Switch and template pattern formation in a discrete reaction-diffusion system inspired by the Drosophila eye.
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.
Pennington
M W
MW
Biophysics Program, The University of Michigan-Ann Arbor, 450 Church St., 48109, Ann Arbor, MI, USA.
Lubensky
D K
DK
D000818:Animals; D004058:Diffusion; D004331:Drosophila melanogaster / Q000166:cytology / Q000235:genetics / Q000378:metabolism*; D004848:Epithelium / Q000378:metabolism; D005123:Eye / Q000166:cytology / Q000378:metabolism*; D053263:Gene Regulatory Networks; D008954:Models, Biological