Non-linear analysis of a model for yeast cell communication

¹ Univ Lyon, CNRS, Université Claude Bernard Lyon 1, UMR5208, Institut Camille Jordan, 69622 Villeurbanne, France
² Univ Lyon, Inria, Université Claude Bernard Lyon 1, CNRS UMR5208, Institut Camille Jordan, 69603 Villeurbanne, France
³ LaMME, UMR 8071 CNRS & Université Évry Val d’Essonne, Évry Cedex, France
⁴ MAP5, CNRS UMR 8145, Université Paris Descartes, 45 rue des Saints Pères, 75006 Paris, France

^* Corresponding author: thomas.lepoutre@inria.fr

Received: 10 April 2019
Accepted: 29 August 2019

Abstract

We study the non-linear stability of a coupled system of two non-linear transport-diffusion equations set in two opposite half-lines. This system describes some aspects of yeast pairwise cellular communication, through the concentration of some protein in the cell bulk and at the cell boundary. We show that it is of bistable type, provided that the intensity of active molecular transport is large enough. We prove the non-linear stability of the most concentrated steady state, for large initial data, by entropy and comparison techniques. For small initial data we prove the self-similar decay of the molecular concentration towards zero. Informally speaking, the rise of a dialog between yeast cells requires enough active molecular transport in this model. Besides, if the cells do not invest enough in the communication with their partner, they do not respond to each other; but a sufficient initial input from each cell in the dialog leads to the establishment of a stable activated state in both cells.