Immune network behavior--I. From stationary states to limit cycle oscillations

Dynamic causal modelling of immune heterogeneity

An interesting inference drawn by some COVID-19 epidemiological models is that there exists a proportion of the population who are not susceptible to infection-even at the start of the current pandemic. This paper introduces a model of the immune response to a virus. This is based upon the same sort of mean-field dynamics as used in epidemiology. However, in place of the location, clinical status, and other attributes of people in an epidemiological model, we consider the state of a virus, B and T-lymphocytes, and the antibodies they generate. Our aim is to formalise some key hypotheses as to the mechanism of resistance. We present a series of simple simulations illustrating changes to the dynamics of the immune response under these hypotheses. These include attenuated viral cell entry, pre-existing cross-reactive humoral (antibody-mediated) immunity, and enhanced T-cell dependent immunity. Finally, we illustrate the potential application of this sort of model by illustrating variational inversion (using simulated data) of this model to illustrate its use in testing hypotheses. In principle, this furnishes a fast and efficient immunological assay-based on sequential serology-that provides a (1) quantitative measure of latent immunological responses and (2) a Bayes optimal classification of the different kinds of immunological response (c.f., glucose tolerance tests used to test for insulin resistance). This may be especially useful in assessing SARS-CoV-2 vaccines.

Message Passing and Metabolism

Active inference is an increasingly prominent paradigm in theoretical biology. It frames the dynamics of living systems as if they were solving an inference problem. This rests upon their flow towards some (non-equilibrium) steady state-or equivalently, their maximisation of the Bayesian model evidence for an implicit probabilistic model. For many models, these self-evidencing dynamics manifest as messages passed among elements of a system. Such messages resemble synaptic communication at a neuronal network level but could also apply to other network structures. This paper attempts to apply the same formulation to biochemical networks. The chemical computation that occurs in regulation of metabolism relies upon sparse interactions between coupled reactions, where enzymes induce conditional dependencies between reactants. We will see that these reactions may be viewed as the movement of probability mass between alternative categorical states. When framed in this way, the master equations describing such systems can be reformulated in terms of their steady-state distribution. This distribution plays the role of a generative model, affording an inferential interpretation of the underlying biochemistry. Finally, we see that-in analogy with computational neurology and psychiatry-metabolic disorders may be characterized as false inference under aberrant prior beliefs.

Stable expression ratios of five pyroptosis-inducing cytokines in the spleen and thymus of mice showed potential immune regulation at the organ level

BACKGROUND

The immune system is one of the most complex regulatory systems in the body and is essential for the maintenance of homeostasis. Despite recent breakthroughs in immunology, the regulation of the immune system and the etiology of autoimmune diseases such as lupus remain unclear. Systemic lupus erythematosus is a systemic autoimmune disease with abnormally and inconsistently expressed pro-inflammatory cytokines. Pyroptosis is a pro-inflammatory form of programmed cell death that is associated with systemic lupus erythematosus. The thymus and spleen are important immune organs involved in systemic lupus erythematosus. Therefore, this study investigated the difference in expression of pyroptosis-inducing pro-inflammatory cytokines between the spleen and thymus in lupus model mice and in control mice, to describe immune regulation at the organ level.

OBJECTIVE

To investigate differences in the expression of pyroptosis-inducing cytokines in the spleen and thymus and to explore immune regulatory networks at the organ level.

METHODS

Two groups of lupus mice and two groups of control mice were utilized for this study. Using the thymus and spleen of experimental animals, mRNA expression levels of five pyroptosis-inducing cytokines (interleukin 1β, interleukin 18, NLRP3, caspase-1 and TNF-α) were determined via quantitative polymerase chain reaction. In addition, tissue distribution of these cytokines was investigated via immunohistochemistry.

RESULTS

All five pyroptosis-inducing inflammatory cytokines showed higher expression in the spleen than in the thymus (p < 0.05). Moreover, the spleen/thymus expression ratios of all five pyroptosis-inducing cytokines were not statistically different between the four experimental groups. Expression of all five cytokines exhibited a stable ratio (spleen/thymus ratios). This distinctive stable spleen/thymus ratio was consistent in all four experimental groups. The stable spleen/thymus ratios of the five inflammatory cytokines were as follows: interleukin 1β (2.02 ± 0.9), interleukin 18 (2.07 ± 1.06), caspase-1 (1.93 ± 0.66), NLRP3 (3.14 ± 1.61) and TNF-α (3.16 ± 1.36). Immunohistochemical analysis showed the cytokines were mainly expressed in the red pulp region of the spleen and the medullary region of the thymus, where immune-activated cells aggregated.

