Calculating rare biophysical events. A study of the milestoning method and simple polymer models.

Performing simulations of large-scale bio-molecular systems has long been one of the great challenges of molecular biophysics. Phenomena, such as the folding and conformational rearrangement of proteins, often takes place over the course milliseconds-to-seconds. The methods of traditional molecular dynamics used to simulate such systems are on the other hand typically limited to giving trajectories of nanosecond-to-microsecond duration. The failure of traditional methods has thus motivated the development of many special purpose techniques that propose to capture the essential characteristics of systems over conventionally inaccessible timescales. This dissertation first focuses on presenting a set of advances made on one such technique, Milestoning. Milestoning gives a statistical procedure for recovering long trajectories of the system based on observations of many short trajectories that start and end on hypersurfaces in the system’s phase space. Justification of the method’s validity typically relies on the assumption that trajectories of the system lose all memory between crossing successive milestones. We start by giving a modified milestoning procedure in which both the memory loss assumption is relaxed and reaction mechanisms are more easily extracted. We follow with numerical examples illustrating the success of new procedure. Then we show how milestoning may be used to compute an experimentally relevant timescale known as the transit time (also known as the reaction path time). Finally, we discuss how time reversal symmetry may be exploited to improve sampling of the trajectory fragments that connect milestones. After discussing milestoning, the dissertation shifts focus to a different way of approaching the problem of simulating long timescales. We consider two polymers models that are sufficiently simple to permit numerical integration of the desired long trajectories of the system. In some limiting cases, we see their simplicity even permits some questions about the dynamcis to be answered analytically. Using these models, we make a series of experimentally verifiable predictions about the dynamics of unfolded polymers.