A traveling-wave solution for bacterial chemotaxis with growth

Abstract

Bacterial cells navigate their environment by directing their movement along chemical gradients. This process, known as chemotaxis, can promote the rapid expansion of bacterial populations into previously unoccupied territories. However, despite numerous experimental and theoretical studies on this classical topic, chemotaxis-driven population expansion is not understood in quantitative terms. Building on recent experimental progress, we here present a detailed analytical study that provides a quantitative understanding of how chemotaxis and cell growth lead to rapid and stable expansion of bacterial populations. We provide analytical relations that accurately describe the dependence of the expansion speed and density profile of the expanding population on important molecular, cellular, and environmental parameters. In particular, expansion speeds can be boosted by orders of magnitude when the environmental availability of chemicals relative to the cellular limits of chemical sensing is high. Analytical understanding of such complex spatiotemporal dynamic processes is rare. Our analytical results and the methods employed to attain them provide a mathematical framework for investigations of the roles of taxis in diverse ecological contexts across broad parameter regimes.

Influence of confinement on the spreading of bacterial populations

“…7C-and thus, the population spreads less effectively. This diffusive scaling of x l is at odds with the prediction of the classic Fisher-KPP model, commonly used to describe growth-driven spreading, that the population spreads at a constant speed as a traveling wave [27,42]. Instead, our finding is consistent with the results of agent-based simulations of a growing population of jammed, incompressible cells [41], which also found ν ≈ 0.5 in the limit of fast nutrient consumption.…”

“…Recent extensions of this model have also considered the case in which nutrient and attractant are separate chemical species, which leads to fundamentally different behavior that would be interesting to explore using our framework in future work [27,42].…”

Chemotactic smoothing of collective migration

“…Together, Equations 1 and 2 provide a continuum model of chemotactic migration that has thus far been successfully used to describe unperturbed E. coli populations ( Keller and Segel, 1971 ; Fu et al, 2018 ; Cremer et al, 2019 ; Bhattacharjee et al, 2021 ). We note that a recently introduced growth-expansion model of chemotactic migration, for which analytical expressions describing chemotactic fronts have been obtained ( Cremer et al, 2019 ; Narla et al, 2021 ), can be thought of as a limit of our model with bacterial growth taken to be independent of the attractant. An interesting direction for future work would be to study the phenomenon of chemotactic smoothing revealed here in the growth-expansion model, similar to a recent analytical study of small-amplitude perturbations in a simplified version of the model considered here ( Alert and Datta, 2021 ).…”

Influence of confinement on the spreading of bacterial populations

“…Instead, the leading edge position progresses as x l ∼ t ν with ν ≈ 0.5 at long times, as shown by the solid curve in Fig 7C —and thus, the population spreads less effectively. This diffusive scaling of x l is at odds with the prediction of the classic Fisher-KPP model, commonly used to describe growth-driven spreading, that the population spreads at a constant speed as a traveling wave [ 27 , 47 ]. Instead, our finding is consistent with the results of agent-based simulations of a growing population of jammed, incompressible cells [ 41 ], which also found ν ≈ 0.5 in the limit of fast nutrient consumption.…”

“…Furthermore, given the experimental conditions, we assume that the cells do not excrete their own chemoattractant or other diffusible signals, as is sometimes the case in low-nutrient conditions and for specific strains. Recent extensions of this model have also considered the case in which nutrient and attractant are separate chemical species, which leads to fundamentally different behavior that would be interesting to explore using our framework in future work [ 27 , 47 ].…”