Size-normalized Robustness of Dpp Gradient in Drosophila Wing Imaginal Disc

Exogenous environmental changes are known to affect the intrinsic characteristics of biological organizms. For instance, the synthesis rate of the morphogen decapentaplegic (Dpp) in a Drosophila wing imaginal disc has been found to double with an increase of 5.9°C in ambient temprerature. If not compensated, such a change would alter the signaling Dpp gradient significantly and thereby the development of thewing imaginal disc. To learn how flies continue to develop "normally" under such an exogenous change, we formulate in this paper a spatially two-dimensional reaction-diffusion system of partial differential equations (PDE) that accounts for the biological processes at work in the Drosophila wing disc essential for the formation of signaling Dpp gradient. By way of this PDE model, we investigate the effect of the apical-basal thickness and antero-posterior span of the wing on the shape of signaling gradients and the robustness of wing development in an altered environment (including an enhanced morphogen synthesis rate). Our principal result is a delineation of the role of wing disc size change in maintaining the magnitude and shape of the signaling Dpp gradient. The result provides a theoretical basis for the observed robustness of wing development, preserving relative but not absolute tissue pattern, when the morphogen synthesis rate is significantly altered. A similar robustness considerqation for simultaneous changes of multiple intrinsic system characteristics is also discussed briefly.