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.