Math. Model. Nat. Phenom.
Volume 7, Number 6, 2012Biological oscillations
|95 - 106
|12 December 2012
An Analysis of Overall Network Architecture Reveals an Infinite-period Bifurcation Underlying Oscillation Arrest in the Segmentation Clock
1 Centro de Investigación y de Estudios
Avanzados del IPN, Depto. de Biomedicina Molecular. Av. Instituto Politécnico Nacional
2508, Col. San Pedro Zacatenco,
2 Centro de Investigación y de Estudios Avanzados del IPN, Unidad Monterrey. Vía del Conocimiento 201, Parque PIIT, CP 66600 Apodaca NL, México
3 Centre for Applied Mathematics in Bioscience and Medicine. 3655 Promenade Sir William Osler McIntyre Medical Building, Room 1123A, Montreal, QC H3G 1Y6, Canada
⋆ Corresponding author. E-mail: firstname.lastname@example.org
Unveiling the mechanisms through which the somitogenesis regulatory network exerts spatiotemporal control of the somitic patterning has required a combination of experimental and mathematical modeling strategies. Significant progress has been made for the zebrafish clockwork. However, due to its complexity, the clockwork of the amniote segmentation regulatory network has not been fully elucidated. Here, we address the question of how oscillations are arrested in the amniote segmentation clock. We do this by constructing a minimal model of the regulatory network, which privileges architectural information over molecular details. With a suitable choice of parameters, our model is able to reproduce the oscillatory behavior of the Wnt, Notch and FGF signaling pathways in presomitic mesoderm (PSM) cells. By introducing positional information via a single Wnt3a gradient, we show that oscillations are arrested following an infinite-period bifurcation. Notably: the oscillations increase their amplitude as cells approach the anterior PSM and remain in an upregulated state when arrested; the transition from the oscillatory regime to the upregulated state exhibits hysteresis; and opposing Fgf8 and RA gradients along the PSM naturally arise in our simulations. We hypothesize that the interaction between a limit cycle (originated by the Notch delayed-negative feedback loop) and a bistable switch (originated by the Wnt-Notch positive cross-regulation) is responsible for the observed segmentation patterning. Our results agree with previously unexplained experimental observations and suggest a simple plausible mechanism for spatiotemporal control of somitogenesis in amniotes.
Mathematics Subject Classification: 92b25 / 92c15 / 92c42
Key words: somitogenesis / vertebrate embryo / delay differential equations / bistability / SNIC / SNIPER bifurcation
© EDP Sciences, 2012
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