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

Stirk, Emily R.; Lythe, Grant; Berg, Hugo van den; Hurst, Gareth A.D.; Molina-Parı́s, Carmen

doi:10.1016/j.mbs.2009.12.004

Cited by 15 publications

(19 citation statements)

References 34 publications

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“…We also make the approximation that heterogeneity is constant with changes in age. For a mathematical study of the impact sp-MHC availability has on clonal diversity, the reader is referred to Stirk et al (20, 21). Our model is a mathematical description of the homeostasis of the naive T cell repertoire, but does not consider stimulation by foreign antigens.…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2013

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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.

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Section: Introductionmentioning

confidence: 99%

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

et al. 2013

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“…For diffusions and other continuous-state processes, a good starting point is Steinsaltz and Evans [140] (but see also Cattiaux et al [22] and Pinsky [119]) and for branching processes there is an excellent recent review by Lambert [95,Section 3]. Whilst many issues remain unresolved, the theory has reached maturity, and the use of quasi-stationary distributions is now widespread, encompassing varied and contrasting areas of application, including cellular automata (Atman and Dickman [9]), complex systems (Collet et al [34]), ecology (Day and Possingham [41], Gosselin [63], Gyllenberg and Sylvestrov [68], Kukhtin et al [89], Pollett [122]) epidemics (Nåsell [106,107,108], Artalejo et al [6,7]), immunology (Stirk et al [141]), medical decision making (Chan et al [24]), physical chemistry (Dambrine and Moreau [37,38], Oppenheim et al [112], Pollett [121]), queues (Boucherie [17], Chen et al [25], Kijima and Makimoto [84]), reliability (Kalpakam and Shahul-Hameed [73], Kalpakam [74], Li and Cao [98,99]), survival analysis (Aalen and Gjessing [1,2], Steinsaltz and Evans [139]) and telecommunications (Evans [53], Ziedins [152]).…”

Section: Modelling Quasi Stationaritymentioning

confidence: 99%

Quasi-Stationary Distributions

2010

Constructive Computation in Stochastic Models With Applications

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This paper contains a survey of results related to quasi-stationary distributions, which arise in the setting of stochastic dynamical systems that eventually evanesce, and which may be useful in describing the long-term behaviour of such systems before evanescence. We are concerned mainly with continuous-time Markov chains over a finite or countably infinite state space, since these processes most often arise in applications, but will make reference to results for other processes where appropriate. Next to giving an historical account of the subject, we review the most important results on the existence and identification of quasi-stationary distributions for general Markov chains, and give special attention to birth-death processes and related models. Results on the question of whether a quasi-stationary distribution, given its existence, is indeed a good descriptor of the long-term behaviour of a system before evanescence, are reviewed as well. The paper is concluded with a summary of recent developments in numerical and approximation methods.

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“…Due to an intrinsically complex multi-factor nature of immune response [41], several mathematical models have investigated stochastic aspects of immune dynamics. They have included, among others, analyses of T cell homeostasis [42] and repertoire [43,44]; the dynamics of T cell activation thresholds [45,46]; T-cell proliferation and activation, including the role of cytokines [47][48][49]; self-tolerance based on regulatory T cells [23]; cytokine-mediated pathogen-induced autoimmunity [36]; T cell recruitment in response to a viral infection [50]; as well as an investigation of how a variable affinity between T cell receptors and MHC-peptide complexes may affect possible outcomes during T cell selection [51]. These models have focused primarily on investigating such aspects of stochastic dynamics as the probability distribution of T cell activation thresholds, or simulations of immune dynamics that are valid for relatively small numbers of cell populations (thus going beyond the mean-field models).…”

Section: Introductionmentioning

confidence: 99%

Quantifying the Role of Stochasticity in the Development of Autoimmune Disease

Nicholson

Blyuss

Fatehi

2020

Cells

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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.

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

Cited by 15 publications

References 34 publications

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

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

Quasi-Stationary Distributions

Quantifying the Role of Stochasticity in the Development of Autoimmune Disease

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