Long-Timescale Kinetics and Mechanisms of Protein Conformational Change

Because biomolecules are complex, high-dimensional systems exhibiting fluctuations ranging from femtoseconds (e.g., bond vibrations) to seconds (e.g., ligand binding) and beyond, understanding their dynamics requires both experimental and theoretical investigation to bridge the timescale gap. Although molecular dynamics (MD) simulations have proven extremely powerful as a "computational microscope" for sampling the equilibrium distributions of biomolecules and probing their atomistic dynamics, many important biomolecular processes such as protein folding, ligand binding, or protein-protein association occur on timescales beyond the reach of direct simulation. MD simulation of slow biological processes such as these would require ~10^12 integration timesteps or more to observe a single millisecond-scale transition event, remaining intractable even for the fastest supercomputers today. In addition, understanding the mechanism of complex conformational changes requires obtaining statistics of these rare transitions between metastable states, compounding the computational requirements. The framework of transition path theory provides one approach to tackling this problem by considering statistics like the committor—which gives the probability of committing the product state before returning to a reactant state and is an ideal reaction coordinate—and the reactive flux, which maps the flow of reactive trajectories. Recently, researchers have developed an approach (known as the dynamical Galerkin approximation, or DGA) that leverages relatively short MD trajectories to compute these kinetic statistics by solving an dynamical operator equation projected onto a basis. The requisite elements in the resulting linear system are estimated via Monte Carlo averages over trajectory data. However, DGA suffers from challenges in constructing appropriately expressive basis sets and in data efficiency. Moreover, for most biological problems of interest, it remains challenging to extract mechanistic insight from the computed kinetic statistics, requiring human intuition to suggest physically informative coordinates that can appropriately describe the reaction dynamics. In this work, I consider two representative biological problems that illustrate the challenges of understanding mechanisms of slow conformational changes through MD simulations: how membrane proteins sense voltage and how metamorphic proteins switch structures. I also propose two ways of remedying the problems associated with DGA: first, by incorporating extra terms in the basis expansion to alleviate non-Markovian effects, and second, by representing the statistics using a neural network and learning them via an inexact iterative scheme based in numerical linear algebra. Together, the methods and applications in this work comprise a unified framework for understanding long-timescale biomolecular dynamics.