Waves for a Hyperbolic Keller–segel Model and Branching Instabilities

Abstract

Recent experiments for swarming of the bacteria Bacillus subtilis on nutrient-rich media show that these cells are able to proliferate and spread out in colonies exhibiting complex patterns as dendritic ramifications. Is it possible to explain this process with a model that does not use local nutrient depletion? We present a new class of models which is compatible with the experimental observations and which predict branching instabilities and does not use nutrient limitation. These conclusions are based on numerical simulations. The most complex of these models is also the biologically most accurate but the essential effects can also be obtained in simplified versions which are amenable to analysis. An example of instability mechanism is the transition from a shock wave to a rarefaction wave in a reduced two by two hyperbolic system.

On the Asymptotic Theory From Microscopic to Macroscopic Growing Tissue Models: An Overview With Perspectives

“…37,38,45,81,83,95 However, in the previous phenomenological model (2.4) one of the most important features in the last years is related to the blow-up of solutions whose connection with real process behaviors needs to be explained, while following Refs. 33 and 43, one should design models where the propagation of cell populations is made by fronts and singularities.…”

Emerging morphologies in round bacterial colonies: comparing volumetric versus chemotactic expansion

“…Instabilities here rely on the introduction of a nonlinear diffusion coefficient (Kawasaki et al 1997;Mimura et al 2000) or to a limitation in the rate of transition from the bacteria active (i.e., motile and proliferative) state to the passive one (Matsushita et al 1998). Such models may also include a chemotactic term (Cerretti et al 2011;Marrocco et al 2010), combining signals from a chemorepellent and a chemoattractant (to prevent overcrowding, while keeping the cells together).…”

Travelling plateaus for a hyperbolic Keller–Segel system with attraction and repulsion: existence and branching instabilities

“…The logistic sensitivity µ c (1 − n/n max ) takes into account a volume filling (or quorum sensing) effect, i.e., a reduction of the cell response to the chemo-attractant (whose concentration is denoted by c), which prevents overcrowding [10,19,20]. The special form at hand has been proposed in [5] as a reduced model for a more detailed system to study complex patterns as the dendritic ramification of Bacillus subtilis, recently obtained with high nutrient experiments in [11,12,14], whereas pattern formation based on local nutrient depletion is also possible [16,8,17]. It includes a chemo-repellent of concentration S (that can be interpreted as the effect of surfactin) with a constant sensitivity µ S .…”

“…The repellent force ∇S can generate surprising dynamics of the plateaus and branching instabilities may occur. This was shown in [5], and the instability could be analyzed for D S small, because the limiting system with D S = 0 can be recast as an hyperbolic system according to a method introduced in [13]. Then, stability/instability of discontinuities can be seen as a transition from shock to rarefaction waves.…”