A qualitative study of an idiotypic cyclic network

Within the network hypothesis proposed by Jerne, the immune response is interpreted as a collective behaviour of different antibody species, interacting through idiotypic recognition. In order to insure the stability of the network, only a few species would be implied in the response to an antigenic challenge.We study a network made up of small cycles of idiotypic units, each element activating the subsequent one and repressing the preceding one. In the recent theoretical models, the kinetics is described by steep sigmoidal functions with a repression threshold lower than the stimulation one. To enable a systematic qualitative analysis of the dynamics, we replace the continuous kinetics by stepfunctions. The antibodies are thus considered as control elements like genes, enzymes or neurones. In order to account for the different thresholds, we use discrete three-level variables.We develop two methods to study the dynamics: the first one, due to Glass, describes the time-evolution of a cycle by a system of piecewise linear (PL) differential equations and the second method is the boolean formalization, applied extensively by Thomas in the field of genetic regulation.These techniques provide complementary informations about the dynamics of the cycle: the PL method establishes a state transition diagram providing all the potential behaviours independently of the parameter values in the model, whereas the purely logical analysis permits a simulation of the trajectories for precise values of the parameters.The state transition diagram presents several steady states. It suggests to interpret the response to an antigenic challenge as a transition from one steady state to another. The multiplicity of the steady states might be associated with the various modes of immune response depending on the doses of antigen injected and on the previous antigenic history of the system.     

From complex spatial dynamics to simple Markov chain models: do predators and prey leave footprints?

In this paper we present a concept for using presence–absence data to recover information on the population dynamics of predator–prey systems. We use a highly complex and spatially explicit simulation model of a predator–prey mite system to generate simple presence–absence data: the number of patches with both prey and predators, with prey only, with predators only, and with neither species, along with the number of patches that change from one state to another in each time step. The average number of patches in the four states, as well as the average transition probabilities from one state to another, are then depicted in a state transition diagram, constituting the "footprints" of the underlying population dynamics. We investigate to what extent changes in the population processes modeled in the complex simulation (i.e. the predator's functional response and the dispersal rates of both species) are reflected by different footprints
The transition probabilities can be used to forecast the expected fate of a system given its current state. However, the transition probabilities in the modeled system depend on the number of patches in each state. We develop a model for the dependence of transition probabilities on state variables, and combine this information in a Markov chain transition matrix model. Finally, we use this extended model to predict the long-term dynamics of the system and to reveal its asymptotic steady state properties.     

Memory but no suppression in low-dimensional symmetric idiotypic networks

We present a new symmetric model of the idiotypic immune network. The model specifies clones of B-lymphocytes and incorporates: (1) influx and decay of cells; (2) symmetric stimulatory and inhibitory idiotypic interactions; (3) an explicit affinity parameter (matrix); (4) external (i.e. non-idiotypic) antigens. Suppression is the dominant interaction, i.e. strong idiotypic interactions are always suppressive. This precludes reciprocal stimulation of large clones and thus infinite proliferation. Idiotypic interactions first evoke proliferation, this enlarges the clones, and may in turn evoke suppression. We investigate the effect of idiotypic interactions on normal proliferative immune responses to antigens (e.g. viruses). A 2-D, i.e. two clone, network has a maximum of three stable equilibria: the virgin state and two asymmetric immune states. The immune states only exist if the affinity of the idiotypic interaction is high enough. Stimulation with antigen leads to a switch from the virgin state to the corresponding immune state. The network therefore remembers antigens, i.e. it accounts for immunity/memory by switching beteen multiple stable states. 3-D systems have, depending on the affinities, 9 qualitatively different states. Most of these also account for memory by state switching. Our idiotypic network however fails to account for the control of proliferation, e.g. suppression of excessive proliferation. In symmetric networks, the proliferating clones suppress their anti-idiotypic suppressors long before the latter can suppress the former. The absence of proliferation control violates the general assumption that idiotypic interactions play an important role in immune regulation. We therefore test the robustness of these results by abandoning our assumption that proliferation occurs before suppression. We thus define an “escape from suppression” model, i.e. in the “virgin” state idiotypic interactions are now suppressive. This system erratically accounts for memory and never for suppression. We conclude that our “absence of suppression from idiotypic interactions” does not hinge upon our “proliferation before suppression” assumption.     

The regulation of resistance to Schistosoma mansoni by auto-anti-idiotypic immunity. III. An analysis of effects on epitopic recognition, idiotypic expression, and anti-idiotypic reactivity at the clonal level.

