Mathematical models of cell signalling in heterogeneous populations

Immune cells express thousands of receptors on their membrane surface to sense their environment and communicate with each other. Receptors bind specifically to extra-cellular molecules called ligands. The binding of a ligand to its receptor initiates an intra-cellular signalling cascade which leads to the control of cellular fate, such as division, death, migration or differentiation. As every cell expresses a different number of receptors, each cell responds differently to a given ligand. First motivated by seemingly paradoxical experimental observations on the interleukin-7/interleukin-7 receptor (IL-7/IL-7R) receptor-ligand system, this thesis investigates how receptor copy numbers impact the cell's response, as measured by the amplitude and the half-maximal effective concentration (or EC50). In particular, deterministic mathematical models of various receptor-ligand systems are developed. For each model, making use of algebraic tools, such as Grobner bases, analytic expressions for the amplitude and the EC50 are computed. Such expressions allow one to identify precisely how a cell's response depends on the receptor core structure, namely receptor chain copy numbers and receptor architecture. They also reduce numerical errors and facilitate parameter inference, as demonstrated by the fitting of two IL-7R models to the motivating experimental data set. The results obtained are generalised to a larger family of receptor-ligand systems, for which the amplitude is computed without the use of advanced algebraic tools. Finally, as the immune system relies on the coordination of many cells to fight pathogens, the complex relationship between the cell population dynamics and the receptor copy number distribution in the cellular population is examined. To this end, agent-based models of increasing complexity, which model the competition for interleukin-2 (IL-2) within the T cell population, are constructed, by adding stochastic cellular events one at a time. A mathematical description of each model is provided, which enables us, when possible, to derive the desired receptor copy number distribution (in this case for the IL-2 receptor).