Nature of the transition state ensemble for protein folding

Transcription

1 Nature of the transition state ensemble for protein folding N. H. Putnam, V.S. Pande, D.S. Rokhsar he ability of a protein to fold rapidly to its unique native state from any of its possible unfolded conformations depends critically on the transition state for folding. A variety of different physical characterizations of this ensemble have been proposed, spanning a spectrum from common partial structures to a broad collection of relatively unrelated conformations. Here we determine and compare the transition state ensemble for three simple but widely used lattice models for proteins. For each polymer we find that the transition state ensemble can be divided into one or two distinct structural classes. he conformations within a class share a common partially folded compact nucleus. With these transition states in hand, we can test the protein engineering method by simulating the folding of a complete set of single-site mutants, and computing the effects of these mutations on folding and unfolding rates to obtain values. April, For many small proteins the cooperative process of folding can be viewed as a two-state unimolecular chemical reaction from an unfolded state (the reactant ) to the native state (the product ), with its rate limiting step governed by a transition state (S). nlike the transition state for a simple chemical reaction, however, the transition state for the folding of a polymer consists of an ensemble of conformations rather than a unique configuration. he nature of this ensemble whether it contains specific, partially folded structures or is a broad collection of largely unrelated conformations is a focus of the debate between the classical and new views of folding []. We have analysed the folding dynamics of three different 7- residue lattice polymers that exhibit the hallmarks of protein folding rapid, reproducible, and spontaneous folding to a unique, sequence-dependent native structure. Lattice chains of this size have been proposed to represent the folding properties of small alpha-helical proteins [], and their two-state equilibrium transitions are well-characterized [ ]. Previous studies of the folding mechanisms of these models, however, have reached contradictory conclusions, with some studies [ ] finding specific nucleation sites and others [ ] supporting a more diverse set of folding nuclei. hese studies used a variety of heuristic methods for identifying putative transition state conformations. Here we compare and contrast the folding pathways of three different models for protein-like heteropolymers, with different degrees of specificity and heterogeneity. he Gō model [] includes only favorable interactions between native contacts, and thus has the highest degree of specificity for the native state. he LC model [] is a three-letter code that includes the competing effects of energetically favorable non-native interactions. Finally, the Miyazawa-Jernigan (MJ) model [] includes both native and non-native interactions, and an amino-acid-like distribution of interaction strengths derived from an analysis of the statistics of known protein structures. (See Methods for more details.) It is now well-established that many thermodynamic properties of protein-like heteropolymers are independent of the details of the model that is used [, 7]. We therefore might expect the common features of the kinetic folding pathways of these diverse models to be robust properties of folding polymers, including proteins. o identify the folding transition state for our model polymers, we appeal to the definition of a transition state as those species (i.e., conformations) that are equally likely to proceed in the forward (folding) or reverse (unfolding) directions [,,]. We determine these conformations by direct computation in our model systems. his method provides an unambiguous characterization of the transition state ensemble (including the statistical weights of the conformations involved). It has the further advantage that it does not require the problematic choice of a reaction coordinate for folding. A variety of computational techniques can then be brought to bear on the structural characterization of the S ensemble, to answer the question: what does the transition state ensemble for folding look like? he principal experimental technique for studying the structure of folding transition states is the -value method pioneered by Fersht et al. []. In these experiments, a collection of single site mutants are constructed, and their folding and unfolding rates are measured. he -value of a mutant is defined by!! " # % &'&'(*),+ -/. &'&'( ),+ () where &&'(,+/7&'(,+/ refers to the wild &'(,+/ type, to the mutant, and. he second approximate equality in Equation, which is the key to interpreting values, assumes that the mutations do not affect (i) the free energy of the unfolded state, (ii) the nature of the S ensemble, and (iii) the prefactors in the Arrhenius rate formula. In principle, if the mutated residue participates in local native structure in the S, the free energy barrier for folding will be altered, but not the barrier for unfolding (since both the native and transition states will have roughly the same free energy shift), and will be near unity; conversely, residues that are unstructured in the &'(:; S+=< will have a value near zero, since then one expects that will be relatively unaffected. Physical Biosciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 7, SA and Department of Physics, niversity of California, Berkeley, CA 7-7, SA Address as of June, : Deparment of Chemistry, Stanford niversity, Stanford, CA

