UBC Theses and Dissertations
Dynamical inference for biological processes through the lens of optimal transport Zhang, Stephen Yimai
This thesis focuses on inference problems involving stochastic dynamics in biological systems. Many biological processes involve a population of cells that collectively undergo changes in their state over time. Experimental techniques for measuring cellular state are fundamentally limited so that the relationship between the immediate state of a cell and its past or future states cannot be directly observed. Inference techniques must be relied upon to reconstruct information about the evolution of individual cells within a population over time. Herein we discuss optimal transport as a mathematical framework for this kind of estimation problem, and build up methodology for inference in both out-of-equilibrium (non-stationary) and equilibrium (stationary) systems. For motivation, we begin with with a preliminary discussion of the nature of noisy dynamical systems arising in biology. We then turn to an overview of optimal transport theory from a computational viewpoint, which will form the mathematical workhorse of the remainder of this thesis. The first scenario we consider deals with a time-course of statistically independent observations drawn from an evolving population. We set up a inference framework modelling the process as a stochastic dynamical system. We identify and discuss the conditions required for identifiability of the underlying dynamics. Using the tools of optimal transport, we develop Global Waddington-OT, a computational method that can infer dynamics by solution of a convex program. We extensively investigate the performance of our method with simulated datasets, and also show positive results with a real-world biological dataset. The second scenario addresses the setting where observations are made from an evolving population at its equilibrium state. Using the arguments developed in the time-course scenario, we find that the necessary assumptions for identifiability carry over. We formulate a computational method StationaryOT that also estimates dynamics by solving a convex program, and investigate its performance with simulated and biological data.
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