Results

In the following, we consider $n$ proteins, whose concentrations are denoted by $x_1, .., x_n$. Generalising the type of system proposed by Monod & Jacob (1961), and the type of equation that was used by Gardner et al. (2000), we suppose that each of these proteins undergoes exponential decay (be it natural or promoted by proteases functioning far from saturation), and that it inhibits the synthesis of other proteins in the switch, with cooperativity $c$ (see below for a discussion of $c$). Cooperativity and repressor site binding characteristics are taken to be the same for all proteins; repression is modelled by a simple, Michaelis-like function. Concentrations $x_1, .., x_n$ are normalised with respect to the Michaelis constant. $\sigma$ denotes the strength of unrepressed protein expression, relative to the exponential decay. The interaction graph of the system is of the type illustrated in Figure 1b. We get the following system of differential equations:

$$\begin{array}{lll} \nonumber \frac{\mathrm{d}x_1}{\mathrm{d}... ...n+\frac{\sigma}{1+x_{1}^{c}+..+x_{n-1}^\mathrm{c}} \end{array}$$

Such a system could have evolved from duplication of genes involved in a bistable switch, and subsequent differentiation of repressor sequence and repressor-binding properties. It is trivial to show that the field defined by the system is inward-pointing on a domain of biological interest. The differential system is difficult to study because there is no algebraic way to solve multivariate polynomial systems involving generic parameters (but it is possible to show the existence of $n$ ss which one would expect, see appendix). However, there are properties one can derive from general considerations, and simulations can be used to study systems with specific values of the parameters. We now illustrate the fact that positive feedback, stemming from the interaction circuits between any single pair of proteins, must outweigh the negative feedback stemming from the exponential decay of the proteins. In the general case, the system has a steady state where $x_1=x_2=..=x_n$; in order to have a switch, such a steady state should clearly be unstable. As shown in the appendix, this requires both the cooperativity $c$ to have a lower bound (determined by the dimension of the system), and the strength of the positive feedback circuits (which are proportional to $\sigma$ at any given point) to be strong enough:

• $$\sigma>(c-(n-1))^{-\frac{1}{c}}+(n-1)(c-(n-1))^{-\frac{c+1}{c}}$$ (2)

  
  

• $$c>n-1$$ (3)

  
  

This theoretical study allows to derive constraints on the system if it is to exhibit switch-like behaviour, but doesn't show that it actually does exhibit such a behaviour. However, computer simulations (data not shown) indeed show that the proposed system, for values of $n$ up to 6, and such that $c$ and $\sigma$ match the conditions detailed above, has $n$ (and no more) stable steady states, each corresponding to a high concentration for one of the proteins, and a low concentration for all the other ones. We expect this result to hold for all values of $n$.