# Positive Autoregulation

### From bio-physics-wiki

Positiv autoregulation (PAR) occurs, when the product of a gene activates its own production. PAR is a common network motif in transcritpton networks but occurs less often in the *E. coli* network than negative autoregulation.

We use separation of time scales as in simple gene regulation and can neglect the delay through the time used to activate the transcription factor, since it is much smaller than the time needed for transcription. Then the production rate of protein $Y$ depends only on the amount of protein $Y$ present, the *higher* the concentration of $Y$ the *higher* the production rate $\beta$, thus the production rate $\beta(Y)$ is a function of $Y$. A good approximation for $\beta(Y)$ is the increasing Hill function.
\begin{align}
\beta(Y)=\frac{\beta_{max} \cdot {Y^*}^n}{K^n+{Y^*}^n}
\end{align}
This leads to a nonlinear ODE that describes the change in concentration of protein $Y$
\begin{align}
\frac{dY}{dt}=\frac{\beta_{max} \cdot {Y^*}^n}{K^n+{Y^*}^n}-\alpha Y
\end{align}
Positive autoregulation leads to bi-stability.

Even if the gene is fully repressed or unactivated, there are some transcripts made. This phenomenon is refered to as **basal gene expression** or leakage. Consider the dynamics with of positive autoregulation with different values of basal gene expression by adding a constant term $\beta_0$ to the Hill function
\begin{align}
\frac{dY}{dt}=\frac{\beta_{max} \cdot {Y^*}^n}{K^n+{Y^*}^n}+\beta_0-\alpha Y
\end{align}
If $\beta_0$ starts high only one stable fixed point occurs at high expression levels. If $\beta_0$ is lowered a saddle knot bifurcation occurs and two new fixed points arise as shown in the picture above. Two stable fixed points (f.p.) and one unstable fixed point in the middle.

positive autoregulation with hight Hill coefficient leads to bi-stability

For a small Hill coefficient ($n=1$) positive autoregulation slows the responsetime.

Since the difference $\frac{\beta_{max} \cdot {Y^*}^n}{K^n+{Y^*}^n}-\alpha Y$ is allways smaller than in simple regulation ($\beta_{max}-\alpha Y$) networks with the PAR motif reach a given steady state expression level slower than in simple regulation.

positive autoregulation with small Hill coefficient slows the responsetime

Further reading:

- Steven H. Strogatz - Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering (Page 90 Ex. 3.7.5 and 243 Ex. 8.1.1)
- Uri Alon - An Introduction to Systems Biology