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

Abstract

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.

A unifying model for the propagation of prion proteins in yeast brings insight into the [PSI+] prion

“…In each case (mother system or daughter system), two 108 different periodic solutions are locally stable, a prion-free periodic solution and a 109 [P SI + ] periodic solution. A similar behavior was proven with a one-dimensional 110 bi-stable equation in [21], but our results are only numerical so far. Those solutions are 111 denoted on the phenotype map in Fig 4, corresponding to the first point in each periodic 112 impulsion.…”

“…The existence and the properties of such solutions are studied 261 theoretically [37,38]. IDEs are used in various modeling fields, including 262 epidemiology [39], population dynamics [40] and more recently T-cell differentiation [21]. 263 In order to model cell division with IDEs, we need to clarify how species are 264 distributed between mother and daughter cells.…”

Model-based assessment of the Role of Uneven Partitioning of Molecular Content on Heterogeneity and Regulation of Differentiation in CD8 T-cell Immune Responses

“…Proposition 1 ( [18]). Assume f AP C = 0, n > 1 and λ T 2 (n − 1) n−1 n > nk T λ T 3 , then equation (3) has exactly three non-negative steady states:…”

“…As discussed in the introduction, the concentration of protein Tbet can be associated to the level of differentiation of an effector CD8 T-cell, with high level of Tbet correlating with fully differentiated effector cell, while low Tbet levels are associated to memory precursor effector cells. Proposition 1 below, reproduced from [18], gives necessary and sufficient conditions to allow bistable behaviour of Tbet concentration.…”

“…In a recent work (Girel and Crauste [18]), we studied how stochastic uneven molecular partitioning, repeated at each cell division, could regulate the effector versus memory cell-fate decision in a CD8 T-cell lineage. To do so, we analysed an impulsive differential equation describing the concentration of the protein Tbet in a CD8 T-cell subject to divisions, where impulses were associated with uneven partitioning of Tbet.…”