Analysis of a non local model for spontaneous cell polarisation

Vincent Calvez (1, 2) , Rhoda Hawkins (3) , Nicolas Meunier (4) , Raphael Voituriez (5)

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées
2 NUMED - Numerical Medicine
3 Department of Mathematics [Bristol]
4 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5
5 LPTMC - Laboratoire de Physique Théorique de la Matière Condensée

In this work, we investigate the dynamics of a non-local model describing spontaneous cell polarisation. It consists in a drift-diffusion equation set in the half-space, with the coupling involving the trace value on the boundary. We characterize the following behaviours in the one-dimensional case: solutions are global if the mass is below the critical mass and they blow-up in finite time above the critical mass. The higher-dimensional case is also discussed. The results are reminiscent of the classical Keller-Segel system in double the dimension. In addition, in the one-dimensional case we prove quantitative convergence results using relative entropy techniques. This work is complemented with a more realistic model that takes into account dynamical exchange of molecular content at the boundary. In the one-dimensional case we prove that blow-up is prevented. Furthermore, density converges towards a non trivial stationary configuration.

Mots clés: Cell polarisation; global existence; blow-up; asymptotic convergence; entropy method; Keller-Segel system

Domaines: Equations aux dérivées partielles [math.AP]

Vincent Calvez, Rhoda Hawkins, Nicolas Meunier, Raphael Voituriez. Analysis of a non local model for spontaneous cell polarisation. 2011. ⟨hal-00594777⟩