Read-out of morphogen gradients with constant amplitude c₀ on a growing domain.

Gradient read-out for a gradient with constant amplitude, c₀, in case of (left column) perfect scaling with λ = k₁ ⋅ L(t); k₁ = 0.11 (ref [10]), (center column) absence of scaling with λ = 20.2 μm (ref [9]), and (right column) imperfect scaling with $\lambda =\sqrt{D2v_{g}L(t)-D2v_g{L}_0}$ with $D2v_g=3\mu m$ and L₀ = 46 μm (ref [20]). (a-c) A morphogen gradient at three timepoints (L(t) = 100 μm (light grey), L(t) = 200 μm (dark grey), L(t) = 300 μm (black)) on a domain that is scaled with respect to the current length of the domain, L(t). Five different concentration thresholds are shown as differently colored lines. (d-f) Trajectories of the positions, ζ_θ, where the different threshold concentration, shown in panels a-c, are attained. The dashed, black horizontal lines mark the time interval that is further analysed in panels j-l. (g-i) The derivative of the relative read-out position with respect to the domain length, $d{\zeta }_{\theta }dL$, evaluated at different final read-out positions. (j-l) The maximal deviation of the relative read-out positions, Δζ, in the interval from 125 μm to 300 μm (marked by dashed, black horizontal lines in panels d-f) for different final read-out positions. The colored dots indicate the final read-out position of the thresholds shown in panels a-c.