The limiting conditional probability distribution in a stochastic model of T cell repertoire maintenance

Quantifying T Cell Cross-Reactivity: Influenza and Coronaviruses

If viral strains are sufficiently similar in their immunodominant epitopes, then populations of cross-reactive T cells may be boosted by exposure to one strain and provide protection against infection by another at a later date. This type of pre-existing immunity may be important in the adaptive immune response to influenza and to coronaviruses. Patterns of recognition of epitopes by T cell clonotypes (a set of cells sharing the same T cell receptor) are represented as edges on a bipartite network. We describe different methods of constructing bipartite networks that exhibit cross-reactivity, and the dynamics of the T cell repertoire in conditions of homeostasis, infection and re-infection. Cross-reactivity may arise simply by chance, or because immunodominant epitopes of different strains are structurally similar. We introduce a circular space of epitopes, so that T cell cross-reactivity is a quantitative measure of the overlap between clonotypes that recognize similar (that is, close in epitope space) epitopes.

Quantifying the Role of Stochasticity in the Development of Autoimmune Disease

In this paper, we propose and analyse a mathematical model for the onset and development of autoimmune disease, with particular attention to stochastic effects in the dynamics. Stability analysis yields parameter regions associated with normal cell homeostasis, or sustained periodic oscillations. Variance of these oscillations and the effects of stochastic amplification are also explored. Theoretical results are complemented by experiments, in which experimental autoimmune uveoretinitis (EAU) was induced in B10.RIII and C57BL/6 mice. For both cases, we discuss peculiarities of disease development, the levels of variation in T cell populations in a population of genetically identical organisms, as well as a comparison with model outputs.

Stochastic dynamics in a time-delayed model for autoimmunity

In this paper we study interactions between stochasticity and time delays in the dynamics of immune response to viral infections, with particular interest in the onset and development of autoimmune response. Starting with a deterministic time-delayed model of immune response to infection, which includes cytokines and T cells with different activation thresholds, we derive an exact delayed chemical master equation for the probability density. We use system size expansion and linear noise approximation to explore how variance and coherence of stochastic oscillations depend on parameters, and to show that stochastic oscillations become more regular when regulatory T cells become more effective at clearing autoreactive T cells. Reformulating the model as an Itô stochastic delay differential equation, we perform numerical simulations to illustrate the dynamics of the model and associated probability distributions in different parameter regimes. The results suggest that even in cases where the deterministic model has stable steady states, in individual stochastic realisations, the model can exhibit sustained stochastic oscillations, whose variance increases as one gets closer to the deterministic stability boundary. Furthermore, in the regime of bi-stability, whereas deterministically the system would approach one of the steady states (or periodic solutions) depending on the initial conditions, due to the presence of stochasticity, it is now possible for the system to reach both of those dynamical states with certain probability. Biological significance of this result lies in highlighting the fact that since normally in a laboratory or clinical setting one would observe a single individual realisation of the course of the disease, even for all parameters characterising the immune system and the strength of infection being the same, there is a proportion of cases where a spontaneous recovery can be observed, and similarly, where a disease can develop in a situation that otherwise would result in a normal disease clearance.

The expanding role of systems immunology in decoding the T cell receptor repertoire

T cells play a crucial role in the immune system's defense against many infectious diseases, including persistent infections for which no effective vaccines currently exist. The T cell component of the adaptive immune system is highly complex involving a constantly evolving landscape of various inter-related T cell populations. These T cell populations are characterized by their phenotypic and functional properties as well as the collection, or repertoire, of T cell receptors (TCR) that mediate T cell recognition of antigenic peptides derived from pathogens. Understanding the various processes and factors that impact the development and evolution of the broader T cell repertoire available to recognize and respond to pathogens and the characteristics of antigen-experienced T cell repertoires associated with effective immune control of pathogens is critical to the rational design of T cell-based vaccines and therapies. In this article we discuss, using examples of recent research, the promise that systems immunology approaches, involving quantitative analysis and mathematical and computational modeling of immunological data, hold for decoding the complex TCR repertoire system in the current era of advancing technologies.

Stochastic Effects in Autoimmune Dynamics

Among various possible causes of autoimmune disease, an important role is played by infections that can result in a breakdown of immune tolerance, primarily through the mechanism of “molecular mimicry”. In this paper we propose and analyse a stochastic model of immune response to a viral infection and subsequent autoimmunity, with account for the populations of T cells with different activation thresholds, regulatory T cells, and cytokines. We show analytically and numerically how stochasticity can result in sustained oscillations around deterministically stable steady states, and we also investigate stochastic dynamics in the regime of bi-stability. These results provide a possible explanation for experimentally observed variations in the progression of autoimmune disease. Computations of the variance of stochastic fluctuations provide practically important insights into how the size of these fluctuations depends on various biological parameters, and this also gives a headway for comparison with experimental data on variation in the observed numbers of T cells and organ cells affected by infection.

