Transition-based RRT for Exploring the Energy Landscape of Biomolecules L´ eonard Jaillet Institut de Rob` otica i Inform` atica Industrial, CSIC-UPC, Barcelona, Spain. Email: [email protected] Juan Cort´ es LAAS-CNRS, Universit´ e de Toulouse, France. Email: [email protected] Abstract—We propose a new method for exploring the con- formational energy landscapes of biomolecules. It combines ideas coming from robotic path planning and from statistical physics. The method constructs a random tree whose expansion is driven by a double strategy. A first bias drives the expansion toward yet unexplored regions. Additionally, Monte-Carlo-like transition tests favor the exploration of low-energy regions. The balance between these two strategies is achieved by a self-adaptive scheme. As a proof of concept, the method has been applied to the alanine dipeptide. I. I NTRODUCTION Characterizing the long-time conformational rearrangements of molecular systems is a hard problem that attracts the interest of scientists from decades ago. Methods based on Molecular Dynamics (MD) and Monte Carlo (MC) algorithms [1, 2] are widely used to address this problem. However, such methods are inefficient for simulating large-amplitude conformation changes between stable conformations, since transitions trough higher-energy barriers are rare events. In recent years, many extensions have been proposed to circumvent this difficulty (e.g. [3, 4]). Despite encouraging progresses, the modeling of long-time conformational changes remains challenging. We propose to apply the recent algorithm T-RRT [5] to efficiently explore the conformational energy landscape of biomolecules. Similarly to MC methods, T-RRT applies small moves and a transition test based on the Metropolis crite- rion. However, instead of generating a single path on the conformational space, it constructs a tree with better coverage properties. Such a data structure avoids the undesired behavior of MC simulation algorithms, which tend to waste time getting back to regions of the space already explored. II. METHOD The core of the T-RRT method inherits from the basic extend-RRT [6]. The same exploration strategy is used to induce a Voronoi bias that implicitly guides the tree expansion toward yet unexplored regions of the space. T-RRT extends the basic RRT principle by integrating a transition test to hinder the tree expansion toward energetically unfavorable regions. Similarly to Monte-Carlo simulations, the acceptance rule of a local move depends on the energy variation ΔE ij between the new 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 p ij defined as: p ij = exp(- ΔEij kT ) if ΔE ij > 0 1 otherwise , where k is the Boltzmann constant, and T the temperature. T is a key parameter since it defines the level of difficulty of a transition for a given energy increment. We propose a reactive scheme to dynamically tune its value according to the information acquired during the exploration. First, T is initialized with a small value. During the search, when the number of consecutive rejections reaches a maximum number, T is multiplied by a given factor. On the contrary, each time an uphill transition test succeeds, T is divided by the same factor. This simple temperature regulation strategy is an effective way to balance the search between unexplored regions and low energy regions. Finally, random samples are discarded if the distance to the nearest node in the tree is smaller that the extension step-size. It can be shown that this simple filtering greatly improves the coverage properties of the tree by avoiding an excessive refinement of already explored regions. III. RESULTS T-RRT has been used to find the minima and the transition paths of the alanine dipeptide, a common benchmark for computational methods in chemical physics. AMBER force field with implicit solvent was used to compute energies. An internal coordinate representation with constant bond lengths and 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 six stable states of the peptide. Figure 2.a shows these minima, superimposed on the energy map on the {φ, ψ} coordinates