2 [ d _ H Z H H [ In practice, -values often lie between zero and one; values less than zero are not uncommon. (For a summary of experimentally measured > values see [].) he interpretation of such values is crucial for using this method to probe the S ensemble []. Since we have characterized the S ensemble for our lattice models, we can examine the meaning of intermediate values directly using the protein engineering method. Previous theoretical studies have used surrogates for [, ], calculation of selected -values [], or values associated with specific contacts rather than sites []. Here we directly simulate the experimental protocol by studying the kinetics of a complete set of single-site mutants. Our results shed light on the relationship between -values and the nature of the transition state ensemble. wo-phase Kinetics. For each model, a sequence was chosen to stabilize a single native structure. o eliminate possible differences in the folding pathway due to structural differences, the same compact native structure (previously used by Socci et al. []) was used for all three models. Each model polymer is fully specified by the free energies of contact between residues? along the chain. (See Methods for more details.) In Figure Column B, these energies are represented in a color code as a function of contact range (?AB@ ) and mean position (?DCE@F GIH ), where? are residue positions along the chain. Each model was studied at its respective midpoint temperature, where the folding transition is cooperative with single rate kinetics, as shown by the exponential distribution of first passage times (Figure Column A). Folding probability and transition states. he folding probability JLKNMGO PL RQ of a conformation Q is defined as the probability that subsequent time evolution from Q leads directly to folding, without first encountering an unfolded state [, ] Operationally, we calculate JLKNMGO PS RQ by performing independent simulations that all start from the same conformation Q. hese trajectories are continued until they either reach the native structure or reach a conformation that is unambiguously unfolded (see Methods); J KMO P RQ is the fraction of these runs that fold. he folding probability provides a straightforward computational assay for the folding status of a polymer conformation. Conformations whose folding probabilities are near unity are committed to folding, and are therefore past the transition state; conformations with J KNMGO P near zero are committed to unfold. he transition state ensemble comprises those conformations that are equally likely to fold or unfold, i.e., with JLKNMGO P H. When these conformations fold directly, they do so rapidly, at rates that are considerably faster than the overall folding rate: (MJ), (Gō), and (LC), times faster than the respective folding rates, as measured by mean first passage times. his is consistent with Shaknovich s kinetic criterion [] for a putative folding nucleus. By considering conformations drawn from a long run which contains many folding and unfolding events, the statistical weight of a conformation in the S ensemble is governed by the Boltzmann distribution. he S ensemble for folding and unfolding are the same, as expected on general statistical mechanical grounds. Figure Column C shows probability densities of J KNMGO P values for each model. As expected for a two-state reaction, conformations with JLKNMGO P near zero or one dominate, and relatively few conformations have JSKNMO P near one-half. his is consistent with the idea that the transition state is a rare, transiently populated state along the reaction pathway. he J KNMO P probability density exhibits several weak local maxima near J KNMGO P IH, which indicates the presence of transiently-populated metastable states. hese J KNMGO P IH conformations are on the folding pathway since they can, by definition, fold to the native state without first returning to the unfolded state. Structure of S conformations. Our central focus is the nature of the conformations that compose the transition state ensemble. Are they very diverse, or do they have common structure? A crude measure of their partial structure can be found by computing the distribution of the number of native contacts. Conformations in the S ensemble -. of each model have an average number of native contacts. Out of total native Z -/. contacts, the MJ transition state has, WVDX YZ X [ -/. (mean standard deviation); for the Gō model, ]\LX X \ -/. ; for the LC model, WVFX ^*Z`_LX [ -.. Simply requiring that is approximately, however, is not a sufficient condition for membership in the transition state ensemble. Column of Figure shows the distribution of JSKNMO P values for those conformations -. that have between Z & -. YIacb. For the MJ model have _ed JSKNMGO