Stochastic compartmental models and CD8+ T cell exhaustion

In this PhD thesis, mathematical models for cell differentiation are presented. Cell differentiation is a widely observed process in cellular biology allowing a small pool of not specialised cells to develop and maintain a bigger population of cells with a specific function. Different mathematical techniques are employed in this thesis, to study cell differentiation process. We propose a time-independent stochastic mathematical model to represent a general differentiation process via a sequence of compartments. Since we are interested in the ultimate fate of the system, we define a discrete-time branching processes and consider the impact, on the final population, of cells passing through only one or multiple compartments. Further, we include time dependency and define a continuous-time Markov chain to analyse cells dynamics along the sequence of compartments over time. Also, we focus on the journey of a single cell over time and compute a number of summary statistics of interest. Moreover, the impact of different types of differentiation events is considered and numerical results inspired by biological applications, mainly related to immunology, are
summarised to illustrate our theoretical approach and methods. In the last Chapter, we focus on a specific cell differentiation process: cells of the immune system have been observed to differentiate towards a dysfunctional state, called exhaustion, during a chronic infection or cancer. One of the aims of this PhD thesis is to shed light into the exhaustion-differentiation process of CD8+
T cells and its reversibility which is a topic of interest for the current and future development of immunotherapies. In particular, based on data collected by the Kaech Lab, several deterministic mathematical models are defined to investigate cells’ trajectory towards the exhausted state as well as the duration of the antigen signal at early time point of stimulation.