Mutual inhibition with autocatalysis, and leak

The model is the same as previously, except that each element has a "leaky" expression, modelled as a constant production term $\alpha$. The equations become:

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

$$\dots$$ (2)

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

The system is interesting only for $c>1$ (see appendix B). If the leak is small, it doesn't have a major effect on the system, except when the cooperativity is close to 1: as shown in appendix B, it is impossible for more than one switch element to be "on", at a much higher level than the leak level $\alpha$.

Even when the cooperativity is close to 1, it is still the case that only one variable at the same time can dominate all others; in order for that to happen, the transcription strength must be sufficiently high. A simulation was performed for a cooperativity of $1.1$, with increasing $\sigma$ (see Figure 2). All switch elements are initially coexpressed, and once $\sigma$ becomes sufficiently high, one switch element is upregulated, and others downregulated.

The same pattern of coexpression followed by exclusive expression can be achieved with a decreasing leak (see Figure 3), with the difference that the level of initial coexpression decreases slightly with time (this level is lower than the relative maximum transcription strength $\sigma$, but higher than the leak $\alpha$). Once the leak has become sufficiently small, exclusive upregulation occurs.

We show in appendix B that our models with mutual inhibition and autocatalysis, with or without leak, always converge to an equilibrium (and thus never oscillate).

Figure 2: Time evolution of the concentrations of 4 switch elements ($x_1$ to $x_4$), for the model with mutual inhibition with autocatalysis, and leak, with the transcription strength $\sigma$ being gradually increased over time. The 4 elements are initially coexpressed at an identical level, which increases with $\sigma$; when $\sigma$ reaches a threshold level, one element is upregulated, and others are downregulated. Parameters in the simulation were $\alpha =2$ and $c=1.1$ Low, random noise was added to allow the system to escape the equilibrium as it became unstable.

Figure 3: Time evolution of the concentrations of 4 switch elements ($x_1$ to $x_4$), for the model with mutual inhibition with autocatalysis, and leak, with the leak level $\alpha$ being gradually decreased over time. The 4 elements are initially coexpressed at identical levels (higher than the leak $\alpha$ because of autocatalysis); when the leak reaches a threshold level, one element is upregulated, and others are downregulated. Note that the scales for the $x_i$ and for $\alpha$ are different by a factor of 11, equal to $c/(c-1)$ in this simulation. Thus, it is impossible for the curve of more than one $x_i$ to be above that of $\alpha$ at equilibrium. Thus, in the boxed region, the system is in the process of responding to the drop in $\alpha$, and not at equilibrium. Parameters in the simulation were $\sigma =100$ and $c=1.1$ Low, random noise was added to allow the system to escape the equilibrium as it became unstable.