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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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