CONCLUSION

The stable spleen/thymus expression ratios of pyroptosis-inducing cytokines indicated that immune organs exhibit strictly regulated functions to maintain immune homeostasis and adapt to the environment.

The role of positive feedback loops involving anti-dsDNA and anti-anti-dsDNA antibodies in autoimmune glomerulonephritis

Autoimmune glomerulonephritis (GN) is a potentially life-threatening renal inflammation occurring in a significant percentage of systemic lupus erythematosus (SLE) patients. It has been suggested that GN develops and persists due to a positive feedback loop, in which inflammation is promoted by the deposition in the kidney of immune complexes (IC) containing double-stranded DNA (dsDNA) and autoantibodies specific to it, leading to cellular death, additional release to circulation of dsDNA, continuous activation of dsDNA-specific autoreactive B cells and further formation of IC. We have recently presented a generic model exploring the dynamics of IC-mediated autoimmune inflammatory diseases, applicable also to GN. Here we extend this model by incorporating into it a specific B cell response targeting anti-dsDNA antibodies-a phenomenon whose occurrence in SLE patients is well-supported empirically. We show that this model retains the main results found for the original model studied, particularly with regard to the sensitivity of the steady state properties to changes in parameter values, while capturing some disease-specific observations found in GN patients which are unaccountable using our previous model. In particular, the extended model explains the findings that this inflammation can be ameliorated by treatment without lowering the level of anti-dsDNA antibodies. Moreover, it can account for the inverse oscillations of anti-dsDNA and anti-anti-dsDNA antibodies, previously reported in lupus patients. Finally, it can be used to suggest a possible explanation to the so-called regulatory role of TLR9, found in murine models of lupus; i.e., the fact that the knockdown of this DNA-sensing receptor leads, as expected, to a decrease in the level of anti-dsDNA antibodies, but at the same time results in a counter-intuitive amplification of the autoreactive immune response and an exacerbated inflammation. Several predictions can be derived from the analysis of the presented model, allowing its experimental verification.

Modeling immune complex-mediated autoimmune inflammation

A number of autoimmune diseases are thought to feature a particular type of self-sustaining inflammation, caused by the deposition of immune complexes (IC) in the inflamed tissue and a consequent activation of local effector cells. The persistence of this inflammation is due to a positive feedback loop, where autoantigen particles released as part of the tissue damage caused by the inflammation stimulate autoreactive B cells, leading to the formation of further immune complexes and their subsequent deposition. We present a mathematical model for the exploration of IC-mediated autoimmune inflammation and its clinical implications. We characterize the possible differences between normal individuals and those susceptible to such inflammation, and show that both random perturbations and bifurcations can lead to disease onset. Our model explains how defects in the mechanisms responsible for cellular debris clearance contribute to the development of disease, in agreement with empirical evidence. Moreover, we show that parameters governing the dynamics of immune complexes, such as their clearance rate, have an even stronger effect in determining the behavior of the system. We demonstrate the existence of hysteresis, implying that once IC-mediated autoimmune inflammation is triggered, its long-term suppression may be difficult to achieve. Our results can serve to guide the development of novel therapies to autoimmune diseases involving this type of inflammation.

A mathematical design of vector vaccine against autoimmune disease

Viruses have been implicated in the initiation, progression, and exacerbation of several human autoimmune diseases. Evidence also exists that viruses can protect against autoimmune disease. Several proposed mechanisms explain the viral effects. One mechanism is "molecular mimicry" which represents a shared immunologic epitope with a microbe and the host. We consider, using a simple mathematical model, whether and how a viral infection with molecular mimicry can be beneficial or detrimental for autoimmune disease. Furthermore, we consider the possibility of development of a vector therapeutic vaccine that can relieve autoimmune disease symptoms. Our findings demonstrate that vaccine therapy success necessitates (i) appropriate immune response function, (ii) appropriate affinities with self and non-self antigen, and (iii) a replicative vector vaccine. Moreover, the model shows that the viral infection can cause autoimmune relapses.

Dynamical properties of autoimmune disease models: Tolerance, flare-up, dormancy

The mechanisms of autoimmune disease have remained puzzling for a long time. Here we construct a simple mathematical model for autoimmune disease based on the personal immune response function and the target cell growth function. We show that these two functions are sufficient to capture the essence of autoimmune disease and can explain characteristic symptom phases such as tolerance, repeated flare-ups and dormancy. Our results strongly suggest that a more complete understanding of these two functions will underlie the development of an effective therapy for autoimmune disease.

