ieeetransactionsonnanobioscience,vol.14,no.5,july2015 …

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IEEE TRANSACTIONS ON NANOBIOSCIENCE, VOL. 14, NO. 5, JULY 2015 1

Characterizing Energy Landscapes of Peptides Usinga Combination of Stochastic Algorithms

Didier Devaurs, Kevin Molloy, Marc Vaisset, Amarda Shehu, Thierry Siméon, and Juan Cortés*

Abstract—Obtaining accurate representations of energy land-scapes of biomolecules such as proteins and peptides is central tothe study of their physicochemical properties and biological func-tions. Peptides are particularly interesting, as they exploit struc-tural flexibility to modulate their biological function. Despite theirsmall size, peptide modeling remains challenging due to the com-plexity of the energy landscape of such highly-flexible dynamicsystems. Currently, only stochastic sampling-based methods canefficiently explore the conformational space of a peptide. In thispaper, we suggest to combine two such methods to obtain a fullcharacterization of energy landscapes of small yet flexible peptides.First, we propose a simplified version of the classical Basin Hop-ping algorithm to reveal low-energy regions in the landscape, andthus to identify the corresponding meta-stable structural states ofa peptide. Then, we present several variants of a robotics-inspiredalgorithm, the Transition-based Rapidly-exploring Random Tree,to quickly determine transition path ensembles, as well as transi-tion probabilities between meta-stable states. We demonstrate thiscombined approach on met-enkephalin.Index Terms—Energy landscape, peptides, stochastic algo-

rithms.

I. INTRODUCTION

G LOBAL thermodynamic and kinetic properties ofmolecules can be extracted from an analysis of their

conformational energy landscapes [1]. In particular, obtainingan accurate representation of a molecule's energy landscape isa significant first step to conducting detailed structure-functionstudies for biomolecules of central importance in the cell, suchas proteins and peptides [2].In this work, we focus on small peptides. Despite their modest

size, they represent in many ways a more challenging settingthan larger proteins. Peptides exhibit high structural flexibility,

Manuscript received March 28, 2015; revised nulldate; accepted March31, 2015. This work has been partially supported by the French Na-tional Research Agency (ANR) under project ProtiCAD (project numberANR-12-MONU-0015). Asterisk indicates corresponding author.D. Devaurs was with the CNRS, LAAS, F-31400 Toulouse, France and also

with the University de Toulouse, LAAS, F-31400 Toulouse, France. He is nowwith the Department of Computer Science, Rice University, Houston, TX 77005USA.K. Molloy, M. Vaisset, and T. Siméon are with the CNRS, LAAS, F-31400

Toulouse, France, and also with University de Toulouse, LAAS, F-31400Toulouse, France.A. Shehu is with the Department of Computer Science, George Mason Uni-

versity, Fairfax, VA 22030 USA.*J. Cortés is with the CNRS, LAAS, F-31400 Toulouse, France, and also with

University de Toulouse, LAAS, F-31400 Toulouse, France.Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TNB.2015.2424597

which enables them to recognize different molecular partners inthe cell, and thus to modulate their biological function [3]. Con-trary to proteins, that are often characterized by a unique nativestate and a funnel-shaped energy landscape, peptides are char-acterized by several meta-stable structural states; their energylandscape may contain a multitude of competitive low-energybasins.The existence of multiple localminima in amolecule's energy

landscape makes it particularly challenging to map this land-scape and reconstruct all the functionally-important regions init. Experimental methods, such as X-ray crystallography or nu-clear magnetic resonance (NMR), cannot reveal such maps, asthey can uncover only few structures at best [4]. It is thereforethe task of computational techniques to obtain detailed repre-sentations of energy landscapes.Currently, only sample-based representations of the energy

landscape can be afforded. Even for small peptides, the spaceof possible conformations is vast, and the effective degrees offreedom needed to represent a conformation are numerous. Thehigh dimensionality of the space is accompanied by a complex(non-linear, non-convex) expression for the conformationalenergy, which is the result of competing local and non-localinter-atomic interactions. Probing this landscape is thereforevery computationally costly. Currently, only stochastic opti-mization techniques provide the right balance between accuracyand computational efficiency [1], [2].Obtaining a representation of an energy landscape can be

divided into two sub-problems: 1) determining meta-stablestructural states (i.e., local energy minima); 2) computing tran-sition paths between the identified states. Both can be addressedby achieving an effective sampling of the conformational space.In this paper, we propose to combine two sampling-based tech-niques to obtain a full characterization of energy landscapes ofsmall yet highly-flexible peptides. First, we present a simplevariant of the Basin Hopping algorithm [5] to sample localminima in a peptide's energy landscape. Local minima are thenorganized via density-based clustering to reveal meta-stablestructural states. Second, we present several variants of arobotics-inspired algorithm, the Transition-based Rapidly-ex-ploring Random Tree (T-RRT) [6], to map out the connectivitybetween these states, thus completing the reconstruction of thepeptide's energy landscape. In particular, we propose a newvariant of T-RRT allowing the computation of transition pathensembles and transition probabilities between meta-stablestructural states. In a preliminary version of this paper [7], wepresented proof-of-concept results on a minimalist peptide: theterminally-blocked alanine. Here we present results on a largerand much more challenging system: met-enkephalin.

1536-1241 © 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

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II. METHODS

The work presented in this paper is motivated by recentstudies showing that robotics-inspired sampling-based al-gorithms provide a good basis for efficient conformationalexploration in computational structural biology [8], [9]. TheTransition-based RRT (T-RRT) algorithm is an example ofsuch algorithms [6], [10]. It is based on the Rapidly-exploringRandom Tree (RRT) [11], a popular path planning algorithmthat can tackle complex problems in high-dimensional spaces.RRT has been successfully used in various disciplines, such asrobotics, aerospace, computer animation, and computationalstructural biology. T-RRT is an extension of RRT involving aprobabilistic transition test based on the Metropolis criterion.Like Metropolis Monte Carlo (MC) methods [12], it appliessmall moves to the molecular system; but, instead of gener-ating a single path over the search space, it constructs a tree,providing a more efficient exploration. Moreover, the treeconstruction is intrinsically biased toward unexplored regionsof the space, and favors expansions in low-energy areas.This work is partly based on an existing variant of T-RRT,

called Multi-T-RRT [13], and on a new extension to it, calledMulti-T-RRT with Cycles. These algorithms are combined withan in-house variant of the Basin Hopping algorithm involvinga simplified minimization process. Together they are used toobtain a complete representation of the energy landscape ofhighly-flexible peptides. Details of these methods are presentedin the next sections.

