Existence and stability of periodic solutions of an impulsive differential equation and application to CD8 T-cell differentiation.

Unequal partitioning of the molecular content at cell division has been shown to be a source of heterogeneity in a cell population. We propose to model this phenomenon with the help of a scalar, nonlinear impulsive differential equation (IDE). To study the effect of molecular partitioning at cell division on the effector/memory cell-fate decision in a CD8 T-cell lineage, we study an IDE describing the concentration of the protein Tbet in a CD8 T-cell, where impulses are associated to cell division. We discuss how the degree of asymmetry of molecular partitioning can affect the process of cell differentiation and the phenotypical heterogeneity of a cell population. We show that a moderate degree of asymmetry is necessary and sufficient to observe irreversible differentiation. We consider, in a second part, a general autonomous IDE with fixed times of impulse and a specific form of impulse function. We establish properties of the solutions of that equation, most of them obtained under the hypothesis that impulses occur periodically. In particular, we show how to investigate the existence of periodic solutions and their stability by studying the flow of an autonomous differential equation. Then we apply those properties to prove the results presented in the first part.