Understanding dynamics in complex systems is challenging because there are many degrees of freedom, and those that are most important for describing events of interest are often not obvious. The leading eigenfunctions of the transition operator are useful for visualization, and they can provide an efficient basis for computing statistics, such as the likelihood and average time of events (predictions). Here, we develop inexact iterative linear algebra methods for computing these eigenfunctions (spectral estimation) and making predictions from a dataset of short trajectories sampled at finite intervals. We demonstrate the methods on a low-dimensional model that facilitates visualization and a high-dimensional model of a biomolecular system. Implications for the prediction problem in reinforcement learning are discussed.

TheCiona intestinalisvoltage-sensing phosphatase (Ci-VSP) is a membrane protein containing a voltage-sensing domain (VSD) that is homologous to VSDs from voltage-activated ion channels responsible for cellular excitability. Two crystal structures of Ci-VSD in putative resting and active conformations suggest a stepwise voltage-sensing mechanism involving translocation and rotation of the S4 helix. Relying on a theoretical framework based upon dynamical operators, we use mechanistic statistics estimated from an ensemble of unbiased molecular dynamics trajectories to elucidate the molecular determinants of the resting-active transition. Sparse regression with a small set of coordinates reveals that activation is primarily governed by the movement of arginine side chains through a central hydrophobic constriction, independent of global S4 movements. While translocation and rotation do provide a meaningful representation of the overall motion of the S4 helix, the analysis indicates that the activation mechanism cannot be accurately described without incorporating more fine-grained details such as the sequential exchange of salt bridges between arginines along S4 and negative countercharges along the S1--S3 helices. The importance of these salt bridges has been previously suggested by structural and functional studies. These results highlight how molecular dynamics simulations can now quantitatively characterize complex and functionally important conformational changes of proteins.