Extended Data Figure 6: Model of two-strain competitive expansion dynamics.

a, The growth-expansion model of bacterial range expansion developed in Ref. 9 and outlined in Extended Data Fig. 5 is extended to several strains of bacteria, whose densities are denoted by ρ_i(x, t), with i ∈ {1,2} for two strains. The different bacterial strains are assumed to consume the same nutrient (n) and grow at the same rate λ(n), in accordance to Monod’s law. They also sense and consume the same signaling molecule, the attractant (a), at the same rate μ(a). The random motion of each bacterial strain is described by a diffusion term, with effective diffusion coefficients D_i for strain i; see Eqs. [S5], [S6]. The spatial attractant profile a(x, t) (resulting from bacterial consumption, Eq. [S8]) leads to directed motion which is modeled by a drift term in Eqs. [S5] and [S6]. The dependence of the drift velocities, v_i, on the attractant profile is given by Eq. [S7], which describes a range of proportional sensing, v ∝ ∂_xa/a for K_I < a < K_A. The magnitude of the chemotactic response is parametrized by the chemotactic coefficient c_i for strain i. Finally, the dynamics of the nutrient and attractant are described by Eqs. [S8] and [S9], with D₀ characterizing molecular diffusion. b, Outcome of the competitive expansion dynamics, showing the density profiles of two strains, 24 hours after inoculating with equal mixture at the origin. Competition is run for one strain resembling the ancestor (black line, with expansion speed $u_{anc}=6mm/h$) and the other strain resembling a mutant, with expansion speed, u_mut, increasing from left to right as indicated above the plots. The experimental data is taken from Extended Data Fig. 4l. c, Increase and the abrupt freezing of the crossover distance over time as observed in simulations (line) and experiments (circles). The expansion speeds of the competing pair are u_anc vs u_mut = 6mm/ℎ vs 7mm/ℎ, with the latter mimicking the expansion speed of strain D; see Extended Data Fig. 3b. Full temporal evolution of the density profiles is shown in Supplementary Video 2. The experimental data is from Extended Data Fig. 4g. d, The chemotactic competition model of panel a is repeated to determine the stability distance d^∗ for various u and several smaller growth rate values, λ (indicated in legend). The results confirm that the linear relation (2) between the stability distance and expansion speed shown in the main text holds for each growth rate simulated. e, The slope u/d^∗(u) in panel d is plotted against the growth rate λ. A linear dependence on λ is seen as predicted by Supplementary Analysis 2. f, Effect of differing growth-rate (x-axis) on the outcome of competition and cross-over distance, for two strains with different chemotactic coefficients (c₁,c₂). Legend shows the ratio c₂/c₁. The main result that the interior is dominated by the slower strain and exterior dominated by the faster strain still holds. The crossover distance separating the different regions of dominance is found to vary smoothly with growth rate differences, but in opposite ways depending on whether the faster growing strain moves faster (red) or slower (green). The observed change of the crossover distances are minimal when growth-rate differences are minimal (e.g., <1% as we observed experimentally for TB, see Extended Data Fig. 1e) but become substantial when growth-rate differences become large. See Supplementary Analysis 3 for details. g, The stability distance d^∗ as shown in panel d, but for an alternative model formulation of chemotactic movement following Taylor & Stocker⁴⁰ with different functions describing sensing and the directed movement along gradients. See Supplementary Analysis 4 for more details. Despite the model changes, a linear relation with growth-rate dependent slopes were still observed. In b-g, the chemotactic coefficient c was varied to modify expansion speeds. The cellular diffusion coefficient, D, was varied accordingly, with D = c/6.25 remaining constant. Simulations with a fixed diffusion coefficient gave similar results. Experiments in b, c were repeated independently three times, obtaining similar results.