Modelling T-cell-mediated suppression dependent on interactions in multicellular conjugates

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.

Myalgic Encephalomyelitis/Chronic Fatigue Syndrome as a Hyper-Regulated Immune System Driven by an Interplay Between Regulatory T Cells and Chronic Human Herpesvirus Infections

Autoimmunity and chronic viral infections are recurrent clinical observations in Myalgic Encephalomyelitis/Chronic Fatigue Syndrome (ME/CFS), a complex disease with an unknown cause. Given these observations, the regulatory CD4+ T cells (Tregs) show promise to be good candidates for the underlying pathology due to their capacity to suppress the immune responses against both self and microbial antigens. Here, we discussed the overlooked role of these cells in the chronicity of Human Herpes Virus 6 (HHV6), Herpes Simplex 1 (HSV1), and Epstein–Barr virus (EBV), as often reported as triggers of ME/CFS. Using simulations of the cross-regulation model for the dynamics of Tregs, we illustrated that mild infections might lead to a chronically activated immune responses under control of Tregs if the responding clone has a high autoimmune potential. Such infections promote persistent inflammation and possibly fatigue. We then hypothesized that ME/CFS is a condition characterized by a predominance of this type of infections under control of Tregs. In contrast, healthy individuals are hypothesized to trigger immune responses of a virus-specific clone with a low autoimmune potential. According to this hypothesis, simple model simulations of the CD4+ T-cell repertoire could reproduce the increased density and percentages of Tregs observed in patients suffering from the disease, when compared to healthy controls. A deeper analysis of Tregs in the pathogenesis of ME/CFS will help to assess the validity of this hypothesis.

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.

Generic, scalable and decentralized fault detection for robot swarms

Robot swarms are large-scale multirobot systems with decentralized control which means that each robot acts based only on local perception and on local coordination with neighboring robots. The decentralized approach to control confers number of potential benefits. In particular, inherent scalability and robustness are often highlighted as key distinguishing features of robot swarms compared with systems that rely on traditional approaches to multirobot coordination. It has, however, been shown that swarm robotics systems are not always fault tolerant. To realize the robustness potential of robot swarms, it is thus essential to give systems the capacity to actively detect and accommodate faults. In this paper, we present a generic fault-detection system for robot swarms. We show how robots with limited and imperfect sensing capabilities are able to observe and classify the behavior of one another. In order to achieve this, the underlying classifier is an immune system-inspired algorithm that learns to distinguish between normal behavior and abnormal behavior online. Through a series of experiments, we systematically assess the performance of our approach in a detailed simulation environment. In particular, we analyze our system’s capacity to correctly detect robots with faults, false positive rates, performance in a foraging task in which each robot exhibits a composite behavior, and performance under perturbations of the task environment. Results show that our generic fault-detection system is robust, that it is able to detect faults in a timely manner, and that it achieves a low false positive rate. The developed fault-detection system has the potential to enable long-term autonomy for robust multirobot systems, thus increasing the usefulness of robots for a diverse repertoire of upcoming applications in the area of distributed intelligent automation.

Combining Theoretical and Experimental Techniques to Study Murine Heart Transplant Rejection

The quality of life of organ transplant recipients is compromised by complications associated with life-long immunosuppression, such as hypertension, diabetes, opportunistic infections, and cancer. Moreover, the absence of established tolerance to the transplanted tissues causes limited long-term graft survival rates. Thus, there is a great medical need to understand the complex immune system interactions that lead to transplant rejection so that novel and effective strategies of intervention that redirect the system toward transplant acceptance (while preserving overall immune competence) can be identified. This study implements a systems biology approach in which an experimentally based mathematical model is used to predict how alterations in the immune response influence the rejection of mouse heart transplants. Five stages of conventional mouse heart transplantation are modeled using a system of 13 ordinary differential equations that tracks populations of both innate and adaptive immunity as well as proxies for pro- and anti-inflammatory factors within the graft and a representative draining lymph node. The model correctly reproduces known experimental outcomes, such as indefinite survival of the graft in the absence of CD4⁺ T cells and quick rejection in the absence of CD8⁺ T cells. The model predicts that decreasing the translocation rate of effector cells from the lymph node to the graft delays transplant rejection. Increasing the starting number of quiescent regulatory T cells in the model yields a significant but somewhat limited protective effect on graft survival. Surprisingly, the model shows that a delayed appearance of alloreactive T cells has an impact on graft survival that does not correlate linearly with the time delay. This computational model represents one of the first comprehensive approaches toward simulating the many interacting components of the immune system. Despite some limitations, the model provides important suggestions of experimental investigations that could improve the understanding of rejection. Overall, the systems biology approach used here is a first step in predicting treatments and interventions that can induce transplant tolerance while preserving the capacity of the immune system to protect against legitimate pathogens.

