A model aided understanding of spot pattern formation in chemotactic E. coli colonies

Abstract

Colonies of mutant E. coli strains, when inoculated in the centre of a plate, form highly symmetric, stable spot patterns. These spot patterns are only observed in chemotactic E. coli strains, i.e. strains that bias their motion so that cell populations move up gradients of a chemical in the cells' local environment. It is an important question whether these patterns are due to genetic control or self-organization. Here we present a macroscopic continuum model of E. coli pattern formation that incorporates cell diffusion, chemotaxis, population growth and conversion to an inactive state. This model satisfactorily reproduces the observed spot patterns, supporting the view that these patterns are indeed a result of self-organization, and allows us to infer plausible minimal mechanisms that generate the observed patterns.

Blow-up of weak solutions to a chemotaxis system under influence of an external chemoattractant

“…In particular, since W (ε) ր W as ε ց 0, we have 1]. This inequality at hand we can now verify that by choosing γ sufficiently large (3.8) is indeed fulfilled.…”

Chemotaxis can prevent thresholds on population density

“…Lemma 26. Let κ ≥ 0, µ > 0, q > n. Then for D > 0 there is some T (D) > 0 such that for any radial symmetric nonnegative u 0 ∈ W 1,q (Ω) fulfilling u 0 W 1,q (Ω) < D there is a unique W 1,q (Ω)-solution (u, v) of (2) in Ω × (0, T (D)). Furthermore, if u 0ε are compatible functions satisfying u 0ε − u 0 W 1,q (Ω) < ε, this solution (u, v) can be approximated by solutions (u ε , v ε ) of (1) (with initial condition u 0ε ) in the following sense:…”

“…30]. However, if this is not the case, any radial solution to (2) with somehow (L p -)large enough initial mass blows up in finite time (Theorem 33). In combination with the fact that solutions to (2) can be obtained as limits of solutions to (1), this yields the announced theorem about nonexistence of thresholds to population density: If µ < 1 and u 0 L p (for p > 1 1−µ ) is large enough, before some time T any threshold on the population density will be exceeded at least at one point by any population that diffuses slowly enough.…”

“…After the following short section which recalls a few basic properties of solutions to the second equation in (1) and equation (1) with ε > 0, in Section 3 we focus our attention on existence of solutions to (1) and estimates yielding a common existence time (Theorem 19) as well as preparing for compactness arguments (by the estimates of Lemma 20 and Corollary 17). Section 4 will be devoted to definition, uniqueness, existence, estimates and a blow-up result for solutions to (2), followed in Section 5 by, finally, the proof of the "no threshhold" theorem 1.…”

Pattern formation in the wake of triggered pushed fronts

“…Many examples of this triggered pattern formation arise in systems with mass-conserving properties. Typical model equations for such systems are the Cahn-Hilliard equation [15,29,30,41], the Keller-Segel model for chemotaxis [1,33], reaction-diffusion systems [32], or phase-field systems [14,18,19]. Other examples arise in ion-bombardment studies [17] and are modeled by Kuramoto-Shivashinsky-type models, while still others arise when studying general osciliatory instabilities using real and complex Ginzburg-Landau models [4,6,7].…”