Abstract
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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Literature
Abbas, A. K. 1988. “A Reassessment of the Mechanisms of Antigen-Specific T-cell-Dependent B-cell Activation.”Immunol. Today 9, 89–94.
Berek, C., G. M. Griffiths and C. Milstein. 1985. “Molecular Events During Maturation of the Immune Response to Oxazolone.”Nature 316, 314–318.
Bernable, R. R., A. Coutinho, C. Martinez-A and P. A. Cazenave. 1981. “Immune Networks. Frequencies of Antibody- and Idiotype-Producing B cell Clones in Various Steady States.”J. exp. Med. 154, 552–556.
Bona, C. A. and B. Pernis 1984. “Idiotypic Networks.” InFundamental Immunology, W. E. Paul (Ed.), pp. 577–592. New York: Raven Press.
Cohn, M. 1986. “The Concept of Functional Idiotypic Network for Immune Regulation Mocks all and Comforts None.”Ann. Immunol. (Inst. Pasteur) 137C, 64–76.
De Boer, R. J. 1983. RIND: Great Integrator Differential Equations. Bioinformatics Group, University of Utrecht, The Netherlands.
—. 1988. “Symmetric Idiotypic Networks: Connectance and Switching, Stability, and Suppression.” InTheoretical Immunology, A. S. Perelson (Ed.), Part Two, pp. 265–289. SFI Studies in the Science of Complexity, Vol. III. Reading, MA: Addison-Wesley.
—. 1989. “Information Processing in Immune Systems: Clonal Selection versus Idiotypic Network Models.” InTheoretical Models for Cell to Cell Signalling, A. Goldbeter (Ed.). London: Academic Press.
— and P. Hogeweg. 1986. “Interactions between Macrophages and T-lymphocytes: Tumor Sneaking Through Intrinsic to Helper T cell Dynamics.”J. theor. Biol. 120, 331–351.
— and —. 1987a. “Immunological Discrimination Between Self and Non-Self by Precursor Depletion and Memory Accumulation.”J. theor. Biol.,124, 343–369.
— and —. 1987b. “Self-Nonself Discrimination due to Immunological Nonlinearities: the Analysis of a Series of Models by Numerical Methods.”IMA J. Math. Appl. Med. Biol. 4, 1–32.
— and —. 1988. “Memory but no Suppression in Low-Dimensional Symmetric Idiotypic Networks.”Bull. math. Biol. 51, 0.
De Boer, R. J. and P. Hogeweg. Submitted “Idiotypic Network Models Incorporating T-B cell Cooperation: Conditions for Percolation.”
Early, P., H. Huang, M. Davis, K. Calame and L. Hood. 1980. “An Immunoglobulin Heavy Chain Variable Region Gene is Generated from Three Segments of DNA:V H,D andJ H.”Cell 19, 981–992.
Eichmann, K. 1974. “Idiotypic Suppression—I. Influence of the Dose and of the Effector Functions of Anti-Idiotypic Antibody on the Production of an Idiotype.”Eur. J. Immunol. 4, 296–302.
— and K. Rajewsky. 1975. “Induction of T and B cell Immunity by Anti-Idiotypic Antibody.”Eur. J. Immunol. 5, 661–666.
Erdos, P. and A. Renyi. 1959. “On the Random Graphs 1, Vol. 6.” Institute of Mathematics University of DeBreceniens, Debrecar, Hungary.
Erdos, P. and A. Renyi, 1960. “On the Random Graphs, Publ. No. 5.” Mathematics Institute of the Hungarian Academy of Science.
Farmer, J. D., N. H. Packard and A. S. Perelson. 1986. “The Immune System, Adaptation, and Machine Learning.”Physica 22D, 187–204.
Goldstein, B. 1988. “Desensitization, histamine release and the aggregation ofIgE on human basophils.” InTheoretical Immunology, A. S. Perelson (Ed.), Part One pp. 3–40. SFI Studies in the Science of Complexity, Vol. II. Reading, MA: Addison-Wesley.
