Mathematical Modeling of T-Cell Experimental Data

T lymphocytes (T cells) are key components of the adaptive immune system. These cells are able to recognize an enormous variety of pathogens thanks to the great specificity of their trans-membrane proteins, the T cell receptors (TCRs). TCR diversity is created during T cell maturation in the thymus by somatic gene-segment rearrangements and random nucleotide additions or deletions. Out of all possible T cell clones bearing specific TCRs, only a small fraction are successfully released in peripheral blood as the result of clonal selection. Among the selected clones, some self-reactive cells with the capacity to induce an auto-immune disease are erroneously released in periphery. To compensate for this functional flaw, the immune system has developed peripheral control mechanisms. One of them are regulatory T cells that are specialized in the control of harmful self-reactive clones. In this thesis, we combine mathematical modeling and experimental data to address immunological questions related to the dynamics of regulatory T cells and to the measurement of the structural diversity of T cell receptors. The dissertation is split into two main parts. In the first part, we model the lifelong dynamics of human regulatory T cells (Tregs). Despite their limited proliferation capacity, Tregs constitute a population maintained over the entire lifetime of an individual. The means by which Tregs sustain a stable pool in vivo are controversial. We define a novel mathematical model that we use to evaluate several biological scenarios about the origins and the proliferation capacity of two subsets of Tregs: precursor CD4+CD25+-CD45RO- and mature CD4+CD25+CD45RO+ cells. The lifelong dynamics of Tregs are described by a set of ordinary differential equations, driven by a stochastic process representing the major immune reactions involving these cells. Most of the parameters are considered as random variables having an a priori distribution. The likelihood of a scenario is estimated using Monte Carlo simulations. The model dynamics are validated with data from human donors of different ages. Analysis of the data led to the identification of two properties of the dynamics: (a) the equilibrium in the CD4+CD25+ Tregs population is maintained over both precursor and mature Tregs pools together, and (b) the ratio between precursor and mature Tregs is inverted in the early years of adulthood. Then, using the model, we identified four biologically relevant scenarios that have the above properties: (1) if the unique source of mature Tregs is the antigendriven differentiation of precursors that acquire the mature profile in the periphery, then the proliferation of Tregs is essential for the development and the maintenance of the pool; if there exist other sources of mature Tregs, such as (2) a homeostatic regulation, (3) a thymic migration, or (4) a peripheral conversion of effectors into Tregs, then the antigen-induced proliferation is not necessary for the development of a stable pool of Tregs. In the second part of the dissertation, we address the general question of TCR diversity by improving the interpretation of AmpliCot, an experimental technique that aims at the diversity measurement of nucleic acid sequences. This procedure has the advantage over other cloning and sequencing techniques of being time- and expense- effective. In short, a fluorescent dye that binds double-stranded DNA is added to a sample of PCR-amplified DNA. The sample is melted, such that the DNA becomes single-stranded, and then re-annealed under stringent conditions. The annealing kinetics, measured in terms of fluorescence intensity, are a function of the diversity and of the concentration of the sample and have been interpreted assuming second order kinetics. Using mathematical modeling, we show that a more detailed model, involving heteroduplex- and transient-duplex formation, leads to significantly better fits of experimental data. Moreover, the new model accounts for the diversity-dependent fluorescence loss that is typically observed. As a consequence, we show that the original method for interpreting the results of AmpliCot experiments should be applied with caution. We suggest alternative methods for diversity extrapolation of a sample.

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