Anterior-posterior patterning of Drosophila wing discs I: A baseline mathematical model.

The wing imaginal disc of Drosophila is one of the commonly used model systems for the studies of patterning, growth, and scaling. Development of the wing disc involves many interacting components as well as a variety of compound processes whose underlying mechanisms are still under investigation. For instance, it remains unclear about how to form compound experimentally-measured patterns of Decapentaplegic (Dpp) type-I receptor Thickveins (Tkv), as well as phosphorylated Mothers Against Dpp (pMad, the latter of which is the indicator of Dpp signaling activities. In this work, we proposed a baseline mathematical model that integrates established experimental facts to investigate the formation of pMad and Tkv gradients. Our model is validated by the accurate reproduction of complex asymmetric profiles of Tkv and pMad in both anterior and posterior compartments of the wing disc. Moreover, using our model as a numerical platform, we examined specific roles played by Engrailed (En), Hedgehog (Hh) and Dpp in the establishment of Tkv and pMad profiles. It turns out that En, Hh, Dpp all play an essential role in the formation of pMad and Tkv patterns. In particular, our proposed model supports the crucial part of the downregulation of Tkv by Dpp. Further, dual negative regulations of Tkv by both Hh and Dpp simultaneously prevent the Dpp signaling from interfering the Hh signaling and expand the effective range of Dpp gradients. Finally, parameter sensitivity was carried out to ensure that our results and conclusions are robust against specific choices of parameter values.