Appendix 8—figure 1. Diffusive wave initiation with Hill function activation.

(A) Wave initiation for cells in 1D with diffusion in 1D. Here, the initiation time $t_{\text{init}}$ is normalized by the characteristic time scale ${\tau }_c=h^2{C}_{\text{th}}/a\rho$ and the initial signaling colony size r_i is normalized by the characteristic length scale $l_c={\left(h^2{C}_{\text{th}}D/a\rho \right)}^{1/2}$. Yellow data points are individual simulations with dashed black guides to the eye connecting the yellow points. The solid yellow line corresponds to Heaviside-like activation and is reproduced from the main text. (B) Same as A, but for cells in 2D and diffusion in 2D. Here, ${\tau }_c=h{C}_{\text{th}}/a\rho$ and $l_c={\left(h{C}_{\text{th}}D/a\rho \right)}^{1/2}$. (C) Same as A, but for cells in 3D and diffusion in 3D. Here, ${\tau }_c=C_{\text{th}}/a\rho$ and $l_c={\left(C_{\text{th}}D/a\rho \right)}^{1/2}$. We note the value of $r_i=\sqrt{3}{l}_c$, the value below which waves fail to initiate for Heaviside-like activation.