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    Please use this identifier to cite or link to this item: https://tkuir.lib.tku.edu.tw/dspace/handle/987654321/112023

    Title: Analysis on Mathematical Models of Somitogenesis in Zebrafish
    Authors: 廖康伶
    Liao, Kang-Ling
    Contributors: 淡江大學數學系
    Keywords: Somitogenesis;Zebrafish;Synchronous oscillation;Oscillation-arrested;Traveling wave pattern
    Date: 2012-03
    Issue Date: 2017-11-09 10:11:51 (UTC+8)
    Abstract: Somitogenesis is a process for the development of somites which are transient,
    segmental structure that lies along the anterior-posterior axis of vertebrate
    embryos. The pattern of somites is governed by the segmentation clock and its
    timing is controlled by the clock genes which undergo synchronous oscillation
    over adjacent cells in the posterior presomitic mesoderm (PSM), oscillation
    slowing down and traveling wave pattern in the traveling wave region, and the
    oscillation-arrested in the determined region. In this dissertation, we analyze
    mathematical models which depict the kinetics of the zebrafish segmentation clock
    genes subject to direct autorepression by their own products under time delay, and
    cell-to-cell interaction through Delta-Notch signaling. Our goal is to elucidate how
    synchronous oscillations are generated for the cells in the posterior PSM, and how
    oscillations are arrested for the cells in the anterior PSM. Moreover, by using the
    information of two-cell system, we construct a non-autonomous lattice model with
    suitable gradients of degradation rates and delays to generate traveling wave
    patterns. First, for delayed system of two coupled cells, a sequential-contracting
    technique is employed to derive the global convergence to the equilibrium, which
    corresponds to the oscillation-arrested for the cells at the determined region.
    Applying the delay Hopf bifurcation theory and the center manifold theorem, we
    derive the criteria for the existence of stable synchronous oscillations for the cells
    at the tail bud of the PSM. Our analysis provides the basic parameter regimes and
    delay magnitudes for stable synchronous, asynchronous oscillation, and
    oscillation-arrested. Next, based on these analytical results, hence the
    understanding of parameter regimes and delay magnitudes corresponding to
    various dynamic phases, we further construct a non-autonomous lattice system and
    design suitable gradients of degradation rates and delays for this lattice model.
    Consequently, the lattice system can generate synchronous oscillation, traveling
    wave pattern, oscillation slowing-down, and oscillation-arrested in each corresponding region in the embryo. We further distinguish between different
    gradient structures which lead to normal and abnormal segmentation respectively
    and connect these structures to the dynamical regimes for the two-cell model. In
    addition, we also study another ODE model and compare the dynamics between
    the delayed model and the ODE model, to learn the pertinence of the modeling.

    Delta-Notch 訊號互相聯繫的影響。我們的目標主要在是闡明:在預定體節中
    合雙細胞的延遲系統,我們使用 sequential-contracting 技巧得到系統全局收斂
    表現的振盪消失。另外,我們再使用 Hopf-分歧理論,以及 center manifold 理
    論和 normal form,得到產生穩定同步週期解的條件。而這個動態行為則對應
    Appears in Collections:[Graduate Institute & Department of Mathematics] Thesis

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