Making fields be inward-pointing

We will show in the following that, given biologically reasonable hypotheses, any field can be altered to be inward-pointing (and thus be subject to Corollary 2), in a way that doesn't add positive circuits. We will consider a system fulfilling two requirements. Firstly, we will take the domain $D$ to contain the cone $\{ x\in D \mathrm{st} \forall i \in I x_i>0\}$, although this is mainly a matter of simplifying the demonstration. Secondly, we will suppose that when a state variable vanishes there is a drive to replenish it, even if only by a very small amount. If we consider a set of genes under transcriptional control of one another, this corresponds biologically to the fact that repression is not "perfect", and that expression is "leaky". This is a well-known phenomenon in prokaryotes, and it has been suggested (Bird, 1995) that higher-order organisms developed new mechanisms to enhance repression because of their increased number of genes; however, it seems that even with mechanisms involving chromatin-structure and DNA methylation, there is still some expression-leakage (Chelly et al., 1991). More generally, specificity in biochemical processes is hardly ever perfect, and always entails a compromise between high precision and energetic price (as illustrated by the proof-reading mechanism in DNA duplication, discussed by Savageau & Freter, 1979). Also, according to the thermodynamic laws describing the binding of molecules, it is impossible to reach a state in which a site (such as an operator site) is genuinely saturated. When the concentration of a molecule approaches 0, the enzymatic mechanisms of its destruction or alteration should be proportional to its concentration, while its basal rate of production "leakage" is not dependent on its concentration. It would thus seem that when the concentration is low enough, there is a drive toward replenishment. If there is however a set of molecules dependent on one another as regards their production, it is possible that there is a steady state where all concentrations vanish; but such a steady state would not be stable, and could be eliminated from the study by restricting the domain of study to the quasi-cone $\{ x\in D \mathrm{st} \forall i \in I x_i>\epsilon\}$, with $\epsilon$ suitably small. These assumptions do not necessarily make $F$ an inward-pointing field, which would probably be too strict a requirement for our result to apply to many biological systems. However, as will be detailed in the appendix, $F$ can be "bent" into an inward-pointing field $G$, while preserving properties related to positive circuits. $G$ is no longer relevant from the biological point of view, but this is not important since it has been bent far enough from the origin to have at least two stable ss, and has no more positive circuits than $F$. Corollary 2 can be applied to $G$, showing that it has a positive circuit, and the same is thus true of $F$. What's more, the way $G$ is built shows that the distance of the positive circuit found for $F$ to the origin is at most $\sqrt{n}$ times the distance of the second most distant stable ss of $F$ to the origin. This demonstration is valid for example for all kinds of systems modelled by fields $F$ involving classical Michaelis-Menten, Hill, or Monod-Wyman-Changeux kinetics.