Figure 4. Application to neutrophil swarming.

(A) Schematic of the simple diffusion model. Here, cells on the target (within r_i) signal distant neighbors by continuously emitting a single signaling molecule. If the neighboring cells have a chemotactic response, they migrate toward the target with some noise – that is, some non-zero angle $\theta$ with respect to the target. Otherwise, they move around with no sustained directionality. (B) Experimental data (color plot) reproduced from Reátegui et al., 2017 showing the information wave front in neutrophil swarming experiments. By tracking the neutrophils in space and time, they observe highly directed motion of the neutrophils towards the target (pink) starting around $t=200$ s. There is a clear boundary in space and time – the information wave front – between the regions where cells migrate toward the target (pink) and jostle around with no particular direction (white and light blue). While a relay theory (black line) is consistent with the convex shape of the information wave front, simple diffusive signaling by only the cells on the target (gray line) is not. The diffusion constants for both models is $D=1.25\times {10}^{-10}$ m²/s. The threshold concentrations for the relay and simple diffusion models are $C_{\text{th}}/a\rho \approx 3.66\times {10}^5$ s/m and $2.91\times {10}^4$ s/m, respectively. The parameters for the relay model are chosen to fit the wave front by eye while the simple diffusion model parameters are chosen to give the same signaling distance at $t=500$ s. (C) Gradients created by signaling relays (black) and simple diffusion (gray) models in panel B. The dashed vertical lines indicate the location of the information wave front. As time increases from left to right, the relay signaling motif gives an information wave that signals cells faster than simple diffusion in the long time limit. Cells within the wave front (to the left of the dashed lines that indicate the wave fronts) experience significantly larger gradients when the cells utilize a relay, which may facilitate efficient chemotaxis.