De Boer R J, Perelson A S, Kevrekidis I G
Utrecht University, The Netherlands.
Bull Math Biol. 1993;55(4):745-80. doi: 10.1007/BF02460672.
We develop a model for the idiotypic interaction between two B cell clones. This model takes into account B cell proliferation, B cell maturation, antibody production, the formation and subsequent elimination of antibody-antibody complexes and recirculation of antibodies between the spleen and the blood. Here we investigate, by means of stability and bifurcation analysis, how each of the processes influences the model's behavior. After appropriate nondimensionalization, the model consists of eight ordinary differential equations and a number of parameters. We estimate the parameters from experimental sources. Using a coordinate system that exploits the pairwise symmetry of the interactions between two clones, we analyse two simplified forms of the model and obtain bifurcation diagrams showing how their five equilibrium states are related. We show that the so-called immune states lose stability if B cell and antibody concentrations change on different time scales. Additionally, we derive the structure of stable and unstable manifolds of saddle-type equilibria, pinpoint their (global) bifurcations and show that these bifurcations play a crucial role in determining the parameter regimes in which the model exhibits oscillatory behavior.
我们构建了一个关于两个B细胞克隆之间独特型相互作用的模型。该模型考虑了B细胞增殖、B细胞成熟、抗体产生、抗体 - 抗体复合物的形成及随后的清除，以及抗体在脾脏和血液之间的再循环。在此，我们通过稳定性和分岔分析来研究每个过程如何影响模型的行为。经过适当的无量纲化处理后，该模型由八个常微分方程和若干参数组成。我们从实验数据中估计这些参数。使用一个利用两个克隆之间相互作用的成对对称性的坐标系，我们分析了模型的两种简化形式，并得到了展示其五个平衡态之间关系的分岔图。我们表明，如果B细胞和抗体浓度在不同时间尺度上发生变化，那么所谓的免疫状态就会失去稳定性。此外，我们推导了鞍型平衡态的稳定和不稳定流形的结构，确定了它们的（全局）分岔，并表明这些分岔在确定模型表现出振荡行为的参数区域中起着关键作用。
