A model for bHLH proteins

Each switch bHLH protein is supposed to bind to a common activator according to the law of mass action, with a binding constant $K_2$, and a total quantity of activator $a_t$. In turn, the heterodimer activates transcription of the corresponding switch protein only, with Hill kinetics and cooperativity 2 (as with cooperativity 1, no interesting equilibria exist, as shown in appendix C). The equations are:

$$\frac{\mathrm{d}x_1}{\mathrm{d}t} =-x_1+\sigma \frac{\left( \frac{a_t x_1}{1+\Sigma_{i=1}^{n}x_i} \right)^2} {K_2 + \left( \frac{a_t x_1}{1+\Sigma_{i=1}^{n}x_i} \right)^2}$$

$$\dots$$ (3)

$$\frac{\mathrm{d}x_n}{\mathrm{d}t} =-x_n+\sigma \frac{\left( \frac{a_t x_n}{1+\Sigma_{i=1}^{n}x_i} \right)^2} {K_2 + \left( \frac{a_t x_n}{1+\Sigma_{i=1}^{n}x_i} \right)^2}$$

These equations simplify to:

$$\frac{\mathrm{d}x_i}{\mathrm{d}t}=-x_i+\sigma \frac{x_i^2}{\alpha D^2 + x_i^2},$$

with $D=1+\Sigma_{i=1}^nx_i$, $\sigma, \alpha=K_{2}/a_{t}^2 \in _{*}^{+}$

Parameter $\alpha$ is a measure of the harshness of the competition between switch elements (it increases if $K_2$ increases, ie if binding to the common activator occurs at higher concentrations, and if $a_t$ diminishes, ie if the quantity of common activator goes down). The value of $\alpha$ is essential in determining the behaviour of the system. As shown in appendix C, and summarised in section 3.3.3, switch elements can coexist only if $\alpha$ is sufficiently low, ie if the competition is not too harsh (the lower $\alpha$, the more switch elements can be non-0 at equilibrium). Figure 5 shows a simulation with $\alpha$ being increased over time; switch elements are sharply turned off as $\alpha$ reaches threshold values. Figure 6 shows how an increase in $\alpha$ leaves only 1 switch element on, which remains exclusively on even if the competition level is relaxed to its original value.

We prove in the appendix that the system always converges to an equilibrium for $\alpha \ge 1/2$; extensive simulations have also shown this to be the case for $\alpha<1/2$.

Figure 5: Time evolution of the concentrations of 4 switch elements ($x_1$ to $x_4$), in the bHLH dimerisation model, with competition parameter $\alpha$ being gradually increased over time. The horizontal lines mark the values $\alpha =1/4^2$, $\alpha =1/3^2$, and $\alpha =1/2^2$. Each time $\alpha$ reaches one of those threshold values, one of the switch elements is repressed. Low, random noise was added to allow the system to escape equilibria as they became unstable. In this simulation $\sigma =100$.

Figure 6: Same as Figure 5, but with a pulse of the competition parameter $\alpha$. The initial conditions are such that the switch elements coexist for low $\alpha$; once $\alpha$ has sufficiently increased, only 1 switch element stays on, and remains on with all others off, even when $\alpha$ is brought back to its initial, low value.