P Hc ; the corresponding numbers are % for Gō and only % for LC. From this distribution, we find -. that in the Gō and LC models, the conformations with that are not transition states are. hus although these conformations contain over % native contacts, these contacts are formed in wrong combinations and/or in conjunction with non-native contacts that prevent them from folding directly. hese conformations therefore lie off-pathway mostly committed to unfold, as shown by the peak near J KNMGO P since J KNMGO P implies that they must unfold first before refolding. For the MJ model, conformations with that are not transition states are more evenly partitioned between those committed to unfold (as in Gō and LC) and conformations that are ). his latter group can highly committed to folding (JSKNMO P be associated with the post-critical nucleus of Shakhnovich et al. []. What structural features distinguish transition states from other conformations? We applied several analysis techniques to characterize the S ensemble computed using J KNMGO P. Both hierarchical clustering algorithms [, ] and principal component analyses [7,] were used to organize conformations into classes. hese techniques are described further in Methods. hese two approaches agree, and provide clear evidence that folding proceeds via compact nuclei that involve a combination of short and long range contacts. Figure summarizes the results of the principal component analysis, and shows aligned conformations sampled from groups determined by the principal component analysis. A single nucleus structure dominates the transition state of the MJ model (with other structures appearing only rarely). Both the Gō and LC models, however, exhibit a pair of distinct parallel folding pathways that proceed through nucleation of native structure at two distinct sites, one of which corresponds to the nucleation site found in the MJ model. Each cluster exhibits a well-structured -.

3 [ t MJ j Go k LC number of events v number of events number of events f f g.. p A B C f h g g i.... contact range g i.. umc ime x f f g g... MC ime x t lm ean chain posn. q( R) r s Equilibrium r r r r r.. m r... oequilibrium r. r r r r... m r. r. Equilibrium t v t w D n Q Q S r r r... r. m n Q Q S r r r... n Q Q S r.m umc ime x r r r r. m r... r r r... r. m Figure : Column A shows the distribution of first passage times for folding for each of the three models. Column B shows a plot of the inter-residue contact energies for each model. he energy of each of the possible contacts between residues? that are not neighbors along the polymer chain is shown in a grayscale in units of x y at the temperature used in the simulations, plotted versus the contact range z?{e@z and the mean chain position?lc}@c GIH. he contact map of the native conformation defines the contact energies in the Gō model. he contact energy map of the LC model shows that all of the native contacts are of the favorable type, and that many non-native contacts of equal strength are possible. In the MJ model, the native contacts tend to be the strongest, but they vary in strength, as do the non-native contacts. Column C shows the probability density of J KNMGO P from equilibrium runs for each model. ransition state conformations, with -. J KMO P IH, are rare conpared to the folded and unfolded states. Column D shows probability densitiess ofj KNMGO P for the Z & ensembles, composed of those conformations which have roughly the same number of native contacts as do transition state conformations. hese distributions show that many of these conformations are not transition states, but ). are committed to folding (J KNMGO P ) or unfolding (J KNMGO P

4 A B C D % % 7 MJ 7 e 7 % % 7 Go 7 e 7 % ƒ % LC 7 7 e ~ 7 Figure : he transition state ensembles of three model proteins are shown. Each row represents a different model: MJ (top), Gō (middle), and LC (bottom). Column B shows the distribution of S conformations projected on the normalized eigenvector corresponding to the second largest eigenvalue of the contact correlation matrix ˆ. (In each model the eigenvector with the largest eigenvalue corresponds to the average properties of the ensemble and does not contain information useful for grouping the conformations. See Methods for details.) Columns A and C each show conformations for each model, selected at random from the left and right sides of the principal component distributions. hese conformations have been rotated, translated and reflected to maximize their overlap with the native structure, and the position of each monomer has been averaged over a short window of simulation time. he overlayed S conformations show that for the MJ model, conformations drawn from the extremes of the principal component distribution are structured in the same part of the chain, and do not constitute distinct structures. In contrast, the other two models show peaks in the distribution which correspond to the specific compact partial structures visible in the overlays. Column D shows calculated values superimposed on the native state conformation. he area of the circle at each position is proportional to the magnitude of the value. Positive values are shown in black, and negative values are shown in gray.

