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Aggregation of cell surface receptors by multivalent ligand can trigger a variety of cellular responses. A well-studied receptor that responds to aggregation is the high affinity receptor for IgE (FcepsilonRI), which is responsible for initiating allergic reactions. To quantify antigen-induced aggregation of IgE-FcepsilonRI complexes, we have developed a method based on multiparameter flow cytometry to monitor both occupancy of surface IgE combining sites and association of antigen with the cell surface. The number of bound IgE combining sites in excess of the number of bound antigens, the number of bridges between receptors, provides a quantitative measure of IgE-FcepsilonRI aggregation. We demonstrate our method by using it to study the equilibrium binding of a haptenated fluorescent protein, 2,4-dinitrophenol-coupled B-phycoerythrin (DNP25-PE), to fluorescein isothiocyanate-labeled anti-DNP IgE on the surface of rat basophilic leukemia cells. The results, which we analyze with the aid of a mathematical model, indicate how IgE-FcepsilonRI aggregation depends on the total concentrations of DNP25-PE and surface IgE. As expected, we find that maximal aggregation occurs at an optimal antigen concentration. We also find that aggregation varies qualitatively with the total concentration of surface IgE as predicted by an earlier theoretical analysis.

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Anti-idiotype of (natural) autoantibodies participate in the regulation of autoantibodies, their idiotypes. Focusing on an idiotype/anti-idiotype pair embedded in an environment such as the central immune system, we start with the experimental fact that the level of anti-idiotypes is low in autoimmune patients but high in healthy individuals, and present a quantitative model. This is then used to develop an adaptive control strategy that induces a transition back to the tolerant, healthy, state and thus offers a vista of treating autoimmune diseases caused by the failure of idiotypic control of autoreactive B cells. The idea is to introduce an antigen or anti-idiotype that binds to the autoantibodies with high affinity, and to determine whether or not a fixed dose is to be injected depending on the autoantibody titer exceeding or not exceeding a threshold. Quantitative criteria are provided. The procedure is the more adaptive in that monitoring the autoantibody titer need only happen every x-th day where x can greatly exceed one. Adaptive control turns out to be robust. The arguments presented here also give a quantitative explanation of why the antigen-autoantibody interaction has to be specific so as to induce a backward transition and why an IVIg treatment therefore does not lead to a permanent improvement.

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We present and analyze a model for the interaction of human immunodeficiency virus type 1 (HIV-1) with target cells that includes a time delay between initial infection and the formation of productively infected cells. Assuming that the variation among cells with respect to this 'intracellular' delay can be approximated by a gamma distribution, a high flexible distribution that can mimic a variety of biologically plausible delays, we provide analytical solutions for the expected decline in plasma virus concentration after the initiation of antiretroviral therapy with one or more protease inhibitors. We then use the model to investigate whether the parameters that characterize viral dynamics can be identified from biological data. Using non-linear least-squares regression to fit the model to simulated data in which the delays conform to a gamma distribution, we show that good estimates for free viral clearance rates, infected cell death rates, and parameters characterizing the gamma distribution can be obtained. For simulated data sets in which the delays were generated using other biologically plausible distributions, reasonably good estimates for viral clearance rates, infected cell death rates, and mean delay times can be obtained using the gamma-delay model. For simulated data sets that include added simulated noise, viral clearance rate estimates are not as reliable. If the mean intracellular delay is known, however, we show that reasonable estimates for the viral clearance rate can be obtained by taking the harmonic mean of viral clearance rate estimates from a group of patients. These results demonstrate that it is possible to incorporate distributed intracellular delays into existing models for HIV dynamics and to use these refined models to estimate the half-life of free virus from data on the decline in HIV-1 RNA following treatment.

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The T-independent B cell response induced by highly multivalent hapten polymer preparations has been studied extensively. The in vitro measured dose-response curve tends to be roughly bell-shaped with the peak response occurring at very low ligand concentrations, between 0.1-1 ng/ml for a variety of different ligands. Furthermore, polymers with more than approximately 10 haptens tend to be stimulatory, whereas polymers with fewer than 10 haptens conjugated, tend to be inhibitory. These observations have been perplexing when viewed within the context of standard theories of receptor ligation by multivalent ligands. We present a new analysis of these previous experiments that reconciles the differences between theory and experiment. From this theory it is concluded that the peak in the observed dose response curve only weakly reflects properties of the ligand and the affinity of surface immunoglobulin for the hapten, but depends strongly on the density of antigen-specific B cells in the culture. The number of responding cells decreases at low ligand concentrations, because cells have to share limiting amounts of ligand and not because of the decreasing probability of receptors and ligands meeting each other. Our theory leads to the same conclusion as made by previous researchers, namely that a minimum number of receptor sites, of the order of 10, need to be bound to a single ligand in order to stimulate a B cell. While this conclusion was based on the lack of immunogenicity of antigens carrying less than a minimum number of haptens, the quantitative results of this study, derived from fitting experimental dose response curves obtained with highly multivalent antigens, provide evidence for the immunon hypothesis that is based upon the degree of receptor aggregation. Our theory also provides quantitative agreement with experimental observations on systems, in which both stimulatory and non-stimulatory polymers are mixed in the same system.

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Idiotypic interactions may be a factor in the selection of the B cell repertoire. Simulations of an evolving idiotypic network where new clones are introduced on a daily basis have shown that macroscopic properties, such as network size and connectivity, attain stationary values despite the rapid turnover of individual clones, indicating that idiotypic networks possess self-organizing properties. Affinities between antibodies were either zero, low (0.1), or high (1.0) in these simulations. It has been suggested that network properties may change when affinities take arbitrary real values. Here we show that the previous results of De Boer & Perelson on network size and connectivity are not changed when affinities take many different, closely spaced values.

