Discussion

In this specific example, because of the simple structure of the network, high values of the weight of the positive feedback circuits does not seem to affect the existence and stability of ss. In other cases, however, positive feedback can have adverse effects above a threshold value (see Cinquin & Demongeot); in those cases lemma 2 should allow to derive useful constraints. Our demonstration of the existence of a positive feedback circuit doesn't give a precise location result, but intuitively it corresponds to the fact that there is a "destabilising force" that pushes the system to amplify small differences when protein concentrations are close to an unstable steady state; such unstable states necessarily exist if there are other stable steady states, and if the field is inward pointing. In the case of the model switch we propose, these unstable steady states would correspond to states in which some of the concentrations are equal (these states have to be unstable if we want switch-like behaviour). Equation 2 from the previous subsection shows in particular that the condition $\sigma>1$ must be met for the system to be switch-like (details not shown); since $\sigma$ is a measure of the unrepressed synthesis rate of each protein (i.e. in the absence of all other proteins) relative to the decay rate, this means that the unrepressed rate of synthesis (providing positive feedback circuits) must be stronger that the rate of decay (providing negative feedback in the system). If the cooperativity $c$ is not an integer, as it tends to its lower bound $n-1$, the synthesis rate $\sigma$ must tend to infinity to achieve instability of the ss where all concentrations are equal. As the cooperativity $c$ tends to infinity, the lower bound on $\sigma$ tends to 1. Equation 3 is a generalisation of a property of bistable switches studied by Gardner et al. (2000), which have been shown to require cooperativity strictly superior to 1 for one of the repressors (that requirement being a specific example of a more general property derived by Cherry & Adler, 2000). Higher dimensional systems of the same type need greater cooperativity to achieve switch-like behaviour. Cooperativity can be achieved in multiple ways, for example by protein multimerisation, or by the presence of multiple binding sites. As the size of the system increases, and thus the minimum cooperativity, it seems that protein multimerisation quickly reaches a practical limit. This fact points to high-dimension multistable switches being more readily implemented in eukaryotes than in prokaryotes, since the eukaryotic mechanism of transcription can integrate the influence of many different cis-factors spread out over a lengthy sequence (Struhl, 1999), and seems to have a natural tendency to cooperativity (Carey, 1998); in eukaryotes, DNA-protein binding could also show cooperativity due to the presence of nucleosomes, and this cooperativity could thus be independent of specific protein-protein interactions (Polach & Widom, 1996). Sophisticated "cis-regulatory information processing" has been identified and modelled by Yuh et al. (2001), and a system in which different cis-regulatory elements were combined to drive additively the expression of a reporter has been engineered by Kirchhamer et al. (1996). These remarks are of course consistent with the fact that sophisticated cellular memory, in the form of cellular differentiation, is a characteristic of eukaryotes.