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The growth of imaginal discs in holometabolous insects is coordinated with larval growth to ensure the symmetrical and proportional development of the adult appendages. In ants, the differential growth of these discs generates distinct castes-the winged male and queen castes and the wingless worker caste. In the hyperdiverse ant genus Pheidole, the worker caste is composed of two morphologically distinct subcastes: small-headed minor workers and larger, big-headed, soldiers. Although these worker subcastes are completely wingless, soldier larvae develop rudimentary forewing discs that function in generating the disproportionate head-to-body scaling and size of soldiers. It remains unclear, however, how rudimentary forewing discs in soldier larvae are coordinated with other imaginal discs. Here we show, using quantitative nano-CT imaging and three-dimensional analyses, that the increase in the volume of the soldier rudimentary forewing discs is coordinated with larval size as well as with the increase in the volume of the leg and eye-antennal (head) discs. However, relative to larval size, we found that when the rudimentary forewing discs appear during the last larval instar, they are relatively smaller but increase in volume faster than that of the head (eye-antennal) and leg discs. These findings show that the rudimentary wing disc in soldier larvae has evolved novel patterns of inter-organ coordination as compared with other insects to generate the big-headed soldier caste in Pheidole. More generally, our study raises the possibility that novel patterns of inter-organ coordination are a general feature of rudimentary organs that acquire novel regulatory functions during development and evolution.

Sei k ein algebraischer Zahlkorper und sei D eine zentrale Divisionsalgebra endlicher Dimension uber k. Der Kern der reduzierten Norm-Abbildung nrd:M(2,D)-> k definiert eine lineare algebraische k-Gruppe G, die im Allgemeinen mit SL_2(D) bezeichnet wird. Eine maximale Ordnung L in D induziert eine arithmetische Untergruppe Gamma=SL_2(L) in SL_2(D). Diese operiert auf einem homogenen Raum X, der durch eine Lie-Gruppe, die G zugeordnet ist, definiert wird. Der Quotientenraum X/Gamma ist nicht kompakt, kann aber in naturlicher Weise, nach Borel und Serre, kompaktifiziert werden. Diese Kompaktifizierung \overline{X}/Gamma erhalt man, indem man endlich viele Randkomponenten e'(P) hinzuuugt, eine fur jede eigentliche parabolische Untergruppe P von G. In dieser Arbeit studieren wir die Anzahl dieser Randkomponenten. Insbesondere betrachten wir, wie diese mit verschiedenen Invarianten der Algebra D zusammenhangen. Der Fall einer total definiten Quaternionenalgebra ist besonders interessant. Des weiteren behandeln wir die Geometrie einer solchen Komponente.

The origin of complex worker-caste systems in ants perplexed Darwin1 and has remained an enduring problem for evolutionary and developmental biology2–6. Ants originated approximately 150 million years ago, and produce colonies with winged queen and male castes as well as a wingless worker caste7. In the hyperdiverse genus Pheidole, the wingless worker caste has evolved into two morphologically distinct subcastes—small-headed minor workers and large-headed soldiers8. The wings of queens and males develop from populations of cells in larvae that are called wing imaginal discs7. Although minor workers and soldiers are wingless, vestiges or rudiments of wing imaginal discs appear transiently during soldier development7,9–11. Such rudimentary traits are phylogenetically widespread and are primarily used as evidence of common descent, yet their functional importance remains equivocal1,12–14. Here we show that the growth of rudimentary wing discs is necessary for regulating allometry—disproportionate scaling—between head and body size to generate large-headed soldiers in the genus Pheidole. We also show that Pheidole colonies have evolved the capacity to socially regulate the growth of rudimentary wing discs to control worker subcaste determination, which allows these colonies to maintain the ratio of minor workers to soldiers. Finally, we provide comparative and experimental evidence that suggests that rudimentary wing discs have facilitated the parallel evolution of complex worker-caste systems across the ants. More generally, rudimentary organs may unexpectedly acquire novel regulatory functions during development to facilitate adaptive evolution. In the ant genus Pheidole the growth of rudimentary wing discs—which influence developmental allometry to produce castes with distinct morphologies—is socially regulated to determine the worker-to-soldier ratio in Pheidole colonies.