To err is robotic, to tolerate immunological: fault detection in multirobot systems

Fault detection and fault tolerance represent two of the most important and largely unsolved issues in the field of multirobot systems (MRS). Efficient, long-term operation requires an accurate, timely detection, and accommodation of abnormally behaving robots. Most existing approaches to fault-tolerance prescribe a characterization of normal robot behaviours, and train a model to recognize these behaviours. Behaviours unrecognized by the model are consequently labelled abnormal or faulty. MRS employing these models do not transition well to scenarios involving temporal variations in behaviour (e.g., online learning of new behaviours, or in response to environment perturbations). The vertebrate immune system is a complex distributed system capable of learning to tolerate the organism's tissues even when they change during puberty or metamorphosis, and to mount specific responses to invading pathogens, all without the need of a genetically hardwired characterization of normality. We present a generic abnormality detection approach based on a model of the adaptive immune system, and evaluate the approach in a swarm of robots. Our results reveal the robust detection of abnormal robots simulating common electro-mechanical and software faults, irrespective of temporal changes in swarm behaviour. Abnormality detection is shown to be scalable in terms of the number of robots in the swarm, and in terms of the size of the behaviour classification space.

Understanding the roles of activation threshold and infections in the dynamics of autoimmune disease

Onset and development of autoimmunity have been attributed to a number of factors, including genetic predisposition, age and different environmental factors. In this paper we discuss mathematical models of autoimmunity with an emphasis on two particular aspects of immune dynamics: breakdown of immune tolerance in response to an infection with a pathogen, and interactions between T cells with different activation thresholds. We illustrate how the explicit account of T cells with different activation thresholds provides a viable model of immune dynamics able to reproduce several types of immune behaviour, including normal clearance of infection, emergence of a chronic state, and development of a recurrent infection with autoimmunity. We discuss a number of open research problems that can be addressed within the same modelling framework.

A stochastic chemical dynamic approach to correlate autoimmunity and optimal vitamin-D range

Motivated by several recent experimental observations that vitamin-D could interact with antigen presenting cells (APCs) and T-lymphocyte cells (T-cells) to promote and to regulate different stages of immune response, we developed a coarse grained but general kinetic model in an attempt to capture the role of vitamin-D in immunomodulatory responses. Our kinetic model, developed using the ideas of chemical network theory, leads to a system of nine coupled equations that we solve both by direct and by stochastic (Gillespie) methods. Both the analyses consistently provide detail information on the dependence of immune response to the variation of critical rate parameters. We find that although vitamin-D plays a negligible role in the initial immune response, it exerts a profound influence in the long term, especially in helping the system to achieve a new, stable steady state. The study explores the role of vitamin-D in preserving an observed bistability in the phase diagram (spanned by system parameters) of immune regulation, thus allowing the response to tolerate a wide range of pathogenic stimulation which could help in resisting autoimmune diseases. We also study how vitamin-D affects the time dependent population of dendritic cells that connect between innate and adaptive immune responses. Variations in dose dependent response of anti-inflammatory and pro-inflammatory T-cell populations to vitamin-D correlate well with recent experimental results. Our kinetic model allows for an estimation of the range of optimum level of vitamin-D required for smooth functioning of the immune system and for control of both hyper-regulation and inflammation. Most importantly, the present study reveals that an overdose or toxic level of vitamin-D or any steroid analogue could give rise to too large a tolerant response, leading to an inefficacy in adaptive immune function.

The benefits of diversity: heterogenous DC populations allow for both immunity and tolerance

The immune system must simultaneously mount a response against foreign antigens while tolerating self. How this happens is still unclear as many mechanisms of immune tolerance are antigen non-specific. Antigen specific immune cells called T-cells must first bind to Immunogenic Dendritic Cells (iDCs) before activating and proliferating. These iDCs present both self and foreign antigens during infection, so it is unclear how the immune response can be limited to primarily foreign reactive T-cells. Regulatory T-cells (Tregs) are known to play a key role in self-tolerance. Although they are antigen specific, they also act in an antigen non-specific manner by competing for space and growth factors as well as modifying DC behavior to help kill or deactivate other T-cells. In prior models, the lack of antigen specific control has made simultaneous foreign-immunity and self-tolerance extremely unlikely. We include a heterogeneous DC population, in which different DCs present antigens at different levels. In addition, we include Tolerogenic DC (tDCs) which can delete self-reactive T-cells under normal physiological conditions. We compare different mathematical models of immune tolerance with and without Tregs and heterogenous antigen presentation. For each model, we compute the final number of foreign-reactive and self-reactive T-cells, under a variety of different situations. We find that even if iDCs present more self-antigen than foreign antigen, the immune response will be primarily foreign-reactive as long as there is sufficient presentation of self-antigen on tDCs. Tregs are required primarily for rare or cryptic self-antigens that do not appear frequently on tDCs. We also find that Tregs can only be effective when we include heterogenous antigen presentation, as this allows Tregs and T-cells of the same antigen-specificity to colocalize to the same set of DCs. Tregs better aid immune tolerance when they can both compete for space and growth factors and directly eliminate other T-cells. Our results show the importance of the structure of the DC population in immune tolerance as well as the relative contribution of different cellular mechanisms.

