Synchronized oscillations in a mathematical model of segmentation in zebrafish - Liao, Kang-Ling; Shih, Chih-Wen; Tseng, Jui-Pin (Nonlinearity, 2012)
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 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). In this paper, we analyze a mathematical model which depicts 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 signalling. 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. For this system of delayed equations, an iteration technique is employed to derive the global convergence to the synchronous equilibrium, which corresponds to the oscillation-arrested. By 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 ranges and delay magnitudes for stable synchronous, asynchronous oscillation and oscillation-arrested. We exhibit how synchronous oscillations are affected by the degradation rates and delays. Extended from the analytic theory, further numerical findings linked to the segmentation process are presented.