Chemotaxis migration and morphogenesis of living colonies

Abstract

Abstract. Development of forms in living organisms is complex and fascinating. Morphogenetic theories that investigate these shapes range from discrete to continuous models, from the variational elasticity to time-dependent fluid approach. Here a mixture model is chosen to describe the mass transport in a morphogenetic gradient: it gives a mathematical description of a mixture involving several constituents in mechanical interactions. This model, which is highly flexible can incorporate many biological processes but also complex interactions between cells as well as between cells and their environment. We use this model to derive a free-boundary problem easier to handle analytically. We solve it in the simplest geometry: an infinite linear front advancing with a constant velocity. In all the cases investigated here as the 3D diffusion, the increase of mitotic activity at the border, nonlinear laws for the uptake of morphogens or for the mobility coefficient, a planar front exists above a critical threshold for the mobility coefficient but it becomes unstable just above the threshold at long wavelengths due to the existence of a Goldstone mode. This explains why sparsely bacteria exhibit dendritic patterns experimentally in opposition to other colonies such as biofilms and epithelia which are more compact. In the most unstable situation, where all the laws: diffusion, chemotaxis driving and chemoattractant uptake are linear, we show also that the system can recover a dynamic stability. A second threshold for the mobility exists which has a lower value as the ratio between diffusion coefficients decreases. Within the framework of this model where the biomass is treated mainly as a viscous and diffusive fluid, we show that the multiplicity of independent parameters in real biologic experimental set-up may explain varieties of observed patterns.

On growth and form ofBacillus subtilisbiofilms

“…Because mass reorganization is strongly hindered in this case, we have developed a biomechanical model for an incompressible biomass where the motion of the cells is overdamped. Note that the biomechanical model formed by (6.3) and (6.4) corresponds to the sharp interface limit of the more general two-phase mixture theory also used in modelling tumour growth [57,58]. A model built upon this theory has also been recently developed to describe the early stages of biofilm growth (when there could be a radial gradient of ECM concentration) [41] but requires a bacterial doubling time of 150 min to fit the experimental data.…”

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

“…6c). However, although disk-like colonies can be mathematically recovered by increasing σ , this would correspond to a ratio between the surface tension of the colony and the friction between the bacteria and the substrate that seems out of a biologically admissible range, for the specific biological values reported for eukaryotic cells (Ben Amar 2013;Ziebert and Aranson 2013). Concerning the parameters set in the simulation in Fig.…”

Re-epithelialization: advancing epithelium frontier during wound healing

“…At extremely low velocities, the cell migration satisfies a Darcy's law , M p being a porosity coefficient equal to the square of the epithelium height divided by a friction coefficient. As shown in [18], this law is deduced in tissues when friction between phases or friction with a substrate balances the hydrostatic part of the elastic stress acting on the cells [18,20]. The wound perturbs the homoeostatic state of the surrounding and a source of morphogens will try to restore an optimal equilibrium value in the aperture c 0 .…”

“…The latter is continuously supplied by an incoming flux from the third dimension normal to the epithelial layer and uptaken by cell receptors. As shown in [18], the thinness of the moving layer transforms the three-dimensional process into a two-dimensional model, where the localized thickness variation at the border contributes to a tension T . In addition, we assume that a strong viscous friction exists between the moving layer and the substrate [19–23], which allows writing a Darcy's law for the average velocity field inside the tissue, while cell–cell interactions and mitosis are transformed into interface boundary conditions in the sharp interface limit.…”