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

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.

The evolution of mathematical immunology

The types of mathematical models used in immunology and their scope have changed drastically in the past 10 years. Classical models were based on ordinary differential equations (ODEs), difference equations, and cellular automata. These models focused on the 'simple' dynamics obtained between a small number of reagent types (e.g. one type of receptor and one type of antigen or two T-cell populations). With the advent of high-throughput methods, genomic data, and unlimited computing power, immunological modeling shifted toward the informatics side. Many current applications of mathematical models in immunology are now focused around the concepts of high-throughput measurements and system immunology (immunomics), as well as the bioinformatics analysis of molecular immunology. The types of models have shifted from mainly ODEs of simple systems to the extensive use of Monte Carlo simulations. The transition to a more molecular and more computer-based attitude is similar to the one occurring over all the fields of complex systems analysis. An interesting additional aspect in theoretical immunology is the transition from an extreme focus on the adaptive immune system (that was considered more interesting from a theoretical point of view) to a more balanced focus taking into account the innate immune system also. We here review the origin and evolution of mathematical modeling in immunology and the contribution of such models to many important immunological concepts.

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--I. From stationary states to limit cycle oscillations

We develop a model for the idiotypic interaction between two B cell clones. This model takes into account B cell proliferation, B cell maturation, antibody production, the formation and subsequent elimination of antibody-antibody complexes and recirculation of antibodies between the spleen and the blood. Here we investigate, by means of stability and bifurcation analysis, how each of the processes influences the model's behavior. After appropriate nondimensionalization, the model consists of eight ordinary differential equations and a number of parameters. We estimate the parameters from experimental sources. Using a coordinate system that exploits the pairwise symmetry of the interactions between two clones, we analyse two simplified forms of the model and obtain bifurcation diagrams showing how their five equilibrium states are related. We show that the so-called immune states lose stability if B cell and antibody concentrations change on different time scales. Additionally, we derive the structure of stable and unstable manifolds of saddle-type equilibria, pinpoint their (global) bifurcations and show that these bifurcations play a crucial role in determining the parameter regimes in which the model exhibits oscillatory behavior.