Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

A Lattice model on somitogenesis of zebrafish

Pages: 2789 - 2814, Volume 17, Issue 8, November 2012      doi:10.3934/dcdsb.2012.17.2789

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Kang-Ling Liao - Department of Applied Mathematics, National Chiao Tung University, Hsinchu 300, Taiwan (email)
Chih-Wen Shih - Department of Applied Mathematics, National Chiao Tung University, Hsinchu 300, Taiwan (email)

Abstract: Somitogenesis is the process of the development of somites which are segmental structure in vertebrate embryos. This process depends on the expression of segmentation clock genes. In this investigation, we consider lattice systems which describe the kinetics of the chief segmentation clock genes in zebrafish under negative feedback regulation with delay through interaction with the Delta-Notch signaling among neighboring cells. We first derive the analytical theories for the oscillation-arrested and synchronous oscillation in an autonomous lattice model. Based on the parameter regimes in the theories, we design suitable gradients of degradation rates and delays in a non-autonomous lattice model. Such a lattice system can generate synchronous oscillations, oscillatory traveling waves, oscillation slowing down, oscillation-arrested, and high-low expression levels. We further distinguish between different gradient structures which lead to normal and abnormal segmentations respectively and connect these structures to the dynamical regimes in the cell-cell model.

Keywords:  Somitogenesis, lattice model, oscillation-arrested, synchronous oscillation, traveling wave.
Mathematics Subject Classification:  Primary: 92B25, 37N25, 34K18, 34K13.

Received: April 2011;      Revised: April 2012;      Available Online: July 2012.