5 ŠΦ Š Φ 7... MJ Go LC f(i) f(i) f(i) Š Φ number of residues MJ Go LC Φ Φ Φ Figure : op row: vs. A?, the probability that a residue is found to be fixed in position in the transition state ensemble. (See Methods for details.) In the MJ plot, residue is off the vertical scale with a value of X YAZŒ[SX _. Bottom row: -value distribution for the MJ, Gō and LC models are shown. Black bars show the distribution of -values of residues with [LX a, which typically lie at the upper end of the distributions. Φ Φ Φ. MJ. Go. LC f(i) f(i) f(i) MJ Go LC number of residues Ž.... Ž Φ Φ Φ Figure : he correlation between structure in the S and is improved š by using a locally-averaged -value, designated, which is enhanced by clusters of large -values. We therefore define œ C GIž Ÿ œ, where the summation is over the neighbors of residue?, = is the number of neighbors it has in the native state, and the prime indicates that negative values are replaced with zero. op row: vs. A? with the least squares linear fit. he linear correlation coeficients ( ) are. (MJ),.77 (Gō), and. (LC). Bottom row: distribution for the MJ, Gō and LC models are shown in light gray. Black bars show the distribution of -values of [SX a residues with. Ž

6 < core or nucleus with largely unstructured tails. he presence of two distinct folding nuclei that do not interconvert implies the existence of two parallel folding pathways in the Gō and LC models. As a control, we have applied the same analysis to other ensembles drawn from the equilibrium simulations (data not shown). nfolded state conformations from each model show little or no common structure, with mean between and. :; For each model, the, J KNMO P d ensemble contains some structures which are low energy misfolded states which have many contacts in common with the native state but which must be unfolded completely before the native state can be reached because of topological constraints []. :; he, JLKNMGO P` H ensembles (the conformations which possess appreciable native structure and are likely to fold directly) reflect the structural features of the transition states in each model. he principal difference between these conformations and those of the S is that in the high J KNMGO P group, the unstructured part of the polymer is positioned close to its native structure. he fact that folding probability depends sensitively on the disposition of unstructured parts of the chain that make no contacts emphasizes the complexity of the transition state ensemble, and the limitations of characterizing conformations solely by which native interactions are present. In their analysis of protein engineering experiments on CI [], Fersht and co-workers have previously pointed out the importance of native-like topology of the chain in transition state conformations. ransition States and -values. Once we have determined the transition state ensemble, we can use it to test the protein engineering method [] for inferring transition state structure by simulating the experimental protocol: for each model, we systematically mutate every residue, and measure the change in folding and unfolding rates to obtain a complete set of -values (see Methods). hese are shown in Fig., column D. For the MJ model, which we have seen has only one transition state class, we find that residues with the largest -values (residues,,,, and ) lie in the structured S-nucleus. hese maximum -values are in the range.-., indicating that even residues that always occupy a specific relative location in the transition state can have appreciably less than unity. he disordered tails of the S, consisting of residues - and -7, have -values close to zero. Interestingly, however, we find that other residues which are well-structured in the S (e.g., residues 7,,, -, and 7) also have small [LX [d -values ( d [SXWY ). he large and negative of residue, which only forms a contact with residue in the native state, results becuse both the folding and unfolding rates are increased when the wild type serine is mutated to alanine. Serrano and co-workers have observed negative -values in three cyclical permutants of an SH domain, and have proposed that they are due to the presence of a non-native interaction in the transition state []. We find that a non-native contact between residues and is present in % of the,7 wildtype sequnce MJ transition state conformations in our ensemble. Residues on the top face of the polymer (in the orientation shown in Fig. ) are not only well-structured in the S, but due to the largely structured nature of the middle horizontal layer have essentially native environments in the S ensemble. Even accepting the approximate equality in eq. () and interpreting a ratio of effective free energies, is evidently influenced by more than local structure variations in the number of accessible conformations (i.e., the entropy) can also play a role. For example, the structuring of critical residues on either side of the short loop - limit the degrees of freedom of the loop itself. When the S ensemble comprises two different nuclei, as in the Gō and LC models, the interpretation of -values becomes more complicated. Well-formed structures that are found in both classes (e.g., residues -) yield -values that are among the largest, but (as in the MJ model) these maximal s are well below unity (.