Past exam of the biological physics course of the University of Oxford / 2017 / 1 / b / Solution

For the strong Hill autorepression response, in the requested approximation, $\dot{Y}=\beta K^n{Y}^{-n}-\alpha Y$. Put $z=Y^{n+1}$. Thenso the formal zero-initial-concentration solution isSetting $Y(T_{1/2})=Y_{\mathrm{st}}/2$ givesThe formal ratio decreases with positive $n$; for $n=1$ it is about $0.208$, and for $n=2$ about $0.0642$. Its large-$n$ asymptote is $2^{-(n+1)}/[(n+1)\log 2]$. This is a nonlinear half-response time, not simply the near-equilibrium relaxation time $1/[(n+1)\alpha ]$.

Approximation qualification. The omitted initial layer is precisely the one the source permits us to ignore: the approximate derivative diverges at $Y=0$, whereas the full Hill equation has initial slope $\beta$. The approximation is not uniform as $n\to \infty$ at fixed $R=\beta /(\alpha K)$. Strong repression already at half of the final concentration needs $R^{1/(n+1)}/2\gg 1$. The actual Hill equation obeys $\dot{Y}\le \beta$, so $T_{1/2}\ge Y_{\mathrm{st}}/(2\beta )$. Its steep-switch limit approaches the logic-response scale in a; the formal vanishing large-$n$ result above cannot be extrapolated past its regime of validity.