ofthe potential correspondingto stableand unstable motion. This decomposition allows us to identifythe fraction of imaginary frequencies as

fu(T) = Zunstable(T)/Z(7), [41

where Z.,table(T) is the configuration integral for the unstablemotion and Z(T) is the total configuration integral. In general,the required configuration integrals for proteins cannot beperformed analytically and one requires numerical methodsfor evaluating fu(T). Nevertheless, a simple normal-modemodel can be used to obtain g(EB) from the calculation off (T).We make two assumptions. (i) The nonlinear modes of theprotein can be adequately described by the corresponding 3None-dimensional normal modes. This appears to be reasonableat least near one of the local minima of the conformationalsubstates as well as near the saddle point. (ii) We approximatethe true potential separating two CS by a piecewise localharmonic potential. The local harmonic approximation to thefree energy surface is likely to be valid (18), even though thedynamics of the protein as a whole is highly anharmonic.These two approximations, which enable us to reduce thecomplex landscape into a series of local harmonic wellsseparated by various barrier heights, can be rationalized if thefluctuations of the protein about an instantaneous position aredue to barrier crossing and not simply stable anharmonicmotion. Within this approximate characterization ofthe actualfree energy landscape, the equilibrium fraction of unstablemodesfu(T, EB), for a particular value ofthe barrier height EB,can be readily calculated. The result (unpublished data) is

e-2e fdxex2

fu(T, EB) = [5]dte-2 + e-2 2rfdxex2

Proc. Natl. Acad. Sci. USA 90 (1993)

Proc. Natl. Acad. Sci. USA 90 (1993) 811

where 42 = 83EB/2. If we assume a continuous distribution ofbarrier heights, the fraction of unstable modes measured at aparticular temperature can be written as

fEfu(T) = |dEsg(EB)fu(T, EB). [6]

By calculatingfu(T) from a molecular dynamics trajectory (ora Monte Carlo simulation) at a series of temperatures, onecan in principle extract an approximate form for the distri-bution of barrier heights g(EB) for the system. The aboveequation, which is a Fredholm integral equation of the firstkind, is a general way of obtaining g(EB). The analyticrepresentation of the kernel fu(T, EB) that we have obtainedusing an idealized caricature of the energy landscape isperhaps the simplest one imaginable. The knowledge ofg(EB)together with DFN(T) gives us a way of quantifying thedynamics of exploration of the complex energy landscape inproteins. The above computational methods were used toanalyze molecular dynamics trajectories at several temper-atures for the S-peptide of RNase A and the RNase A-3'-UMP enzyme-product complex.

RESULTS AND DISCUSSIONWe have calculated the nonbonded force metric at severaltemperatures for the enzyme-product complex by using anumber of molecular dynamics trajectories. The reason forcomputing dFN(t) is the following: proteins adopt well-defined three-dimensional structures because residues farapart in the protein (nonbonded residues) would prefer to beclose in the configuration space. Thus, the initiation offolding is determined by the time scale required for estab-lishing the nonbonded contacts (19). The relaxation of non-bonded forces, probed by dFN(t) for times on the order of 100psec, can yield useful insights into the dynamics of earlyevents in the establishment of these contacts. Fig. 1 shows aplot of dFN(t) as a function of time for the enzyme-productcomplex at several temperatures. Initial configurations forthe a and 83 states were taken as endpoints of a 50-psectrajectory at a higher temperature. This figure shows thatthere is a rapid initial convergence followed by a slow, longtime decay. We have analyzed the initial decay that occurs attimes less than -10 psec. This analysis shows that there is aspectrum of relaxation times as indicated by the presence ofseveral distinct slopes in the plot of the reciprocal of dFN(t).This shows that even in the 10-psec regime the proteinsamples several different CS that are separated by smallbarriers. These CS would correspond to the substructure inthe tier organization of substates proposed by Frauenfelder etal. (2). The short time exploration of substates is due to thepeptide exploring an essentially local harmonic potentialminimum and the relatively constant interactions along themain chain, which are nearly independent of the peptideconformation.For times greater than =15 psec (see Fig. 1), we notice a

rather slow decay in dFN(t). Because dFN(t) probes thedynamics of two independent trajectories that map ontodistinct CS, this implies that there is a bottleneck that isovercome only on time scales greater than those explored byour simulations. This is suggestive of the existence of anorganization of CS into tiers of minima separated by barriersof increasing height. The exploration of various substatesdepends on the temperature, and this is illustrated by ana-lyzing dFN(t) in terms of the properties of the individualtrajectories. The decomposition of the force metric intofluctuation metric and the cross terms provides an interestingcontrast between the low- and high-temperature dynamics.We find that at 40 K the independent trajectories are frozen