A. Basin HoppingThe Basin Hopping (BH) algorithm is a popular method for

sampling local minima of an energy landscape. It was origi-nally introduced to obtain the Lennard-Jones minima of smallatom clusters [5]. Recently, BH has gained new attention to pre-dict protein structure [14], and to find intermediate structuresof chemical reactions [15]. The method consists of repeatedlyapplying a structural perturbation followed by an energy min-imization, which yields a trajectory of minima. The result is a(discrete) coarse-grained representation of the energy landscapethat can be seen as a collection of interpenetrating staircases.Our implementation of BH (presented in Algorithm 1) differs

from the classical one in that it does not involve local, gradient-based minimizations, but relies on simple Monte-Carlo-based(MC-based) minimizations. The algorithm follows a randomrestart procedure performing several rounds, each one startingfrom a conformation randomly sampled in the search space.Every round builds a trajectory of minima by performing a suc-cession of structural perturbations followed by MC-based min-imizations. Every MC-based minimization starts from a con-formation obtained by performing a large-amplitude perturba-tion of the minimum reached at the previous step, or from therandom sample, in the first step. An MC-based minimizationis an iterative succession of small-amplitude perturbations. Ateach iteration, the perturbed conformation replaces the previousone if the Metropolis criterion is satisfied. More precisely, adownhill move in the energy landscape is always accepted. Anuphill move is accepted or rejected based on the probability

(where is the Boltzmann constant), whichdecreases exponentially with the energy variation for

a given temperature , where and are the energies ofthe old and the perturbed state, respectively. Every MC-basedminimization produces a low-energy conformation that we calla minimum in a minor abuse of language. It is compared to theminimum obtained in the previous iteration, and accepted or re-jected also based on the result of a Metropolis-like transitiontest. Different temperatures can be used for the transition test in-side the local energy minimization procedure and for the one ateach iteration of BH (usually, the former being much lower thanthe latter). Every round produces what we call a milestone: theminimum (along the trajectory) having the lowest energy. Notethat, in order to speedup computation, a round can be stoppedbased on a consecutive number of rejections, similarly to theMC-based minimization procedure.All the milestones (or theminima) produced by BH have to be

grouped to provide a comprehensible list of meta-stable struc-tural states. This clustering can be done in several ways. In thiswork, we have applied a density-based clustering technique thathas been shown to provide good results for small peptides [16].

B. Multi-T-RRT With Cycle-AdditionThe Rapidly-exploring Random Tree (RRT) algorithm [11]

is a well-known path planning method in robotics. It can dealwith complex problems by performing an efficient exploration,even in high-dimensional search spaces. Starting from an ini-tial conformation , RRT iteratively constructs a tree thattends to rapidly expand over the conformational space . Thenodes and edges of correspond to states (i.e., molecular con-formations) and small-amplitude moves between states, respec-tively. At each iteration of the tree construction, a conforma-tion is randomly sampled in . Then, an extension toward

is attempted, starting from its nearest neighbor , in. This means performing a linear interpolation between

and , at a distance equal to the extension step-size , from. If the extension succeeds, a new conformation is

added to and an edge is built between and . Thecriteria on when to stop the exploration can be reaching a giventarget conformation , a given number of nodes in the tree,a given number of iterations, or a given running time.

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DEVAURS et al.: CHARACTERIZING ENERGY LANDSCAPES OF PEPTIDES USING A COMBINATION OF STOCHASTIC ALGORITHMS 3

The Transition-based RRT (T-RRT) algorithm is a variant ofRRT developed to explore a conformational space while takingthe conformational energy into account [6], [10]. T-RRT ex-tends RRT by integrating a stochastic transition test used to eval-uate the local move from to based on their respectiveenergies and , aiming to favor the exploration of low-en-ergy regions of the space. This transition test is based on theMetropolis criterion, as explained for BH in the previous sub-section.The level of selectivity of this transition test is controlled

by the temperature : low temperatures limit the expansion togentle slopes of the energy landscape, and high temperatures en-able it to climb steep slopes. The basic MC method, as well asBH, consider constant temperature. In contrast, is a self-adap-tive parameter of the T-RRT algorithm. After each accepteduphill move, is decreased to avoid over-exploring high-en-ergy regions: it is divided by , where

is the energy difference between the highest-energy and the lowest-energy conformations in the graph builtso far. After each rejected uphill move, is increased to facili-tate the exploration and avoid being trapped in a local energyminimum: it is multiplied by , where isthe temperature increase rate. The pseudo-code of the T-RRTtransitionTest is presented in Algorithm 2.The Multi-T-RRT algorithm is a multiple-tree variant of

T-RRT [13]. Instead of building a single tree rooted at someinitial conformation, the idea is to build trees rooted atgiven conformations , . The pseudo-code of theMulti-T-RRT is presented in Algorithm 3. At each iteration,a tree is chosen for expansion in a round-robin fashion.Then, an extension is attempted toward a randomly sampledconformation , starting from its nearest neighbor , in. If the extension succeeds, the new conformation is

added to , and connected to . Then, we search for theconformation , which is the closest to within all treesother than . If the distance between and is lessthan or equal to the extension step-size , is linked to andmerged with , the tree to which belongs. In that case,the number of trees is decreased by 1. The space explorationcontinues until all trees are merged into a single one or anotherstopping criterion (number of nodes, number of expansions,running time) is met.A drawback of theMulti-T-RRT is that it returns a single path

connecting each pair of initial conformations, possibly not beingthe most likely transition path. To address this issue, we proposea new extension to T-RRT, based on a cycle-addition procedure.Starting from the tree produced by the Multi-T-RRT, the idea

is to allow the space exploration to continue, and to add edgesleading to the creation of cycles. This enables us to construct agraph from which several paths can be extracted between twogiven conformations. The pseudo-code of this T-RRT with Cy-cles is shown in Algorithm 4. It differs from T-RRT only in that,after every successful extension, an edge is added betweenand each conformation in the graph that is not already con-nected to , and whose distance to is less than or equalto the extension step-size . The stopping criteria can involvethe number of iterations or the running time, as well as a con-vergence test.As several paths may exist in the graph between two given

conformations, a quality criterion is required to compare paths.This is achieved by associating weights with the edges of ,based on the notion of mechanical work [10]. More precisely,the weight of the directed edge connecting and is equalto , i.e., to the positive energy variationbetween and . This constitutes the amount of energy thathas to be added to the molecule for the transition from to

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to happen. Note that using the mechanical work as a qualitycriterion requires to create two directed edges between and, instead of creating a single undirected edge. Finally, given

two conformations in , the best (directed) path linking themis defined as the one minimizing the mechanical work, and isobtained using Dijkstra's algorithm.In this work, we use the mechanical work as a path-quality

criterion because it has been shown that T-RRT tends to gen-erate paths having a low mechanical work [10]. Furthermore, inmany situations, the mechanical work can assess the quality of apath better than a simple criterion such as the integral of the costalong the path [10]. Other criteria could be considered, such asthe minimum resistance (linked to the MaxFlux algorithm [17],[18]) or the maximum flux [19]. However, finding which crite-rion is the most relevant is out of the scope of this paper.

