Negative autoregulation ($\dot{X}=\beta f(X)-\alpha X,f^{\prime }(X)<0$)

A gene product reduces its own production. This negative feedback can stabilize its abundance and shorten the approach to a regulated level. In the strong logic approximation for gene regulation, production switches off near a self-repression threshold; a smooth repression law avoids artificial infinitely rapid switching at that threshold.

At a Hill equation equilibrium with $s=(Y/K{)}^n$, differentiating $\beta =\alpha Y(1+s)$ gives the displayed logarithmic sensitivity. It approaches one without effective feedback and $1/(n+1)$ under strong repression. Threshold pinning in the logic approximation for gene regulation instead gives approximately zero production sensitivity while the threshold is held fixed.

Neglecting the one in a strongly repressed Hill equation gives $\dot{Y}=\beta K^n{Y}^{-n}-\alpha Y$. The substitution $z=Y^{n+1}$ makes this a linear ordinary differential equation. It yields $Y_{\mathrm{st}}=(\beta K^n/\alpha {)}^{1/(n+1)}$ and half-response time $-\log (1-2^{-(n+1)})/[(n+1)\alpha ]$. The singular initial slope is an artifact of the approximation, whose range is not uniform at arbitrarily large $n$ and fixed production/degradation ratio.