Idiotypic regulation of B cell differentiation.

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.

Capacity of a model immune network.

The capacity of a model immune network in terms of the number of different antigens that can be vaccinated against without any memory lost is computed and tested by numerical simulations. We also investigate memory loss and failure to vaccinate due to overcrowding the network with too many antigens. The computations are done for two different strategies for proliferation, one implying all the antigen specific clones and the second one being more thrifty.

Tolerance to hormones and receptors in an idiotypic network model.

Using a simple mathematical model of the Jerne network, we investigate the conditions of antigen presentation that result in tolerance to pairs of antigens with complementary shapes, such as hormones and receptors. This study is motivated by the prevalence of auto-immune diseases involving hormones or neurotransmitters and their respective receptors as self-antigens. The model shows that, in order to ensure full tolerance to coupled antigens in those conditions that ensure tolerance to isolated antigens, both antigens have to be presented simultaneously in similar conditions. This result suggests a possibility for preventing some auto-immune diseases.

Dynamics and topology of idiotypic networks.

Jerne's idiotypic network was previously modelled using simple proliferation dynamics and a homogeneous tree as a connection structure. The present paper studies analytically and numerically the genericity of the previous results when the network connection structure is randomized, e.g., with loops and varying connection intensities. The main feature of the dynamics is the existence of different localized attractors that can be interpreted in terms of vaccination and tolerance. This feature is preserved when loops are added to the network, with a few exceptions concerning some regular lattices. Localized attractors might be destroyed by the introduction of a continuous distribution of connection intensities. We conclude by discussing possible modifications of he elementary model that preserve localization of the attractors and functionality of the network.

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 Boer et 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.

Window automata analysis of population dynamics in the immune system.

A formalism based on window automata is proposed as a method to analyse complex population dynamics. The method is applied to a model of the immune network (Weisbuch, G.et al., 1990.J. theor. Biol. 146, 483-499), and used to predict which attractor the system reaches after antigenic stimulation, as a function of the parameters. The attractors of the dynamics are interpreted in terms of immune conditions such as vaccination or tolerance. Scaling laws that define the regimes in the parameter space corresponding to the specific attractor reached under antigenic stimulation are derived.

Localized memories in idiotypic networks.

The present paper investigates conditions under which immunological memory can be maintained by stimulatory idiotypic network interactions. The paper was motivated by the work of (De Boer & Hogeweg, 1989b, Bull. math. Biol. 51, 381-408.) which claimed that idiotypic memory is not possible because of percolation within the network. Here we reinvestigate the issue of percolation using both the previous model and a simpler one (Weisbuch, 1990, J. theor. Biol. 143, 507-522.) that allows analytic analysis. We focus on network topologies in which each Ab1 is connected to several Ab2s, which in turn are connected to several Ab3s. It is demonstrated that, for a considerable range of parameters, both models account for the existence of localized memory-states in which only the Ab1 and the Ab2 clones are activated and the clones of the Ab3 level remain virgin. The existence of localized memory-states seems to contradict the previous percolation result. This discrepancy will be shown to depend on the system dynamics. By simulation we explore the parameter regimes for which one finds percolation and those for which localized memory-states exists. We show that the conditions required for attaining the localized memory-state are considerably more stringent than those required for its existence and local stability. We conclude that both localized memory and percolation are possible in stimulatory idiotypic networks.

A shape space approach to the dynamics of the immune system.

A simple mathematical model of Jerne's immune network is proposed where interactions among idiotypes are set according to their location in a shape space. Although the number of interacting idiotypes is potentially infinite, the simplicity of the model makes it possible to compute the attractors of the dynamics, to define the regions in the four parameters space related to the dynamical behavior and to predict the scaling law giving the number of different antigens that can be presented to the network without triggering dangerous instabilities. It is shown that only a low connectivity regime is safe for the immune network.

[A model of the evolution of species on three levels, based on the global properties of Boolean networks]. / Un modèle de l'évolution des espèces à trois niveaux, basé sur les propriétés globales des réseaux booléens.

Population dynamics of organisms characterized by a six loci genome with four alleles obeys a differential system in which fitness is obtained as a global property of a boolean network. This system exhibits two different regimes: accelerated evolution and punctuated equilibria.

Polyelectrolyte theory of charged-ligand binding to nucleic acids.

The Poisson-Boltzmann framework provides simple and sound formulae for evaluation of the interactions of polynucleotides with charged ligands. The theory is based on the observation that shape-dependent effects are small. Hence the results for the charged plane may be used as a first approximation for polyelectrolytes. The counterion distribution and the extent of counterion or ligand binding is derived on the basis of a sum-rule for counterion concentrations, combined with the mass-action law. Calculations require no more than a pocket calculator even in the case of mixed-salt solutions. Significant qualitative results are given, and applied to the problems of site-binding and of repressor-DNA interaction.