Auto-anti-idiotypic mechanisms can regulate the protective immune response against Schistosoma mansoni. Anti-idiotypic responses were stimulated by immunization of mice either with nonspecifically induced lymphoblasts, produced with Con A, or with Ag-induced lymphoblasts bearing specific idiotypic receptors. The effect of the induced anti-idiotypic response upon clonotypic cellular reactivity was assessed in vitro through the suppression of antigen-mediated blast transformation by cloned T cells and in vivo by suppression of resistance to S. mansoni and delayed-type hypersensitivity responses against specific Ag. Differential regulation of humoral immune responses was studied at the levels of specific epitopic recognition, the expression of specific Id, and the production of anti-idiotypic responses directed against mAb bearing specific Id. Anti-idiotypic sensitization resulted in variable (10 to 90%) suppression of the immune response to discrete antigenic epitopes, the expression of specific idiotypic phenotypes, and anti-idiotypic, antiparatopic responses against T cell clonotypes and antibody idiotypic phenotypes. In vitro admixture and in vivo challenge studies resulted in consonant differential suppression. Thus idiotypic regulation can mold the fine specificities of the protective immune response to S. mansoni at the clonal level and may provide an approach to optimize the expression and assessment of resistance.     

Modelling the spatio-temporal dynamics of multi-species host-parasitoid interactions: heterogeneous patterns and ecological implications

A mathematical model of the spatio-temporal dynamics of a two host, two parasitoid system is presented. There is a coupling of the four species through parasitism of both hosts by one of the parasitoids. The model comprises a system of four reaction-diffusion equations. The underlying system of ordinary differential equations, modelling the host-parasitoid population dynamics, has a unique positive steady state and is shown to be capable of undergoing Hopf bifurcations, leading to limit cycle kinetics which give rise to oscillatory temporal dynamics. The stability of the positive steady state has a fundamental impact on the spatio-temporal dynamics: stable travelling waves of parasitoid invasion exhibit increasingly irregular periodic travelling wave behaviour when key parameter values are increased beyond their Hopf bifurcation point. These irregular periodic travelling waves give rise to heterogeneous spatio-temporal patterns of host and parasitoid abundance. The generation of heterogeneous patterns has ecological implications and the concepts of temporary host refuge and niche formation are considered.     

The enhancement of stability of p53 in MTBP induced p53–MDM2 regulatory network

We have modeled an MTBP-MDM2–p53 regulatory network by integrating p53–MDM2 autoregulatory model (Proctor and Gray, 2008) with the effect of a cellular protein MTBP (MDM2 binding protein) which is allowed to bind with MDM2 (Brady et al., 2005). We study this model to investigate the activation of p53 and MDM2 steady state levels induced by MTBP protein under different stress conditions. Our simulation results in three approaches namely deterministic, Chemical Langevin equation and stochastic simulation of Master equation show a clear transition from damped limit cycle oscillation to fixed point oscillation during a certain time period with constant stress condition in the cell. This transition is the signature of transition of p53 and MDM2 levels from activated state to stabilized steady state levels. We present various phase diagrams to show the transition between unstable and stable states of p53 and MDM2 concentration levels and also their possible relations among critical value of the parameters at which the respective protein level reach stable steady states. In the stochastic approach, the dynamics of the proteins become noise induced process depending on the system size. We found that this noise enhances the stability of the p53 steady state level.     