Kinetics of T lymphocyte responses to persistent antigens

Long term sequential study of immune responses in the same individuals is difficult from the time commitment required and the problem of maintaining enough subjects to provide for comparative analysis. We closely studied one hundred women with silicone mammary devices through cross sectional analysis up to 26 years post implantation and a similar sample of women to 6 years post explantation. The T cell index, calculated from tritiated thymidine incorporation during lymphoblast transformation, rose to a post implant peak at 10.5-12.0 years, falling progressively over the next 14.0-15.5 years to values indicative of probable immune quiescence. Post explantation, the index rose over the first 3 years and then sharply declined to within the range for unexposed controls. The shape of these time curves contains considerable information referent cell dynamics for both stimulatory and inhibitory factors and for demonstrating net group effects, appropriate to analysis in the cross sectional perspective. When a subset of four women was studied frequently and sequentially up to 8 years, an internal oscillatory pattern emerged, focusing attention on both the stimulatory and the inhibitory aspects of long term clonal expansion. IL-2 has stimulatory and inhibitory properties at different levels of production and is considered a prime candidate as the essential cytokine. The equations have details, however, which require exploration beyond any such provisional conclusion. The analytic process was aided by normalization of oscillatory data to eliminate subject variability and by Pareto optimization to assess the trend shown by normalization. Pareto analysis revealed two minimally coordinated oscillations, one over time and the other along net clonal expansion or decline of the siloxane specific T lymphocyte clone. The segments of the time related oscillation greatly exceeded the reaction times of cytokines currently known to be active in T cell regulation. Although the ultimate controlling factor(s) may be cytokine or chemokine combinations, the data are compatible with some more basic regulatory factor(s) of cell integrity, including limits on the number of cell divisions which can be sustained in long term immunopathic lesions, among other processes.

A mathematical method for investigating dynamic behavior of an idiotype network of the immune system. The time minimum optimal control theory

We proposed a mathematical method to investigate an integrated property of an idiotype immune network under the time minimum optimal control. The transient changes of amounts of B cell receptor bound antibodies and immune complex in the network system were expressed by detailed differential equations. The rate constant for binding the second Fab arm of antibody was set as a function of coulombic repulsive force to express the influence of redistribution of electrical charges in the ligand-receptor molecular complex. We proposed time minimum optimal control strategy as an organizing principle for rapid reactions of the immune system. Based on the rigorous mathematical foundations of the optimal control theory, we determined the differential equations for co-state variables for the state variables to compute the time minimum transient changes in the amount of the species. Biological parameters in the immune reactions were utilized from the reported experimental data. Numerical computation disclosed that influence of changes in a rate constant extended to all the species of the network. Changes in a rate constant in a different B cell system reinforced the collaborations among the idiotypes and lead them to set in motion the ejection of the antigen. Simulation of reported experimental data by the present method was successful. There were, however, some inevitable dissociations between reported experimental data and computed results. The present method will be available for evaluating the time minimum reaction of the immune network system.

A new bell-shaped function for idiotypic interactions based on cross-linking

Most recent models of the immune network are based upon a phenomenological log bell-shaped interaction function. This function depends on a single parameter, the "field," which is the sum of all ligand concentrations weighted by their respective affinities. The typical behavior of these models is dominated by percolation, a phenomenon in which a local stimulus spreads globally throughout the network. The usual reason for employing a log bell-shaped interaction function is that B cells are activated by cross-linking of their surface immunoglobulin receptors. Here we formally derive a new phenomenological log bell-shaped function from the chemistry of receptor cross-linking by bivalent ligand. Specifying how this new function depends on the ligand concentrations requires two fields: a binding field and a cross-linking field. When we compare the activation functions for ligand-receptor pairs with different affinities, the one-field and the two-field functions differ markedly. In the case of the one-field activation function, its graph is shifted to increasingly higher concentration as the affinity decreases but keeps its width and height. In the case of the two-field activation function, the graph of a low-affinity interaction is nested within the graphs of all higher-affinity interactions. We show that this difference in the relations among activation functions for different affinities radically changes the network behavior. In models that described B cell proliferation using the one-field activation function, network behavior was dominated by low-affinity interactions. Conversely, in our new model, the high-affinity interactions are the most significant. As a consequence, percolation is no longer the only typical network behavior.

A simple model for the statistics of events in idiotypic networks

A simple random graph model of idiotypic networks is introduced: this model allows (1) to evaluate the stability of the network dynamics' fixed points, and (2) to compute the statistics of events triggered in response to the arrival of new molecules (metadynamics) using a dynamic mean-field approximation based on the theory of branching processes. It is shown that (1) the network dynamics is unlikely to have many stable fixed points in a strict sense, but that (2) the reorganizations which the network undergoes owing to the metadynamics are always subcritical if plausible figures are injected into the model. In other words the distance between successive (unstable or weakly stable) fixed points is relatively small, so that the overall behavior is stable.

