Figure 1. Asymptotic relay dynamics of cells in 2D with diffusion in 2D.

(A) Schematic illustrating the diffusive relay motif. Cells (pink with purple nucleus) release a signaling molecule that diffuses (blue clouds). They do so when the local concentration exceeds a threshold, $C_{\text{th}}$. This gives rise to a diffusive wave with wave speed v. (B) Snapshot concentration profiles. Asymptotic theory (Equation (6), black lines) and numerical simulation of Equation (2) (red dots, details of the numerical methods can be found in Materials and methods) are in good agreement and show outward-propagating waves. Here, $D={10}^{-10}$ m²/s, ${\displaystyle v=2}$ µm/s, and $h{C}_{\text{th}}/a\rho =D/v^2$. Numerical simulations assume that a cell colony of size $r_i=4D/v$ (dashed vertical line) centered at the origin starts signaling $t=0$. (C) Numerical wave speed as measured at $t=100D/v^2$ (markers) agrees well with theory (Equation (5), black line) as we independently vary D (circles) and $h{C}_{\text{th}}/a\rho$ (diamonds) relative to the panel B values (red circle and diamond).