Multibody minimum-energy trajectory with applications to protein folding

This work addresses the optimal control of multibody systems being actuated with control forces in order to find a dynamically feasible minimum-energy trajectory of the system. The optimal control problem and its constraints are integrated in a discrete version of the equation of motion allowing the minimization of system energy with respect to a discrete state and control trajectory. The work is centred on a specific type of openchain multibody system, with strong local propensity, where the overall system kinematics is described essentially by the torsion around the links that connect rigid bodies. The coupling between the rigid body motion, and the optimal conformation is described as an elastic band of replicas of the original system with different conformations. The band forces are used to control system’s motion directly, reflecting the influence of the system energy field on its conformation, using for that the Nudged-Elastic Band method. Here the equation of motion of the multibody grid are solved by using the augmented Lagrangean method. In this context, if a feasible minimum-energy trajectory of the original system exists it is a stationary state of the extended system. This approach is applied to the folding of a single chain protein. Protein chains can be seen as multibody systems described as open-loop chains in which the rigid bodies represent clusters of atoms, that are linked together by bounds modelled as kinematics joints. Since peptide planes remain relatively rigid during protein motion and its overall dynamics is described by the local propensities of their backbone dihedral angles. In this contribution the presented method to find a transition state pathway between two conformations is tested. The use of the method in two types of protein coerce-grained models are discussed, i.e., when the system components linked together by revolute joints, for the second case we allowed some flexibility on the bodies structure. In both study cases a statistical potential field defined using a database with protein native conformations is used.