Gottwald, B. A. and G. Wanner. 1981. “A Reliable Rosenbrock Integrator for Stiff Differential Equations.”Computing 26, 355–360.
Gray, D. 1988. “Is the Survival of Memory B cells Dependent on the Oerisistence of Antigen?” InAdv. Exp. Med. Biol., in press.
Grossman, Z. 1982. “Recognition of Self and Regulation of Specificity at the Level of Cell Populations.”Immunol. Rev. 79, 119–138.
Gunther, N. and G. W. Hoffmann. 1982. “Qualitative Dynamics of a Network Model of Regulation of the Immune System: a Rationale for theIgM toIgG switch.”J. theor. Biol. 94, 815–855.
Hardt, D. A., A. L. Wang, L. L. Pawlak and A. Nisonoff. 1972. “Suppression of Idiotypic Specificities in Adult Mice by Administration of Anti-Idiotypic Antibody.”J. exp. Med. 135, 1293–1299.
Hebb, D. O. 1949.The Organization of Behavior. New York: Wiley.
Hoffmann, G. W. 1975. “A Theory of Regulation and Self Non-Self Discrimination in an Immune Network.”Eur. J. Immunol. 5, 638–647.
— 1979. “A Mathematical Model of the Stable States of a Network Theory of Self-Regulation.” InSystems Theory in Immunology, C. Bruni, G. Doria, G. Koch and R. Strom (Eds), Vol. 32, pp. 239–257. Lecture Notes in Biomathematics. Berlin: Springer.
— 1980. On Network Theory and H-2 Restriction.” InComtemporary Topics Immunobiology, N. L. Warner (Ed.), Vol. 11, pp. 185–226. New York: Plenum Press.
— 1986. “A Neural Network Model Based on the Analogy with the Immune System.”J. theor. Biol. 122, 33–67.
Holland, J. H. 1986. “Escaping Brittleness: the Possibilities of General Purpose Learning Algorithms Applied to Parallel Rule-Based Systems.” InMachine Learning: An Artificial Intelligence Approach, R. S. Michalski, J. G. Carbonell and T. M. Mitchell (Eds), Vol. II, pp. 593–623. Los Altos: Morgan Kauffman.
Holmberg, D., S. Forsgen, F. Ivars and A. Coutinho. 1984. “Reactions AmongIgM Antibodies Derived from Normal Neonatal Mice.”Eur. J. Immunol. 14, 435–441.
—, G. Wennerstrom, L. Andrade and A. Coutinho. 1986. “The High Idiotypic Connectivity of “Natural” Newborn Antibodies is not found in the Adult Mitogen-Reactive B Cell Repertoires.”Eur. J. Immunol. 16, 82–87.
— 1987. “High Connectivity, Natural Antibodies Preferentially use 7183 and QUPC 52V H Families”Eur. J. Immunol. 17, 399–403.
Hopfield, J. J. and D. W. Tank. 1986. “Computing with Neural Circuits: a Model.”Science 233, 625–633.
Irvine, D. H. and M. A. Savageau. 1985a. “Network Regulation of the Immune Response: Alternative Control Points for Suppressor Modulation of Effector Lymphocytes.”J. Immunol. 134, 2100–2116.
Jerne, N. K. 1974. “Towards a Network Theory of the Immune System.”Ann. Immunol. (Inst. Pasteur) 125C, 373–389.
— 1984. “Idiotypic Networks and Other Preconceived Ideas.”Immunol. Rev. 79, 5–24.
Kauffman, S. A. 1986. “Autocatalytic Sets of Proteins.”J. theor. Biol. 119, 1–24.
Langman, R. E. and M. Cohn. 1986. “The ‘Complete’ Idiotypic Network is an Absurd Immune System.”Immunol. Today 7, 100–101.
Lawler, A. M., P. Sin and P. J. Gearhart. 1987. “Adult B-cell Repertoire is Biased Toward Two Heavy-Chain Variable-Region Genes that Rearrange Frequently in Fetal pre-B cells.”Proc. Natn. Acad. Sci. U.S.A. 84, 2454–2458.