A basic mathematical model of the immune response

Interaction of the immune system with a target population of, e.g., bacteria, viruses, antigens, or tumor cells must be considered as a dynamic process. We describe this process by a system of two ordinary differential equations. Although the model is strongly idealized it demonstrates how the combination of a few proposed nonlinear interaction rules between the immune system and its targets are able to generate a considerable variety of different kinds of immune responses, many of which are observed both experimentally and clinically. In particular, solutions of the model equations correspond to states described by immunologists as "virgin state," "immune state" and "state of tolerance." The model successfully replicates the so-called primary and secondary response. Moreover, it predicts the existence of a threshold level for the amount of pathogen germs or of transplanted tumor cells below which the host is able to eliminate the infectious organism or to reject the tumor graft. We also find a long time coexistence of targets and immune competent cells including damped and undamped oscillations of both. Plausibly the model explains that if the number of transformed cells or pathogens exeeds definable values (poor antigenicity, high reproduction rate) the immune system fails to keep the disease under control. On the other hand, the model predicts apparently paradoxical situations including an increased chance of target survival despite enhanced immune activity or therapeutically achieved target reduction. A further obviously paradoxical behavior consists of a positive effect for the patient up to a complete cure by adding an additional target challenge where the benefit of the additional targets depends strongly on the time point and on their amount. Under periodically pulsed stimulation the model may show a chaotic time behavior of both target growth and immune response. (c) 1995 American Institute of Physics.

Memory capacity in large idiotypic networks

Many models of immune networks have been proposed since the original work of Jerne [1974, Ann. Immun. (Inst. Pasteur)125C, 373-389]. Recently, a limited class of models (Weisbuch et al., 1990, J. theor. Biol 146, 483-499) have been shown to maintain immunological memory by idiotypic network interactions. We examine generalizations of these models when the networks are both large and highly connected to study their memory capacity, i.e., their ability to account for immunization to a large number of random antigens. Our calculations show that in these minimal models, random connectivities with continuously distributed affinities reduce the memory capacity to essentially nil.

On the maternal transmission of immunity: a 'molecular attention' hypothesis

Maternally-derived antibodies can provide passive protection to their offspring. More subtle phenomena associated with maternal antibodies concern their influence in shaping the immune repertoire and priming the neonatal immune response. These phenomena suggest that maternal antibodies play a role in the education of the neonatal immune system. The educational effects are thought to be mediated by idiotypic interactions among antibodies and B cells in the context of an idiotypic network. This paper proposes that maternal antibodies trigger localized idiotypic network activity that serves to amplify and translate information concerning the molecular shapes of potential antigens. The triggering molecular signals are contained in the binding regions of the antibody molecules. These antibodies form complexes and are taken up by antigen presenting cells or retained by follicular dendritic cells and thereby incorporated into more traditional cellular immune memory mechanisms. This mechanism for maternal transmission of immunity is termed the molecular attention hypothesis and is contrasted to the dynamic memory hypothesis. Experiments are proposed that may help indicate which models are more appropriate and will further our understanding of these intriguing natural phenomena. Finally, analogies are drawn to attention in neural systems.

Immune networks modeled by replicator equations

In order to evaluate the role of idiotypic networks in the operation of the immune system a number of mathematical models have been formulated. Here we examine a class of B-cell models in which cell proliferation is governed by a non-negative, unimodal, symmetric response function f (h), where the field h summarizes the effect of the network on a single clone. We show that by transforming into relative concentrations, the B-cell network equations can be brought into a form that closely resembles the replicator equation. We then show that when the total number of clones in a network is conserved, the dynamics of the network can be represented by the dynamics of a replicator equation. The number of equilibria and their stability are then characterized using methods developed for the study of second-order replicator equations. Analogies with standard Lotka-Volterra equations are also indicated. A particularly interesting result of our analysis is the fact that even though the immune network equations are not second-order, the number and stability of their equilibria can be obtained by a superposition of second-order replicator systems. As a consequence, the problem of finding all of the equilibrium points of the nonlinear network equations can be reduced to solving linear equations.

Complex behaviours of AB model describing idiotypic network

A simple chemical model of the idiotypic network of immune systems, namely the AB model, has been developed by De Boer et al. The complexity of the system, such as the steady states, periodic oscillations and chaotic motions, has been examined by the authors mentioned above. In the present paper, the periodic motions and chaotic behaviours exhibited by the system are intuitively described. To clarify in which parameter domains concerned the system exhibits periodic oscillations and in which parameter domains the system demonstrates chaotic behaviours the Lyapounov exponent is explored. To characterize the strangeness of the attractors, the fractal dimension problem is worked out.

