Analysis of the bHLH model

Without cooperativity in transcriptional activation by the bHLH dimer, there is only one stable steady-state:

$$\dot{x_i}=x_i \left( -1 +\frac{\sigma}{\alpha \left( 1+\Sigma_{j=1}^n x_j \right) +x_i} \right)$$

If at some steady state $k$ variables are on and share a common value $\bar{x}$ (variables at a steady state, if not 0, share a common value),

$$1=\frac{\sigma}{k\alpha \bar{x}+\bar{x} +\alpha}$$

$$\bar{x}=\frac{\sigma-\alpha}{k\alpha+1},$$

and if $x_p(t_0)=0$,

$$J_{p,p}=\left( -1 + \frac{\sigma}{k\alpha \bar{x} + \alpha} \right)$$

$$J_{p,p}=\frac{\sigma-\alpha}{\alpha \left( k\sigma +1 \right)}>0,$$

and $J_{p,l}=0$ for $p \ne l$, proving the unstability of the steady state.

In the following, it is assumed that transcriptional activation occurs with cooperativity 2, and the steady-state equations become

$$\bar{x}_i=\sigma \frac{\bar{x}_i^2} {\left(\frac{D}{a_t}\right)^2 K_2 + \bar{x}_i^2}$$

$$\forall i, \alpha D^2 + \bar{x_i}^2=\sigma \bar{x_i}$$ (9)