Many genes have been identified as driving cellular differentiation, but because of their complex interactions, the understanding of their collective behaviour requires mathematical modelling. Intriguingly, it has been observed in numerous developmental contexts, and particularly hematopoiesis, that genes regulating differentiation are initially co-expressed in progenitors despite their antagonism, before one is upregulated and others downregulated. We characterise conditions under which 3 classes of generic "master regulatory networks", modelled at the molecular level after experimentally-observed interactions (including bHLH protein dimerisation), and including an arbitrary number of antagonistic components, can behave as a "multi-switch", directing differentiation in an all-or-none fashion to a specific cell-type chosen among more than 2 possible outcomes. bHLH dimerisation networks can readily display coexistence of many antagonistic factors when competition is low (a simple characterisation is derived). Decision-making can be forced by a transient increase in competition, which could correspond to some unexplained experimental observations related to Id proteins; the speed of response varies with the initial conditions the network is subjected to, which could explain some aspects of cell behaviour upon reprogramming.The coexistence of antagonistic factors at low levels, early in the differentiation process or in pluripotent stem cells, could be an intrinsic property of the interaction between those factors, not requiring a specific regulatory system
