Switch-like responses are an essential aspect of the dynamics of signaling networks, and are expected to be crucial in mediating cellular differentiation, a process during which one cell-type is chosen and all others excluded, in an all-or-none fashion. Such responses have been documented experimentally (Xiong, 2003), and bistable switches have been thoroughly characterized from a mathematical point of view (Cherry, 2000). It is particularly noteworthy that in the course of cellular differentiation, many antagonistic genes are often co-expressed early-on despite their antagonism, before one gradually takes over (as discussed by Cinquin, 2005). bHLH proteins form a large family, which has been shown to have a crucial role in numerous instances of commitment to specific lineages and differentiation (Massari, 2000). It has been shown that models of bHLH networks can account particularly well for the co-expression of antagonistic genes early in the differentiation process (Cinquin, 2005), although that study was limited to networks whose elements had identical biochemical parameters.
Here we generalize the study to a wider set of bHLH networks, and show that the simple rule which sets a limit to the number of genes which can be co-expressed, depending on the rate of competition in the network, still holds. The relaxed assumptions allow us to illustrate this rule in a context where differentiation outcome is specified by tuning the individual parameters of the elements in the network.