Figure 6: Relative contributions of diffusion and advection on ligand transport in the Drosophila wing disc model.

(a) A small clone that solely contributes to the observed growth (no growth outside the clone) has only minor impact on gradient scaling as long as the total domain size grows linearly. Blue vertical lines indicate the initial boundaries of the clone; red vertical lines indicate the boundaries of the clone in the final domain. (b) A small non-growing clone does not influence scaling as long as the total domain size grows linearly. Blue vertical lines indicate the initial boundaries of the clone; red vertical lines indicate the boundaries of the clone in the final domain. (c) Linear tip growth (no advection) only slightly affects the quality of scaling. (a–c) SE refers to the scaling error (equation (2)) between the gradients at t=24 h and t=90 h (0: perfect scaling; 1: no scaling; for details, see Methods). (d) The influence of advection of the bound (and therefore advected) ligand species is very small compared with the diffusion of the free ligand, suggesting that the influence of advection is negligible in the wing disc and scaling is mainly obtained by a diffusion-based pre-steady-state expansion of the gradient. The calculation was carried out at the final time point, t=90 h. (e) The measured length of the exponential gradient, λ(t), at different lengths, L(t), of the wing disc (red dots)¹⁵ can be fitted with a square-root function of domain length (which itself is linearly related to time), thus supporting a diffusion-based pre-steady-state gradient expansion (for details, see Methods). (f) Gradients do not scale on an exponentially growing domain. SE refers to the scaling error (equation (2)) between the gradient at t=40 h and t=90 h (0: perfect scaling; 1: no scaling; for details, see Methods). Note that the exponentially growing domain at 40 h has the same length as the linearly growing domain at 24 h. For further details, see Supplementary Note 2.