-.). Comparing all three models, the -value of residue 7 is largest in the LC polymer, even though it has the least structure in the S for this model. It appears that mutating residue 7 increases the folding rate by destabilizing off-pathway structures rather than by stabilizing the transition state. Other unusual -values can also be found. A detailed study of scenarios which give rise to such unusual values can be found in []. he correlations of with transition state structure are exhibited more quantitatively in Fig., which shows scatter plots of mutant at residue? vs. A? a measure of the probability that a residue? participates in persistent structure in the S ensemble. here is a broad general trend in all three models for an increase in with increasing S structure. Also shown in Figure is the distribution of -values for each of the models. he distributions are broad, like that found for CI [, ]. he residues that are structured in the transition state more than a[cb of the time are indicated in these distributions by dark shading. Evidently these structured residues tend to have the largest s, and vice versa. While an individual high does not necessarily imply structure of that residue in the transition state, clusters of high s can be used to infer S structure []. his is illustrated in figure, which shows that coarse-graining each residue s -values by averaging it with those of its neighbors in the native structure improves the correlation between -value and. Discussion. From our analysis and comparison of the transition states of three different models for heteropolymer folding, we can draw three main conclusions. First, we find that the transition state ensemble for folding comprises a small number (one or two) of well-defined core structures. Previous investigations of lattice models for proteins have yelded a range of characterizations of the transition state for folding. hese include the results obtained by Shakhnovich and coworkers, which indicate that for -mer MJ sequences optimized for fast folding, the formation of a specific set of core contacts is the bottleneck of the folding transition []. hirumalai et al. have found evidence that folding proceeds not through the formation of a single core structure, but through multiple folding nuclei. [] Onuchic and coworkers have argued that structure is first nucleated at many delocalized sites []. Several explanations are possible for these differing interpretations of similar (and in some cases identical) models. Each study

7 ª _ uses different criteria for the identification of transition state conformations or folding nuclei. Socci et al. used the number of native contacts to identify the transition state ensemble of LC model. We have shown here that of these conformations, a substantial fraction (7%) are in fact committed to unfold, and represent off-pathway structures, since they have a significant number of native contacts but must still unfold. Another % are committed to fold. Including these conformations in an analysis of the transition state ensemble complicates the analysis and may mask the existence of specific sets of core contacts in the folding transition state. Our results show that, depending on the interaction model, folding can be initiated in a single or multiple (two) distinct compact core structures, each of which consists of a specific combination of contacts. In the MJ model, where we observe a single class of transition state structure, our results are consistent with the specific nucleus model of Shakhnovich et al.. In the other two models we observe two distinct nucleation sites which are parallel folding pathways in the sense that individual folding trajectories (transits between the folded and unfolded states) include conformations belonging to only one of the classes. Experimental evidence has been found that parallel folding pathways may be important in the folding of hen egg white lysozyme and dihydrofolate reductase []. We have also found that the disposition of unstructured parts of the chain affect a conformation s folding probability. his observation may explain some of discrepancy between the analyses of hirumalai et al. and Shakhnovich et al. Shakhnovich and coworkers identified specific sets of contacts present in the last time steps of lattice polymer folding simulations [], and inferred the existence of a specific folding nucleus and showed that the presence of these contacts should be sufficient for rapid folding. However, hirumalai et al. found that conformations generated at high temperature containing