Immune network behavior—II. From oscillations to chaos and stationary states

Abstract

Two types of behavior have been previously reported in models of immune networks. The typical behavior of simple models, which involve B cells only, is stationary behavior involving several steady states. Finite amplitude perturbations may cause the model to switch between different equilibria. The typical behavior of more realistic models, which involve both B cells and antibody, consists of autonomous oscillations and/or chaos. While stationary behavior leads to easy interpretations in terms of idiotypic memory, oscillatory behavior seems to be in better agreement with experimental data obtained in unimmunized animals. Here we study a series of models of the idiotypic interaction between two B cell clones. The models differ with respect to the incorporation of antibodies, B cell maturation and compartmentalization. The most complicated model in the series has two realistic parameter regimes in which the behavior is respectively stationary and chaotic. The stability of the equilibrium states and the structure and interactions of the stable and unstable manifolds of the saddle-type equilibria turn out to be factors influencing the model's behavior. Whether or not the model is able to attain any form of sustained oscillatory behavior, i.e. limit cycles or chaos, seems to be determined by (global) bifurcations involving the stable and unstable manifolds of the equilibrium states. We attempt to determine whether such behavior should be expected to be attained from reasonable initial conditions by incorporating an immune response to an antigen in the model. A comparison of the behavior of the model with experimental data from the literature provides suggestions for the parameter regime in which the immune system is operating.

The evolution of mathematical immunology

“…If in the past, mathematical models in immunology were limited to a few groups, such as the Los Alamos T‐10 group led by Perelson and his trainees (8, 15, 82, 131–134) and Nowak (23, 135–140), there are now current practices in all fields of immunology. Mathematical models that were once a minor branch based on theoretical considerations are now associated with most domains of molecular immunology and are often driven by precise experimental results.…”

The Concept of Nonlinearity in Complex Systems

“…Further studies have shown that on the level of cells nonlinear dynamics can also be observed [24,[40][41][42][43][44][45][46][47][48]. Up to now, there is only few information concerning the probable relevance of complex cell reactions to trauma and shock responses such as chemotaxis, leukocyte accumulation, and granulocyte migration [44,46].…”

“…Am J Physiol 1995 [12] Decrease of the complexity of arterial blood pressure control in conscious dogs by baroreceptor denervation Int J Microcirc Clin Exp 1997 [87] Vasomotor activity in microvascular perfusion Immune system Bull Math Biol 1993 [41,45] Nonlinear behavior in B-cell production of antibodies Microbiology Int J Clin Pharmacol Ther 1995 Nonlinear approach for a better understanding [88] of the pharmacokinetics and -dynamics of aminoglycosides matical techniques are phenomenological and attempt to assess the qualitative character of a system's dynamics. Some of the techniques allow short-range predictions but without attempting to provide an understanding of the biological or physical mechanisms that ultimately govern the behavior of the system.…”