Waves of Autocrine Signaling in Patterned Epithelia

C. B. Muratov¹ ; S. Y. Shvartsman²

¹ Department of Mathematical Sciences, New Jersey Institute of Technology Newark, NJ 07102, USA
² Department of Chemical Engineering and Lewis Sigler Institute for Integrative Genomics Princeton University, Princeton, NJ 08544, USA

Mathematical modelling of natural phenomena, Tome 5 (2010) no. 5, pp. 46-63.

Résumé

A biophysical model describing long-range cell-to-cell communication by a diffusible signal mediated by autocrine loops in developing epithelia in the presence of a morphogenetic pre-pattern is introduced. Under a number of approximations, the model reduces to a particular kind of bistable reaction-diffusion equation with strong heterogeneity. In the case of the heterogeneity in the form of a long strip a detailed analysis of signal propagation is possible, using a variational approach. It is shown that under a number of assumptions which can be easily verified for particular sets of model parameters, the equation admits a unique (up to translations) variational traveling wave solution. A global bifurcation structure of these solutions is investigated in a number of particular cases. It is demonstrated that the considered setting may provide a robust developmental regulatory mechanism for delivering chemical signals across large distances in developing epithelia.

Détail

DOI : 10.1051/mmnp/20105504

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