Pattern formation in one- and two-dimensional shape-space models of the immune system.

A large-scale model of the immune network is analyzed, using the shape-space formalism. In this formalism, it is assumed that the immunoglobulin receptors on B cells can be characterized by their unique portions, or idiotypes, that have shapes that can be represented in a space of a small finite dimension. Two receptors are assumed to interact to the extent that the shapes of their idiotypes are complementary. This is modeled by assuming that shapes interact maximally whenever their coordinates in the space-space are equal and opposite, and that the strength of interaction falls off for less complementary shapes in a manner described by a Gaussian function of the Euclidean "distance" between the pair of interacting shapes. The degree of stimulation of a cell when confronted with complementary idiotypes is modeled using a log bell-shaped interaction function. This leads to three possible equilibrium states for each clone: a virgin, an immune, and a suppressed state. The stability properties of the three possible homogeneous steady states of the network are examined. For the parameters chosen, the homogeneous virgin state is stable to both uniform and sinusoidal perturbations of small amplitude. A sufficiently large perturbation will, however, destabilize the virgin state and lead to an immune reaction. Thus, the virgin system is both stable and responsive to perturbations. The homogeneous immune state is unstable to both uniform and sinusoidal perturbations, whereas the homogeneous suppressed state is stable to uniform, but unstable to sinusoidal, perturbations. The non-uniform patterns that arise from perturbations of the homogeneous states are examined numerically. These patterns represent the actual immune repertoire of an animal, according to the present model. The effect of varying the standard deviation sigma of the Gaussian is numerically analyzed in a one-dimensional model. If sigma is large compared to the size of the shape-space, the system attains a fixed non-uniform equilibrium. Conversely if sigma is small, the system attains one out of many possible non-uniform equilibria, with the final pattern depending on the initial conditions. This demonstrates the plasticity of the immune repertoire in this shape-space model. We describe how the repertoire organizes itself into large clusters of clones having similar behavior. These results are extended by analyzing pattern formation in a two-dimensional (2-D) shape-space.(ABSTRACT TRUNCATED AT 400 WORDS)     

A graph model for the evolution of specificity in humoral immunity

The immune system protects the body against health-threatening entities, known as antigens, through very complex interactions involving the antigens and the system's own entities. One remarkable feature resulting from such interactions is the immune system's ability to improve its capability to fight antigens commonly found in the individual's environment. This adaptation process is called the evolution of specificity. In this paper, we introduce a new mathematical model for the evolution of specificity in humoral immunity, based on Jerne's functional, or idiotypic, network. The evolution of specificity is modeled as the dynamic updating of connection weights in a dynamic graph whose nodes are related to the network's idiotypes. At the core of this weight-updating mechanism are the increase in specificity caused by clonal selection and the decrease in specificity due to the insertion of uncorrelated idiotypes by the bone marrow. As we demonstrate through numerous computer experiments, for appropriate choices of parameters the new model correctly reproduces, in qualitative terms, several immune functions.     

Evolution of specificity in an immune network

A dynamic antigen response of the immune network is discussed, based on shape-space modelling. The present model extends the shape-space modelling by introducing the evolution of specificity of idiotypes. When the amount of external antigen increases, a measure of stability of the immune network is lost and thus the network can respond to the antigen. It is shown that specific and non-specific responses emerge as a function of antigen amounts. A specific response is observed with a fixed-point attractor, and a non-specific response is observed with a chaotic attractor for the lymphocyte population dynamics. The network topology also changes between fixed-point and chaotic attractors. For some antigen amounts, chaotic attractors will vanish or become long-lived super-transient states. A dynamic bell-shaped response function will thus emerge. The relevance of long-lived chaotic transient states embedded in fixed-point attractors is discussed with respect to immune functions.     