III. RESULTS AND DISCUSSION

In a preliminary version of this paper [7], we presented re-sults provided by the combination of BH and Multi-T-RRT tocharacterize the conformational energy landscape of the termi-nally-blocked alanine. The energy landscape of this minimalistpeptide, which involves several local minima connected bymul-tiple pathways, can be projected on a single and meaningful2-dimensional map. Thus, it is a very good benchmark systemto illustrate the performance of computational methods in phys-ical chemistry. Results on the terminally-blocked alanine are notpresented again here because of space limitations. For details onthis illustrative test system, the interested reader is referred to[7].Here, we present results on met-enkephalin, a pentapeptide

with sequence: Tyr-Gly-Gly-Phe-Met. Despite its relativelysmall size, studying met-enkephalin is challenging from astructural point-of-view because of its great conformationalvariability. Met-enkephalin is still the focus of a large amountof work (e.g., [20], [21]). This endogenous opioid peptide is ofsignificant interest due to its important roles in physiologicalprocesses, mostly related to pain and depression [22].

A. Met-Enkephalin Model and Settings

In this work, we consider an internal-coordinate represen-tation of the molecule, assuming constant bond lengths andbond angles. Therefore, the conformational parameters are the

dihedral angles of the backbone of each amino-acidresidue plus the dihedral angles of the side-chains, whichadds up to 23 degrees of freedom. We assume that all theangles but the angles can take values in the full angularrange . As the peptide bond torsions are knownto undergo only small variations, the angles are allowed tovary only up to 10 from the planar trans conformation.Measuring the distance between two conformations is re-

quired to select neighbor nodes in RRT-based algorithms andfor clustering. Here we use a weighted -norm in the spaceof the and backbone dihedral angles (excluding the firstand the last angle of the molecular chain), because they are themost relevant variables describing conformations of peptides.The weight of each dihedral angle within the norm is chosen as

the Euclidian distance between its rotation axis and the farthestatom to it. In other words, greater weights are assigned to angleswhose rotation results in larger conformational variation.To illustrate the performance of the algorithms, we project

their output on Ramachandran maps (i.e., maps) of thethree middle residues (Gly2, Gly3, Phe4). Other representa-tions are possible, such as projections on the first componentsprovided by dimensionality reduction techniques (e.g., PCA,Isomap), or on pairs of structural descriptors (e.g., RMSD toa reference structure vs. end-to-end distance). However, webelieve that a small set of maps is a clearer representationto identify conformational regions of a small peptide. Each 2-Dmap was generated using an exhaustive search procedure, byvarying both dihedral angles with a 10 step-size and findingthe lowest-energy conformation corresponding to eachpair using an MC-based minimization procedure, such as theone involved in the BH algorithm. Note that such a computa-tionally expensive procedure is only used here to visualize andvalidate our results.To compute conformational energy values, we use an

in-house implementation of the AMBER parm96 force-field[23] with an implicit representation of the solvent using theGeneralized Born approximation. All programs have been runon a singe core of an Intel© Xeon© CPU E5-2650 at 2.00 GHz.

B. Local Energy Minima of Met-Enkephalin

We have used the Basing Hopping (BH) algorithm, as de-scribed in Section II-A, to sample low-energy conformations ofmet-enkephalin. The parameters of BH were set as follows: 500rounds were performed, with each round executing a maximumof 1000 MC-based minimizations. The number of consecutiverejections to estimate the convergence of each local minimiza-tion and of each BH round was set to 100. The large-amplitudeperturbations in the inner loop of the BH algorithm affected onlythe and dihedral angles, with a maximum step-size of 2 ra-dians. The small-amplitude perturbations performed during anMC-based minimization affected all variables, with a maximumstep-size of 0.2 radians. Only one angle was perturbed at eachiteration. The temperatures for the Metropolis-like transitiontests at each iteration of BH and in the MC-based minimizationmethod were set to 1000 K and 0.01 K, respectively.The 232 440 minima and 500 milestones produced by the BH

algorithm are projected in the three Ramachandran maps shownin Fig. 1.1 The algorithm required about 24 CPU hours. Notehowever that this computing time can be significantly reducedby using a more efficient implementation of the energy func-tion, and by means of parallel computation (parallelization ofthemain loop in Algorithm 1 is straightforward). Computationalefficiency issues are further discussed in Section IV. Note alsothat the use of a more sophisticated, gradient-based local min-imization method would help speeding up the performance ofBH.The plots in Fig. 1 show that the minima provide a good cov-

erage of all the low-energy regions, the milestones being more

1Color figures are available in the online version of this manuscript.

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Fig. 1. Sampling the energy landscape of met-enkephalin using the Basin Hopping algorithm. The black markers represent the sampled minima and the whitemarkers represent the milestones. The four large violet symbols indicate representative states identified by clustering (see Table I).

TABLE IPROBABILITIES OF DIRECT TRANSITIONS AND THE MINIMAL-WORK PATHS BETWEEN THE FOUR IDENTIFIED MINIMA USING THE MULTI-T-RRT METHOD WITH

CYCLES

Fig. 2. Projection of 100 path between a folded and unfolded states of met-enkephalin computed with the Multi-T-RRT algorithm.

concentrated in a few areas. The density-based clustering algo-rithm identified 8 clusters, some of which are not clearly sepa-rated. The clusters containing the lowest-energy minima corre-spond to folded conformations ofmet-enkephalin, whereas clus-ters of higher-energy minima contain stretched conformationsof the peptide. Intermediate conformations are grouped into sev-eral medium-energy clusters. These results are consistent withthose presented in other related work on met-enkephalin.Here we take a closer look at results presented in some recent

work providing a meaningful representation of the energy

landscape of met-enkephalin through a Markov state model(MSM) built fromMolecular Dynamics simulations (MD) [21].Even though this other approach applies a different energymodel (GROMOS96 force field), a different conformationalsampling method (MD) and a different clustering algorithm(hybrid kcenter-kmedoid), results show striking similarities toours. Results in [21] suggest that four out of the eight backbonedihedral angles involved in our metric are sufficient to charac-terize the conformational space of met-enkephalin. These fourangles are the angles of Gly2, Gly3, and Phe4, and the

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angle of Gly3: . The authors define a 4-signcode to characterize conformations with respect to the values(positive or negative) of these four angles. The two regionsmost frequently sampled by MD correspond to conformationsof type and . As can be seen in Fig. 1,a large number of the BH milestones are found in these tworegions. Interestingly, these two classes are correctly identifiedamong the clusters of BH milestones, and they contain thelowest-energy conformations. The class , which isalso significantly sampled by MD, is well represented in ourresults. Only a few milestones appear in the region ,although results in [21] suggest a relatively more importantpopulation of this type of conformation. This is probably dueto differences in the force-fields. The highest-energy cluster ofBH milestones contains stretched conformations of both types

and merged together. We refer to thiscluster as .To illustrate the application of the Multi-T-RRT, presented in

the next subsection, we have selected the lowest energy confor-mation of each class , , , and

. The first two correspond to folded conformations, thelast one is a stretched/unfolded conformation and the third oneis an intermediate conformation. Other intermediate states iden-tified by clustering are not considered in the following analysis.Table I shows a structural representation of these four confor-mations (only the backbone is represented for clarity), togetherwith the corresponding 4-sign code and the potential energy. Ageometric symbol is associated to each conformation and usedin the tables and Ramachandran maps to identify these confor-mations.

