Extended Data Fig. 5: Population expansion without attractant is quantitatively captured by Fisher–Kolmogorov dynamics.

The Fisher–Kolmogorov dynamics is a canonical model to describe the dynamics of expanding populations^19,20. It has been successfully used to investigate the expansion and evolution of non-moving bacteria at the front of dense bacterial colonies^{46,71,72,73,74}. Here, we probe the Fisher–Kolmogorov dynamics and its validity to describe swimming bacteria. The Fisher–Kolmogorov dynamics is driven by population growth and undirected random motion (diffusion)^32,33. To compare the predictions of Fisher–Kolmogorov dynamics to the expansion of a bacterial population in the absence of a chemoattractant, we thus independently quantified growth rates and cellular diffusion for cells homogeneously distributed in soft agar (a, b). We then compared the observed migration speed and the density profile of the migrating population (for growth on glycerol as the sole carbon source, as in Fig. 2d, top) with the Fisher–Kolmogorov predictions (c, d). a, Quantification of growth by measuring the temporal density increase of a homogeneously distributed population in agar (Extended Data Fig. 4a–c, Supplementary Text 1.4). Spatially averaged density increased exponentially with growth rate λ = 0.59 h⁻¹ for densities <0.1 OD₆₀₀. For higher densities, the growth rate decreased but this regime is not important for the propagation of the front where density is low. b, Diffusion and drift of cells homogeneously distributed in soft agar. Analysis of recorded cell movement confirms the variance of position displacement to increase linearly in time (orange symbols) with diffusion constant D = 41.5 μm² s⁻¹ (linear fit of var(x) = 2DΔt). In comparison, the average displacement of cells (purple symbols) and the calculated drift (⟨Δx/Δt⟩, purple line) are small, indicating the absence of directed chemotactic movement. Data show average over three independent repeats (Extended Data Fig. 3, Supplementary Text 1.5). c, d, Front and spatiotemporal dynamics of an expanding population. c, Comparison of predicted expansion speed with the observed propagation of the population front. Position of the front R(t) was determined from the observed cellular densities (threshold OD₆₀₀ < 0.005); it increased linearly in time, that is, R(t) = u_obs t with a speed u_obs = 0.62 mm h⁻¹. Dashed line denotes predicted expansion speed calculated as \({u}_{{\rm{FK}}}=2\sqrt{\lambda \times D}=0.59\,{\rm{mm}}\,{{\rm{h}}}^{-1}\). d, Density profile of the population front. Observed density profile can be fitted to an exponential dependence \(\rho (r,t)\sim {{\rm{e}}}^{-{k}_{{\rm{o}}{\rm{b}}{\rm{s}}}(r-R(t))}\) with k_obs ≈ 1.2 mm⁻¹. Dashed line indicates the slope of the exponential density profile predicted by the Fisher–Kolmogorov equation: \({k}_{{\rm{F}}{\rm{K}}}=\sqrt{\lambda /D}=1.99\,{{\rm{m}}{\rm{m}}}^{-1}\). The discrepancy is likely to result from the low spatial resolution of the very sharp density drop; the exponential dependence of the experimental profile is defined by just three points. All experiments were conducted once with strain HE274 (wild type), using glycerol as the carbon source (no additional attractant, glycerol cannot be sensed). Growth and cell-tracking experiments were performed with uniform cell mixture in saturating glycerol conditions (40 mM). Expansion experiments were performed with 1 mM glycerol.

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