UBC Theses and Dissertations
Exploring the use of spectral seriation to uncover dynamics in embryonic development : a geometric and probabilistic approach Zendehboodi, Roomina
Understanding the dynamics of embryonic development is crucial to finding treatments for conditions such as aging and cancer. The development of an embryo can be represented as a curve in the Wasserstein space, and to construct this curve, static snapshots of gene expression profiles are obtained at n selected time points. Since the measurement techniques for obtaining these snapshots are destructive, we have to infer the developmental trajectory using a series of static snapshots of gene expression profiles taken at different time points t₁, t₂,...tₙ. To obtain these snapshots, multiple embryos are allowed to develop until each of the desired time points is reached, and the gene expression profile is then captured. However, to reconstruct the curve we need to know which embryo had reached which developmental stage; this information is lost during the measurements. To overcome this, a pairwise similarity function between profiles can be defined, and the profiles can be arranged so that the more similar they are, the closer they are placed together. This is part of a larger class of problems known as the “seriation” problem. In this thesis, the feasibility of using the “spectral seriation” method proposed by Atkins et al. is investigated to recover the order of the profiles based on their similarity, which enables the construction of the curve. The gene expression profile of an embryo can be seen as a probability measure on a compact set. Although the exact measures are unknown, they can be approximated empirically using m samples. In this thesis, we demonstrate that, under reasonable assumptions and with sufficient time points and samples per time point, the spectral seriation method can be effective in sequencing the data. Additionally, we provide tools to determine the number of time points and samples per time point needed to achieve a desired error bound. Furthermore, we investigate how the geometric properties of the curve representing the embryonic development can affect our ability to sequence the data.
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