Cell-extracellular matrix interaction and the mechanical properties of cell nucleus have been demonstrated to play a fundamental role in cell movement across fibre networks and micro-channels and then in the spread of cancer metastases. The lectures will be aimed at presenting several mathematical models dealing with such a problem, starting from modelling cell adhesion mechanics to the inclusion of influence of nucleus stiffness in the motion of cells, through continuum mechanics, kinetic models and individual cell-based models.[-]

At the individual scale, bacteria as E. coli move by performing so-called run-and-tumble movements. This means that they alternate a jump (run phase) followed by fast re-organization phase (tumble) in which they decide of a new direction for run. For this reason, the population is described by a kinetic-Botlzmann equation of scattering type. Nonlinearity occurs when one takes into account chemotaxis, the release by the individual cells of a chemical in the environment and response by the population.

These models can explain experimental observations, fit precise measurements and sustain various scales. They also allow to derive, in the diffusion limit, macroscopic models (at the population scale), as the Flux-Limited Keller-Segel system, in opposition to the traditional Keller-Segel system, this model can sustain robust traveling bands as observed in Adler's famous experiment.

Furthermore, the modulation of the tumbles, can be understood using intracellular molecular pathways. Then, the kinetic-Boltzmann equation can be derived with a fast reaction scale. Long runs at the individual scale and abnormal diffusion at the population scale, can also be derived mathematically.[-]

In this talk, we present the construction of PDE model describing the evolution of microalgae or bacteria interacting together and in interaction with their environment. These models are based on the mixture theory and are coupled with reaction-diffusion equations or fluid mechanics equations. We begin with the description of micro-algae biofilms in fountains, then the description of photosynthetic biofilms of micro-algae producing lipids and finally, the evolution of gut microbiota in interaction with the colon rheology.[-]

We study the global existence of the parabolic-parabolic Keller–Segel system in $\mathbb{R}^{d}$. We prove that initial data of arbitrary size give rise to global solutions provided the diffusion parameter $\tau$ is large enough in the equation for the chemoattractant. This fact was observed before in the two-dimensional case by Biler, Guerra and Karch (2015) and Corrias, Escobedo and Matos (2014). Our analysis improves earlier results and extends them to any dimension d ≥ 3. Our size conditions on the initial data for the global existence of solutions seem to be optimal, up to a logarithmic factor in $\tau$ , when $\tau\gg 1$: we illustrate this fact by introducing two toy models, both consisting of systems of two parabolic equations, obtained after a slight modification of the nonlinearity of the usual Keller–Segel system. For these toy models, we establish in a companion paper finite time blowup for a class of large solutions.[-]