A minimal mechanism for bacterial pattern formation

Abstract

Colonies of Escherichia coli or Salmonella typhimurium form geometrically complex patterns when exposed to, or feeding on, intermediates of the tricarboxylic acid (TCA) cycle. In response to the TCA cycle intermediate, the bacteria secrete aspartate, a potent chemo-attractant. As a result, the cells form highdensity aggregates arranged in striking regular patterns. The simplest are temporary spots formed in a liquid medium by both E. coli and S. typhimurium. In semi-solid medium S. typhimurium forms concentric rings arising from a low-density bacterial lawn, which are either continuous or spotted, whereas E. coli forms complex patterns arising from a dense swarm ring, including interdigitated spots (also called sun£ower spirals), radial spots, radial stripes and chevrons. We present a mathematical model that captures all three of the pattern-forming processes experimentally observed in both E. coli and S. typhimurium, using a minimum of assumptions.

A user’s guide to PDE models for chemotaxis

“…Note that the receptor-binding model formally reduces to the minimal model in the limit α → 0. The above "receptor" sensitivity law has been derived and applied in numerous models for chemotaxis [29,55,59,93,94,100]. By including further details of the signal transduction process, other forms of sensitivity dependence can be deduced: if co-operative binding occurs, then the concentration of bound receptors may more reasonably by described as a Hill function φ(v) = βv n αv n +1 ; the effects of internalisation of receptor-signal complexes has been incorporated by Sherratt et al [95,96]; an extension to multiple chemical species has been investigated in Painter et al [82]; memory effects have been included in a model by Boon and Herpigny [8].…”

Active Brownian particles

“…It is a continuous model of partial differential equations (PDEs) for the dynamics of the bacterial density ρ(r, t) and the concentration(s) of the involved chemical agent(s) c(r, t) (see e.g. [346,226,240,161]). …”

New Interior Penalty Discontinuous Galerkin Methods for the Keller–Segel Chemotaxis Model

“…A finite-volume, [21], and a finite-element, [34], methods have been proposed for a simpler version of the Keller-Segel model, ρ t + ∇ · (χρ∇c) = ∆ρ, ∆c − c + ρ = 0, in which the equation for concentration c has been replaced by an elliptic equation using an assumption that the chemoattractant concentration c changes over much smaller time scales than the density ρ. A fractional step numerical method for a fully time-dependent chemotaxis system from [41] has been proposed in [42]. However, the operator splitting approach may not be applicable when a convective part of the chemotaxis system is not hyperbolic, which is a generic situation for the original Keller-Segel model as it was shown in [12], where the finitevolume Godunov-type central-upwind scheme was derived for (1.1) and extended to some other chemotaxis and haptotaxis models.…”