Hexagonal Patterning of the Insect Compound Eye: Facet Area Variation, Defects, and Disorder

Abstract

The regular hexagonal array morphology of facets (ommatidia) in the Drosophila compound eye is accomplished by regulation of cell differentiation and planar cell polarity during development. Mutations in certain genes disrupt regulation, causing a breakdown of this perfect symmetry, so that the ommatidial pattern shows onset of disorder in the form of packing defects. We analyze a variety of such mutants and compare them to normal (wild-type), finding that mutants show increased local variation in ommatidial area, which is sufficient to induce a significant number of defects. A model formalism based on Voronoi construction is developed to predict the observed correlation between ommatidium size variation and the number of defects, and to study the onset of disorder in this system with statistical tools. The model uncovers a previously unknown large-scale systematic size variation of the ommatidia across the eye of both wild-type and mutant animals. Such systematic variation of area, as well as its statistical fluctuations, are found to have distinct effects on eye disorder that can both be quantitatively modeled. Furthermore, the topological order is also influenced by the internal structure of the ommatidia, with cells of greater relative mechanical stiffness providing constraints to ommatidial deformation and thus to defect generation. Without free parameters, the simulation predicts the size-topology correlation for both wild-type and mutant eyes. This work develops formalisms of size-topology correlation that are very general and can be potentially applied to other cellular structures near the onset of disorder.

Universal Features of Metastable State Energies in Cellular Matter

“…The diagnostics of material properties and their spatial distribution (in industrial applications) or the occurrence and distribution of pathological changes (in biological tissues) is aided by these findings. The geometric information used here can be further combined with topological statistics [29,34,42], which is the subject of ongoing work [43].…”

Mechanics of epithelial tissue formation

“…We find that the touching boundaries of the (proto)cells correspond closely to a Voronoi tessellation of their nuclei, an effect that becomes more pronounced after cellularization. Although Voronoi tessellations have occasionally been used to describe cellular patterns in epithelial tissues [21,22,23,24,25,26,27], to the best of our knowledge, the fact that the nuclei are located at the centers of the corresponding Voronoi cells has not been shown previously. Tessellations have also been used as a basis for mechanical modeling of cellular tissues, especially in vertex models where forces act on the vertices of a lattice [23,28,29,30,31,32,33,34].…”

Characterization of the genetic architecture underlying eye size variation withinDrosophila melanogasterandDrosophila simulans

“…Ommatidia number is determined during the third larval instar as the morphogenetic furrow moves from posterior to anterior across the retinal field to trigger the differentiation of rows of ommatidia in their hexagonal pattern (reviewed in FRANKFORT and MARDON 2002;KUMAR 2011;KUMAR 2012). The final size of ommatidia is determined later during the pupal stage as these facets become fully differentiated, although much less is known about the specification of ommatidia size than ommatidia number (CAGAN and READY 1989;VANDENDRIES et al 1996;MILLER and CAGAN 1998;FICHELSON et al 2012;KIM et al 2016).…”