Mutual inhibition with autocatalysis

Each switch element is supposed to undergo non-regulated degradation (modeled as exponential decay, with an arbitrary speed 1), and transcription/translation with a relative speed $\sigma$. Each element positively auto-regulates itself, and represses expression of others, with a cooperativity $c$. Calling $x_i$ the concentration of each switch element, the corresponding equations are, for $n$ elements:

$$\frac{\mathrm{d}x_1}{\mathrm{d}t} =-x_1+\frac{\sigma x_{1}^{c}}{1+\Sigma_{i=1}^{n} x_{i}^\mathrm{c}}$$

$$\dots$$ (1)

$$\frac{\mathrm{d}x_n}{\mathrm{d}t} =-x_n+\frac{\sigma x_{n}^{c}}{1+\Sigma_{i=1}^{n} x_{i}^\mathrm{c}}$$

The analysis is restricted to $c\ge1$, as there is only one steady state (0) if $c<1$. The results presented in appendix A show that it is possible for one switch element to be on and others off (for $\sigma>2$), but that if there is some cooperativity in the system (ie $c>1$), it is impossible for more than 1 element to be on at the same time. On the contrary, if there is no cooperativity ($c=1$), simulations show that a multitude of equilibria exist and are stable (switch elements in the "on" state can even coexist at different concentrations). Thus, the multistability behaviour of this system is tunable only by changes in the strength of the cooperativity, a mechanism which seems to be of unlikely biological relevance.