A mathematical perspective on CD4(+) T cell quorum-sensing

We analyse a mathematical model of the peripheral CD4(+) T cell population, based on a quorum-sensing mechanism, by which an optimum number of regulatory T cells can be established and maintained. We divide the population of a single T cell receptor specificity into four pools: naive, IL-2 producing, IL-2 non-producing, and regulatory CD4(+) T cells. Proliferation, death and differentiation of cells are introduced as transition probabilities of a stochastic Markov model, with the assumption that the amount of IL-2 available to CD4(+) T cells is proportional to the size of the population of IL-2 producing CD4(+) T cells. We explore the population dynamics both in the absence and in the presence of specific antigen. We study the establishment of the peripheral CD4(+) T cell pool from thymic output in the absence of antigen, and its return to homeostasis after an immune challenge, by steady state analysis of the deterministic approximation. The number of regulatory T cells at steady state is greater in the presence of antigen than in its absence. We also consider the stochastic dynamics of the model after an immune challenge, in particular the behaviour leading to ultimate extinction of the IL-2 producing and regulatory T cell populations.

A mathematical model of immune activation with a unified self-nonself concept

The adaptive immune system reacts against pathogenic nonself, whereas it normally remains tolerant to self. The initiation of an immune response requires a critical antigen(Ag)-stimulation and a critical number of Ag-specific T cells. Autoreactive T cells are not completely deleted by thymic selection and partially present in the periphery of healthy individuals that respond in certain physiological conditions. A number of experimental and theoretical models are based on the concept that structural differences discriminate self from nonself. In this article, we establish a mathematical model for immune activation in which self and nonself are not distinguished. The model considers the dynamic interplay of conventional T cells, regulatory T cells (Tregs), and IL-2 molecules and shows that the renewal rate ratio of resting Tregs to naïve T cells as well as the proliferation rate of activated T cells determine the probability of immune stimulation. The actual initiation of an immune response, however, relies on the absolute renewal rate of naïve T cells. This result suggests that thymic selection reduces the probability of autoimmunity by increasing the Ag-stimulation threshold of self reaction which is established by selection of a low number of low-avidity autoreactive T cells balanced with a proper number of Tregs. The stability analysis of the ordinary differential equation model reveals three different possible immune reactions depending on critical levels of Ag-stimulation: a subcritical stimulation, a threshold stimulation inducing a proper immune response, and an overcritical stimulation leading to chronic co-existence of Ag and immune activity. The model exhibits oscillatory solutions in the case of persistent but moderate Ag-stimulation, while the system returns to the homeostatic state upon Ag clearance. In this unifying concept, self and nonself appear as a result of shifted Ag-stimulation thresholds which delineate these three regimes of immune activation.

Mathematical Models of the Impact of IL2 Modulation Therapies on T Cell Dynamics

Several reports in the literature have drawn a complex picture of the effect of treatments aiming to modulate IL2 activity in vivo. They seem to promote either immunity or tolerance, probably depending on the specific context, dose, and timing of their application. Such complexity might derive from the pleiotropic role of IL2 in T cell dynamics. To theoretically address the latter possibility, our group has developed several mathematical models for Helper, Regulatory, and Memory T cell population dynamics, which account for most well-known facts concerning their relationship with IL2. We have simulated the effect of several types of therapies, including the injection of: IL2; antibodies anti-IL2; IL2/anti-IL2 immune-complexes; and mutant variants of IL2. We studied the qualitative and quantitative conditions of dose and timing for these treatments which allow them to potentiate either immunity or tolerance. Our results provide reasonable explanations for the existent pre-clinical and clinical data, predict some novel treatments, and further provide interesting practical guidelines to optimize the future application of these types of treatments.