Qualitative dynamics of a network model of regulation of the immune system: A rationale for the IgM to IgG switch

A mathematical model has been developed that simulates some of the main features of a network theory of regulation of the immune system. According to the network viewpoint, the V regions (idiotypes) on antibodies and lymphocytes are self-antigens, to which other lymphocytes of the system can respond specifically, just as they respond to foreign antigens. The resultant couplings between the lymphocytes are considered to be basic for the regulation of the system.The present mathematical model simulates the interactions between cells that recognize the antigen (“positive cells”), and “negative cells” that have receptors that specifically recognize the V regions of the positive cells. The mathematical model incorporates only the interactions that are postulated to be important in the four steady states of the theory, and includes neither the antigen nor any accessory (“A”) cells. The effects of both antigen-specific and anti-idiotypic T and B cells are included, as well as antigen-specific and anti-idiotypic T cell factors, and the two main classes of antibodies. The model is a first order autonomous ordinary differential equation in two variables. We describe a geometric technique that gives strong information on the model, without explicitly solving the ordinary differential equation. This technique proves to be powerful in permitting us to systematically scan the parameter space of the model. The detailed analysis leads to support for the idea that the model provides a rationale for the switch observed in the immune system from the production of one major class of antibody (IgM) to the other major class (IgG). The analysis also leads to a new, previously unsuspected possibility for the nature of the suppressed state within the context of the postulates of the symmetrical network theory.     

Complementary idiotypy in the regulation of the immune response.

A new concept is presented for the interactions of two complementary antibodies in the immune response. These antibodies bind to each other by means of their variable sequence determinants and therefore are designated as complementary idiotypes. Under certain conditions, both complementary idiotypes are produced by the same animal at the same time. An idiotype can drastically affect the expression of the complementary idiotype in the animal, inducing a peripheral quench effect of antibody-binding activity and a central effect on the immunocompetent cell, which produces the complementary idiotype. It is proposed that complementary idiotypes might be induced during every immune response, thus playing an essential role in the regulation of the immune response.     

Development of an Idiotypic Network in Shape Space

Based upon the shape-space formalism, a model of an idiotypic network including both bound and free immunoglobulins is simulated. Our point of interest is the network development in the context of self antigens. The investigations are organized around simulations initiated by various spatial configurations of antigens; the behavior of the system with respect to antigens is analyzed in terms of morphogenetic processes occurring in the shape space. For certain values of the parameters, the network expands by traveling waves. The resulting spatial pattern is a partition of the shape space into zones where introduction of an antigen entails an infinite growth of the clones binding to it, and into zones where, on the contrary, the anti-antigen idiotypes decrease. Among the parameter combinations tested, some produce a partition that remains static whereas others produce a partition that changes in time. For other values of the parameters, the patterns generated do not partition shape space into zones; in these cases, it is observed that the system systematically explodes when an antigen is present.     

Role of self carriers in the immune response and tolerance. VIII. Suppression of the MOPC-104E idiotypic response by idiotype-modified self, but not by dextran-modified self

We have investigated the suppression of the anti-dextran B1355S immune response using our model of modified self. The anti-dextran response is idiotypically well defined in BALB/c mice. This system enables us to examine the contribution of various predominant idiotypes to the antibody response under conditions of suppression by antigen or by idiotype-specific suppressor cells. Our results demonstrate that the total anti-dextran response can be inhibited by pretreatment of animals with dextran-coupled syngeneic spleen cells; however, the representation of major idiotypes constituting this response are not reduced in percentage. In contrast, pretreatment of mice with MOPC-104E-coupled spleen cells leads to a specific suppression of the private IdI-104E idiotype. The total anti-dextran response remains unchanged, as well as proportions of other major idiotypes known (IdI-588 and IdX). This suppression is mediated by Thy-1.2+, Lyt-2.2+ T cells, as demonstrated by adoptive transfer assays. This system will allow the molecular dissection of the regulation of an idiotypically well-defined system for the suppression by either antigen- or idiotype-specific suppressor T cells.     

