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A Cayley tree immune network model with antibody dynamics

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Abstract

A Cayley tree model of idiotypic networks that includes both B cell and antibody dynamics is formulated and analysed. As in models with B cells only, localized states exist in the network with limited numbers of activated clones surrounded by virgin or near-virgin clones. The existence and stability of these localized network states are explored as a function of model parameters. As in previous models that have included antibody, the stability of immune and tolerant localized states are shown to depend on the ratio of antibody to B cell lifetimes as well as the rate of antibody complex removal. As model parameters are varied, localized steady-states can break down via two routes: dynamically, into chaotic attractors, or structurally into percolation attractors. For a given set of parameters percolation and chaotic attractors can coexist with localized attractors, and thus there do not exist clear cut boundaries in parameter space that separate regions of localized attractors from regions of percolation and chaotic attractors. Stable limit cycles, which are frequent in the two-clone antibody B cell (AB) model, are only observed in highly connected networks. Also found in highly connected networks are localized chaotic attractors. As in experiments by Lundkvistet al. (1989.Proc. natn. Acad. Sci. U.S.A. 86, 5074–5078), injection ofAb 1 antibodies into a system operating in the chaotic regime can cause a cessation of fluctuations ofAb 1 andAb 2 antibodies, a phenomenon already observed in the two-clone AB model. Interestingly, chaotic fluctuations continue at higher levels of the tree, a phenomenon observed by Lundkvistet al. but not accounted for previously.

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Literature

  • Bhattacharya-Chatterjee, M., S. Mukerjee, W. Biddle, K. A. Foon and H. Köhler. 1990. Murine monoclonal anti-idiotype antibody as a potential network antigen for human carcino-embryonic antigen.J. Immunol. 145, 2758–2765.

    Google Scholar 

  • Celada, F. 1971. The cellular basis of immunological memory.Prog. Allergy 15, 223–267.

    Google Scholar 

  • Celada, F. and P. E. Seiden 1992. A computer model of cellular interactions in the immune system.Innunol. Today 13, 56–62.

    Article  Google Scholar 

  • Coutinho, A. 1989. Beyond clonal selection and network.Immunol. Rev. 110, 63–87.

    Article  Google Scholar 

  • De Boer, R. J. 1988. Symmetric idiotypic networks: connectance and switching, stability, and suppression. In:Theoretical Immunology, Part Two,SFI Studies in the Sciences of Complexity. A. S. Perelson (Ed.), pp. 265–289. Redwood City, CA: Addison-Wesley.

    Google Scholar 

  • De Boer, R. J. and A. S. Perelson. 1991. Size and connectivity as emergent properties of a developing immune network.J. theor. Biol. 149, 381–424.

    Google Scholar 

  • De Boer, R. J., I. G. Kevrekidis and A. S. Perelson. 1990. A simple idiotypic network model with complex dynamics.Chem engng Sci. 45, 2375–2382.

    Article  Google Scholar 

  • De Boer, R. J., A. U. Neumann, A. S. Perelson, L. A. Segel and G. Weisbuch. 1993c. Recent approaches to immune networks. InMathematics Applied to Biology and Medicine. J. Demongoet and V. Capasso (Eds.) Winnipeg: Wuerz (in press).

    Google Scholar 

  • De Boer, R. J., L. A. Segel and A. S. Perelson. 1992. Pattern formation in one and two-dimensional shape space models of the immune system.J. theor. Biol. 155, 295–233.

    Article  Google Scholar 

  • De Boer, R. J., A. S. Perelson and I. G. Kevrekidis. 1993a. Immune network behavior I: From stationary states to limit cycle oscillations.Bull. math. Biol. 55, 745–780.

    Article  MATH  Google Scholar 

  • De Boer, R. J., A. S. Perelson and I. G. Kevrekidis. 1993b. Immune network behavior II: From oscillations to chaos and stationary states.Bull. math. Biol. 55, 781–816.

    Article  MATH  Google Scholar 

  • Doedel, E. J. 1981. AUTO: a program for the bifurcation analysis of autonomous systems.Cong. Num. 30, 265–285.

    MATH  MathSciNet  Google Scholar 

  • Farmer, J. D., N. H. Packard and A. S. Perelson. 1986. The immune system, adaptation, and machine learning.Physica 22D, 187–204.

    MathSciNet  Google Scholar 

  • Fortmann, T. E. and K. L. Hitz. 1977.An Introduction to Linear Control Systems, p. 170. New York: Marcel Dekker, Inc.

    MATH  Google Scholar 

  • Hiernaux, J. R., P. J. Baker and C. DeLisi. 1982. Oscillatory immune response to lipopolysaccharide. InRegulation of Immune Response Dynamics. C. DeLisi and J. Hiernaux (Eds), Vol. 1, pp. 121–136, Boca Raton, FL: CRC Press.

    Google Scholar 

  • Hoffmann, G. W. 1975. A theory of regulation and self-nonself discrimination in an immune network.Eur. J. Immunol. 5, 638–647.

