Symmetric idiotypic networks : connectance and switching, stability, and suppression

We present a network model that incorporates: 1) symmetric idiotypic interactions, 2) an explicit affinity parameter (matrix), 3) external (i.e., non idiotypic) antigens, 4) idiotypic stimulation at low population densities, and 5) idiotypic suppression at high densities. Such an idiotypic network of two clones has three stable states: a virgin state, i.e., an equilibrium between the normal influx and turnover of cells, and two immune states (one for each clone), which are maintained by idiotypic interactions. In its immune state, a clone suppresses its idiotypic partner and immediately rejects antigen. Introduction of antigen into the virgin state causes a state switch to the corresponding immune state: antigens are thus remembered, i.e., the network displays memory. This symmetric network cannot account for suppression of proliferating clones. Clones that proliferate suppress their anti-idiotypic "suppressors" long before these have grown large enough to become suppressive. This is a consequence of symmetry: asymmetric versions of our model do account for suppression. We here assume that proliferation precedes suppression; if the reverse is assumed (i.e., suppression), the model cannot account for either memory or suppression. We conclude that the model incorporating proliferation before suppression is superior. We next analyse 50-dimensional (50-D) networks of this same model. The network connectance crucially determines the behavior of the network. Only weakly connected networks know a 50-D virgin state in which all clones are in a "resting" state. Switching behavior only occurs in weakly connected systems. The stability of the respective states reached by the systems first decreases, but later increases when connectance increases. Most importantly, highly connected systems are highly unresponsive, i.e., most clones are suppressed; hence, most antigens expand progressively. We conclude that only weakly connected networks have, immunologically, reasonable behavior