Regulation of clones responding to α 1 → 3 dextran: I. Individual variation in the expression of idiotypes

Balb/c mice were immunized with dextran B1355S and assayed for serum antibodies and direct plaque-forming cells. The earliest detectable anti-dextran response occurred in 2-week-old animals. Anti-idiotypic antisera against MOPC-104E and J558 were raised in A/He mice and rendered individual idiotype specific by cross-absorption. When the amounts of MOPC-104E and J558 idiotypes in immune sera and the PFC response of individual mice were analyzed, the ratio of both idiotypes were found highly variable in all tested animals. This observed individual variability in the expression of two major idiotypes in the Balb/c response to dextran B1355S provides an experimental basis for investigating the mechanisms regulating idiotype expression.     

The use of idiotypes as markers for antibody variable regions in the rabbit.

This article reviews the use of idiotypes as variable region genetic markers in the rabbit. Topics discussed include reagents and assays for the detection of idiotypes, evidence concerning the association of idiotypes with specific antigen binding sites on antibodies, and the inheritance of idiotypes in the rabbit. Several points are emphasized in this review. First, interpretation of idiotypic phenomenons are strongly dependent on the reagents and assays employed. Second, while strong evidence exists that a given idiotype is a marker for a specific antigen binding site, exceptions to this association have been reported. Third, the inheritance of identical or similar idiotypes has been demonstrated in several instances, but it is not always demonstrable, perhaps because an idiotype is a complex phenotype. Several reasons for this complexity are pointed out. Fourth, idiotypes are linked to group a allotypes and VL subgroups but exceptions to these associations have been described for antibodies isolated from single individuals. The significance of these exceptions is discussed. Current areas of interest in rabbit idiotypy include the relationship of idiotypes to other V region markers, the genetics of idiotypes specific to H or L chains, and the relationships among the idiotypes of antibodies isolated from a single rabbit.     

Modeling adaptive biological systems

During the evolution of many systems found in nature, both the system composition and the interactions between components will vary. Equating the dimension with the number of different components, a system which adds or deletes components belongs to a class of dynamical systems with a finite dimensional phase space of variable dimension. We present two models of biochemical systems with a variable phase space, a model of autocatalytic reaction networks in the prebiotic soup and a model of the idiotypic network of the immune system. Each model contains characteristic meta-dynamical rules for constructing equations of motion from component properties. The simulation of each model occurs on two levels. On one level, the equations of motion are integrated to determine the state of each component. On a second level, algorithms which approximate physical processes in the real system are employed to change the equations of motion. Models with meta-dynamical rules possess several advantages for the study of evolving systems. First, there are no explicit fitness functions to determine how the components of the model rank in terms of survivability. The success of any component is a function of its relationship to the rest of the system. A second advantage is that since the phase space representation of the system is always finite but continually changing, we can explore a potentially infinite phase space which would otherwise be inaccessible with finite computer resources. Third, the enlarged capacity of systems with meta-dynamics for variation allows us to conduct true evolution experiments. The modeling methods presented here can be applied to many real biological systems. In the two studies we present, we are investigating two apparent properties of adaptive networks. With the simulation of the prebiotic soup, we are most interested in how a chemical reaction network might emerge from an initial state of relative disorder. With the study of the immune system, we study the self-regulation of the network including its ability to distinguish between species which are part of the network and those which are not.     

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.     

Effects of selective neutrality on the evolution of molecular species

We introduce a model of evolution on a fitness landscape possessing a tunable degree of neutrality. The model allows us to study the general properties of molecular species undergoing neutral evolution. We find that a number of phenomena seen in RNA sequence-structure maps are present also in our general model. Examples are the occurrence of ''common'' structures that occupy a fraction of the genotype space which tends to unity as the length of the genotype increases, and the formation of percolating neutral networks that cover the genotype space in such a way that a member of such a network can be found within a small radius of any point in the space. We also describe a number of new phenomena that appear to be general properties of systems possessing selective neutrality. In particular, we show that the maximum fitness attained during the adaptive walk of a population evolving on such a fitness landscape increases with increasing degree of neutrality, and is directly related to the fitness of the most fit percolating network.     