Mechanisms Underlying CD4+ Treg Immune Regulation in the Adult: From Experiments to Models

To maintain immunological balance the organism has to be tolerant to self while remaining competent to mount an effective immune response against third-party antigens. An important mechanism of this immune regulation involves the action of regulatory T-cell (Tregs). In this mini-review, we discuss some of the known and proposed mechanisms by which Tregs exert their influence in the context of immune regulation, and the contribution of mathematical modeling for these mechanistic studies. These models explore the mechanisms of action of regulatory T cells, and include hypotheses of multiple signals, delivered through simultaneous antigen-presenting cell (APC) conjugation; interaction of feedback loops between APC, Tregs, and effector cells; or production of specific cytokines that act on effector cells. As the field matures, and competing models are winnowed out, it is likely that we will be able to quantify how tolerance-inducing strategies, such as CD4-blockade, affect T-cell dynamics and what mechanisms explain the observed behavior of T-cell based tolerance.

Stochastic approximation to the T cell mediated specific response of the immune system

We develop a stochastic model to study the specific response of the immune system. The model is based on the dynamical interaction between Regulatory and Effector CD4+ T cells in the presence of Antigen Presenting Cells inside a lymphatic node. At a mean field level the model predicts the existence of different regimes where active, tolerant, or cyclic immune responses are possible. To study the model beyond mean field and to understand the specific responses of the immune system we use the Linear Noise Approximation and show that fluctuations due to finite size effects may strongly alter the mean field scenario. Moreover, it was found that the existence of a certain characteristic frequency for the fluctuations. All the analytical predictions were compared with simulations using Gillespie's algorithm.

T cell state transition produces an emergent change detector

We model the stages of a T cell response from initial activation to T cell expansion and contraction using a system of ordinary differential equations. Results of this modeling suggest that state transitions enable the T cell population to detect change and respond effectively to changes in antigen stimulation levels, rather than simply the presence or absence of antigen. A key component of the system that gives rise to this emergent change detector is initial activation of naïve T cells. The activation step creates a barrier that separates the long-term, slow dynamics of naïve T cells from the short-term, fast dynamics of effector T cells. This separation allows the T cell population to compare current, up-to-date changes in antigen levels to long-term, steady state levels. As a result, the T cell population responds very effectively to sudden shifts in antigen levels, even if the antigen were already present prior to the change. This feature provides a mechanism for T cells to react to rapidly expanding sources of antigen stimulation, such as viruses, while maintaining tolerance to constant or slowly fluctuating sources of stimulation, such as healthy tissue during growth. In addition to modeling T cell activation, we also formulate a model of the proliferation of effector T cells in response to the consumption of positive growth signal, secreted throughout the T cell response. We discuss how the interaction between T cells and growth signal generates an emergent threshold detector that responds preferentially to large changes in antigen stimulation while ignoring small ones. As a final step, we discuss how the de novo generation of adaptive regulatory T cells during the latter phase of the T cell response creates a negative feedback loop that controls the duration and magnitude of the T cell response. Hence, the immune network continually adjusts to a shifting baseline of (self and non-self) antigens, and responds primarily to abrupt changes in these antigens rather than merely their presence or absence.

Dynamics of immunological models

We analyse the effect of the regulatory T cells (Tregs) in the local control of the immune responses by T cells. We obtain an explicit formula for the level of antigenic stimulation of T cells as a function of the concentration of T cells and the parameters of the model. The relation between the concentration of the T cells and the antigenic stimulation of T cells is an hysteresis, that is unfold for some parameter values. We study the appearance of autoimmunity from cross-reactivity between a pathogen and a self antigen or from bystander proliferation. We also study an asymmetry in the death rates. With this asymmetry we show that the antigenic stimulation of the Tregs is able to control locally the population size of Tregs. Other effects of this asymmetry are a faster immune response and an improvement in the simulations of the bystander proliferation. The rate of variation of the levels of antigenic stimulation determines if the outcome is an immune response or if Tregs are able to maintain control due to the presence of a transcritical bifurcation for some tuning between the antigenic stimuli of T cells and Tregs. This behavior is explained by the presence of a transcritical bifurcation.

Diversity and plasticity of Th cell types predicted from regulatory network modelling

Alternative cell differentiation pathways are believed to arise from the concerted action of signalling pathways and transcriptional regulatory networks. However, the prediction of mammalian cell differentiation from the knowledge of the presence of specific signals and transcriptional factors is still a daunting challenge. In this respect, the vertebrate hematopoietic system, with its many branching differentiation pathways and cell types, is a compelling case study. In this paper, we propose an integrated, comprehensive model of the regulatory network and signalling pathways controlling Th cell differentiation. As most available data are qualitative, we rely on a logical formalism to perform extensive dynamical analyses. To cope with the size and complexity of the resulting network, we use an original model reduction approach together with a stable state identification algorithm. To assess the effects of heterogeneous environments on Th cell differentiation, we have performed a systematic series of simulations considering various prototypic environments. Consequently, we have identified stable states corresponding to canonical Th1, Th2, Th17 and Treg subtypes, but these were found to coexist with other transient hybrid cell types that co-express combinations of Th1, Th2, Treg and Th17 markers in an environment-dependent fashion. In the process, our logical analysis highlights the nature of these cell types and their relationships with canonical Th subtypes. Finally, our logical model can be used to explore novel differentiation pathways in silico.

