Waves of Autocrine Signaling in Patterned Epithelia

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

^* Corresponding author. E-mail: muratov@njit.edu

Abstract

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.