    Google Scholar 

  • Hoffmann, G. W. 1982. The application of stability criteria in evaluating network regulation models. InRegulation of Immune Response Dynamics. C. DeLisi and J. Hiernaux (Eds), Vol. 1, pp. 137–162. Boca Baton, FL: CRC Press.

    Google Scholar 

  • Holmberg, D. 1987. High connectivity, natural antibodies preferentially use 7183 and QUPC52 V-H families.Eur. J. Immunol. 17, 399–403.

    Google Scholar 

  • Holmberg, D., S. Forsgen, F. Ivars and A. Coutinho. 1984. Reactions among IgM antibodies derived from normal neonatal mice.Eur. J. Immunol. 14, 435–441.

    Google Scholar 

  • Holmberg, D., Å. Andersson, L. Carlson and S. Forsgen. 1989. Establishment and functional implications of B-cell connectivity.Immunol. Rev. 110, 89–103.

    Article  Google Scholar 

  • Hooykaas, H., R. Benner, J. R. Pleasants and B. S. Wostmann. 1984. Isotypes and specificities of immunoglobulins produced by germ-free mice fed chemically defined “antigen-free” diet.Eur. J. Immunol.,14, 1127–1130.

    Google Scholar 

  • Huang, J.-Y., R. E. Ward and H. Köhler. 1988. Biological mimicry of antigenic stimulation: analysis of thein vivo antibody responses induced by monoclonal anti-idiotypic antibodies.Immunol. 63, 1–8.

    Google Scholar 

  • Jerne, N. K. 1974. Towards a network theory of the immune system.Ann. Immunol. (Inst. Pasteur) 125C, 373–389.

    Google Scholar 

  • Jerne, N. K., J. Roland and P. A. Cazenave. 1982. Recurrent idiotypes and internal images.EMBO J. 1, 243–247.

    Google Scholar 

  • Kearney, J. F. and M. Vakil. 1986. Idiotype-directed interactions during ontogeny play a major role in the establishment of the B cell repertoire.Immunol. Rev. 94, 39–50.

    Article  Google Scholar 

  • Kearney, J. F., M. Vakil and N. Nicholson. 1987. Non-random VH gene expression and idiotype anti-idiotype expression in early B cells. In:Evolution and Vertebrate Immunity: the Antigen Receptor and MHC Gene Families. G. Kelsoe and D. Schulze (Eds.), pp. 373–389. Austin, TX: University of Texas Press.

    Google Scholar 

  • Köhler, H. 1991. The servant idiotype network.Scand. J. Immunol. 33, 495–497.

    Article  Google Scholar 

  • Köhler, H., H. L. Cheng, A. K. Sood, M. McNamara-Ward, J. H. Huang, R. E. Ward and T. Kieber-Emmons. 1986. On the mechanism of internal image vaccines. In:Idiotypes. M. Reichlin and J. D. Capra (Eds), pp. 179–190. Orlando: Academic Press.

    Google Scholar 

  • Köhler, H., T. Kieber-Emmons, S. Srinivasan, S. Kaveri, W. J. W. Morrow, S. Muller, C. Kang and S. Raychaudhuri. 1989. Revised immune network concepts.Clin. Immunol. Immunopathol. 52, 104.

    Article  Google Scholar 

  • Lundkvist, I., A. Coutinho, F. Varela and D. Holmberg. 1989. Evidence for a functional idiotypic network amongst natural antibodies in normal mice.Proc. natn Acad. Sci. U.S.A. 86, 5074–5078.

    Article  Google Scholar 

  • Neumann, A. and G. Weisbuch. 1992a. Window automata analysis of population dynamics in the immune system.Bull. math. Biol. 54, 21–44.

    MATH  Google Scholar 

  • Neumann, A. U. and G. Weisbuch. 1992b. Dynamics and topology of immune networks.Bull. math. Biol. 54, 699–726.

    Article  MATH  Google Scholar 

  • Novotný, J., M. Handschumacher and R. E. Bruccoleri. 1987. Protein antigenicity: a static surface property.Immunol. Today 8, 26–31.

    Article  Google Scholar 

  • 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.

    Google Scholar 

  • Perelson, A. S. 1989. Immune network theory.Immunol. Rev. 110, 5–36.

    Article  Google Scholar 

  • Perelson, A. S. and C. DeLisi. 1980. Receptor clustering on a cell surface. I. Theory of receptor cross-linking by ligands bearing two chemically identical functional groups.Math. Biosci. 48, 71–110.

    Article  MATH  MathSciNet  Google Scholar 

  • Perelson, A. S. and G. Weisbuch. 1992. Modeling immune reactivity in secondary lymphoid organs.Bull. math. Biol. 54, 649–672.

    Article  MATH  Google Scholar 

  • Raychaudhuri, S., C. Kang, S. Kaveri, T. Kieber-Emmons and H. Köhler. 1990. Tumor idiotype vaccines VII. Analysis and correlation of structural, idiotypic, and biological properties of protective and nonprotective Ab2.J. Immunol. 145, 760–767.