C. Transition Paths of Met-Enkephalin

We have applied the Multi-T-RRT algorithm to quicklydiscover many transition paths between the aforementionedmeta-stable states of met-enkephalin, presented in Table??[PLEASE CLARIFY]. We have used the followingparameter settings: The extension step-size is set to 2.0, whichmeans that, in the worst case, the maximum displacement ofan atom between two conformations connected by an edge isof 2 . Note however that this distance is usually significantlysmaller. At the beginning of the exploration, we impose theprobability of accepting an energy increment of 0.1 kcal/molto be approximately 50% by using the Boltzmann constant

together with an initial temperatureto 70 K. The temperature increase rate is set to 0.1.Starting from four roots at the given states, Multi-T-RRT re-

turns a single tree in approximately 2 minutes. Paths connectingthe minima are extracted for analysis. To study the diversityof the transition paths of met-enkephalin, we run Multi-T-RRT100 times. As an illustrative example, paths between the folded

and unfolded states are visualized on thesame Ramachandran maps in Fig. 2. The average cost for thetransition from to is 49.8 (MW) and in thereverse direction is 44.6 (MW), with a standard deviation acrossall the runs of 14.5. We observe that all of these paths avoid thehigh-energy regions in all projections. The paths are extremelydiverse, although the standard deviation of their cost is not very

high. This indicates that the energy landscape between theseminima is relatively flat. One can see in the figure that a fewpaths go across saddle regions corresponding to values ofclose to zero. This shows that Multi-T-RRT is able to effectivelyexplore vast portions of the conformational space.

D. Capturing Distinct Transitions of Met-Enkephalin

We have employed Multi-T-RRT with Cycles to computetransition probabilities between all pairs of energy minima formet-enkephalin. We have executed the method 100 times witha maximal running time of 10 minutes as stopping condition.Fig. 3 shows the projection of one of the resulting graphs, whichcovers all the low-energy regions on the Ramachandran mapsand constructs connections through transition regions. To es-timate transition probabilities amongst the 100 executions, wecount how many runs yield a graph from which a direct transi-tion path can be extracted between a given pair of minima. Moreprecisely, we consider that a run produces a direct transition pathbetween two minima if the best path (with respect to the cost interms of mechanical work) in the graph between them does notgo through another minimum. Table I shows the probabilitiesof direct transitions between minima and the costs (average andstandard deviation) associated to them. We can observe that di-rect transitions between all pairs of states are highly probable inmost cases. Since the mechanical work is an additive function,we can also compute the cost of indirect paths going through asequence of minima from the values in Table I. Table II presentsthe costs of different pathways from one of the two folded states

and to the unfolded state . In theaforementioned related work on met-enkephalin [21], differentcosts are also associated to alternative transition pathways fromfolded to unfolded states based on the Markov state model gen-erated form MD. Although our cost rankings do not match per-fectly to those in [21], most likely because of differences in thelandscapes inferred by the different force-fields, the overall con-clusion is the same: direct transitions from folded states to un-folded states are more probable than transitions going throughintermediate states.

IV. CONCLUSION

In this paper, we have presented a methodology to exploreand characterize the energy landscape of flexible peptides. Thismethodology combines variants of two stochastic, sampling-based algorithms: Basin Hopping (BH) and Transition-basedRapidly-exploring Random Tree (T-RRT). A simplified versionof the BH algorithm, where local, gradient-based energy mini-mization steps are replaced by simple Monte-Carlo-based min-imization steps, achieves a relevant exploration of the energylandscape yielding numerous samples around energy minima.This leads to a quick determination of meta-stable structuralstates, which can be used as a starting point for the analysis ofconformational transitions. The multiple-tree version of T-RRTis very fast at generating transition paths between a set of states.Running this algorithm several times produces a good descrip-tion of the transition path ensembles. Finally, Multi-T-RRTwithCycles yields diverse transition paths in a single run of the al-gorithm.

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Fig. 3. Projection of a graph obtained by a single run of Multi-T-RRT with Cycles starting from four meta-stable states of met-enkephalin.

TABLE IICOSTS OF TRANSITION PATHWAYS FROM FOLDED TO UNFOLDED STATES

Results on met-enkephalin presented in this paper, andresults on the terminally-blocked alanine described in a prelim-inary version [7], show that the combination of BH and T-RRTquickly produces a meaningful representation of the energylandscape of small yet highly-flexible peptides. The identifiedmeta-stable states and the transition pathways between themare comparable to those obtained with other, more expensivecomputational methods. Our results also illustrate the fact thatstochastic algorithms can compete with MD-based approachesin providing accurate and insightful findings about flexiblebiomolecules. Nevertheless, this is an early work, and manyaspects still have to be improved and further investigatedin terms of theory and implementation. Directions of futurework include exploiting the graph produced by a single run ofthe Multi-T-RRT with Cycles to describe transition path andtransition state ensembles. This will allow us to make better useof computational resources, as opposed to aggregating pathsextracted from several runs of the Multi-T-RRT. In addition,Markov-based transition-step analysis can be conducted on thegraph produced by one or more runs of T-RRT. This analysisallows one to estimate the stability of each computed state, andprovides a rigorous basis for the designation of a state as stableor semi-stable.We also aim to improve the implementation in order to effi-

ciently deal with larger peptides and proteins. We have initiatedsome experiments with larger systems to identify the efficiencyof our algorithms. One of our current test systems is chignolin,an artificial “mini-protein” composed of 10 residues. Chignolinpresents an interesting energy landscape that has been inves-tigated with other methods [24]. Our first tests with chignolinshow that our methods are still applicable in practice to larger

systems, although computing time increases significantly.When running the Multi-T-RRT using four conformations asinput (one folded, one unfolded and two intermediate statesidentified by BH), computing a graph connecting them takesabout 100 minutes on average. To reduce the computationtime, it will be necessary to carefully implement the underlyingmethods in the algorithms, in particular the nearest neighborsearch and energy computation. We are also working on theparallelization of Multi-T-RRT, building on our previous expe-rience with the basic RRT algorithm [25].

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[13] D. Devaurs, M. Vaisset, T. Siméon, and J. Cortés, “A multi-tree ap-proach to compute transition paths on energy landscapes,” in Proc.AIRMCB Workshop, 2013.

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[16] M. Kim, S.-H. Choi, J. Kim, K. Choi, J.-M. Shin, S.-K. Kang, Y.-J.Choi, and D. Jung, “Density-based clustering of small peptide confor-mations sampled from a molecular dynamics simulation,” J. Chem. Inf.Model., vol. 49, no. 11, pp. 2528–2536, 2009.

[17] S. Huo and J. Straub, “The MaxFlux algorithm for calculating vari-ationally optimized reaction paths for conformational transitions inmany body systems at finite temperature,” J. Chem. Phys., vol. 107,no. 13, pp. 5000–5006, 1997.

[18] S. Huo and J. Straub, “Direct computation of long time processes inpeptides and proteins: Reaction path study of the coil-to-helix transitionin polyalanine,” Proteins: Struct., Funct., Genetics, vol. 36, no. 2, pp.249–261, 1999.

