Further studies on the problem of immune network modelling

In a previous work we have analysed a family of antibody and B-cell network models (basic AB models) of the immune system. This analysis focused principally on the physiological interpretation of their parameters. Our approach consisted in building a detailed and general mathematical model (referred to as the GIB model) and then simplifying it formally to a version (named the RIB model) that belongs to the family of AB models, but which is more general than the basic AB models. From that study it was clear that some of the assumptions necessary to simplify the GIB model into the RIB one, as well as to recover the basic AB models from the RIB one, are quite unrealistic from a physiological point of view. All this raised the issue of the reliability, or even the heuristic value, of theoretical studies based on current network models for experimental immunologists. One approach to clarify this issue is to ask whether the unrealism of the assumptions implicit in the RIB and AB models entails qualitatively different behaviours between them compared to the GIB one. We initiate here such a work by performing a comparative study of a two-clone system of the AB and RIB models, and a variant of the GIB model in which the different molecular compartments were merged into a single one (labelled IGB model). Because all those models rely critically on certain B-cell activation functions, which constitute the core of an implicit model of individual B-cell reactivity or "local rules", we focused the present numerical study, to a great extent, on two parameters determining those activation functions (Hill coefficient and thresholds). Our results indicate that: (1) the RIB and IGB models display in general a much larger diversity of steady states than the AB models; (2) only under a very restricted parameter regime did all studied models behave similarly; (3) the parameter regime under which the AB and IGB models, but not the RIB one, behave similarly is still rather restricted through not as much as in (2); and (4) even relatively small quantitative changes (within reasonable values) in the postulated "local rules" can induce very large quantitative changes in the behaviour of the AB and RIB models but not the IGB model. In the light of the present results, we discuss the need of postulating a set of "local rules" solidly based on experimental evidence as a necessary condition for the reliability of current network models.