PIPS: Path Integral Stochastic Optimal Control for Path Sampling in Molecular Dynamics

We consider the problem of \textit{Sampling Transition Paths}: Given two metastable conformational states of a molecular system, \eg\ a folded and unfolded protein, we aim to sample the most likely transition path between the two states. Sampling such a transition path is computationally expensive due to the existence of high free energy barriers between the two states. To circumvent this, previous work has focused on simplifying the trajectories to occur along specific molecular descriptors called Collective Variables (CVs). However, finding CVs is non trivial and requires chemical intuition. For larger molecules, where intuition is not sufficient, using these CV-based methods biases the transition along possibly irrelevant dimensions. In this work, we propose a method for sampling transition paths that considers the entire geometry of the molecules. We achieve this by relating the problem to recent works on the Schr\"odinger bridge problem and stochastic optimal control. Using this relation, we construct a \emph{path integral} method that incorporates important characteristics of molecular systems such as second-order dynamics and invariance to rotations and translations. We demonstrate our method on commonly studied protein structures like Alanine Dipeptide, and also consider larger proteins such as Polyproline and Chignolin.