[19] R. Zhao, J. Shen, and R. Skeel, “Maximum flux transition paths ofconformational change,” J. Chem. Theory Comput., vol. 6, no. 8, pp.2411–2423, 2010.

[20] F. Sicard and P. Senet, “Reconstructing the free-energy landscape ofmet-enkephalin using dihedral principal component analysis and well-tempered metadynamics,” J. Chem. Phys., vol. 138, no. 23, 2013.

[21] R. Banerjee and R. I. Cukier, “Transition paths of met-enkephalin frommarkov state modeling of a molecular dynamics trajectory,” J. Phys.Chem. B, vol. 118, no. 11, pp. 2883–2895, 2014.

[22] D. Sauriyal, A. Jaggi, and N. Singh, “Extending pharmacological spec-trum of opioids beyond analgesia: Multifunctional aspects in differentpathophysiological states,” Neuropeptides, vol. 45, pp. 175–188, 2011.

[23] P. Kollman, R. Dixon, W. Cornell, T. Fox, C. Chipot, and A. Pohorille,“The development/application of a “minimalist” organic/biochemicalmolecular mechanic force field using a combination of ab initio calcu-lations and experimental data,” in Computer Simulation of Biomolec-ular Systems, ser. Computer Simulations of Biomolecular Systems, W.van Gunsteren, P. Weiner, and A. Wilkinson, Eds. Dordrecht, TheNetherlands: Springer, 1997, vol. 3, pp. 83–96.

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IEEE TRANSACTIONS ON NANOBIOSCIENCE, VOL. 14, NO. 5, JULY 2015 1

Characterizing Energy Landscapes of Peptides Usinga Combination of Stochastic Algorithms

Didier Devaurs, Kevin Molloy, Marc Vaisset, Amarda Shehu, Thierry Siméon, and Juan Cortés*

Abstract—Obtaining accurate representations of energy land-scapes of biomolecules such as proteins and peptides is central tothe study of their physicochemical properties and biological func-tions. Peptides are particularly interesting, as they exploit struc-tural flexibility to modulate their biological function. Despite theirsmall size, peptide modeling remains challenging due to the com-plexity of the energy landscape of such highly-flexible dynamicsystems. Currently, only stochastic sampling-based methods canefficiently explore the conformational space of a peptide. In thispaper, we suggest to combine two such methods to obtain a fullcharacterization of energy landscapes of small yet flexible peptides.First, we propose a simplified version of the classical Basin Hop-ping algorithm to reveal low-energy regions in the landscape, andthus to identify the corresponding meta-stable structural states ofa peptide. Then, we present several variants of a robotics-inspiredalgorithm, the Transition-based Rapidly-exploring Random Tree,to quickly determine transition path ensembles, as well as transi-tion probabilities between meta-stable states. We demonstrate thiscombined approach on met-enkephalin.Index Terms—Energy landscape, peptides, stochastic algo-

rithms.

I. INTRODUCTION

G LOBAL thermodynamic and kinetic properties ofmolecules can be extracted from an analysis of their

conformational energy landscapes [1]. In particular, obtainingan accurate representation of a molecule's energy landscape isa significant first step to conducting detailed structure-functionstudies for biomolecules of central importance in the cell, suchas proteins and peptides [2].In this work, we focus on small peptides. Despite their modest

size, they represent in many ways a more challenging settingthan larger proteins. Peptides exhibit high structural flexibility,

Manuscript received March 28, 2015; revised nulldate; accepted March31, 2015. This work has been partially supported by the French Na-tional Research Agency (ANR) under project ProtiCAD (project numberANR-12-MONU-0015). Asterisk indicates corresponding author.D. Devaurs was with the CNRS, LAAS, F-31400 Toulouse, France and also

with the University de Toulouse, LAAS, F-31400 Toulouse, France. He is nowwith the Department of Computer Science, Rice University, Houston, TX 77005USA.K. Molloy, M. Vaisset, and T. Siméon are with the CNRS, LAAS, F-31400

Toulouse, France, and also with University de Toulouse, LAAS, F-31400Toulouse, France.A. Shehu is with the Department of Computer Science, George Mason Uni-

versity, Fairfax, VA 22030 USA.*J. Cortés is with the CNRS, LAAS, F-31400 Toulouse, France, and also with

University de Toulouse, LAAS, F-31400 Toulouse, France.Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TNB.2015.2424597

which enables them to recognize different molecular partners inthe cell, and thus to modulate their biological function [3]. Con-trary to proteins, that are often characterized by a unique nativestate and a funnel-shaped energy landscape, peptides are char-acterized by several meta-stable structural states; their energylandscape may contain a multitude of competitive low-energybasins.The existence of multiple localminima in amolecule's energy

landscape makes it particularly challenging to map this land-scape and reconstruct all the functionally-important regions init. Experimental methods, such as X-ray crystallography or nu-clear magnetic resonance (NMR), cannot reveal such maps, asthey can uncover only few structures at best [4]. It is thereforethe task of computational techniques to obtain detailed repre-sentations of energy landscapes.Currently, only sample-based representations of the energy

landscape can be afforded. Even for small peptides, the spaceof possible conformations is vast, and the effective degrees offreedom needed to represent a conformation are numerous. Thehigh dimensionality of the space is accompanied by a complex(non-linear, non-convex) expression for the conformationalenergy, which is the result of competing local and non-localinter-atomic interactions. Probing this landscape is thereforevery computationally costly. Currently, only stochastic opti-mization techniques provide the right balance between accuracyand computational efficiency [1], [2].Obtaining a representation of an energy landscape can be

divided into two sub-problems: 1) determining meta-stablestructural states (i.e., local energy minima); 2) computing tran-sition paths between the identified states. Both can be addressedby achieving an effective sampling of the conformational space.In this paper, we propose to combine two sampling-based tech-niques to obtain a full characterization of energy landscapes ofsmall yet highly-flexible peptides. First, we present a simplevariant of the Basin Hopping algorithm [5] to sample localminima in a peptide's energy landscape. Local minima are thenorganized via density-based clustering to reveal meta-stablestructural states. Second, we present several variants of arobotics-inspired algorithm, the Transition-based Rapidly-ex-ploring Random Tree (T-RRT) [6], to map out the connectivitybetween these states, thus completing the reconstruction of thepeptide's energy landscape. In particular, we propose a newvariant of T-RRT allowing the computation of transition pathensembles and transition probabilities between meta-stablestructural states. In a preliminary version of this paper [7], wepresented proof-of-concept results on a minimalist peptide: theterminally-blocked alanine. Here we present results on a largerand much more challenging system: met-enkephalin.