these core contacts do not fold much faster than randomly selected high temperature conformations []. Second, broad distributions of -values have been used to argue in favor of delocalized folding nuclei by Onuchic and coworkers []. his interpretation has been questioned by Fersht et al. []. Our analysis of the -distribution of the MJ model (Fig.), however, supports the latter view, since although the distribution is rather broad and not bimodal, the transition state is evidently described by a single nucleus (see Fig. ). hus, the broad, unimodal distributions of values found in proteins (e.g., CI) do not preclude the possibility of a single specific nucleus. Since the distributions for the Gō and LC models are also broad, one cannot use such distributions to discriminate between one and two distinct nuclei. Finally, by numerically evaluating -values for our model polymers, we have explored the correlation between the structure of the transition state and this popular experimental probe. For the MJ model, which is perhaps the most protein-like of the three polymers we considered, the maximal values are near /. Residues with large are likely to be structured in the transition state. Interestingly, however, we find that small does not imply the absence of structure in the S. While individual -values are problematic, clusters of high -values correlate well with the computationally derived S structures []. Acknowledgements. We thank W. Kesser, E. Stephens and S.-L. olley for helpful discussions and S. Marqusee for a critical reading of the manuscript. his work was supported by a grant from Lawrence Berkeley National Laboratory (LDRD--7) and by a National Science Foundation graduate research fellowship (NHP). his project used recources of the National Energy Research Scientific Computing Center, which is supported by the Office of Energy Research of the. S. Department of Energy. Methods. Lattice Monte Carlo Simulation. All three models represent a small protein by a lattice polymer of 7 residues on an infinite, three-dimensional cubic lattice. Each polymer has the same native structure, which was introduced previously by Socci et al. []. he internal free energy of conformation Q (which incorporates polymer-polymer, polymer-solvent, and solvent-solvent free energies) in each model is given by «Q * ž± ² Q³ () where the contact matrix Q is if residues? are on neighboring sites of the lattice, and zero otherwise. he interaction matrix ˆ is the change in free energy achieved by placing residues of type µ and µfœ in contact, and µf is the residue type at position? along the chain. he matrix ˆ differs for each of the three models, and is the only part of the simulation that does. he Gō model is a simplified model for proteins in which only pairs of residues that make a contact in the native structure interact; all other contacts are energetically neutral. Every native contacts is taken to have the same free energy gain. hus there are 7 model amino :/ " =¹ acid types, with a sequence #Q*.? and For the three letter code model of Onuchic et al. [], there are three residue types which we identify (arbitrarily) as hydrophobic (H), polar (P), or neutral (N). Like types attract with energy ; different types attract with an energy to mimic the tendency of proteins to collapse non-specifically. We use the sequence HPHPPPNPHNPHPHPHNHNPHNHHNHP which was previously studied in ref. []. We have also studied variants of this model for which the non-specific attraction is eliminated or converted into a repulsion; the results of these studies are qualitatively similar to those discussed in the text (data not shown). he MJ interaction model has twenty residue types interacting with their nearest neighbors according to a potential of mean force derived by Miyazawa and Jernigan from a study of the protein structure database []. sing an evolutionary algorithm [, ] we designed a sequence (QFHIQVMGRWLEFNNAD- MIKSNYSEL) to have the same native conformation as the LC model. Dynamics are simulated by a standard Metropolis process in which local changes in the polymer are attempted and accepted according to the Boltzmann weight of the new conformation. For more details, see ref. []. his dynamics has two important properties: (a) it satisfies detailed balance, and recovers thermodynamic equilibrium at long times, and (b) reproduces the known 7