Unreasonable implications of reasonable idiotypic network assumptions

We first analyse a simple symmetric model of the idiotypic network. In the model idiotypic interactions regulate B cell proliferation. Three non-idiotypic processes are incorporated: (1) influx of newborn cells; (2) turnover of cells: (3) antigen. Antigen also regulates proliferation. A model of 2 B cell populations has 3 stable equilibria: one virgin, two immune. The twodimensional system thus remembers antigens, i.e. accounts for immunity. By contrast, if an idiotypic clone proliferates (in response to antigen), its anti-idiotypic partner is unable to control this. Symmetric idiotypic networks thus fail to account for proliferation regulation. In high-D networks we run into two problems. Firstly, if the network accounts for memory, idiotypic activation always propagates very deeply into the network. This is very unrealistic, but is an implication of the “realistic” assumption that it should be easier to activate all cells of a small virgin clone than to maintain the activation of all cells of a large (immune) clone. Secondly, graph theory teaches us that if the (random) network connectance exceeds a threshold level of one interaction per clone, most clones are interconnected. We show that this theory is also applicable to immune networks based on complementary matching idiotypes. The combination of the first “percolation” result with the “interconnectancr” result means that the first stimulation of the network with antigen should eventually affect most of the clones. We think this is unreasonable. Another threshold property of the network connectivity is the existence of a virgin state. A gradual increase in network connectance eliminates the virgin state and thus causes an abrupt change in network behaviour. In contrast to weakly connected systems, highly connected networks display autonomous activity and are unresponsive to external antigens. Similar differences between neonatal and adult networks have been described by experimentalists. The robustness of these results is tested with a network in which idiotypic inactivation of a clone occurs more generally than activation. Such “long-range inhibition” is known to promote pattern formation. However, in our model it fails to reduce the percolation, and additionally, generates semi-chaotic behaviour. In our network, the inhibition of a clone that is inhibiting can alter this clone into a clone that is activating. Hence “long-range inhibition” implies “long-range activation”, and idiotypic activation fails to remain localized. We next complicate this model by incorporating antibody production. Although this “antibody” model statically accounts for the same set of equilibrium points, it dynamically fails to account for state switching (i.e. memory). The switching behaviour is disturbed by the autonomous slow decay of the (long-lived) antibodies. After antigenic triggering the system now performs complex cyclic behaviour. Finally, it is suggested that (idiotypic) formation of antibody complexes can play only a secondary role in the network. In conclusion, our results cast doubt on the functional role of a profound idiotypic network. The network fails to account for proliferation regulation, and if it accounts for memory phenomena, it “explodes” upon the first encounter with antigen due to extensive percolation.     

Antiidiotypic networks

Jerne envisions the immune system as a web of V domains that constitutes an idiotypic network. He thinks that regulatory processes governed by idiotypic interactions can explain the generation of the various immune states. We discuss a few models that furnish information about the possible configuration of this immune network: closed or open ended. It appears that closed configurations only can generate stable immune states. Moreover, we cite some experimental data in favor of the network hypothesis. We show how they can lead to propose the structure of functional regulatory circuits whose cellular and molecular interactions are mediated by idiotypic recognition processes.     

Logical and symbolic analysis of robust biological dynamics

Logical models provide insight about key control elements of biological networks. Based solely on the logical structure, we can determine state transition diagrams that give the allowed possible transitions in a coarse grained phase space. Attracting pathways and stable nodes in the state transition diagram correspond to robust attractors that would be found in several different types of dynamical systems that have the same logical structure. Attracting nodes in the state transition diagram correspond to stable steady states. Furthermore, the sequence of logical states appearing in biological networks with robust attracting pathways would be expected to appear also in Boolean networks, asynchronous switching networks, and differential equations having the same underlying structure. This provides a basis for investigating naturally occurring and synthetic systems, both to predict the dynamics if the structure is known, and to determine the structure if the transitions are known.     

Central immune system,the self and autoimmunity

We model how auto-reactiveB cells are kept under control by an idiotypic network. Autoimmunity occurs when the control is broken by an infection or not achieved through an abnormal ontogenetic evolution. We describe the idiotypic network, viz., the central immune system, by idiotype-anti-idiotype pairs which are coupled to a set of highly connected clones, which interact with each clone of the network. Some clones of the central immune system recognize self-antigen. We find a huge variety of fixed points which can be classified as tolerant, autoimmune, and neutral states according to the concentration of the auto-reactive antibody. Most significant are auto-reactive clones which are a member of an idiotype-anti-idiotype pair. In a healthy individual, an autoimmune disease is induced by an antigen infection which triggers a transition from a tolerant to an autoimmune state. Autoimmunity is induced more readily by an antigen coupling to theanti-idiotype than by one interacting with the auto-reactive clone itself. We indicate a possible therapy which simply reverses the processes that have lead to the autoimmune disease. In the early development of the central immune system its highly connected, core part serves to draw the more specific clones of idiotype-anti-idiotype pairs into the network. In order to avoid autoimmunity in ontogenetic evolution the anti-idiotype of an auto-reactive clone must be formed in advance by a sufficiently long period of time. Thus, a well ordered succession of the appearance of the more specific clones is required.     

Stability and bifurcations in a model of antigenic variation in malaria

We examine the properties of a recently proposed model for antigenic variation in malaria which incorporates multiple epitopes and both long-lasting and transient immune responses. We show that in the case of a vanishing decay rate for the long-lasting immune response, the system exhibits the so-called “bifurcations without parameters” due to the existence of a hypersurface of equilibria in the phase space. When the decay rate of the long-lasting immune response is different from zero, the hypersurface of equilibria degenerates, and a multitude of other steady states are born, many of which are related by a permutation symmetry of the system. The robustness of the fully symmetric state of the system was investigated by means of numerical computation of transverse Lyapunov exponents. The results of this exercise indicate that for a vanishing decay of long-lasting immune response, the fully symmetric state is not robust in the substantial part of the parameter space, and instead all variants develop their own temporal dynamics contributing to the overall time evolution. At the same time, if the decay rate of the long-lasting immune response is increased, the fully symmetric state can become robust provided the growth rate of the long-lasting immune response is rapid. This work was partially supported by the ATRJVVO grant from the James Martin 21st Century School, University of Oxford.     