1536-1241 © 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

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II. METHODS

The work presented in this paper is motivated by recentstudies showing that robotics-inspired sampling-based al-gorithms provide a good basis for efficient conformationalexploration in computational structural biology [8], [9]. TheTransition-based RRT (T-RRT) algorithm is an example ofsuch algorithms [6], [10]. It is based on the Rapidly-exploringRandom Tree (RRT) [11], a popular path planning algorithmthat can tackle complex problems in high-dimensional spaces.RRT has been successfully used in various disciplines, such asrobotics, aerospace, computer animation, and computationalstructural biology. T-RRT is an extension of RRT involving aprobabilistic transition test based on the Metropolis criterion.Like Metropolis Monte Carlo (MC) methods [12], it appliessmall moves to the molecular system; but, instead of gener-ating a single path over the search space, it constructs a tree,providing a more efficient exploration. Moreover, the treeconstruction is intrinsically biased toward unexplored regionsof the space, and favors expansions in low-energy areas.This work is partly based on an existing variant of T-RRT,

called Multi-T-RRT [13], and on a new extension to it, calledMulti-T-RRT with Cycles. These algorithms are combined withan in-house variant of the Basin Hopping algorithm involvinga simplified minimization process. Together they are used toobtain a complete representation of the energy landscape ofhighly-flexible peptides. Details of these methods are presentedin the next sections.

A. Basin HoppingThe Basin Hopping (BH) algorithm is a popular method for

sampling local minima of an energy landscape. It was origi-nally introduced to obtain the Lennard-Jones minima of smallatom clusters [5]. Recently, BH has gained new attention to pre-dict protein structure [14], and to find intermediate structuresof chemical reactions [15]. The method consists of repeatedlyapplying a structural perturbation followed by an energy min-imization, which yields a trajectory of minima. The result is a(discrete) coarse-grained representation of the energy landscapethat can be seen as a collection of interpenetrating staircases.Our implementation of BH (presented in Algorithm 1) differs

from the classical one in that it does not involve local, gradient-based minimizations, but relies on simple Monte-Carlo-based(MC-based) minimizations. The algorithm follows a randomrestart procedure performing several rounds, each one startingfrom a conformation randomly sampled in the search space.Every round builds a trajectory of minima by performing a suc-cession of structural perturbations followed by MC-based min-imizations. Every MC-based minimization starts from a con-formation obtained by performing a large-amplitude perturba-tion of the minimum reached at the previous step, or from therandom sample, in the first step. An MC-based minimizationis an iterative succession of small-amplitude perturbations. Ateach iteration, the perturbed conformation replaces the previousone if the Metropolis criterion is satisfied. More precisely, adownhill move in the energy landscape is always accepted. Anuphill move is accepted or rejected based on the probability

(where is the Boltzmann constant), whichdecreases exponentially with the energy variation for

a given temperature , where and are the energies ofthe old and the perturbed state, respectively. Every MC-basedminimization produces a low-energy conformation that we calla minimum in a minor abuse of language. It is compared to theminimum obtained in the previous iteration, and accepted or re-jected also based on the result of a Metropolis-like transitiontest. Different temperatures can be used for the transition test in-side the local energy minimization procedure and for the one ateach iteration of BH (usually, the former being much lower thanthe latter). Every round produces what we call a milestone: theminimum (along the trajectory) having the lowest energy. Notethat, in order to speedup computation, a round can be stoppedbased on a consecutive number of rejections, similarly to theMC-based minimization procedure.All the milestones (or theminima) produced by BH have to be

grouped to provide a comprehensible list of meta-stable struc-tural states. This clustering can be done in several ways. In thiswork, we have applied a density-based clustering technique thathas been shown to provide good results for small peptides [16].

B. Multi-T-RRT With Cycle-AdditionThe Rapidly-exploring Random Tree (RRT) algorithm [11]

is a well-known path planning method in robotics. It can dealwith complex problems by performing an efficient exploration,even in high-dimensional search spaces. Starting from an ini-tial conformation , RRT iteratively constructs a tree thattends to rapidly expand over the conformational space . Thenodes and edges of correspond to states (i.e., molecular con-formations) and small-amplitude moves between states, respec-tively. At each iteration of the tree construction, a conforma-tion is randomly sampled in . Then, an extension toward

is attempted, starting from its nearest neighbor , in. This means performing a linear interpolation between

and , at a distance equal to the extension step-size , from. If the extension succeeds, a new conformation is

added to and an edge is built between and . Thecriteria on when to stop the exploration can be reaching a giventarget conformation , a given number of nodes in the tree,a given number of iterations, or a given running time.

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The Transition-based RRT (T-RRT) algorithm is a variant ofRRT developed to explore a conformational space while takingthe conformational energy into account [6], [10]. T-RRT ex-tends RRT by integrating a stochastic transition test used to eval-uate the local move from to based on their respectiveenergies and , aiming to favor the exploration of low-en-ergy regions of the space. This transition test is based on theMetropolis criterion, as explained for BH in the previous sub-section.The level of selectivity of this transition test is controlled

by the temperature : low temperatures limit the expansion togentle slopes of the energy landscape, and high temperatures en-able it to climb steep slopes. The basic MC method, as well asBH, consider constant temperature. In contrast, is a self-adap-tive parameter of the T-RRT algorithm. After each accepteduphill move, is decreased to avoid over-exploring high-en-ergy regions: it is divided by , where

is the energy difference between the highest-energy and the lowest-energy conformations in the graph builtso far. After each rejected uphill move, is increased to facili-tate the exploration and avoid being trapped in a local energyminimum: it is multiplied by , where isthe temperature increase rate. The pseudo-code of the T-RRTtransitionTest is presented in Algorithm 2.The Multi-T-RRT algorithm is a multiple-tree variant of

T-RRT [13]. Instead of building a single tree rooted at someinitial conformation, the idea is to build trees rooted atgiven conformations , . The pseudo-code of theMulti-T-RRT is presented in Algorithm 3. At each iteration,a tree is chosen for expansion in a round-robin fashion.Then, an extension is attempted toward a randomly sampledconformation , starting from its nearest neighbor , in. If the extension succeeds, the new conformation is

added to , and connected to . Then, we search for theconformation , which is the closest to within all treesother than . If the distance between and is lessthan or equal to the extension step-size , is linked to andmerged with , the tree to which belongs. In that case,the number of trees is decreased by 1. The space explorationcontinues until all trees are merged into a single one or anotherstopping criterion (number of nodes, number of expansions,running time) is met.A drawback of theMulti-T-RRT is that it returns a single path

connecting each pair of initial conformations, possibly not beingthe most likely transition path. To address this issue, we proposea new extension to T-RRT, based on a cycle-addition procedure.Starting from the tree produced by the Multi-T-RRT, the idea

is to allow the space exploration to continue, and to add edgesleading to the creation of cycles. This enables us to construct agraph from which several paths can be extracted between twogiven conformations. The pseudo-code of this T-RRT with Cy-cles is shown in Algorithm 4. It differs from T-RRT only in that,after every successful extension, an edge is added betweenand each conformation in the graph that is not already con-nected to , and whose distance to is less than or equalto the extension step-size . The stopping criteria can involvethe number of iterations or the running time, as well as a con-vergence test.As several paths may exist in the graph between two given

conformations, a quality criterion is required to compare paths.This is achieved by associating weights with the edges of ,based on the notion of mechanical work [10]. More precisely,the weight of the directed edge connecting and is equalto , i.e., to the positive energy variationbetween and . This constitutes the amount of energy thathas to be added to the molecule for the transition from to

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to happen. Note that using the mechanical work as a qualitycriterion requires to create two directed edges between and, instead of creating a single undirected edge. Finally, given

two conformations in , the best (directed) path linking themis defined as the one minimizing the mechanical work, and isobtained using Dijkstra's algorithm.In this work, we use the mechanical work as a path-quality

criterion because it has been shown that T-RRT tends to gen-erate paths having a low mechanical work [10]. Furthermore, inmany situations, the mechanical work can assess the quality of apath better than a simple criterion such as the integral of the costalong the path [10]. Other criteria could be considered, such asthe minimum resistance (linked to the MaxFlux algorithm [17],[18]) or the maximum flux [19]. However, finding which crite-rion is the most relevant is out of the scope of this paper.