8 Rouse dynamics of a fluctuating polymer. Initial conditions are randomly selected self-avoiding conformations. Folding probability calculations. was calculated at H [[[ step intervals in long simulation runs as he folding probability JLKNMGO P described in the text. For the purposes of JLKMO P calculations, a conformation is deemed unfolded if it has fewer than a specified number of native contacts (typically seven to nine). For each model this cutoff is chosen to be the typical number of native contacts of unfolded conformations in thermal equilibrium. We find that the nature of the S ensembles are insensitive to this cutoff (data not shown). Principal component analysis. o determine whether distinct classes exist, we diagonalize the contact correlation matrix ˆ for each S ensemble. he labels º and º»œ on this matrix run over all contacts (native and nonnative) that are possible for a lattice 7-mer. he matrix ˆ is the probability that contacts º and º»œ have the same status (i.e., are either both formed or both not formed) in the transition state ensemble. he eigenvectors of this matrix provide canonical coordinates for plotting the distribution of conformations by defining directions onto which the -bit contact state vector of each conformation can be projected. If there are discrete structural classes, we expect one nonzero eigenvalue for each, and that the distribution of conformations, when projected on these principal components, have clearly sparated peaks. A broad distribution of structures will produce enough non-zero eigenvalues to span the space of structures present, and the probability distribution in these dimensions will be broad. his method is similar to the molecule optimal dynamic coordinates method of García in which the correlation matrix of residue positions is diagonalized to find canonical dynamical coordinates [7]. Cluster Analysis. Complete-, mean- and single-link hierarchical clustering algorithms [] were applied to the S and control ensembles of the various models using Hamming distance between both native and complete contact maps as metrics on the space of conformations. he results were an independent classification of the S conformations in to groups consistent with that determined by the principal component analysis. Analysis. Simulated -values were obtained by determining the folding and unfolding rates at the wild-type midpoint temperature for complete sets of single-site mutants. In the MJ model, residues were mutated to alanine, or to glycine in positions in which the # sequence is alanine. In the LC model, all possible single-site mutants were made (i.e., N ¼ H or P, H ¼ P or N, P ¼ N or H). For each residue the average value of for each of the two possible mututants is reported. For the Gō model, residues were mutated to a non-interacting species. Rates were determined by the inverse mean folding (unfolding) time computed from a minimum of a[[ folding and a[[ unfolding events. From these rates, is computed using Equation. Probability that a residue is fixed in the S. he probability A? is calculated by measuring the RMS fluctuation of residue? over a short interval of simulation time (% of the mean-firstpassage time) for each conformation in the S ensemble sample. if A? is the S ensemble average of the quantity ½,¾N =? xà7á%? lattice constant. * d[lxây and ½,¾N =? [ if xà7á%? Äà [LXÂY in units of the. D. V. Laurents and R. L. Baldwin. Protein folding: Matching theory and experiment. Biophys. J., 7:,.. J. N. Onuchic, P. G. Wolynes, Z. Luthey-Shulten, and N. D. Socci. owards and outline of the topography of a realistic protein folding funnel. Proc. Natl. Acad. Sci., :,.. V. S. Pande, A. Y. u Grosberg, and. anaka. Heteropolymer freezing and design: owards physical models of protein folding. Reviews of Modern Physics, in press,.. K. A. Dill and H. S. Chan. From levinthal to pathways and funnels. Nature Struct. Biol., :, 7.. H. S. Chan and K. A. Dill. Protein folding in the landscape perspective: Chevron plots and non-arrhenius kinetics. Proteins: Struct. Funct. Genet., :,.. V. I. Abkevich, A. M. Gutin, and E. I. Shakhnovich. Specific nucleus as the transition state for protein folding: evidence from the lattice model. Biochemistry, :,. 7. E. I. Shakhnovich, V. I. Abkevich, and O. Ptitsyn. Conserved residues and the mechanism of protein folding. Nature, 7:,.. V. S. Pande and D. S. Rokhsar. Folding pathway of a lattice model for proteins. Proc. Natl. Acad. Sci., :7,.. Eugene I. Shakhnovich. Folding nucleus: specific or multiple? insights from lattice models and experiments. Fold. Des., :R R,.. N. D. Socci, J. N. Onuchic, and P. G. Wolynes. Diffusive dynamics of the reaction coordinate for protein folding funnels. J. Chem. Phys., :,.. J. N. Onuchic, N. D. Socci, Z. Luthey-Schulten, and P. G. Wolynes. Protein folding funnels: the nature of the transition state ensemble. Fold. Des., :,.. D. K. Klimov and D. hirumalai. Lattice models for proteins reveal multiple folding nuclei for nucleation-collapse mechanism. J. Mol. Biol., :7,.. D. hirumalai and D. K. Klimov. Fishing for folding nuclei in lattice models of proteins. Fold. Des., :R R,... Lazaridis and M. Karplus. new view of protein folding reconciled with the old through multiple unfolding simulations. Science, 7:, 7.. N. Go. heoretical studies of protein folding. Annu Rev Biophys Bioeng, :,.. S. Miyazawa and R. L. Jernigan. Estimation of effective interresidue contact energies from protein crystal structures: Quasi-chemical approximation. Macromol., :,.