.-Z 4 - r

,- i'__ __'-

0 il 1SC C

FIG. 1. Reciprocal of the normalized nonbonded force metric,dFN(0)/dFN(t), as a function of time for the RNase A-3'-UMPenzyme-product complex. Thin line, data at 40 K; thick line, data at120 K; open circles, data at 240 K; solid circles, data at 300 K.

into a particular CS and fluctuate about the average structurewithout making any significant dihedral angle transitions.This implies that the initial conditions for the two trajectoriesreside in separate CS and transitions between these and othersubstates occur on a much longer time scale than explored byour simulations. On the other hand, at 300 K we find that thetwo trajectories mix and the barriers between the CS canindeed be overcome on time scales on the order of 75 psec.As a quantitative estimate ofthe average barriers between theCS that are sampled, we have computed the temperaturedependence of DFN(T) using the scaling relation given in Eq.3. The average effective barrier height can be estimated fromthe relation Toexp(AF*/kBT) = DFN. If T0 is taken to be =1psec we estimate AF* to be 2.2-3.5 kcal/mol. These valuesfall in the range of activation energies computed by Czer-minski and Elber for model peptides (7).The results for the force metric can also be used to show

that the long time relaxation in the protein is strongly tied toinfrequent events such as dihedral angle relaxation (unpub-lished data). To further explore the nature of the relaxationprocess, we have computed the distribution ofbarrier heightsg(EB) for the S-peptide by calculating the instantaneousnormal-mode spectrum for eight temperatures ranging from40 K to 500 K. At low temperatures, the eigenfrequencies arereal and the density of states agrees with the 0 K normal-mode spectrum. As the temperature is increased, the numberof imaginary eigenfrequencies (corresponding to unstablemodes) increases and the center of the imaginary lobe of thespectrum shifts to higher values. Fig. 2 shows the fraction ofunstable modesfu(T) computed as a function of temperature.The fraction ofunstable modes is zero at 0K and quickly risesthrough 100 K, at which there is a break leading to a weakerslope through higher temperatures.The behavior offu(T) can be interpreted in terms of the

model leading to Eqs. 4-6. A large number of modes in thepeptide will remain stable even at the highest temperature.Other modes, such as those localized in dihedral angle spaceand the low frequency, continuum-like global fluctuations,are imagined to move over a roughly periodic potential withbarriers distributed according to g(EB). Physically, we knowthat the quick rise infu(T) even at low temperatures indicates

Biophysics: Straub and Thirumalai

-el

I

812 Biophysics: Straub and Thirumalai

2

1.8

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

50 100 150 200 250 300 350 400 450 500T, (K)

0 2 4 6 8EB, kcal/mol

10 12 14

FIG. 2. Fraction of unstable modes as a function of temperaturecalculated from a 75-psec constant energy trajectory ofthe S peptide.Shown for comparison is the fit to f.(T) using Eq. 6 (solid circles).

significant anharmonicity in the lowest frequency normalmodes and the presence ofmany low barriers where EB 100

K. The slower rise at higher temperatures indicates thepresence of many high barriers and the rate of increase infu(T) at these temperatures will depend on the nature ofg(EB). The physical insight provided by the changes infu(T)with temperature leads us to postulate the following equationfor g(EB):

g(EB) = aO(EB - Elow) + bEseeEB/EK. [7]

There is a constant density of low energy barriers for EB <E10w [written in terms of the Heavyside function e(E)] andPoisson distribution of higher energy barriers. The parame-ters for our model are a = 0.5 and E1ow = 0.2 kcal/mol, andb = 0.2 and Eo = 1 kcal/mol, respectively, which we derivedfrom fittingfu(T) by Eqs. 5, 6, and 7 for a periodic piecewiseparabolic potential. Our equation for g(EB) with the abovechoice of parameters solves the integral Eq. 5 adequatelyover the entire temperature range. This is shown in Fig. 2,where a comparison between fu(T) computed by MD simu-lations and that computed by evaluating the right-hand sideof Eq. 5 is given.A plot of g(EB) for the S-peptide along with a distribution

of energy barriers for the tetrapeptide Ala-Val-Ala-Ala com-puted by Choi and Elber (20) is shown in Fig. 3. Thetetrapeptide data are plotted only with a resolution of 1kcal/mol and show a large number of low energy barriers, EB< 1 kcal/mol, in agreement with our result. The two curvesfor larger values of EB show qualitatively similar behavior,although our results indicate a significantly smaller number ofhigh energy barriers. In all likelihood, this is because we havecomputedfu(T) only for limited values of T, and inversion offu(T) to obtain g(Es), for larger EB would require data atsignificantly higher temperatures. Nevertheless, given thedifferences in the two systems, it appears that the Poissondistribution ofbarrier heights is a robust feature ofthe energylandscape in proteins. This is further corroborated by notingthat similar distributions have been found experimentally inheme proteins (8). The major consequences of the dynamicsof proteins (i.e., nonexponential relaxation) readily followfrom this distribution of activation energies.