III. RESULTS AND DISCUSSION

In a preliminary version of this paper [7], we presented re-sults provided by the combination of BH and Multi-T-RRT tocharacterize the conformational energy landscape of the termi-nally-blocked alanine. The energy landscape of this minimalistpeptide, which involves several local minima connected bymul-tiple pathways, can be projected on a single and meaningful2-dimensional map. Thus, it is a very good benchmark systemto illustrate the performance of computational methods in phys-ical chemistry. Results on the terminally-blocked alanine are notpresented again here because of space limitations. For details onthis illustrative test system, the interested reader is referred to[7].Here, we present results on met-enkephalin, a pentapeptide

with sequence: Tyr-Gly-Gly-Phe-Met. Despite its relativelysmall size, studying met-enkephalin is challenging from astructural point-of-view because of its great conformationalvariability. Met-enkephalin is still the focus of a large amountof work (e.g., [20], [21]). This endogenous opioid peptide is ofsignificant interest due to its important roles in physiologicalprocesses, mostly related to pain and depression [22].

A. Met-Enkephalin Model and Settings

In this work, we consider an internal-coordinate represen-tation of the molecule, assuming constant bond lengths andbond angles. Therefore, the conformational parameters are the

dihedral angles of the backbone of each amino-acidresidue plus the dihedral angles of the side-chains, whichadds up to 23 degrees of freedom. We assume that all theangles but the angles can take values in the full angularrange . As the peptide bond torsions are knownto undergo only small variations, the angles are allowed tovary only up to 10 from the planar trans conformation.Measuring the distance between two conformations is re-

quired to select neighbor nodes in RRT-based algorithms andfor clustering. Here we use a weighted -norm in the spaceof the and backbone dihedral angles (excluding the firstand the last angle of the molecular chain), because they are themost relevant variables describing conformations of peptides.The weight of each dihedral angle within the norm is chosen as

the Euclidian distance between its rotation axis and the farthestatom to it. In other words, greater weights are assigned to angleswhose rotation results in larger conformational variation.To illustrate the performance of the algorithms, we project

their output on Ramachandran maps (i.e., maps) of thethree middle residues (Gly2, Gly3, Phe4). Other representa-tions are possible, such as projections on the first componentsprovided by dimensionality reduction techniques (e.g., PCA,Isomap), or on pairs of structural descriptors (e.g., RMSD toa reference structure vs. end-to-end distance). However, webelieve that a small set of maps is a clearer representationto identify conformational regions of a small peptide. Each 2-Dmap was generated using an exhaustive search procedure, byvarying both dihedral angles with a 10 step-size and findingthe lowest-energy conformation corresponding to eachpair using an MC-based minimization procedure, such as theone involved in the BH algorithm. Note that such a computa-tionally expensive procedure is only used here to visualize andvalidate our results.To compute conformational energy values, we use an

in-house implementation of the AMBER parm96 force-field[23] with an implicit representation of the solvent using theGeneralized Born approximation. All programs have been runon a singe core of an Intel© Xeon© CPU E5-2650 at 2.00 GHz.

B. Local Energy Minima of Met-Enkephalin

We have used the Basing Hopping (BH) algorithm, as de-scribed in Section II-A, to sample low-energy conformations ofmet-enkephalin. The parameters of BH were set as follows: 500rounds were performed, with each round executing a maximumof 1000 MC-based minimizations. The number of consecutiverejections to estimate the convergence of each local minimiza-tion and of each BH round was set to 100. The large-amplitudeperturbations in the inner loop of the BH algorithm affected onlythe and dihedral angles, with a maximum step-size of 2 ra-dians. The small-amplitude perturbations performed during anMC-based minimization affected all variables, with a maximumstep-size of 0.2 radians. Only one angle was perturbed at eachiteration. The temperatures for the Metropolis-like transitiontests at each iteration of BH and in the MC-based minimizationmethod were set to 1000 K and 0.01 K, respectively.The 232 440 minima and 500 milestones produced by the BH

algorithm are projected in the three Ramachandran maps shownin Fig. 1.1 The algorithm required about 24 CPU hours. Notehowever that this computing time can be significantly reducedby using a more efficient implementation of the energy func-tion, and by means of parallel computation (parallelization ofthemain loop in Algorithm 1 is straightforward). Computationalefficiency issues are further discussed in Section IV. Note alsothat the use of a more sophisticated, gradient-based local min-imization method would help speeding up the performance ofBH.The plots in Fig. 1 show that the minima provide a good cov-

erage of all the low-energy regions, the milestones being more

1Color figures are available in the online version of this manuscript.

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Fig. 1. Sampling the energy landscape of met-enkephalin using the Basin Hopping algorithm. The black markers represent the sampled minima and the whitemarkers represent the milestones. The four large violet symbols indicate representative states identified by clustering (see Table I).

TABLE IPROBABILITIES OF DIRECT TRANSITIONS AND THE MINIMAL-WORK PATHS BETWEEN THE FOUR IDENTIFIED MINIMA USING THE MULTI-T-RRT METHOD WITH

CYCLES

Fig. 2. Projection of 100 path between a folded and unfolded states of met-enkephalin computed with the Multi-T-RRT algorithm.

concentrated in a few areas. The density-based clustering algo-rithm identified 8 clusters, some of which are not clearly sepa-rated. The clusters containing the lowest-energy minima corre-spond to folded conformations ofmet-enkephalin, whereas clus-ters of higher-energy minima contain stretched conformationsof the peptide. Intermediate conformations are grouped into sev-eral medium-energy clusters. These results are consistent withthose presented in other related work on met-enkephalin.Here we take a closer look at results presented in some recent

work providing a meaningful representation of the energy

landscape of met-enkephalin through a Markov state model(MSM) built fromMolecular Dynamics simulations (MD) [21].Even though this other approach applies a different energymodel (GROMOS96 force field), a different conformationalsampling method (MD) and a different clustering algorithm(hybrid kcenter-kmedoid), results show striking similarities toours. Results in [21] suggest that four out of the eight backbonedihedral angles involved in our metric are sufficient to charac-terize the conformational space of met-enkephalin. These fourangles are the angles of Gly2, Gly3, and Phe4, and the

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angle of Gly3: . The authors define a 4-signcode to characterize conformations with respect to the values(positive or negative) of these four angles. The two regionsmost frequently sampled by MD correspond to conformationsof type and . As can be seen in Fig. 1,a large number of the BH milestones are found in these tworegions. Interestingly, these two classes are correctly identifiedamong the clusters of BH milestones, and they contain thelowest-energy conformations. The class , which isalso significantly sampled by MD, is well represented in ourresults. Only a few milestones appear in the region ,although results in [21] suggest a relatively more importantpopulation of this type of conformation. This is probably dueto differences in the force-fields. The highest-energy cluster ofBH milestones contains stretched conformations of both types

and merged together. We refer to thiscluster as .To illustrate the application of the Multi-T-RRT, presented in

the next subsection, we have selected the lowest energy confor-mation of each class , , , and

. The first two correspond to folded conformations, thelast one is a stretched/unfolded conformation and the third oneis an intermediate conformation. Other intermediate states iden-tified by clustering are not considered in the following analysis.Table I shows a structural representation of these four confor-mations (only the backbone is represented for clarity), togetherwith the corresponding 4-sign code and the potential energy. Ageometric symbol is associated to each conformation and usedin the tables and Ramachandran maps to identify these confor-mations.

