Deterministic and stochastic naive T cell population dynamics: symmetric and asymmetric cell division

Abstract

There is growing evidence that asymmetric cell division has a key role in fate decision during T cell responses, however it is unknown if this process also has a role in the normal homeostasis of the naive T cell population. In order to explore asymmetric cell division, we develop a mathematical model in which naive T cells exist in one of two states: a resting and a cycling state. We consider three variants of the model, differing in the state of daughter cells produced by cell division, and study the steady states and time evolution of the populations in each case. We begin with a deterministic model: two coupled ordinary differential equations for the two cell populations (resting and cycling). We then introduce a stochastic generalization of the model and show, by means of the appropriate moment generating function, that the equations of the deterministic model are those of the mean sizes of the populations. The stochastic model allows us to explore questions the deterministic model fails to address, such as extinction of the population and expected lifetimes of a naive T cell clone, when the number of cells is small. Finally, we consider a multi-variate Markov process, in which each cell is classified according to its generation, or number of divisions, up to a maximum number of generations, consistent with the detection limit of fluorescent labelling techniques. Immunologically, this is a challenging question to address, and our results indicate that the mathematical models developed allow us to discern between symmetric and asymmetric division scenarios. These results can be, in principle, experimentally tested and we provide a brief description of the experimental procedure that will allow to determine the relative role of symmetric versus asymmetric cell division in naive T cell homeostasis, without directly observing this process in vivo.

Keywords:

• immunology
• T-cell dynamics
• stochastic processes

Notes

Notes

1. A T cell clonotype is the subset of all T cells that express identical TCR molecules on their membranes.

2. Division is asymmetric if the two daughter cells have different phenotypes. In this article, asymmetric division is as follows: one daughter cell belongs, immediately after birth, to the resting naive T cell pool, whereas the second daughter cell belongs to the cycling phenotype.

3. In this article we model the naive T cell population as a whole, thus the inherent heterogeneity of the naive T cell pool, due to TCR diversity of the different clonotypes, is being neglected. Models of naive T cell homeostasis that include TCR diversity also exist 33–35.

4. Carboxyfluorescein succinimidyl ester (CFSE) is a fluorescent cell-staining dye. The dye can be used to monitor lymphocyte proliferation, both in vitro and in vivo, due to the halving of CFSE fluorescence within daughter cells at each cell division 25.

5. Here a state is defined as either resting or cycling and by its generation: g ∈ {0, 1, … , G − 1, G}.