斑馬魚體節生成之數學模型分析

Abstract

體節是脊椎動物胚胎中沿著前後軸發育的環狀結構，體節生成指的是體節發育的整個過程。而體節的形態與形成時機，分別由細胞分裂時鐘以及時鐘基因所控制。其中，在預定體節中胚層中的後端，相鄰細胞間的時鐘基因的基因表現會產生同步振盪。之後，在預定體節中胚層區域中的前端，也就是旅行波區域，基因表現的振盪會開始減緩，並且出現旅行波圖案。最後，在預定體節中胚層的前端，振盪會完全停止，並且細胞會開始轉變成體節。在這篇博士論文中，我們主要討論的是用來刻畫斑馬魚體節生成的數學模型，這些模型主要是刻畫時鐘基因的動力學方程。包含了：細胞經由自己的基因表現產物所產生的自我抑制現象、過程中的時間延遲、以及細胞之間經由Delta-Notch 訊號互相聯繫的影響。我們的目標主要在是闡明：在預定體節中胚層中的後端，基因表現是如何產生出同步振盪；以及，預定體節中胚層的前端，是經由什麼機制讓基因表現的振盪停止。另外，我們也建構出刻畫耦合多細胞的非自治網格模型，用來產生出正常的旅行波圖案。首先，對於耦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到平衡點的條件。其中，這個動態行為對應到在預定體節中胚層前端的基因表現的振盪消失。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到，在預定體節中胚層中的後端所產生的同步基因表現振盪。而我們的數學分析提供了：基因表現的穩定同步振盪、穩定非同步振盪、振盪消失的參數條件，以及基因表現過程中的時間延遲的大小估計。利用耦合雙細胞的延遲系統之理論結果，我們進一步構造用來刻畫耦合多細胞的非自治網格模型。經由設計適當的基因表現衰退率，與時間延遲所具有的梯度變化，讓此模型可以成功的產生出：在預定體節中胚層中的後端的同步振盪、旅行波區域的旅行波、以及預定體節中胚層前端的振盪消失現象。我們更將這些參數變化的梯度分類，以得到正常與不正常的基因表現所對應的參數範圍。另外，我們也討論常微分方程型式的模型，並且比較這幾種不同類型的模型在動態上的差異。

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.