Dynamics of a class of immune networks. II. Oscillatory activity of cellular and humoral components

Simulations show that for a certain range of its free parameters, a model of the immune network including both cellular and humoral components is capable of self-sustained oscillations. The model also possesses a number of fixed points which appear to be unstable. These results, taken together, suggest the hypothesis that the immune network may be able to sustain a non-degenerate diversity of active clones which are actively connected to each other only on condition that the activities of these clones are oscillatory.     

Idiotypic networks incorporating T-B cell co-operation. The conditions for percolation

Previous work was concerned with symmetric immune networks of idiotypic interactions amongst B cell clones. The behaviour of these networks was contrary to expectations. This was caused by an extensive percolation of idiotypic signals. Idiotypic activation was thus expected to affect almost all (greater than 10(7] B cell clones. We here analyse whether the incorporation of helper T cells (Th) into these B cell models could cause a reduction in the percolation. Empirical work on idiotypic interactions between Th and B cells however, would suggest that two different idiotypic Th models should be developed: (1) a Th which recognises native B cell idiotypes, i.e. a non-MHC-restricted "ThId" model, and (2) a "classical" MHC-restricted helper T cell model. In the ThId model, the Th-B cell interaction is symmetric. A 2-D model of a Th and a B cell clone that interact idiotypically with each other accounts for various equilibria (i.e. one virgin and two immune states). Introduction of antigen does indeed lead to a state switch from the virgin to the immune state; such a system is thus able to "remember" its exposure to antigen. Idiotypic signals do however, percolate in ThId models via these "B-Th-B-Th" pathways: proliferating Th and B cell clones that interact idiotypically, will always activate each other reciprocally. In the MHC-restricted Th model, Th-B interactions are asymmetric. Because the B cell idiotypes are processed and subsequently presented by MHC molecules, the Th receptor and the native B cell receptor are not expected to be complementary. Thus the Th and the B cells are unable to activate each other reciprocally, and a 2-D Th-B cell model cannot account for idiotypic memory. In contrast to the ThId model, idiotypic activation cannot percolate via "B-Th-B-Th" interactions. Due to the assymmetry idiotypic activation stops at the first Th level. A Th clone cannot activate a subsequent B cell clone: if the B cells recognise the Th cells, they see idiotype but get no help; if the Th cells see the B cells, the B cells are helped but see no idiotype. The percolation along "B-B-B" pathways in these two models is next analysed. Two B cells clones, each helped by one Th clone, are connected by a symmetric idiotypic interaction. It turns out that in both models the second (i.e. anti-idiotypic) B cells (B2) never proliferate.(ABSTRACT TRUNCATED AT 400 WORDS)     

A neural network model based on the analogy with the immune system

The similarities between the immune system and the central nervous system lead to the formulation of an unorthodox neural network model. The similarities between the two systems are strong at the system level, but do not seem to be so striking at the level of the components. A new model of a neuron is therefore formulated, in order that the analogy can be used. The essential feature of the hypothetical neuron is that it exhibits hysteresis at the single neuron level. A network of N such neurons is modelled by an N-dimensional system of ordinary differential equations, which exhibits almost 2N attractors. The model has a property that resembles free will. A conjecture concerning how the network might learn stimulus-response behaviour is described. According to the conjecture, learning does not involve modifications of the strengths of synaptic connections. Instead, stimuli ("questions") selectively applied to the network by a "teacher" can be used to take the system to a region of the N-dimensional phase space where the network gives the desired stimulus-response behaviour. A key role for sleep in the learning process is suggested. The model for sleep leads to prediction that the variance in the rates of firing of the neurons associated with memory should increase during waking hours, and decrease during sleep.     