C. Transition Paths of Met-Enkephalin

We have applied the Multi-T-RRT algorithm to quicklydiscover many transition paths between the aforementionedmeta-stable states of met-enkephalin, presented in Table??[PLEASE CLARIFY]. We have used the followingparameter settings: The extension step-size is set to 2.0, whichmeans that, in the worst case, the maximum displacement ofan atom between two conformations connected by an edge isof 2 . Note however that this distance is usually significantlysmaller. At the beginning of the exploration, we impose theprobability of accepting an energy increment of 0.1 kcal/molto be approximately 50% by using the Boltzmann constant

together with an initial temperatureto 70 K. The temperature increase rate is set to 0.1.Starting from four roots at the given states, Multi-T-RRT re-

turns a single tree in approximately 2 minutes. Paths connectingthe minima are extracted for analysis. To study the diversityof the transition paths of met-enkephalin, we run Multi-T-RRT100 times. As an illustrative example, paths between the folded

and unfolded states are visualized on thesame Ramachandran maps in Fig. 2. The average cost for thetransition from to is 49.8 (MW) and in thereverse direction is 44.6 (MW), with a standard deviation acrossall the runs of 14.5. We observe that all of these paths avoid thehigh-energy regions in all projections. The paths are extremelydiverse, although the standard deviation of their cost is not very

high. This indicates that the energy landscape between theseminima is relatively flat. One can see in the figure that a fewpaths go across saddle regions corresponding to values ofclose to zero. This shows that Multi-T-RRT is able to effectivelyexplore vast portions of the conformational space.

D. Capturing Distinct Transitions of Met-Enkephalin

We have employed Multi-T-RRT with Cycles to computetransition probabilities between all pairs of energy minima formet-enkephalin. We have executed the method 100 times witha maximal running time of 10 minutes as stopping condition.Fig. 3 shows the projection of one of the resulting graphs, whichcovers all the low-energy regions on the Ramachandran mapsand constructs connections through transition regions. To es-timate transition probabilities amongst the 100 executions, wecount how many runs yield a graph from which a direct transi-tion path can be extracted between a given pair of minima. Moreprecisely, we consider that a run produces a direct transition pathbetween two minima if the best path (with respect to the cost interms of mechanical work) in the graph between them does notgo through another minimum. Table I shows the probabilitiesof direct transitions between minima and the costs (average andstandard deviation) associated to them. We can observe that di-rect transitions between all pairs of states are highly probable inmost cases. Since the mechanical work is an additive function,we can also compute the cost of indirect paths going through asequence of minima from the values in Table I. Table II presentsthe costs of different pathways from one of the two folded states

and to the unfolded state . In theaforementioned related work on met-enkephalin [21], differentcosts are also associated to alternative transition pathways fromfolded to unfolded states based on the Markov state model gen-erated form MD. Although our cost rankings do not match per-fectly to those in [21], most likely because of differences in thelandscapes inferred by the different force-fields, the overall con-clusion is the same: direct transitions from folded states to un-folded states are more probable than transitions going throughintermediate states.

IV. CONCLUSION

In this paper, we have presented a methodology to exploreand characterize the energy landscape of flexible peptides. Thismethodology combines variants of two stochastic, sampling-based algorithms: Basin Hopping (BH) and Transition-basedRapidly-exploring Random Tree (T-RRT). A simplified versionof the BH algorithm, where local, gradient-based energy mini-mization steps are replaced by simple Monte-Carlo-based min-imization steps, achieves a relevant exploration of the energylandscape yielding numerous samples around energy minima.This leads to a quick determination of meta-stable structuralstates, which can be used as a starting point for the analysis ofconformational transitions. The multiple-tree version of T-RRTis very fast at generating transition paths between a set of states.Running this algorithm several times produces a good descrip-tion of the transition path ensembles. Finally, Multi-T-RRTwithCycles yields diverse transition paths in a single run of the al-gorithm.

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Fig. 3. Projection of a graph obtained by a single run of Multi-T-RRT with Cycles starting from four meta-stable states of met-enkephalin.

TABLE IICOSTS OF TRANSITION PATHWAYS FROM FOLDED TO UNFOLDED STATES

Results on met-enkephalin presented in this paper, andresults on the terminally-blocked alanine described in a prelim-inary version [7], show that the combination of BH and T-RRTquickly produces a meaningful representation of the energylandscape of small yet highly-flexible peptides. The identifiedmeta-stable states and the transition pathways between themare comparable to those obtained with other, more expensivecomputational methods. Our results also illustrate the fact thatstochastic algorithms can compete with MD-based approachesin providing accurate and insightful findings about flexiblebiomolecules. Nevertheless, this is an early work, and manyaspects still have to be improved and further investigatedin terms of theory and implementation. Directions of futurework include exploiting the graph produced by a single run ofthe Multi-T-RRT with Cycles to describe transition path andtransition state ensembles. This will allow us to make better useof computational resources, as opposed to aggregating pathsextracted from several runs of the Multi-T-RRT. In addition,Markov-based transition-step analysis can be conducted on thegraph produced by one or more runs of T-RRT. This analysisallows one to estimate the stability of each computed state, andprovides a rigorous basis for the designation of a state as stableor semi-stable.We also aim to improve the implementation in order to effi-

ciently deal with larger peptides and proteins. We have initiatedsome experiments with larger systems to identify the efficiencyof our algorithms. One of our current test systems is chignolin,an artificial “mini-protein” composed of 10 residues. Chignolinpresents an interesting energy landscape that has been inves-tigated with other methods [24]. Our first tests with chignolinshow that our methods are still applicable in practice to larger

systems, although computing time increases significantly.When running the Multi-T-RRT using four conformations asinput (one folded, one unfolded and two intermediate statesidentified by BH), computing a graph connecting them takesabout 100 minutes on average. To reduce the computationtime, it will be necessary to carefully implement the underlyingmethods in the algorithms, in particular the nearest neighborsearch and energy computation. We are also working on theparallelization of Multi-T-RRT, building on our previous expe-rience with the basic RRT algorithm [25].

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