turkmath.org

Measured scalar quantities are proxies to probe the underlying physical and biological processes of interest. In my talk we examine two problems: in the first, we explore how physical boundaries affect the evolution of passive tracers in the environment. We consider the evolution of a decaying passive scalar in the presence of a Gaussian white noise fluctuating shear flow. We establish the short, intermediate, and long time symmetry properties of the single point statistics for this random passive scalar. In the next problem, we focus on data-driven modeling for rapid single molecule dynamics. First, I will present our Bayesian approach relying on the sticky hidden Markov model (HMM) to interpret time traces provided by the ultra stable optical trap experiments for the analysis of RNA Polymerase (RNAP) transcription elongation dynamics. Next, I will elaborate more on pitfalls in HMM. HMMs are workhorses of biophysics in time series analysis. However, underlying processes evolve continuously in time. Provided that observations are not obtained at high rates, then HMMs become inappropriate in analyzing these processes. Here, we introduce Markov jump processes (MJPs) for this application and provide a comparison between MJPs and HMMs via our numerical experiments for the observation models. We find that as MJPs allow us to estimate rare transitions, they appear to learn fast transitions very well even those occurring faster than the measurement timescales. These same transitions are missed for HMMs case when the data acquisition rate is insufficiently high. These are joint work with my doctoral advisors, postdoctoral mentor and postdoctoral colleague.