T lymphocytes play a key role in the regulation of the immune response in mammals. Various T-helper subtypes (Th1, Th2, Th17, Treg,…) have been identified over the years, characterised by the expression of specific transcription factors and cytokines, which have a critical influence on the selection of different immune responses, driving pro-inflammatory or allergic responses, promoting alternative antibody classes, or preventing (auto)immunity by inhibiting the activation and proliferation of other cells. To gain insight into the heterogeneity and the plasticity of late T-helper lineages, we have built an integrated model of the regulatory network and signalling pathways controlling Th cell differentiation. Relying on a logical modelling framework, we have performed a systematic series of simulations to assess the effects of heterogeneous environments on Th cell differentiation. We have identified stable states corresponding to canonical Th1, Th2, Th17 and Treg subtypes, but also to hybrid cell types co-expressing combinations of Th1, Th2, Treg and Th17 markers in an environment-dependent fashion. Our analysis highlights the nature of these cell types and their relationships with canonical Th subtypes.

Lifelong dynamics of human CD4+CD25+ regulatory T cells: insights from in vivo data and mathematical modeling

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.

Emergent group dynamics governed by regulatory cells produce a robust primary T cell response

The currently accepted paradigm for the primary T cell response is that effector T cells commit to autonomous developmental programs. This concept is based on several experiments that have demonstrated that the dynamics of a T cell response is largely determined shortly after antigen exposure and that T cell dynamics do not depend on the level and duration of antigen stimulation. Another experimental study has also shown that T cell responses are robust to variations in antigen-specific precursor frequency. Various mathematical models have corroborated the first result that programmed T cell responses are insensitive to the level of antigen stimulation. However, this paper proposes that programmed responses do not entirely explain the robustness of T cell dynamics to variations in precursor frequency. This work studies the hypothesis that the dynamics of a T cell response may also be governed by a feedback loop involving adaptive regulatory cells rather than by intrinsic developmental programs. We formulate two mathematical models based on T cell developmental programs. In one model, effector cells undergo a fixed number of divisions before dying. In the second model, effector cells live for a fixed time during which they may divide. The study of these models suggests that developmental programs are not sufficiently robust as they produce an immune response that directly scales with precursor frequencies. Consequently, we derive a third model based on the principle that adaptive regulatory T cells develop in the course of an immune response and suppress effector cells. Our simulations show that this feedback mechanism responds robustly over a range of at least four orders of magnitude of precursor frequencies. We conclude that the proliferation program paradigm does not entirely capture the observed robustness of T cell responses to variations in precursor frequency. We propose an alternative mechanism by which the primary T cell response is governed by an emergent group dynamic and not by individual T cell programs.

Optimal number of regulatory T cells

The adaptive immune system of a vertebrate may attack its own body, causing autoimmune diseases. Regulatory T cells suppress the activity of the autoreactive effector T cells, but they also interrupt normal immune reactions against foreign antigens. In this paper, we discuss the optimal number of regulatory T cells that should be produced. We make the assumptions that some self-reactive immature T cells may fail to interact with their target antigens during the limited training period and later become effector T cells causing autoimmunity, and that regulatory T cells exist that recognize self-antigens. When a regulatory T cell is stimulated by its target self-antigen on an antigen-presenting cell (APC), it stays there and suppresses the activation of other naive T cells on the same APC. Analysis of the benefit and the harm of having regulatory T cells suggests that the optimal number of regulatory T cells depends on the number of self-antigens, the severity of the autoimmunity, the abundance of pathogenic foreign antigens, and the spatial distribution of self-antigens in the body. For multiple types of self-antigen, we discuss the optimal number of regulatory T cells when the self-antigens are localized in different parts of the body and when they are co-localized. We also examine the separate regulation of the abundances of regulatory T cells for different self-antigens, comparing it with the situation in which they are constrained to be equal.