Stochastic descriptors to study the fate and potential of naive T cell clonotypes in the periphery

The population of naive T cells in the periphery is best described by determining both its T cell receptor diversity, or number of clonotypes, and the sizes of its clonal subsets. In this paper, we make use of a previously introduced mathematical model of naive T cell homeostasis, to study the fate and potential of naive T cell clonotypes in the periphery. This is achieved by the introduction of several new stochastic descriptors for a given naive T cell clonotype, such as its maximum clonal size, the time to reach this maximum, the number of proliferation events required to reach this maximum, the rate of contraction of the clonotype during its way to extinction, as well as the time to a given number of proliferation events. Our results show that two fates can be identified for the dynamics of the clonotype: extinction in the short-term if the clonotype experiences too hostile a peripheral environment, or establishment in the periphery in the long-term. In this second case the probability mass function for the maximum clonal size is bimodal, with one mode near one and the other mode far away from it. Our model also indicates that the fate of a recent thymic emigrant (RTE) during its journey in the periphery has a clear stochastic component, where the probability of extinction cannot be neglected, even in a friendly but competitive environment. On the other hand, a greater deterministic behaviour can be expected in the potential size of the clonotype seeded by the RTE in the long-term, once it escapes extinction.

How many TCR clonotypes does a body maintain?

We consider the lifetime of a T cell clonotype, the set of T cells with the same T cell receptor, from its thymic origin to its extinction in a multiclonal repertoire. Using published estimates of total cell numbers and thymic production rates, we calculate the mean number of cells per TCR clonotype, and the total number of clonotypes, in mice and humans. When there is little peripheral division, as in a mouse, the number of cells per clonotype is small and governed by the number of cells with identical TCR that exit the thymus. In humans, peripheral division is important and a clonotype may survive for decades, during which it expands to comprise many cells. We therefore devise and analyse a computational model of homeostasis of a multiclonal population. Each T cell in the model competes for self pMHC stimuli, cells of any one clonotype only recognising a small fraction of the many subsets of stimuli. A constant mean total number of cells is maintained by a balance between cell division and death, and a stable number of clonotypes by a balance between thymic production of new clonotypes and extinction of existing ones. The number of distinct clonotypes in a human body may be smaller than the total number of naive T cells by only one order of magnitude.

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The number of T cells of one clonotype is an integer.

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The history of a clonotype starts with release from the thymus, and ends with extinction.

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Competition and cross-reactivity are included in a natural way.

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The average number of cells per clonotype, in a human body, is only of order 10.

Assessing T cell clonal size distribution: a non-parametric approach

Clonal structure of the human peripheral T-cell repertoire is shaped by a number of homeostatic mechanisms, including antigen presentation, cytokine and cell regulation. Its accurate tuning leads to a remarkable ability to combat pathogens in all their variety, while systemic failures may lead to severe consequences like autoimmune diseases. Here we develop and make use of a non-parametric statistical approach to assess T cell clonal size distributions from recent next generation sequencing data. For 41 healthy individuals and a patient with ankylosing spondylitis, who undergone treatment, we invariably find power law scaling over several decades and for the first time calculate quantitatively meaningful values of decay exponent. It has proved to be much the same among healthy donors, significantly different for an autoimmune patient before the therapy, and converging towards a typical value afterwards. We discuss implications of the findings for theoretical understanding and mathematical modeling of adaptive immunity.

Mathematical Model of Naive T Cell Division and Survival IL-7 Thresholds

We develop a mathematical model of the peripheral naive T cell population to study the change in human naive T cell numbers from birth to adulthood, incorporating thymic output and the availability of interleukin-7 (IL-7). The model is formulated as three ordinary differential equations: two describe T cell numbers, in a resting state and progressing through the cell cycle. The third is introduced to describe changes in IL-7 availability. Thymic output is a decreasing function of time, representative of the thymic atrophy observed in aging humans. Each T cell is assumed to possess two interleukin-7 receptor (IL-7R) signaling thresholds: a survival threshold and a second, higher, proliferation threshold. If the IL-7R signaling strength is below its survival threshold, a cell may undergo apoptosis. When the signaling strength is above the survival threshold, but below the proliferation threshold, the cell survives but does not divide. Signaling strength above the proliferation threshold enables entry into cell cycle. Assuming that individual cell thresholds are log-normally distributed, we derive population-average rates for apoptosis and entry into cell cycle. We have analyzed the adiabatic change in homeostasis as thymic output decreases. With a parameter set representative of a healthy individual, the model predicts a unique equilibrium number of T cells. In a parameter range representative of persistent viral or bacterial infection, where naive T cell cycle progression is impaired, a decrease in thymic output may result in the collapse of the naive T cell repertoire.

Specific T-cell activation in an unspecific T-cell repertoire

T-cells are a vital type of white blood cell that circulate around our bodies, scanning for cellular abnormalities and infections. They recognise disease-associated antigens via a surface receptor called the T-cell antigen receptor (TCR). If there were a specific TCR for every single antigen, no mammal could possibly contain all the T-cells it needs. This is clearly absurd and suggests that T-cell recognition must, to the contrary, be highly degenerate. Yet highly promiscuous TCRs would appear to be equally impossible: they are bound to recognise self as well as non-self antigens. We review how contributions from mathematical analysis have helped to resolve the paradox of the promiscuous TCR. Combined experimental and theoretical work shows that TCR degeneracy is essentially dynamical in nature, and that the T-cell can differentially adjust its functional sensitivity to the salient epitope, "tuning up" sensitivity to the antigen associated with disease and "tuning down" sensitivity to antigens associated with healthy conditions. This paradigm of continual modulation affords the TCR repertoire, despite its limited numerical diversity, the flexibility to respond to almost any antigenic challenge while avoiding autoimmunity.