A dynamical model of ommatidial crystal formation

Abstract

The crystalline photoreceptor lattice in the Drosophila eye is a paradigm for pattern formation during development. During eye development, activation of proneural genes at a moving front adds new columns to a regular lattice of R8 photoreceptors. We present a mathematical model of the governing activator-inhibitor system, which indicates that the dynamics of positive induction play a central role in the selection of certain cells as R8s. The "switch and template" patterning mechanism we observe is mathematically very different from the well-known Turing instability. Unlike a standard lateral inhibition model, our picture implies that R8s are defined before the appearance of the complete group of proneural cells. The model reproduces the full time course of proneural gene expression and accounts for specific features of the refinement of proneural groups that had resisted explanation. It moreover predicts that perturbing the normal template can lead to eyes containing stripes of R8 cells. We observed these stripes experimentally after manipulation of the Notch and scabrous genes. Our results suggest an alternative to the generally assumed mode of operation for lateral inhibition during development; more generally, they hint at a broader role for bistable switches in the initial establishment of patterns as well as in their maintenance.imaginal disc | neural fate specification R egular patterns of cell fate appear widely in biology, and it has long been a major challenge to explain how such de novo patterning occurs during development. The R8 photoreceptor lattice in the Drosophila melanogaster eye is a particularly striking example. The adult Drosophila eye comprises ∼750 ommatidia, each composed of photoreceptors and support cells, packed in a crystalline array (1). These ommatidia are founded by R8 photoreceptors, whose orderly alignment takes shape in the wake of the morphogenetic furrow (MF) that moves from posterior to anterior across the eye imaginal disc during the third larval instar ( Fig. 1) (2). The emergence of a solitary R8 from within a proneural group of competent cells has parallels in many examples of neural and neuronal fate specification (3,4). This selection of a neural precursor through lateral inhibition is generally believed to involve an instability whereby feedback loops reinforce random fluctuations to choose one among a collection of equivalent cells (5-7). Here, we present a mathematical model of R8 selection and spacing that suggests a different scenario. This model turns out to generate patterns through a "switch and template" mechanism at whose heart is a cell-autonomous, bistable switch that takes one state in the R8s and another in the surrounding, undifferentiated cells. Whether a cell switches from undifferentiated to R8 depends primarily on the balance between inductive and inhibitory signals emanating from more mature cells to its posterior; direct inhibitory interactions among neighboring cells play a secondary role, preventing the appearance of superfluous R8s after the ini...

Regulation of neuronal differentiation at the neurogenic wavefront

“…Therefore, the conditions at the wavefront are expected to have relevant consequences for the final pattern of neuronal differentiation. Although the importance of static boundary conditions at the borders of a pattern-forming tissue have received some theoretical notice (Honda et al, 1990;Collier et al, 1996;Murciano et al, 2002;Meir et al, 2002;Plahte, 2001), moving wavefronts of lateral inhibition have only recently come to attention (Owen, 2002;Plahte and Øyehaug, 2007;Pennington and Lubensky, 2010;O'Dea and King, 2011;Lubensky et al, 2011). These studies show how a neurogenic wavefront can sweep across a field of identical cells and leave behind a pattern of different cell types.…”

Notch-mediated lateral inhibition regulates proneural wave propagation when combined with EGF-mediated reaction diffusion

“…Although we cannot exclude this possibility, the above results suggest that the fourcomponent model with Notch-mediated lateral inhibition is essentially consistent with proneural wave progression in silico and in vivo and that the wave acceleration phenotype is equivalent to the classical neurogenic phenotype of Notch mutant clones (2). Although nonlinear regulations of Notch signaling were implemented in the previous models (5,6,(17)(18)(19), the equation for N in the four-component model was sufficient to reproduce all of the experimental conditions described above. feedback (Fig.…”

“…Therefore, many of the theoretical studies regard the mode of Notch action during wave propagation as lateral inhibition. Previous mathematical modeling has revealed essential roles for Notch signaling in the formation of a salt and pepper pattern of photoreceptor neurons (17)(18)(19)(20). Although it was suggested that Dl restricts wave propagation during eye development (17), Notch signaling is also required for undifferentiated cells to acquire a proneural state before the wave progression in Significance Notch-mediated lateral inhibition regulates binary cell fate choice, resulting in salt and pepper patterns during various developmental processes.…”

Lateral inhibition and neurogenesis: novel aspects in motion

“…Such self-regulated progression ensures patterning is occurring in a small moving region, in between non-neurogenic tissue and already patterned tissue. From a theoretical point of view, moving wavefronts of lateral inhibition are receiving increasing attention (Owen, 2002;Plahte and Øyehaug, 2007;Pennington and Lubensky, 2010;Lubensky et al, 2011;O'Dea and King, 2011;O'Dea and King, 2012;Formosa-Jordan et al, 2012;Simakov and Pismen, 2013). The advantage of a moving wavefront versus a neurogenic region surrounded by a fixed boundary remains to be elucidated.…”