Modeling the role of IL-2 in the interplay between CD4+ helper and regulatory T cells: assessing general dynamical properties

Mathematical models accounting for well-known evidences relating to the dynamics of interleukin 2, helper and regulatory T cells are presented. These models extend an existent model (the so-called cross-regulation model of immunity), by assuming IL-2 as the growth factor produced by helper cells, but used by both helper and regulatory cells to proliferate and survive. Two model variants, motivated by current literature, are explored. The first variant assumes that regulatory cells suppress helper cells by limiting IL-2 production and consuming the available IL-2; i.e. they just trigger competition for IL-2. The second model variant adds to the latter competitive mechanism the direct inhibition of helper cells activation by regulatory cells. The extended models retain key dynamical features of the cross-regulation model. But such reasonable behavior depends on parameter constraints, which happen to be realistic and lead to interesting biological discussions. Furthermore, the introduction of IL-2 in these models breaks the local/specific character of interactions, providing new properties to them. In the extended models, but not in the cross-regulation model, the response triggered by an antigen affects the response to other antigens in the same lymph node. The first model variant predicts an unrealistic coupling of the immune reactions to all the antigens in the lymph node. In contrast, the second model variant allows the coexistent of concomitant tolerant and immune responses to different antigens. The IL-2 derived from an ongoing immune reaction reinforces tolerance to other antigens in the same lymph node. Overall the models introduced here are useful extensions of the cross-regulation formalism. In particular, they might allow future studies of the effect of different IL-2 modulation therapies on CD4+ T cell dynamics.

How do regulatory T cells work?

CD4⁺ T cells are commonly divided into regulatory T (Treg) cells and conventional T helper (Th) cells. Th cells control adaptive immunity against pathogens and cancer by activating other effector immune cells. Treg cells are defined as CD4⁺ T cells in charge of suppressing potentially deleterious activities of Th cells. This review briefly summarizes the current knowledge in the Treg field and defines some key questions that remain to be answered. Suggested functions for Treg cells include: prevention of autoimmune diseases by maintaining self-tolerance; suppression of allergy, asthma and pathogen-induced immunopathology; feto-maternal tolerance; and oral tolerance. Identification of Treg cells remains problematic, because accumulating evidence suggests that all the presently-used Treg markers (CD25, CTLA-4, GITR, LAG-3, CD127 and Foxp3) represent general T-cell activation markers, rather than being truly Treg-specific. Treg-cell activation is antigen-specific, which implies that suppressive activities of Treg cells are antigen-dependent. It has been proposed that Treg cells would be self-reactive, but extensive TCR repertoire analysis suggests that self-reactivity may be the exception rather than the rule. The classification of Treg cells as a separate lineage remains controversial because the ability to suppress is not an exclusive Treg property. Suppressive activities attributed to Treg cells may in reality, at least in some experimental settings, be exerted by conventional Th cell subsets, such as Th1, Th2, Th17 and T follicular (Tfh) cells. Recent reports have also demonstrated that Foxp3⁺ Treg cells may differentiate in vivo into conventional effector Th cells, with or without concomitant downregulation of Foxp3.

A population dynamics analysis of the interaction between adaptive regulatory T cells and antigen presenting cells

Background

Regulatory T cells are central actors in the maintenance of tolerance of self-antigens or allergens and in the regulation of the intensity of the immune response during infections by pathogens. An understanding of the network of the interaction between regulatory T cells, antigen presenting cells and effector T cells is starting to emerge. Dynamical systems analysis can help to understand the dynamical properties of an interaction network and can shed light on the different tasks that can be accomplished by a network.

Methodology and Principal Findings

We used a mathematical model to describe a interaction network of adaptive regulatory T cells, in which mature precursor T cells may differentiate into either adaptive regulatory T cells or effector T cells, depending on the activation state of the cell by which the antigen was presented. Using an equilibrium analysis of the mathematical model we show that, for some parameters, the network has two stable equilibrium states: one in which effector T cells are strongly regulated by regulatory T cells and another in which effector T cells are not regulated because the regulatory T cell population is vanishingly small. We then simulate different types of perturbations, such as the introduction of an antigen into a virgin system, and look at the state into which the system falls. We find that whether or not the interaction network switches from the regulated (tolerant) state to the unregulated state depends on the strength of the antigenic stimulus and the state from which the network has been perturbed.

Conclusion/Significance

Our findings suggest that the interaction network studied in this paper plays an essential part in generating and maintaining tolerance against allergens and self-antigens.

How regulatory CD25+CD4+ T cells impinge on tumor immunobiology: the differential response of tumors to therapies

Aiming to get a better insight on the impact of regulatory CD25(+)CD4(+) T cells in tumor-immunobiology, a simple mathematical model was previously formulated and studied. This model predicts the existence of two alternative modes of uncontrolled tumor growth, which differ on their coupling with the immune system, providing a plausible explanation to the observation that the development of some tumors expand regulatory T cells whereas others do not. We report now the study of how these two tumor classes respond to different therapies, namely vaccination, immune suppression, surgery, and their different combinations. We show 1) how the timing and the dose applied in each particular treatment determine whether the tumor will be rejected, with or without concomitant autoimmunity, or whether it will continue progressing with slower or faster pace; 2) that both regulatory T cell-dependent and independent tumors are equally sensitive to vaccination, although the former are more sensitive to T cell depletion treatments and are unresponsive to partial surgery alone; 3) that surgery, suppression, and vaccination treatments, can synergistically improve their individual effects, when properly combined. Particularly, we predict rational combinations helping to overcome the limitation of these individual treatments on the late stage of tumor development.

