Figure 3: The scaling mechanism reproduces the Dpp gradient characteristics from the Drosophila wing imaginal disc quantitatively.

(a) The simulated Dpp network for the Drosophila wing imaginal disc. For details, see main text and Methods. (b) The parameters that affect the amount of receptor that is present outside affect the length of the gradient. Increasing the receptor production by twofold results in higher but shorter gradients (red), while decreasing receptor production by twofold results in wider but lower gradients (blue), as observed in experiments³². The black line shows the standard gradient. (c) The simulated expression profile of the Tkv receptor (black line) and of the Dpp ligand (red line). (d,e) The simulated gradient profiles of total Dpp (d) expand with the growing domain in a way that (e) the rescaled gradient profiles overlay on the rescaled domain. SE refers to the scaling error (equation (2)) between the gradients at t=24 h and t=90 h. Time points shown (light to dark grey): 24, 46, 68, 90 h. (f) The length of the exponential gradient, λ(t), is about linearly related to the length of the domain, L(t), and matches the measured values (red)¹⁵. (g) The fraction of ligand that is unbound, bound to receptors on the membrane, or to internalized receptors matches the measured values¹⁵. The fractions were calculated based on the total concentration of each species in the whole posterior compartment. (h) With the exception of very early stages, the maximal value of the gradient, , and the length of the domain, L(t), are related according to a powerlaw, c₀(t)=L(t)^β (black line); the blue line shows the fitted line with β=1.18; the linear fit was carried out between 10 and 90 h. (i) The exponent β of the power law depends on the relative speed, with which the ligand production zone and the target domain expand, and the measured β is reproduced as long as the two domains expand together as observed in the experiment¹⁵. The relative expansion of the ligand-producing domain only has a very small effect on the length of the gradient, λ/L, and scaling is still observed if the ligand-producing domain does not expand.