9 7. E. I. Shakhnovich. Modeling protein folding: the beauty and power of simplicity. Fold. Des., :,.. J. A. McCammon and M. Karplus. Dynamics of activated processes in globular proteins. Proc. Natl. Acad. Sci., 7:, 7.. R. Du, V. S. Pande, A. Y. Grosberg,. anaka, and E. I. Shakhnovich. On the transition coordinate for protein folding. J. Chem. Phys., :,.. A. R. Fersht. Characterizing transition states in protein folding: an essential step in the puzzle. Curr. Opin. Struct. Biol., :7,.. Alan Fersht. Structure and Mechanism in Protein Science: A Guide to Enzyme Catalysis and Protein Folding, pages. Freeman, New York,.. A. R. Fersht, L. S. Itzhaki, N. F. ElMasry, and J. M. Matthews. Single versus parallel pathways of protein folding and fractional formation of structure in the transition state. Proc. Natl. Acad. Sci., :,.. A. Li and V. Daggett. Characterization of the transition state of protein unfolding by use of molecular dynamics: chymotrypsin inhibitor. Proc. Natl. Acad. Sci., :,.. A. M. Gutin, V. I. Abkevich, and E. I. Shakhnovich. A protein engineering analysis of the transition state for protein folding: simulation in the lattice model. Fold. Des., :,.. Brian Everitt. Cluster Analysis. Halsted, London,.. Mary E. Karpen, Douglas J. obias, and Charles L. Brooks III. Statistical clustering techniques for the analysis of long molecular dynamics trajectories: Analysis of.-ns trajectories of ypgdv. Biochemistry, :,. 7. H. Nymeyer, A. E. García, and J. N. Onuchic. Folding funnels and frustration in off-lattice minimalist protein landscapes. Proc. Natl. Acad. Sci., :,.. Christopher M. Bishop. Neural Netowrks for Pattern Recognition, pages. Clarendon, Oxford,.. L. S. Itzhaki, D. E. Otzen, and A. R. Fersht. he structure of the transition state of chymotrypsin inhibitor analyzed by protein engineering methods: evidence for a nucleationcondensation mechanism for protein folding. J. Mol. Biol., :,.. A. R. Viguera, L. Serrano, and M. Wilmanns. Different folding transitions states may resuly in the same native structure. Nature Struct. Biol., :7,.. D. E. Otzen, L. S. Itzhaki, N. F. elmasry, S. E. Jackson, and A. R. Fersht. Structure of the transition state for the folding/unfolding of the barley chymotrypsin inhibitor and its implications for mechanisms of protein folding. Proc. Natl. Acad. Sci., :,.. C. M. Dobson, P. A. Evans, and S. E. Radford. nderstanding how proteins fold: the lysozyme story so far. IBS, : 7,.. E. I. Shakhnovich and A. M. Gutin. Engineering of stable and fast-folding sequences of model proteins. Proc. Natl. Acad. Sci., :7,.. V. S. Pande, A. Y. Grosberg, and. anaka. Folding thermodynamics and kinetics of imprinted renaturable heteropolymers. J. Chem. Phys., : 7,.