Kang Ling Liao Analysis on Mathematical Models of Somitogenesis in Zebrafish
Kang-Ling Liao (Mathematical Biosciences Institute, The Ohio State University)
(February 14, 2013 10:20 AM - 11:15 AM)
Somitogenesis is a process for the development of somites which are transient, segmental structure that lies along the anterior-posterior axis (AP 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 anterior PSM, called determined region.
In this talk, I will focus on 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. First, for a basic two-cell system with delays, a sequential-contracting technique is employed to derive the global convergence to the equilibrium. This scenario corresponds to the oscillation-arrested for the cells in the determined region. Applying the delay Hopf bifurcation theory, the center manifold theorem, and the normal form method, we derive the criteria for the existence of stable synchronous oscillations for the cells in the tail bud of the PSM. These analytical results can be extended to a specific N-cell model. Hence, we provide an explanation for how synchronous oscillations are generated for the cells in the posterior PSM and how oscillations are arrested for the cells in the anterior PSM. Based on these results of two-cell system, we further construct a non-autonomous lattice delayed system to generate synchronous oscillation, traveling wave pattern, oscillation slowing-down, and oscillation-arrested in each corresponding region in the embryo to fit the experimental observations.