How regulatory CD25+CD4+T cells impinge on tumor immunobiology? On the existence of two alternative dynamical classes of tumors

Aiming to get a better insight on the impact of regulatory CD25(+)CD4(+) T cells in tumor immunobiology, a simple mathematical model was formulated and studied. This model is an extension of a previous model for the dynamics of autoreactive regulatory cells and effector cells that interact upon their co-localized activation at the antigen presenting cells (APCs). It assumes that tumor growth stimulates the activation and migration to the adjacent lymph node of fresh APCs loaded with tumor antigens. These APCs stimulate the growth of both effector and regulatory T cells, which may then migrate to the tumor site and induce tumor cell destruction. Our results predict the existence of two alternative dynamic modes of unbounded tumor growth. In the first mode, the tumor induces the expansion of effector T cells that outcompete regulatory T cells, but nevertheless fail to control the tumor. In the second mode, the tumor induces a balanced expansion of both effector and regulatory T cells, which prevents the tumor from being destroyed by the immune cells. Tumors characterized by a high specific growth rate, low immunogenicity, and that are relatively resistant to T cell destructive functions, will grow in the first mode; conversely, tumors that have a slow specific growth rate, that are immunogenic, and/or that are more sensitive to destruction by T cells will grow in the second mode. Overall, this result provides a simple explanation to the fact that the development of some tumors expands regulatory T cells while others do not, predicting how some key dynamical properties of the tumor determine either one or the other type of behavior.

When three is not a crowd: a Crossregulation model of the dynamics and repertoire selection of regulatory CD4+ T cells

Regulatory CD4(+) T cells, enriched in the CD25 pool of healthy individuals, mediate natural tolerance and prevent autoimmune diseases. Despite their fundamental and potential clinical significance, regulatory T (T(R)) cells have not yet been incorporated in a coherent theory of the immune system. This article reviews experimental evidence and theoretical arguments supporting a model of T(R) cell dynamics, uncovering some of its most relevant biological implications. According to this model, the persistence and expansion of T(R) cell populations depend strictly on specific interactions they make with antigen-presenting cells (APCs) and conventional effector T (T(E)) cells. This three-partner crossregulation imposes that T(R) cells feed on the specific autoimmune activities they suppress, with implications ranging from their interactions with other cells to their repertoire selection in the periphery and in the thymus, and to the relationship between these cells and the innate immune system. These implications stem from the basic prediction that the peripheral dynamics sort the CD4(+) T-cell repertoire into two subsets: a less diverse set of small clones of autoreactive effector and regulatory cells that regulate each other's growth, and a more diverse set of barely autoreactive T(E) cell clones, whose expansion is limited only by APC availability. It is argued that such partitioning of the repertoire sets the ground for self-non-self discrimination.

Modeling regulation mechanisms in the immune system

We develop a mathematical framework for modeling regulatory mechanisms in the immune system. The model describes dynamics of key components of the immune network within two compartments: lymph node and tissue. We demonstrate using numerical simulations that our system can eliminate virus-infected cells, which are characterized by a tendency to increase without control (in absence of an immune response), while tolerating normal cells, which are characterized by a tendency to approach a stable equilibrium population. We experiment with different combinations of T cell reactivities that lead to effective systems and conclude that slightly self-reactive T cells can exist within the immune system and are controlled by regulatory cells. We observe that CD8+ T cell dynamics has two phases. In the first phase, CD8+ cells remain sequestered within the lymph node during a period of proliferation. In the second phase, the CD8+ population emigrates to the tissue and destroys its target population. We also conclude that a self-tolerant system must have a mechanism of central tolerance to ensure that self-reactive T cells are not too self-reactive. Furthermore, the effectiveness of a system depends on a balance between the reactivities of the effector and regulatory T cell populations, where the effectors are slightly more reactive than the regulatory cells.

Regulatory T cell adjustment of quorum growth thresholds and the control of local immune responses

The consequences of regulatory T cell (Treg) inhibition of interleukine 2 secretion is examined by mathematical modelling. We demonstrate that cytokine dependent growth exhibits a quorum T cell population threshold that determines if immune responses develop on activation. Secretion inhibition manipulates the growth dynamics and effectively increases the quorum threshold, i.e. to develop immune responses a higher number of T cells need to be activated. Thus Treg induced secretion inhibition can provide a mechanism for tissue specific regulation of the balance between suppression (control) and immune responses, a balance that can be varied at the local tissue level through the regulation of the local active Treg population size. However, nonspecific inhibition is prone to escape of initially controlled autoimmune T cells through cross reactivity to pathogens and bystander proliferation on unrelated immune responses.

