Mathematical Biology Seminar: Synchronized Oscillation for Segmentation Clock Gene of Zebrafish

Somitogenesis is a process for the development of somites which are transient, segmental structures that lie along the anterior-posterior axis of vertebrate embryos. The pattern of somites is traced out by the ``segmentation clock genes" which undergo synchronous oscillation over adjacent cells. In this presentation, we analyze the dynamics for a model on zebrafish segmentation clock-genes which are subject to direct autorepression by their own products under time delay, and cell-to-cell interaction through Delta-Notch signaling. For this system of delayed equations, we present an ingenious iteration approach to derive the global synchronization and global convergence to the unique synchronous equilibrium. On the other hand, by applying the delay Hopf bifurcation theory and the method of normal form, we derive the criteria for the existence of stable synchronous oscillations. Our analysis provides the basic range of parameters and delay magnitudes for stable synchronous, asynchronous oscillation, and oscillation-arrested dynamics. Based on the derived criteria, further numerical findings on the dynamics which are linked to the biological phenomena are explored for the considered system.