Impossibility of multiple steady states in a family of active membrane transport models

To study the dynamical behavior of active membrane transport models, Vieira and Bisch proposed a complex chemical network (model 3) with two cycles. One cycle involves monomers as pump units while the other cycle uses dimers. In their work, the stoichiometric network analysis was used to study the stability of steady states and the bifurcation analysis was done through numerical methods. They concluded that the possibility of multiple steady states in the model 3 could not be discarded. Here, a zero eigenvalue analysis is applied to prove the impossibility of multiple positive steady states in the model 3. (A positive steady state is one for which all species have positive concentrations.) Moreover, the result is generalized to its family networks. Received: 6 April 1998 / Revised version: 16 October 1998 / Accepted: 28 October 1998     

Network regulation of the immune response: modulation of suppressor lymphocytes by alternative signals including contrasuppression

In the preceding paper we demonstrated that comparison of alternative designs for the immune network can be used to examine the functional significance of specified interactions in normal immune responses. In this paper we examine mathematically the functional significance of three interactions affecting the production of suppressor lymphocytes involved in regulation of normal immune responses. The interactions examined in detail are 1) antigenic stimulation of the production of suppressor lymphocytes, 2) idiotypic stimulation of the production of suppressor lymphocytes, and 3) antigenic inhibition of the production of suppressor lymphocytes (i.e., contrasuppression). The results of our analysis suggest that an immune system with only antigenic stimulation of suppressor production is less effective than a system with both antigenic and idiotypic stimulation of suppressor production on the basis of all of the criteria examined in this study. In turn, the latter system is less effective than a system with only idiotypic stimulation of suppressor production. Furthermore, a system with both idiotypic stimulation and antigenic inhibition of suppressor production can be equal or superior to a system with only idiotypic stimulation of suppressor production on the basis of the same criteria. Similar conclusions hold for the comparison of systems in which regulation by the suppressor lymphocytes of interest is exerted upon production of effector molecules rather than upon production of effector lymphocytes, and also for the comparison of systems in which interactions affecting the production of suppressor factors are of interest.     

Salmonella fecal shedding and immune responses are dose- and serotype- dependent in pigs

Despite the public health importance of Salmonella infection in pigs, little is known about the associated dynamics of fecal shedding and immunity. In this study, we investigated the transitions of pigs through the states of Salmonella fecal shedding and immune response post-Salmonella inoculation as affected by the challenge dose and serotype. Continuous-time multistate Markov models were developed using published experimental data. The model for shedding had four transient states, of which two were shedding (continuous and intermittent shedding) and two non-shedding (latency and intermittent non-shedding), and one absorbing state representing permanent cessation of shedding. The immune response model had two transient states representing responses below and above the seroconversion level. The effects of two doses [low (0.65×10(6) CFU/pig) and high (0.65×10(9) CFU/pig)] and four serotypes (Salmonella Yoruba, Salmonella Cubana, Salmonella Typhimurium, and Salmonella Derby) on the models' transition intensities were evaluated using a proportional intensities model. Results indicated statistically significant effects of the challenge dose and serotype on the dynamics of shedding and immune response. The time spent in the specific states was also estimated. Continuous shedding was on average 10-26 days longer, while intermittent non-shedding was 2-4 days shorter, in pigs challenged with the high compared to low dose. Interestingly, among pigs challenged with the high dose, the continuous and intermittent shedding states were on average up to 10-17 and 3-4 days longer, respectively, in pigs infected with S. Cubana compared to the other three serotypes. Pigs challenged with the high dose of S. Typhimurium or S. Derby seroconverted on average up to 8-11 days faster compared to the low dose. These findings highlight that Salmonella fecal shedding and immune response following Salmonella challenge are dose- and serotype-dependent and that the detection of specific Salmonella strains and immune responses in pigs are time-sensitive.     