Inverse correlation between the incidences of autoimmune disease and infection predicted by a model of T cell mediated tolerance

The contribution of pathogenic infections to the etiology of autoimmune diseases remains one of the outstanding problems in immunology. According to the classical concept of antigen mimicry, a direct correlation between the incidence of autoimmunity and infections would be expected. This view is supported by a few examples of autoimmune disorders, which are documented as being caused by infection with particular pathogens. In contrast, there are several experimental animal models where infection appears to prevent the onset of autoimmunity. Moreover, some epidemiological studies suggest an inverse correlation between the incidence of autoimmunity and infections in human populations. Here we propose a solution to this puzzle based on a theoretical model of tolerance driven by regulatory T cells. The concepts here developed delineate the conditions predicting an inverse correlation between the incidence of autoimmunity and exposition to common infections, and those in which antigen mimicry and inflammation of target organs have a role in the etiology of specific autoimmune disorders.

Tolerance and immunity in a mathematical model of T-cell mediated suppression

Regulatory CD4+CD25+ T cells play a major role in natural tolerance to body components and therefore are relevant to understand the self-non-self discrimination by the immune system. The most pressing theoretical question, regarding the fact that these regulatory cells perform their function through linked recognition of the APCs, is how this "non-specific" mechanism permits a proper balance between tolerance and immunity that is compatible with an effective self-non-self discrimination. To tackle this issue, we develop a numerical simulation, which extends a previous mathematical model of T-cell-mediated suppression to include the thymic generation and the peripheral dynamics of many T cell clones. This simulation can mimic the capacity of the immune system to establish natural tolerance to self-antigens and reliably mount immune responses to foreign antigens. Natural tolerance is based on ubiquitous and constitutive self-antigens, which select and sustain clones of specific regulatory cells. Immune responses to foreign antigens are only achieved if they displace the self-antigens from the APCs, leading to a loss of the regulatory cells, and/or if the foreign antigen introduction entails a sharp increase in the total number of APCs. Meaningful behavior is obtained even if differentiation of regulatory cells in the thymus is antigen non-specific, but requires that a minimum number of new T cells enter the periphery per unit of time, and that the repertoire is selected so that anti-self-affinities are within a proper interval. We conclude that positive selection is required to generate a sufficiently high frequency of self-antigen specific regulatory cells that reliably mediate natural tolerance. Negative selection is required to avoid the emergence at the periphery of very high affinity anti-self-regulatory cells that will make the tolerant state so robust that it could no be broken by the introduction of a foreign antigen. This result highlights the importance of repertoire selection in dominant tolerance proposing a novel role for the processes of positive and negative selection within this framework.

Computational immunology: The coming of age

The explosive growth in biotechnology combined with major advances in information technology has the potential to radically transform immunology in the postgenomics era. Not only do we now have ready access to vast quantities of existing data, but new data with relevance to immunology are being accumulated at an exponential rate. Resources for computational immunology include biological databases and methods for data extraction, comparison, analysis and interpretation. Publicly accessible biological databases of relevance to immunologists number in the hundreds and are growing daily. The ability to efficiently extract and analyse information from these databases is vital for efficient immunology research. Most importantly, a new generation of computational immunology tools enables modelling of peptide transport by the transporter associated with antigen processing (TAP), modelling of antibody binding sites, identification of allergenic motifs and modelling of T-cell receptor serial triggering.

Three-cell interactions in T cell-mediated suppression? A mathematical analysis of its quantitative implications

Aiming to further our understanding of T cell-mediated suppression, we investigate the plausibility of the hypothesis that regulatory T cells suppress other T cells (target cells), while both cells are conjugated with one APC. We use a mathematical model to analyze the proliferation inhibition scored during in vitro suppression assays. This model is a radical simplification of cell culture reality, assuming that thymidine incorporation is proportional to the number of target cells that would instantaneously form conjugates with APCs that are free of regulatory cells. According to this model the inhibition index should be mainly determined by the number of regulatory cells per APC and should be insensitive to the number of target cells. We reanalyzed several published data sets, confirming this expectation. Furthermore, we demonstrate that the instantaneous inhibition index has an absolute limit as a function of the number of regulatory cells per APC. By calculating this limit we find that the model can explain the data under two non-mutually exclusive conditions. First, only approximately 15% of APCs used in the suppression assays form conjugates with T cells. Second, the growth of the regulatory cell population depends on the target cells, such that the number of regulatory cells per APC increases when they are cocultured with target cells and overcomes its limit. However, if neither of these testable conditions is fulfilled, then one could conclude that suppression in vitro does not require the formation of multicellular conjugates.