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Publication# Mathematical Modeling of T-Cell Experimental Data

Résumé

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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Irina Baltcheva, Jean-Yves Le Boudec

Despite their limited proliferation capacity, regulatory T cells (T(regs)) constitute a population maintained over the entire lifetime of a human organism. The means by which T(regs) sustain a stable pool in vivo are controversial. Using a mathematical model, we address this issue by evaluating several biological scenarios of the origins and the proliferation capacity of two subsets of T(regs): precursor CD4(+)CD25(+)CD45RO(-) and mature CD4(+)CD25(+)CD45RO(+) cells. The lifelong dynamics of T(regs) are described by a set of ordinary differential equations, driven by a stochastic process representing the major immune reactions involving these cells. The model dynamics are validated using data from human donors of different ages. Analysis of the data led to the identification of two properties of the dynamics: (1) the equilibrium in the CD4(+)CD25(+)FoxP3(+)T(regs) population is maintained over both precursor and mature T(regs) pools together, and (2) the ratio between precursor and mature T(regs) is inverted in the early years of adulthood. Then, using the model, we identified three biologically relevant scenarios that have the above properties: (1) the unique source of mature T(regs) is the antigen-driven differentiation of precursors that acquire the mature profile in the periphery and the proliferation of T(regs) is essential for the development and the maintenance of the pool; there exist other sources of mature T(regs), such as (2) a homeostatic density-dependent regulation or (3) thymus- or effector-derived T(regs), and in both cases, antigen-induced proliferation is not necessary for the development of a stable pool of T(regs). This is the first time that a mathematical model built to describe the in vivo dynamics of regulatory T cells is validated using human data. The application of this model provides an invaluable tool in estimating the amount of regulatory T cells as a function of time in the blood of patients that received a solid organ transplant or are suffering from an autoimmune disease.

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Noise analysis applied to nuclear reactor physics is a powerful tool to investigate a reactor's kinetic parameters, and more generally underlying physical processes determining core behavior. The kinetic parameters are the coefficients of simplified time-dependent neutron population equations, the so-called point kinetics equations. Experimentally determined kinetic parameters aid to validate codes and to potentially evaluate nuclear data. This thesis focuses on improvements to the kinetic parameter and uncertainty quantification, the experimental techniques, and the direct simulation of noise experiments. In particular, the notion of spatial effects, i.e. effects that render point kinetics assumptions inaccurate for noise measurements, was investigated. This was achieved by drawing on experiments conducted in the zero power reactor CROCUS.
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The present paper is the third and last part of an investigation on what determines reliability in fuel cell model identification. In continuation to the effect of experimental design (Part I) and a process method for stochastic calculation of a model's parameters (Part II), this paper concentrates on the assessment of a model validation. Four criteria are examined. The fit of the model's output to experimental data, the determinant of the covariance matrix of the parameters, the determinant of their correlation matrix, and the product of their variances. As regards the fit to the data, results show that this is mainly a function of the number of measurement points. Repetitions do not seem to improve the average of the fit significantly, but it does improve its variation. For the other three criteria, which are also mathematically linked, results show a counterbalance between them, leading to the conclusion that they cannot be optimized simultaneously. This happens especially between the determinants of the covariance and the correlation matrices.