Modeling immune reactivity in secondary lymphoid organs

Models of the dynamical interactions important in generating immune reactivity have generally assumed that the immune system is a single well-stirred compartment. Here we explicitly take into account the compartmentalized nature of the immune system and show that qualitative conclusions, such as the stability of the immune steady state, depend on architectural details. We examine a simple model idiotypic network involving only two types of B cells and antibody molecules. We show, for model parameters used by De Boeret al. (1990,Chem. Eng. Sci. 45, 2375–2382), that the immune steady state is unstable in a one compartmental model but stable in a two compartment model that contains both a lymphoid organ, such as the spleen, and the circulatory system. This work was performed under the auspices of the U.S. Department of Energy.     

Dynamics of commensalistic systems with self- and cross-inhibition

Up to three stable steady states are possible in a simple commensalistic system, taking place in an open-loop mixed reactor when the growth rates of the two species are inhibited by the substrates they prey on (Self-inhibition). Two stable states are possible in a system with noncompetitive inhibition of the species by the substrate they are not preying on (cross-inhibition). A large number of steady states as well as oscillatory states are possible when both self- and cross-inhibition are strong. Multiplicity of steady states is also possible in a reactor with biomas recirculation for these kinetics. Yet, the latter is more stable than the open-loop reactor in the sense that the domain of steady-state multiplicity is narrower. The stability of steady states and the dynamics of the systems for each of the investigated kinetics are summarized in a qualitative phase plane. The importance of the analysis for improving the selectively and yield of the system and for predicting the response of the system to changes in the operating conditions, is discussed.     

Immunologic control of tumors by in vivo Fc gamma receptor-targeted antigen loading in conjunction with double-stranded RNA-mediated immune modulation

Despite the expression of non-self or neo-epitopes, many tumors such as lymphoid malignancies or cancers induced by oncogenic viruses are able to gradually overcome the immune defense mechanisms and spread. Using a preclinical model of hematological malignancy, we show that Ig-associated idiotypic determinants are recognized by the immune system in a fashion that results in immune deviation, allowing tumor progression and establishment of metastases. Using gene-targeted mice, we show that anti-idiotypic MHC class I-restricted immunity is promoted by ITAM motif (ITAM+) FcgammaR, but kept in check by ITIM motif (ITIM+) FcgammaRIIB-mediated mechanisms. In addition to interfering with the functionality of ITIM+ FcgammaR, effective anti-idiotypic and antitumoral immunity can be achieved by FcgammaR-targeted delivery of epitope in conjunction with administration of stimulatory motifs such as dsRNA, correcting the ineffective response to idiotypic epitopes. The immune process initiated by FcgammaR-mediated targeting of epitope together with dsRNA, resulted in control of tumor growth, establishment of immune memory and protection against tumors bearing antigenic variants. In summary, targeted delivery of MHC class I-restricted epitopes via ITAM+ FcgammaR, in conjunction with use of TLR-binding immune stimulatory motifs such as dsRNA, overcomes suboptimal responses to idiotypic determinants and may constitute a novel approach for the treatment of a broad range of malignancies. Finally, the results shed light on the mechanisms regulating the idiotypic network and managing the diversity associated with immune receptors.     

Oscillations in Biochemical Reaction Networks Arising from Pairs of Subnetworks

Biochemical reaction models show a variety of dynamical behaviors, such as stable steady states, multistability, and oscillations. Biochemical reaction networks with generalized mass action kinetics are represented as directed bipartite graphs with nodes for species and reactions. The bipartite graph of a biochemical reaction network usually contains at least one cycle, i.e., a sequence of nodes and directed edges which starts and ends at the same species node. Cycles can be positive or negative, and it has been shown that oscillations can arise as a result of either a positive cycle or a negative cycle. In earlier work it was shown that oscillations associated with a positive cycle can arise from subnetworks with an odd number of positive cycles. In this article we formulate a similar graph-theoretic condition, which generalizes the negative cycle condition for oscillations. This new graph-theoretic condition for oscillations involves pairs of subnetworks with an even number of positive cycles. An example of a calcium reaction network with generalized mass action kinetics is discussed in detail.     

Stable oscillations in mathematical models of biological control systems

Summary Oscillations in a class of piecewise linear (PL) equations which have been proposed to model biological control systems are considered. The flows in phase space determined by the PL equations can be classified by a directed graph, called a state transition diagram, on anN-cube. Each vertex of theN-cube corresponds to an orthant in phase space and each edge corresponds to an open boundary between neighboring orthants. If the state transition diagram contains a certain configuration called a cyclic attractor, then we prove that for the associated PL equation, all trajectories in the regions of phase space corresponding to the cyclic attractor either (i) approach a unique stable limit cycle attractor, or (ii) approach the origin, in the limitt→∞. An algebraic criterion is given to distinguish the two cases. Equations which can be used to model feedback inhibition are introduced to illustrate the techniques.