Energy Minimization on Manifolds for Docking Flexible Molecules

Abstract

In this paper we extend a recently introduced rigid body minimization algorithm, defined on manifolds, to the problem of minimizing the energy of interacting flexible molecules. The goal is to integrate moving the ligand in six dimensional rotational/translational space with internal rotations around rotatable bonds within the two molecules. We show that adding rotational degrees of freedom to the rigid moves of the ligand results in an overall optimization search space that is a manifold to which our manifold optimization approach can be extended. The effectiveness of the method is shown for three different docking problems of increasing complexity. First we minimize the energy of fragment-size ligands with a single rotatable bond as part of a protein mapping method developed for the identification of binding hot spots. Second, we consider energy minimization for docking a flexible ligand to a rigid protein receptor, an approach frequently used in existing methods. In the third problem we account for flexibility in both the ligand and the receptor. Results show that minimization using the manifold optimization algorithm is substantially more efficient than minimization using a traditional all-atom optimization algorithm while producing solutions of comparable quality. In addition to the specific problems considered, the method is general enough to be used in a large class of applications such as docking multidomain proteins with flexible hinges. The code is available under open source license (at http://cluspro.bu.edu/Code/Code_Rigtree.tar), and with minimal effort can be incorporated into any molecular modeling package.

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“…Second, we have used a pre‐calculated structure‐dependent rotamer library that had relatively few rotamers for most side chains . Third, we have developed a manifold based local energy minimization algorithm . The method was based on a new representation of the search space, and increased the efficiency of local energy minimization within the steps of the MCM method .…”

“…Third, we have developed a manifold based local energy minimization algorithm . The method was based on a new representation of the search space, and increased the efficiency of local energy minimization within the steps of the MCM method . These innovations enabled us to test refinement by MCM for a substantial number of complexes .…”

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“…The space of conformations ψ is a nonlinear manifold and the parameterization of the rotations corresponds to a projection from a (flat) tangent space (in which y is defined) to the manifold itself, projecting straight lines on the tangent space map onto geodesics of the manifold. We refer the reader to the Supplement and related papers 25 , 26 for a more detailed discussion of these spaces.…”

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“…In conventional molecular docking simulations, the structures of ligands are treated as flexible, while the structures of proteins are treated as rigid [ 33 , 34 ]. The lack of flexibility in protein structures can hinder the search for the proper binding conformation.…”