Figure 1: Dynamic scaling of morphogen gradients on growing domains.

(a) According to the French Flag model, concentration thresholds determine differential cell fate (blue, white and red) in a tissue. (b) To maintain proportions in differently sized tissue, gradients need to lengthen on longer domains. (c) Both the height, c₀, and the length of the measured¹⁵ Dpp gradient (equation (1)) increase on the growing wing imaginal disc of length L(t). (d) The measured¹⁵ normalized Dpp gradients, c(t)/c₀, of length λ(t) overlay on a rescaled domain, x/L(t). (e) The spreading of a morphogen gradient on a growing domain as a result of diffusion, advection and dilution in the absence of any reactions. Morphogens enter the domain at the left-hand side according to a flux boundary condition. The boundary at the right-hand side is impermeable, that is, we have zero-flux boundary conditions. For details, see Methods. (f) The normalized simulated gradient profiles from e overlay on the rescaled domain, that is, the gradients scale with domain size (cyan line: 24 h, blue line: 57 h, black line: 90 h). 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). In all panels, the horizontal axis reports the position on the domain and the vertical axis reports concentration, as indicated.