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A Cayley tree immune network model with antibody dynamics

R W Anderson1, A U Neumann, A S Perelson

  • 1Theoretical Biology and Biophysics, Los Alamos National Laboratory, NM 87545.

Bulletin of Mathematical Biology
|November 1, 1993
PubMed
Summary

This study models idiotypic networks with B cells and antibodies, revealing localized states can become chaotic or percolate. Parameter stability depends on B cell and antibody lifetimes, with complex interactions observed.

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Area of Science:

  • Immunology
  • Theoretical Biology
  • Network Dynamics

Background:

  • Idiotypic networks, crucial for immune regulation, involve B cells and antibodies.
  • Previous models explored B cell dynamics, but integrating antibody dynamics adds complexity.
  • Localized states in immune networks represent stable configurations of immune cells and antibodies.

Purpose of the Study:

  • To formulate and analyze a Cayley tree model of idiotypic networks incorporating both B cell and antibody dynamics.
  • To investigate the existence and stability of localized network states under varying model parameters.
  • To explore the breakdown of localized states into chaotic or percolation attractors and their coexistence.

Main Methods:

  • Development of a Cayley tree model simulating B cell and antibody interactions.

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  • Analysis of network states, including localized, chaotic, and percolation attractors.
  • Exploration of parameter space to determine conditions for state stability and transitions.
  • Comparison of model predictions with experimental observations, such as antibody injection effects.

    • Main Results:

    • Localized states, characterized by activated clones amidst virgin ones, were identified.
    • The stability of localized states depends on the ratio of antibody to B cell lifetimes and antibody complex removal rates.
    • Localized steady-states can transition to chaotic or percolation attractors, which can coexist.
    • Stable limit cycles and localized chaotic attractors were observed in highly connected networks.
    • Antibody injection in a chaotic regime can cease fluctuations, consistent with experimental findings.

    Conclusions:

    • The model provides insights into the complex dynamics of idiotypic networks, including B cell and antibody interactions.
    • Parameter stability is sensitive to B cell and antibody lifetimes and clearance rates.
    • The coexistence of different attractor types challenges clear-cut boundaries in network behavior.
    • The model accounts for observed phenomena like chaotic fluctuation cessation upon antibody injection.