FIG. 3. Distribution of barrier heights g(EB) extracted from a fitof the temperature dependence of f.(T). For comparison, we alsoshow the distribution of adiabatic barrier heights calculated directlyfor the tetrapeptide Ala-Val-Ala-Ala by Choi and Elber (20).

The calculations presented here show the importance ofthe CS picture in the dynamics of proteins. The majorobservation is that even on relatively short time scales thereare several CS that are explored, which implies the existenceof small barrier heights (a few hundredths kcal/mol). Thebehavior of dFN(t) at the longest time examined here showsthat barriers on the order of 2-3 kcal/mol are encountered.These results together with the normal-mode analysis fordetermination of g(EB) show that the statistical characteris-tics of the energy landscape responsible for the complexdynamics in proteins are manifested in the presence ofa widedistribution of activation barriers. The results presentedhere, when combined with similar previous theoretical find-ings on model peptides (6, 7) and experimental studies onheme proteins (8), can be used to assert that the Poissondistribution for g(EB), which is one of the characteristics ofa complex energy landscape, may be a "universal" aspect ofall proteins. The range of barrier heights may in fact benecessary for the flexibility and enzymatic function of pro-teins and nucleic acids.

The authors are grateful to Peter G. Wolynes and the anonymousreferees for useful comments. We want to thank Ron Elber forproviding unpublished results on the tetrapeptide. This work wassupported in part by a grant from the National Science Foundationand by the Camille and Henry Dreyfuss Foundation.

1. Frauenfelder, H., Sligar, S. G. & Wolynes, P. G. (1991) Sci-ence 254, 1598-1603.

2. Frauenfelder, H., Parak, F. & Young, R. D. (1988) Annu. Rev.Biophys. Chem. 17, 451-479.

3. Berendzen, J. & Braunstein, D. (1990) Proc. NatI. Acad. Sci.USA 87, 1-5.

4. Elber, R. & Karplus, M. (1987) Science 235, 318-321.5. Noguti, T. & Go, N. (1989) Proteins Struct. Funct. Genet. 5,

97-103.6. Czerminski, R. & Elber, R. (1989) Proc. NatI. Acad. Sci. USA

86, 6963-6967.7. Czerminski, R. & Elber, R. (1990)J. Chem. Phys. 92, 5580-5601.8. Austin, R. H., Beeson, K. W., Eisenstein, L., Frauenfelder,

H. & Gunsalus, J. C. (1975) Biochemistry 14, 5355-5373.9. Steinbach, P. J., Ansari, A., Berendzen, J., Braunstein, D.,

Chu, K., Cowen, B. R., Ehrenstein, D., Frauenfelder, H.,

0.08

0.07

0.06

0.05

0.04

0.03

0.02

0.01

Sa eptide-Ala-Val-Ala-Ala-

Proc. Nad. Acad Sci. USA 90 (1993)

Biophysics: Straub and Thirumalai

Johnson, J. B., Lamb, D. C., Luck, S., Mourant, J. R., Nien-haus, G. U., Ormos, P., Phillips, R., Xie, Z. & Young, R. D.(1991) Biochemistry 30, 3988-4001.

10. Brooks, B. R., Bruccoleri, R. E., Olafson, B. D., States, D. J.,Swaminathan, S. & Karplus, M. (1983)J. Comp. Chem. 4,187-217.

11. Thirumalai, D., Mountain, R. D. & Kirkpatrick, T. R. (1989)Phys. Rev. A 39, 3563-3574.

12. Thirumalai, D. & Mountain, R. D. (1990) Phys. Rev. A Gen.Phys. 42, 4574-4587.

13. Rahman, A., Mandell, M. & McTague, J. P. (1976) J. Chem.Phys. 64, 1564-1568.

Proc. Natl. Acad. Sci. USA 90 (1993) 813

14. La Violette, R. A. & Stillinger, F. H. (1985) J. Chem. Phys. 83,4079-4085.

15. Rosenberg, R. O., Thirumalai, D. & Mountain, R. D. (1989) J.Phys. Cond. Matter 1, 2109-2114.

16. Seeley, G. & Keyes, T. (1989) J. Chem. Phys. 91, 5581-5586.17. Xu, B. C. & Stratt, R. M. (1990) J. Chem. Phys. 92, 1923-1935.18. Hall, R. W. & Wolynes, P. G. (1987) J. Chem. Phys. 86,

2943-2948.19. Guo, Z., Thirumalai, D. & Honeycutt, J. D. (1992) J. Chem.

Phys. 97, 525-535.20. Choi, C. (1992) Ph.D. thesis (Univ. of Illinois, Chicago).