Extended Data Fig. 6: Different models of chemotaxis-driven migration.
From: Chemotaxis as a navigation strategy to boost range expansion
To illustrate the difference among various models of chemotactic expansion, we show here simulation results of four different models. a, The classical model proposed by Keller and Segel34 creates a self-generated attractant gradient owing to attractant consumption by the migrating population. It neglects cell growth (that is, λ = 0 in equation (3) in Fig. 3a), resulting in conservation of the total number of bacteria. It also assumes that the attractant gradient could be detected with infinite precision, such that log-sensing (Weber’s law75) can be implemented by cells down to arbitrary low attractant concentrations, (that is, equation (4) with a− = 0). The latter biologically unrealistic assumption introduces a singularity that pushes all bacteria forward at a steady migration speed, which is determined by the number of cells in the population, the conserved quantity. b, The model introduced by Novick-Cohen and Segel36 fixed the singularity in the Keller–Segel model by imposing a minimal concentration for the sensing of attractant gradient (that is, equation (4) with a− > 0). Owing to the lack of cell growth, the total number of bacteria is still conserved. In this model, the density of the front bulge decays over time because once bacteria diffuse out of the front, they lose the chemotactic gradient and cannot catch up with the front. The reduction in front density reduces the migration speed, which decays steadily towards zero. c, Model including cell growth that depends on attractant concentration (nutrient = attractant). Owing to growth, population size increases over time. However, as the attractant (nutrient) is mostly consumed at the front, there is not much growth behind the front and the trailing region behind the front is mostly flat. This scenario has been realized and analysed experimentally18; see Extended Data Fig. 8 for model details and discussion. d, The GE formulated in this study (Fig. 3a), including the chemotactic effect of an attractant, together with cell growth supplied by a major nutrient source. Front propagation of cells by chemotaxis is coupled to steady growth in the trailing region (see main text and Extended Data Fig. 9). Parameter values for all models are provided in Supplementary Table 8. For simplicity, simulations shown here were solved in one dimension (non-radial). Green lines denote bacteria density, blue lines denote attractant (or sole nutrient) concentration, brown lines denote concentration of nutrients (in addition to the attractant), purple lines show local drift (equation (4)).
