  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). In a first part, we consider 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. A second part is dedicated to the analysis of
the convexity of this flow. Finally, we apply those results to an IDE
describing the concentration of the protein Tbet in a CD8 T-cell, where
impulses are associated to cell division, to study the effect of molecular
partitioning at cell division on the effector/memory cell-fate decision in a
CD8 T-cell lineage. We show that the degree of asymmetry in the molecular
partitioning can affect the process of cell differentiation and the
phenotypical heterogeneity of a cell population.
