Transition-based RRT for Exploringthe Energy Landscape of Biomolecules
Leonard JailletInstitut de Robotica i Informatica Industrial, CSIC-UPC, Barcelona, Spain. Email: [email protected]
Juan CortesLAAS-CNRS, Universite de Toulouse, France. Email: [email protected]
Abstract— We propose a new method for exploring the con-formational energy landscapes of biomolecules. It combines ideascoming from robotic path planning and from statistical physics.The method constructs a random tree whose expansion is drivenby a double strategy. A first bias drives the expansion towardyet unexplored regions. Additionally, Monte-Carlo-like transitiontests favor the exploration of low-energy regions. The balancebetween these two strategies is achieved by a self-adaptivescheme. As a proof of concept, the method has been appliedto the alanine dipeptide.
I. INTRODUCTION
Characterizing the long-time conformational rearrangementsof molecular systems is a hard problem that attracts the interestof scientists from decades ago. Methods based on MolecularDynamics (MD) and Monte Carlo (MC) algorithms [1, 2] arewidely used to address this problem. However, such methodsare inefficient for simulating large-amplitude conformationchanges between stable conformations, since transitions troughhigher-energy barriers are rare events. In recent years, manyextensions have been proposed to circumvent this difficulty(e.g. [3, 4]). Despite encouraging progresses, the modeling oflong-time conformational changes remains challenging.
We propose to apply the recent algorithm T-RRT [5] toefficiently explore the conformational energy landscape ofbiomolecules. Similarly to MC methods, T-RRT applies smallmoves and a transition test based on the Metropolis crite-rion. However, instead of generating a single path on theconformational space, it constructs a tree with better coverageproperties. Such a data structure avoids the undesired behaviorof MC simulation algorithms, which tend to waste time gettingback to regions of the space already explored.
II. METHOD
The core of the T-RRT method inherits from the basicextend-RRT [6]. The same exploration strategy is used toinduce a Voronoi bias that implicitly guides the tree expansiontoward yet unexplored regions of the space. T-RRT extends thebasic RRT principle by integrating a transition test to hinderthe tree expansion toward energetically unfavorable regions.Similarly to Monte-Carlo simulations, the acceptance rule of alocal move depends on the energy variation ∆Eij between thenew state and its parent. This test is based on the Metropolis
Fig. 1. Alanine dipeptide and its seven conformational parameters.
criterion, with a transition probability pij defined as:
pij ={exp(−∆Eij
kT ) if ∆Eij > 01 otherwise
,
where k is the Boltzmann constant, and T the temperature.T is a key parameter since it defines the level of difficultyof a transition for a given energy increment. We propose areactive scheme to dynamically tune its value according tothe information acquired during the exploration. First, T isinitialized with a small value. During the search, when thenumber of consecutive rejections reaches a maximum number,T is multiplied by a given factor. On the contrary, eachtime an uphill transition test succeeds, T is divided by thesame factor. This simple temperature regulation strategy isan effective way to balance the search between unexploredregions and low energy regions. Finally, random samples arediscarded if the distance to the nearest node in the tree issmaller that the extension step-size. It can be shown that thissimple filtering greatly improves the coverage properties of thetree by avoiding an excessive refinement of already exploredregions.
III. RESULTS
T-RRT has been used to find the minima and the transitionpaths of the alanine dipeptide, a common benchmark forcomputational methods in chemical physics. AMBER forcefield with implicit solvent was used to compute energies. Aninternal coordinate representation with constant bond lengthsand bond angles was considered. Thus, the conformational pa-rameters are the seven dihedral angles represented in Figure 1.
The exploration yielded six minima that fit well the sixstable states of the peptide. Figure 2.a shows these minima,superimposed on the energy map on the {φ, ψ} coordinates
Fig. 2. Minima and transition paths computed by T-RRT for the alaninedipeptide, projected on the {φ, ψ} energy map.
(computed explicitly for analyzing results). This result showsthe capacity of T-RRT to find multiple minima in high-dimensional landscapes. Then, T-RRT was applied to computepaths between several pairs of minima, passing through thesaddle-points of the landscape (Figure 2.b-d). Several runs ofT-RRT permitted to capture a variety of possible transitionpaths, with the associated probabilities.
IV. CONCLUSION
We propose a new method for exploring conformationalenergy landscapes. It can be applied to find the main energyminima as well as to compute transition paths between pairsof minima. A simple benchmark has been used to validate theapproach. However, the framework is general, and extensionsof T-RRT will permit to treat more complex systems.
REFERENCES
[1] A. Leach, Molecular Modelling: Principles and Applications, Secondedition, Prentice Hall, 2001.
[2] D. Frenkel and B. Smit, Understanding molecular simulation: Fromalgorithms to applications, Second edition, Academic Press, 2002.
[3] S. A. Adcock and J. A. McCammon, “Molecular dynamics: Survey ofmethods for simulating the activity of proteins,” Chem Rev., vol. 106,no. 5, pp. 1589–1615, 2006.
[4] F. A. Escobedo, E. E. Borrero, and J. C. Araque, “Transition pathsampling and forward flux sampling. applications to biological systems,”Journal of Physics: Condensed Matter, vol. 21, no. 33, p. 333101, 2009.[Online]. Available: http://stacks.iop.org/0953-8984/21/i=33/a=333101
[5] L. Jaillet, J. Cortes, and T. Simeon, “Sampling-based path planning onconfiguration-space costmaps,” IEEE Transactions on Robotics, 2009, inpress.
[6] S. M. LaValle and J. J. Kuffner, “Rapidly-exploring random trees :Progress and prospects,” in Algorithmic and Computational Robotics:New Directions, B. Donald, K. Lynch, and D. Rus, Eds. Boston: A.K.Peters, 2001, pp. 293–308.