    Google Scholar 

  • Richter, P. H. 1975. A network theory of the immune system.Eur. J. Immunol. 5, 350–354.

    Google Scholar 

  • Romball, C. G. and W. O. Weigle. 1982. Cyclic antibody production in immune regulation. In:Regulation of Immune Response Dynamics. C. DeLisi and J. Hiernaux (Eds), Vol. 1, pp. 9–26. Boca Raton, FL CRC Press.

    Google Scholar 

  • Segel, L. A. and A. S. Perelson. 1988. Computations in shape space. A new approach to immune network theory. In:Theoretical Immunology, Part Two,SFI Studies in the Sciences of Complexity. A. S. Perelson (Ed.), pp. 321–343. Redwood City, CA: Addison-Wesley.

    Google Scholar 

  • Segel, L. A. and A. S. Perelson. 1989. Shape space analysis of immune networks. In:Cell to Cell Signaling: From Experiments to Theoretical Models. A. Goldbeter (Ed.), pp. 273–283. New York: Academic Press.

    Google Scholar 

  • Segel, L. A. and A. S. Perelson. 1991. Exploiting the diversity of time scales in the immune system: A B-cell antibody model.J. statist. Phys. 63, 1113–1131.

    Article  Google Scholar 

  • Sparrow, C. 1982. The Lorenz equations: bifurcations, chaos and strange attractors.Applied Mathematical Sciences, Vol. 41. New York: Springer-Verlag.

    MATH  Google Scholar 

  • Spouge, J. L. 1986. Increasing stability with complexity in a system composed of unstable subsystems.J. math. Anal. Appli. 118, 502–518.

    Article  MATH  MathSciNet  Google Scholar 

  • Stewart, J. and F. J. Varela. 1989. Exploring the meaning of connectivity in the immune network.Immunol. Rev. 110, 37–61.

    Article  Google Scholar 

  • Stewart, J. and F. J. Varela. 1990. Dynamics of a class of immune networks. II. Oscillatory activity of cellular and humoral components.J. theor. Biol. 144, 103–115.

    MathSciNet  Google Scholar 

  • Stewart, J. and F. J. Varela. 1991. Morphogenesis in shape-space: Elementary meta-dynamics in a model of the immune network.J. theor. Biol. 154, 477–498.

    Google Scholar 

  • Sulzer, B., A. U. Neumann, J. L. van Hemmen and U. Behn. 1993. Memory in idiotypic networks due to competition between proliferation and differentiation.Bull. math. Biol. 55, 1133–1182.

    Article  MATH  Google Scholar 

  • Taylor, M. A. and I. G. Kevrekidis. 1990. Interactive AUTO: A graphical interface for AUTO86. Technical Report, Department of Chemical Engineering, Princeton University.

  • Varela, F. J. and A. Coutinho. 1991. Second generation immune networks.Immunol. Today 12, 159–166.

    Google Scholar 

  • Varela, F., A. Andersson, G. Dietrich, A. Sundblad, D. Holmberg, M. Kazatchkine and A. Coutinho. 1991. The population dynamics of natural antibodies in normal and autoimmune individuals.Proc. natn. Acad. Sci. U.S.A. 88, 5917–5921.

    Article  Google Scholar 

  • Weigle, W. O. 1975. Cyclical production of antibody as a regulatory mechanism in the immune response.Adv. Immunol. 21, 87–111.

    Article  Google Scholar 

  • Weinand, R. G. 1990. Somatic mutation, affinity maturation and the antibody repertoire: a computer model.J. theor. Biol. 143, 343–382.

    Google Scholar 

  • Weisbuch, G. 1990. A shape space approach to the dynamics of the immune system.J. theor. Biol. 143, 507–522.

    MathSciNet  Google Scholar 

  • Weisbuch, G., R. J. De Boer and A. S. Perelson. 1990. Localized memories in idiotypic networks.J. theor. Biol. 146, 483–499.

    Google Scholar 

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Authors and Affiliations

  1. Theoretical Biology and Biophysics, Los Alamos National Laboratory, 87545, Los Alamos, NM, U.S.A.

    Russell W. Anderson

  2. Santa Fe Institute, 1660 Old Pecos Trail, 87501, Santa Fe, NM, U.S.A.

    Avidan U. Neumann

  3. Theoretical Biology and Biophysics, Los Alamos National Laboratory, 87545, Los Alamos, NM, U.S.A.

    Alan S. Perelson

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  1. Russell W. Anderson
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  2. Avidan U. Neumann
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  3. Alan S. Perelson
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Anderson, R.W., Neumann, A.U. & Perelson, A.S. A Cayley tree immune network model with antibody dynamics. Bltn Mathcal Biology 55, 1091–1131 (1993). https://doi.org/10.1007/BF02460701

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  • DOI: https://doi.org/10.1007/BF02460701

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