#### Vol. 6, No. 1-4, 2011

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Size-normalized robustness of $\mathrm{Dpp}$ gradient in Drosophila wing imaginal disc

### Arthur D. Lander, Qing Nie, Benjamin Vargas and Frederic Y. M. Wan

Vol. 6 (2011), No. 1-4, 321–350
##### Abstract

Exogenous environmental changes are known to affect the intrinsic characteristics of biological organisms. 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 $5.{9}^{\circ }$C in ambient temperature. If not compensated, such a change would alter the signaling Dpp gradient significantly and thereby the development of the wing 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 anteroposterior 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 consideration for simultaneous changes of multiple intrinsic system characteristics is also discussed briefly.

 Dedicated to Charles and Marie-Louise Steele for their years of outstanding management of IJSS and JoMMS,and for their more than forty years of warm friendship with the last author.
##### Keywords
morphogen gradients, robust development, environmental changes, Dpp