Genetic control of the immune response to staphylococcal nuclease. IX. Recombination between genes determining BALB/c antinuclease idiotypes and the heavy chain allotype locus.

The genetic linkage relationship of two antinuclease idiotypes produced by the BALB/c strain was investigated in the backcross (BALB/c x CB.20) X CB.20. These two idiotypes were detected by Lewis rat anti-idiotypic antisera prepared against affinity-purified A/J and SJL antinuclease antibodies, termed the A/J and SJL idiotypes, respectively. Both idiotypes were found to be linked to the IgCHa immunoglobulin heavy chain allotype locus. There was, however, a high frequency of recombination observed between both markers and the IgCHa locus, with eight of 83 backcross animals recombinant for the A/J idiotype and five of 83 recombinant for the SJL idiotype. All such recombinant animals were IgCHb/b homozygotes that had gained one or both idiotypes. These results are consistent with a genetic map of VHr region genes in the BALB/c strain in which genes determining the SJL idiotype are closer to the IgCHa allotype locus than are genes determining the A/J idiotype. This high frequency of recombination may indicate that the chromosome segment containing VH region genes is very large or that it has structural features that promote recombination.     

Characterization of immune complex components by dot blot analysis.

A method is described for the characterization of immune complex components by dot blot analysis. After isolation by chromatographic techniques and precipitation with polyethylene glycol, immune complexes were dissociated in 0.1 M phosphate (pH 2) and bound to a nitrocellulose membrane in a dot blot unit. Biotinylated probes were then used to identify the following immune complex components: specific antigens, biologically active antibodies, antibody isotypes, antibody subclasses, antibody idiotypes, and rheumatoid factors. This nonradioactive procedure takes less than 2 h to perform and has been used to analyze immune complexes isolated from sera (rabbit and human) and synovial fluid (human).     

T15 idiotype expression in the murine response to phosphorylcholine is actively regulated by genes linked to the Igh-C locus

In this report, we demonstrate the existence of a regulatory system that can determine the relative expression of individual clones (idiotypes) within the total immune response to PC. Studies of Igh-C-congenic mice and backcross analysis demonstrate that this regulation is controlled by genes linked to the immunoglobulin heavy chain locus. The regulation acts in a trans fashion. The regulatory genes of one parental allele can control the expression of idiotopes coded for by both parental alleles. This regulation does not appear to affect the quantity of the response, but could affect the quality of the response, depending on the potential repertoire and avidity of the individual clones available to respond to a given antigenic challenge.     

Persistence of viral infections on the population level explained by an immunoepidemiological model

We consider a mathematical model of viral spread in a population based on an immune response model embedded in an epidemic network model. The immune response model includes virus load and effector and memory T cells with two possible outcomes depending on parameters: (a) virus clearance and establishment of immune memory and (b) establishment of a non-zero viral presence characterized with increased T-cell concentrations. Isolated individuals can have different immune system parameters and, after a primary infection, can either return to the infection-free state or develop persistent or chronic infection. When individuals are connected in the network, they can reinfect each other. We show that the virus can persist in the epidemic network for indefinite time even if the whole population consists of individuals that are able to clear the virus when isolated from the network. In this case a few individuals with a relatively weak immune response can maintain the infection in the whole population. These results are in contrast to implications of classical epidemiological models that a viral epidemic will end if there is no influx of new susceptibles and if individuals can become immune after infection.     

Parallel sets and the internal image of antigen within the idiotypic network

Antibodies carrying the internal image of antigens and antibodies with different specificities sharing idiotypes, which were called parallel sets, have been looked on as mere curiosities or odd by-products of antibody diversity. New findings supporting the existence of these antibodies in various experimental systems were presented. In addition, molecular studies of parallel sets in an arsonate system suggested that the genes encoding for these antibodies originate from germ line genes but use a different JH segment. The functions of these antibodies within the idiotype network were discussed.