Memory in idiotypic networks due to competition between proliferation and differentiation

A model employing separate dose-dependent response functions for proliferation and differentiation of idiotypically interacting B cell clones is presented. For each clone the population dynamics of proliferating B cells, non-proliferating B cells and free antibodies are considered. An effective response function, which contains the total impact of proliferation and differentiation at the fixed points, is defined in order to enable an exact analysis. The analysis of the memory states is restricted in this paper to a two-species system. The conditions for the existence of locally stable steady states with expanded B cell and antibody populations are established for various combinations of different field-response functions (e.g. linear, saturation, log-bell functions). The stable fixed points are interpreted as memory states in terms of immunity and tolerance. It is proven that a combination of linear response functions for both proliferation and differentiation does not give rise to stable fixed points. However, due to competition between proliferation and differentiation saturation response functions are sufficient to obtain two memory states, provided proliferation precedes differentiation and also saturates earlier. The use of log-bell-shaped response functions for both proliferation and differentiation gives rise to a "mexican-hat" effective response function and allows for multiple (four to six) memory states. Both a primary response and a much more pronounced secondary response are observed. The stability of the memory states is studied as a function of the parameters of the model. The attractors lose their stability when the mean residence time of antibodies in the system is much longer than the B cells' lifetime. Neither the stability results nor the dynamics are qualitatively changed by the existence of non-proliferating B cells: memory states can exist and be stable without non-proliferating B cells. Nevertheless, the activation of non-proliferating B cells and the competition between proliferation and differentiation enlarge the parameter regime for which stable attractors are found. In addition, it is shown that a separate activation step from virgin to active B cells renders the virgin state stable for any choice of biologically reasonable parameters.

A Cayley tree immune network model with antibody dynamics

A Cayley tree model of idiotypic networks that includes both B cell and antibody dynamics is formulated and analysed. As in models with B cells only, localized states exist in the network with limited numbers of activated clones surrounded by virgin or near-virgin clones. The existence and stability of these localized network states are explored as a function of model parameters. As in previous models that have included antibody, the stability of immune and tolerant localized states are shown to depend on the ratio of antibody to B cell lifetimes as well as the rate of antibody complex removal. As model parameters are varied, localized steady-states can break down via two routes: dynamically, into chaotic attractors, or structurally into percolation attractors. For a given set of parameters percolation and chaotic attractors can coexist with localized attractors, and thus there do not exist clear cut boundaries in parameter space that separate regions of localized attractors from regions of percolation and chaotic attractors. Stable limit cycles, which are frequent in the two-clone antibody B cell (AB) model, are only observed in highly connected networks. Also found in highly connected networks are localized chaotic attractors. As in experiments by Lundkvist et al. (1989. Proc. natn. Acad. Sci. U.S.A. 86, 5074-5078), injection of Ab1 antibodies into a system operating in the chaotic regime can cause a cessation of fluctuations of Ab1 and Ab2 antibodies, a phenomenon already observed in the two-clone AB model. Interestingly, chaotic fluctuations continue at higher levels of the tree, a phenomenon observed by Lundkvist et al. but not accounted for previously.

Immune network behavior--II. From oscillations to chaos and stationary states

Two types of behavior have been previously reported in models of immune networks. The typical behavior of simple models, which involve B cells only, is stationary behavior involving several steady states. Finite amplitude perturbations may cause the model to switch between different equilibria. The typical behavior of more realistic models, which involve both B cells and antibody, consists of autonomous oscillations and/or chaos. While stationary behavior leads to easy interpretations in terms of idiotypic memory, oscillatory behavior seems to be in better agreement with experimental data obtained in unimmunized animals. Here we study a series of models of the idiotypic interaction between two B cell clones. The models differ with respect to the incorporation of antibodies, B cell maturation and compartmentalization. The most complicated model in the series has two realistic parameter regimes in which the behavior is respectively stationary and chaotic. The stability of the equilibrium states and the structure and interactions of the stable and unstable manifolds of the saddle-type equilibria turn out to be factors influencing the model's behavior. Whether or not the model is able to attain any form of sustained oscillatory behavior, i.e. limit cycles or chaos, seems to be determined by (global) bifurcations involving the stable and unstable manifolds of the equilibrium states. We attempt to determine whether such behavior should be expected to be attained from reasonable initial conditions by incorporating an immune response to an antigen in the model. A comparison of the behavior of the model with experimental data from the literature provides suggestions for the parameter regime in which the immune system is operating.