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

TL;DR

This paper investigates a discrete reaction-diffusion model inspired by Drosophila eye patterning, revealing how switch and template mechanisms lead to robust hexagonal gene expression patterns through combined analytic and numerical methods.

Contribution

It introduces a novel discrete reaction-diffusion model demonstrating pattern formation via switch and template mechanisms, with detailed analysis of pattern robustness and parameter dependence.

Findings

Patterns form reliably with proper timescale separation and strong self-activation.

Derived expressions for front speed and pattern wavelength in nonlinear regime.

Pattern final state depends on initial conditions and parameters, unlike linear instability models.

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 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.