 Solution

For the parameter sensitivity of autorepressed expression, unregulated production has $Y_{\mathrm{st}}=\beta /\alpha$, so its logarithmic sensitivity is one. In the strong Hill approximation, $Y_{\mathrm{st}}=(\beta K^n/\alpha {)}^{1/(n+1)}$, with $K,\alpha ,n$ held fixed. HenceThe ideal logic level $Y_{\mathrm{st}}\simeq K$ instead gives $S\simeq 0$, because changing production primarily changes the switching duty fraction, not the threshold. Finite Hill repression has a residual production sensitivity.

The exact Hill steady-state relation makes the comparison continuous. Write $s=(Y_{\mathrm{st}}/K{)}^n$; then $\beta =\alpha Y_{\mathrm{st}}(1+s)$ and $d\beta /d{Y}_{\mathrm{st}}=\alpha [1+(n+1)s]$. ThusIt tends to one for weak repression and to $1/(n+1)$ for strong repression. The calculation concerns a local relative parameter change, not the response to an arbitrary large production perturbation.