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We study the equilibrium binding properties of multivalent ligands to cell surface receptors. We examine the effects of cell density and number of receptors per cell, that is, receptor concentration, on ligand binding. These parameters can significantly affect the formation of receptor aggregates and cross-links. We then use our general results to show that ligand-induced cell proliferation may be self-limiting, since ligand depletion reduces the signal received by individual cells once the cell population has expanded. We discuss the concept of avidity and show its limitations. As a specific example, we examine the binding of haptenated polymers to B cells and reinterpret experiments related to the immunon theory of B-cell activation.

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We analyze a model for the reversible cross-linking of cell surface receptors by a collection of bivalent ligands with different affinities for the receptor as would be found in a polyclonal anti-receptor serum. We assume that the amount of cross-linking determines, via a monotonic function, the rate at which cells become activated and divide. In addition to the density of receptors on the cell surface, two quantities, the binding field and the cross-linking field, are needed to characterize the cross-linking curve, i.e., the equilibrium concentration of cross-linked receptors plotted as a function of the total ligand site concentration. The binding field is the sum of all ligand site concentrations weighted by their respective binding affinities, and the cross-linking field is the sum of all ligand site concentrations weighted by the product of their respective binding and cross-linking affinity and the total receptor density. Assuming that the cross-linking affinity decreases if the binding affinity decreases, we find that the height of the cross-linking curve decreases, its width narrows, and its center shifts to higher ligand site concentrations as the affinities decrease. Moreover, when we consider cross-linking-induced proliferation, we find that there is a minimum cross-linking affinity that must be surpassed before a clone can expand. We also show that under many circumstances a polyclonal antiserum would be more likely than a monoclonal antibody to lead to cross-linking-induced proliferation.

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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.

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We study the equilibrium properties of idiotypically interacting B cell clones in the case where only the differentiation of B cells is affected by idiotypic interactions. Furthermore, we assume that clones may recognize and be stimulated by self antigen in the same fashion as by anti-antibodies. For idiotypically interacting pairs of non-autoreactive clones we observe three qualitatively different dynamical regimes. In the first regime, at small antibody production an antibody-free fixed point, the virgin state, is the only attractor of the system. For intermediate antibody production, a symmetric activated state replaces the virgin state as the only attractor of the system. For large antibody production, finally, the symmetric activated state gives way to two asymmetric activated states where one clone suppresses the other clone. If one or both clones in the pair are autoreactive there is no virgin state. However, we still observe the switch from an almost symmetric activated state to two asymmetric activated states. The two asymmetric activated states at high antibody production have profoundly different implications for a self antigen which is recognized by one of the clones of the pair. In the attractor characterized by high autoantibody concentration the self antigen is attacked vigorously by the immune system while in the opposite steady state the tiny amount of autoantibody hardly affects the self antigen. Accordingly, we call the first state the autoimmune state and the second the tolerant state. In the tolerant state the autoreactive clone is down-regulated by its anti-idiotype providing an efficient mechanism to prevent an autoimmune reaction. However, the antibody production required to achieve this anti-idiotypic control of autoantibodies is rather large.

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We model how auto-reactive B 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 the anti-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.

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The comb size and comb colour of 84 hens (white and brown Hisex), living in battery cages or at free range were investigated. The main question was: does the comb size and colour correlate with the keeping systems i.e. with the quality and intensity of light and the ambient temperature? The results confirm these hypotheses and also show the genetic influence. The crest size of Hisex white hens is larger and paler compared with those of Hisex brown hens. This tendency is more distinct under the conditions of artificial light, less light and higher temperature of the environment in battery cage systems. Comb size and colour are used as a criterium for health and egg production of the hens. The combs are very important for the hen's health for the regulation of the body temperature and for absorbing light and also for the social structure of the flock; the comb form is an essential characteristic to identify the individual animal.

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Memory to antigenic challenge of the immune system is described as a synergy of two components: cycles of interacting B cells in a dynamic equilibrium which store an internal image of an antigen, and long-lived memory B cells which stabilize the cycle that generates them. Small cycles are most relevant to the immune system's memory. The network is globally stable and supports Jerne's idea that suppression is important. Our model allows for exponential increase of antigens during the initial stage of infection. It has a number of stable fixed points, viz the virgin state, the healthy immunized state, and a state of chronic infection, the last occurring if the antigen is virulent enough. Numerical simulations show a difference between primary and secondary response and exhibit both predator-prey and intracycle oscillations. In the case of a chronic infection, the simulations suggest a specific stimulation therapy triggered by repeatedly injecting the antigens, thus making the infection acute. An optimal therapy is indicated.

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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.

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According to Hebb's postulate for learning, information presented to a neural net during a learning session is stored in synaptic efficacies. Long-term potentiation occurs only if the postsynaptic neuron becomes active in a time window set up by the presynaptic one. We carefully interpret and mathematically implement the Hebb rule so as to handle both stationary and dynamic objects such as single patterns and cycles. Since the natural dynamics contains a rather broad distribution of delays, the key idea is to incorporate these delays in the learning session. As theory and numerical simulations show, the resulting procedure is surprisingly robust and faithful. It also turns out the pure Hebbian learning is by selection: the network produces synaptic representations that are selected according to their resonance with the input percepts.

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A subject who spoke essentially in "telegraphic" English, leaving out most articles and auxiliary verbs, was trained to use a particular sentence form that included the articles and verbs to describe a set of standardized pictures. The subject used the trained sentence form to describe the trained pictures, and in addition, use of the sentence form generalized to sets of untrained and novel stimuli. When the trained sentence form was changed, the subject used the new form to describe both training and generalization stimuli. When the original correct form of response was retrained, the subject once again used the trained sentence form to respond to both training and generalization trials.