IntroductionRelated WorkOverviewEstimating the TSE Using Stochastic Roadmap SimulationA Simplified Folding ModelConstructing the Stochastic Conformational RoadmapComputing $\textrm{P}_\mathrm{fold}$Estimating the TSEAn Example on a Synthetic Energy LandscapePredicting Folding RatesMethodsResultsAccuracy in Estimating the TSEPredicting $\Phi$-ValuesMethodsResults on $\Phi$-Value PredictionResults on the Order of Native Structure FormationConclusionThe List of Proteins Used for TestingPredicting Experimental Quantities in Protein FoldingKinetics Using Stochastic Roadmap SimulationTsung-Han Chiang1, Mehmet Serkan Apaydin2, Douglas L. Brutlag3,David Hsu1, and Jean-Claude Latombe31National University of Singapore, Singapore 117543, Singapore2Dartmouth College, Hanover, NH 03755, USA3Stanford University, Stanford, CA 94305, USAAbstract. This paper presents a new method for studying protein folding kinet-ics. It uses the recently introduced Stochastic Roadmap Simulation (SRS) methodto estimate the transition state ensemble (TSE) and predict the rates and Φ-valuesfor protein folding. The new method was tested on 16 proteins. Comparison withexperimental data shows that it estimates the TSE much more accurately thanan existing method based on dynamic programming. This leads to better folding-rate predictions. The results on Φ-value predictions are mixed, possibly due to thesimple energy model used in the tests. This is the first time that results obtainedfrom SRS have been compared against a substantial amount of experimental data.The success further validates the SRS method and indicates its potential as a gen-eral tool for studying protein folding kinetics.1 IntroductionProtein folding is a crucial biological process in nature. Starting out as a long, linearchain of amino acids, a protein molecule remarkably configures itself, or folds,intoa unique three-dimensional structure, called the native state, in order to perform vitalbiological functions. There are two separate, but related problems in protein folding:structure prediction and folding kinetics. In the former problem, we are only interestedin predicting the final three-dimensional structure, i.e., the native state, attained in thefolding process. In the latter problem, we are interested in the folding process itself,e.g., the kinetics and the mechanism of folding. We have at least two important reasonsfor studying the folding process. First, better understanding of the folding process willhelp explain why and how proteins misfold and find therapies for debilitating diseasessuch as Alzheimer’s disease or Creutzfeldt-Jakob (“mad cow”) disease. Second, thiswill aid in the development of better algorithms for structure prediction.In this work, we apply computational methods to study the kinetics of protein fold-ing, specifically, to predict the folding rates and the Φ-values. The folding rate measureshow fast a protein evolves from an unfolded state to the native state. The Φ-value mea-sures the extent to which a residue of a protein attains its native conformation when theprotein is in the transition state of the folding process. Performing such computationalstudies was once very difficult, due to a lack of good models of protein folding, a lack ofefficient computational methods to predict experimental quantities based on theoreticalA. Apostolico et al. (Eds.): RECOMB 2006, LNBI 3909, pp. 410–424, 2006.c Springer-Verlag Berlin Heidelberg 2006Predicting Experimental Quantities in Protein Folding Kinetics Using SRS 411models, and a lack of detailed experimental results to validate the predictions. However,important advances have been made in recent years. On the theoretical side, the energylandscape theory [4, 7] offers a global view of protein folding in microscopic detailsbased on statistical physics. It hypothesizes that proteins fold in a multi-dimensionalenergy funnel by following a myriad of pathways, all leading to the same native state.On the experimental side, residue-specific measurements of the folding process (see,e.g., [14]) provide detailed experimental data to validate theoretical predictions.Our work takes advantage of these developments. To compute the folding rate andΦ-values of a protein, we first estimate the transition state ensemble (TSE), which isa set of high-energy protein conformations that limits the folding rate. We use the re-cently introduced Stochastic Roadmap Simulation (SRS) method [3] on a folding en-ergy landscape proposed in [12]. SRS samples the protein conformational space andbuilds a directed graph, called the stochastic conformational roadmap. The nodes ofthe roadmap represent sampled protein conformations, and the edges represent transi-tions between the conformations. The roadmap compactly encodes a huge number offolding pathways and captures the stochastic nature of the folding process. Using theroadmap, we can efficiently compute the folding probability (Pfold) [8] for each sam-pled conformation in the roadmap and decide which conformations belong to the TSE.Finally, we estimate folding rates and Φ-values using the set of conformations in theTSE.We tested our method on 16 proteins with sizes ranging from 56 to 128 residuesand validated the results against experimental data. The results show that our methodpredicts folding rates with accuracy better than an existing method based on dynamicprogramming (DP) [12]. In the following, this existing method will be called the DPmethod, for lack of a better name. More importantly, our method provides a much morediscriminating estimate of the TSE: our estimate of the TSE contains less than 10%of all sampled conformations, while the estimate by the DP method contains 85–90%.The more accurate estimate better reveals the composition of the TSE and makes ourmethod more suitable for studying the mechanisms of protein folding. For Φ-valueprediction, the accuracy of our method varies among the proteins tested. The resultsare comparable to those obtained from the DP method, but both methods need to beimproved in accuracy to be useful in practice.From a methodology point of view, this is the first time that results based on Pfoldvalues computed by SRS were compared against substantial amount of experimentaldata. Earlier work on SRS compared it with Monte Carlo simulation and showed thatSRS is faster by several orders of magnitude [3]. The comparison with experimentaldata serves as a test of the methodology, and the success further validates
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