Extended Data Fig. 9: Scaling analysis of expansion dynamics and illustration of the stochastic migration process.
From: Chemotaxis as a navigation strategy to boost range expansion
a, The exponential trailing region of the density profile is fixed by the cell growth rate λ and expansion speed u of the front. Because cells in the trailing region do not experience drift (Extended Data Fig. 3g), the apparent ‘movement’ of the trailing region at the same speed as the front bulge is possible only if it has an exponential profile, ρ(r,t) ~ ek(r − ut), with k = λ/u. b, Scaling of the expansion speed with model parameters. According to the GM model (Fig. 3a), the density peak at the propagating front is determined by a balance between cell growth and back diffusion (Fig. 4b). Using a crude scaling analysis to capture this balance, we can obtain (approximately) the quantitative determinants of the propagating speed. Consider a sharply peaked density bulge at the front, with peak density ρpeak and width w. The number of cells contained in the peak region, Npeak, is given by the relation in equation (E9.a). Cell birth rate, λNpeak, is balanced by the back-diffusion flux, which is approximated as Dρpeak/w, leading to the relation in equation (E9.b). To relate to the migration speed u, we note that around the density peak the drift speed v is nearly maximal (Fig. 4a), and equation (4) becomes \({v}_{max}\approx {\chi }_{0}\frac{{\rm{d}}}{{\rm{d}}x}\,\mathrm{ln}(a)\). In the scaling approach, we take u ~ vmax and the approximation \(\frac{{\rm{d}}}{{\rm{d}}x}\,\mathrm{ln}(a)\sim 1/w\) leading to the relation in equation (E9.c). Combining equations (E9.a) and (E9.c), we obtain equation (E9.d) with the expansion speed increasing with the square-root of the growth rate λ. Note the χ0/D factor appearing as a prefactor in the expression for u, which is responsible for the increase in the expansion speed in the presence of chemotaxis with respect to the Fisher–Kolmogorov dynamics (Extended Data Fig. 5) and for the dynamics with the attractant being the sole nutrient (Extended Data Fig. 8). c, d, Scaling results are confirmed by simulations of the GE model when varying growth rate λ (c) and chemotactic coefficient χ0 (d). e, To further illustrate the intricate dynamics at the front of the expanding population, we performed stochastic agent-based simulations looking at the trajectories of single cells. Shown here are cell trajectories for a few selected cells located within the population front (pioneers) at time t = 6.5 h. Bottom, 38 trajectories with colour indicating the time the trajectory escaped from the front and cells switched from being pioneers to being settlers, which grow and colonize localities behind the front. Red circles indicate cell division events. Highlighted area (cyan) denotes front region with aspartate concentration in the range a− < a < a+. Top, position distribution of all simulated trajectories (1,000) at time t = 8 h. See Supplementary Text 3 for details.
