Eur. Phys. J. E (2010) 33: 129-148
https://doi.org/10.1140/epje/i2010-10647-6

Regular Article

Switch and template pattern formation in a discrete reaction-diffusion system inspired by the Drosophila eye

¹ Biophysics Program, The University of Michigan-Ann Arbor, 450 Church St., 48109, Ann Arbor, MI, USA
² Physics Department, The University of Michigan-Ann Arbor, 450 Church St., 48109, Ann Arbor, MI, USA

^* e-mail: dkluben@umich.edu

Received: 3 January 2010
Revised: 2 June 2010
Accepted: 21 July 2010
Published online: 23 September 2010

Abstract

We examine a spatially discrete reaction-diffusion model based on the interactions that create a periodic pattern in the Drosophila eye imaginal disc. This model is known to be capable of generating a regular hexagonal pattern of gene expression behind a moving front, as observed in the fly system. In order to better understand the novel “switch and template” mechanism behind this pattern formation, we present here a detailed study of the model's behavior in one dimension, using a combination of analytic methods and numerical searches of parameter space. We find that patterns are created robustly, provided that there is an appropriate separation of timescales and that self-activation is sufficiently strong, and we derive expressions in this limit for the front speed and the pattern wavelength. Moving fronts in pattern-forming systems near an initial linear instability generically select a unique pattern, but our model operates in a strongly nonlinear regime where the final pattern depends on the initial conditions as well as on parameter values. Our work highlights the important role that cellularization and cell-autonomous feedback can play in biological pattern formation.