Previous results

Plathe et al. (1995) published the first attempt to show an existence result for positive circuits; however their demonstration does not take into consideration the case of a coefficient of $J_F$ vanishing without changing sign, and excludes without biological discussion the case of coefficients changing signs concomitantly. Snoussi derived independently a demonstration formalising that of Plathe et al., without going into the detail of non-constant sign of the Jacobian. Gouze proposed another demonstration in which he also assumed that $\mathrm{sgn}(J_{F})$ was constant throughout $D$, and that $D$ was convex, assumptions on which his demonstration crucially depended; under the supplementary assumptions that the vanishing set of $F$ comprised 2 or more isolated points of $D$, he showed that the interaction graph of $F$ had a positive circuit. The hypothesis that $\mathrm{sgn}(J_{F})$ is constant is a very restrictive one: it means that partial derivatives of $F$ may not cancel anywhere in $D$, i.e. that if variable $i$ exerts a positive (respectively negative) influence on variable $j$ for some state of the system, that influence remains positive (respectively negative) for all states of the system. An example of a situation in which the sign of $J_F$ is not constant is that of the well-studied arabinose operon in E. coli: as a monomer, araC represses transcription of the arabinose operon but as a dimer, it activates transcription. The presence of arabinose induces dimerisation of araC, and thus induces transcription of the operon. If one considers total araC and arabinose concentrations as well as operon transcription as state variables, transcription increases as [araC] increases in the presence of arabinose, but decreases as [araC] increases in the absence of arabinose; the influence of [araC] on transcription thus changes from positive in the presence of arabinose to negative in the absence of arabinose. More generally, it is possible to build systems for which the sign of the Jacobian matrix is not constant by considering a gene under transcriptional control of two binding sites for the same protein, one site having a high affinity for the regulating protein and activating transcription, and the other site having a lower affinity for the regulating protein and repressing transcription, as is the case for the TATA-Binding Protein of Acanthamoeba (Ogbourne & Antalis, 1998), or for the $\lambda$ repressor in E. coli (Ptashne, 1986). A similar situation can also be met with proteins having an activity-enhancing and an activity-repressing site for the same molecule, as is the case for example for phosphofructokinase with ATP. It is possible to enhance Gouze's and Snoussi's theorems by only supposing that $\mathrm{sgn}(J_{F})$ is constant in a convex neighbourhood of a segment joining two stable ss, but it is hard to think of justifications for such a hypothesis. In the following, we thus attempt to relax the constant-sign hypothesis. In order to do so, we have to introduce a new condition, but for which we have a natural, biological justification.