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

Gunther, Nicholas; Hoffmann, Geoffrey W.

doi:10.1016/0022-5193(82)90080-7

Cited by 42 publications

(23 citation statements)

References 15 publications

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“…In the virgin state idiotypic interactions are therefore low or absent. This contrasts strongly with the assumptions of previous verbal (Jerne, 1974) and mathematical (Hoffmann, 1979;Gunther and Hoffmann, 1982) models; in these models it is assumed that idiotypic interactions are suppressive in the virgin state. We demonstrated previously however that our models are more powerful (De Boer, 1988;De Boer and Hogeweg, 1988).…”

Section: I Influx/effluxmentioning

confidence: 81%

Unreasonable implications of reasonable idiotypic network assumptions

Boer

Hogeweg

1989

Bltn Mathcal Biology

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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 "interconnectance" 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 ca...

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Section: I Influx/effluxmentioning

confidence: 81%

Unreasonable implications of reasonable idiotypic network assumptions

Boer

Hogeweg

1989

Bltn Mathcal Biology

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“…Hoffmann et aL (Hoffmann, 1975(Hoffmann, , 1979(Hoffmann, , 1980Gunther & Hoffmann, 1982;Hoffmann et al, 1988) also developed "release of suppression" models. The basic idea of these models is to consider only the Abl and the Ab2 populations, which were called "plus" and "minus".…”

Section: Introductionmentioning

confidence: 99%

Localized memories in idiotypic networks

Weisbuch¹,

Boer

Perelson

1990

Journal of Theoretical Biology

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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 Abl 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 Ab 1 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.

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“…We thus have a 2-D network comparable to those analysed before [Hoffmann, 1979;Gunther & Hoffmann, 1982;De Boer, 1988;De Boer & Hogeweg, 1989b], except for the fact that here we study an explicit Th-B interaction. Here we analyse two such Th-B networks: the ThId model ( Fig.…”

Section: Resultsmentioning

confidence: 71%

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

Boer¹,

Hogeweg²

1989

Journal of Theoretical Biology

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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 (> 107) 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

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

Cited by 42 publications

References 15 publications

Unreasonable implications of reasonable idiotypic network assumptions

Unreasonable implications of reasonable idiotypic network assumptions

Localized memories in idiotypic networks

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

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