Martinez-A, C., P. Pereira, M. L. Toribo, M. A. R. Marcos, A. Bandeira, A. De la Hera, C. Marquez, P-A. Cazenave and A. Coutinho. 1988. “The Participation of B cells and Antibodies in the Selection and Maintenance of T cell Repertoires.”Immunol. Rev. 101, 191–215.
Melchers, F. and J. Anderson. 1986. “Factors Controlling the B-cell Cycle.”Ann. Rev. Immunol. 4, 13–36.
NAG. 1984. Numerical Algorithms Group, Oxford, U.K.
Novotny, J., M. Handschumacher and R. E. Bruccoleri. 1987. “Protein Antigenicity: a Static Surface Property.”Immunol. Today 8, 26–31.
Pereira, P., L. Forni, E. L. Larsson, M. Cooper, C. Heusser and A. Coutinho. 1986. “Autonomous Activation of B and T cells in Antigen-Free Mice.”Eur. J. Immunol. 16, 685–688.
Perelson, A. S. 1988. “Towards a Realistic Model of the Immune Network.” InTheoretical Immunology, A. S. Perelson (Ed.), Part Two, pp. 377–401. SFI Studies in the Science of Complexity Vol. III. Reading, MA: Addison-Wesley.
— 1984. “Some Mathematical Models of Receptor Clustering by Multivalent Ligands.” InCell Surface Dynamics: Concepts and Models, A. S. Perelson, C. DeLisi, and F. W. Wiegel (Eds), pp. 223–275. New York: Marcel Dekker.
Pollok, B. A., A. S. Bhown and J. F. Kearny. 1982. “Structural and Biological Properties of a Monoclonal Auto-Anti-(Anti-idiotype) Antibody.”Nature 299, 447–449.
Trenker, E. and R. Riblet. 1975. “Induction of Antiphosorylcholine Antibody Formation by Anti-Idiotypic Antibodies.”J. exp. Med. 142, 1121–1132.
Segel, L. A. and A. S. Perelson. 1988. “Computation in Shape Space: A New Approach to Immune Network Theory.” InTheoretical Immunology, A. S. Perelson (Ed.), Part Two, pp. 321–343. SFI Studies in the Science of Complexity, Vol. III. Reading, MA: Addison-Wesley.
Segel, L. A. and A. S. Perelson 1989. “Explanation of a Paradoxical Instability Caused by Relatively Short Range Inhibition.”SIAM J. appl. Math, in press.
Urbain, J. 1986. “Idiotypic Networks: a Noisy Background or a Breakthrough in Immunological Thinking? The Broken Mirror Hypothesis.”Ann. Immunol. (Inst. Pasteur),137C, 57–64.
Vakil, M. and J. F. Kearny. 1986. “Functional Characterization of Monoclonal Auto-Anti-Idiotype Antibodies Isolated from the Early B cell Repertorie of BALB/c Mice.”Eur. J. Immunol. 16, 1151–1158.
Varela, F. J., A. Coutinho, B. Dupire and N. N. Vaz. 1988. “Cognitive Networks: Immune, Neural, and Otherwise.” InTheoretical Immunology, A. S. Perelson (Ed.), Part Two, pp. 359–375. SFI Studies in the Science of Complexity, Vol. III. Reading, MA: Addison-Wesley.
Vieira, P. and K. Rajewsky. 1988. “The Half-Lives of Serum Immunologlobulins in Adult Mice.”Eur. J. Immunol. 18, 313–316.
Weisbuch, G. 1989. Proceedings of “Theories of Immune Networks” Workshop. Jerusalem, May 1988. InLecture Notes in Biomathematics, in press.
Wikler, M., J-D. Franssen, C. Collignon, O. Leo, B. Mariamé, P. Van de Walle, D. De Groote and J. Urbain. 1979. “Idioptypic Regulation of the Immune System.”J. exp. Med. 150, 184–195.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
De Boer, R.J., Hogeweg, P. Unreasonable implications of reasonable idiotypic network assumptions. Bltn Mathcal Biology 51, 381–408 (1989). https://doi.org/10.1007/BF02460115
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02460115