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

Start both systems from zero and compare their times to reach half of their own steady concentrations. Without regulation,In the strong logic approximation for gene regulation, production is initially fully on and the concentration rises towards the repression threshold. The regulated level is pinned near $Y_{\mathrm{st}}\simeq K$, so it reaches $K/2$ before switching:The faster relative response comes from allowing high initial production while keeping the final abundance low. Negative autoregulation can reduce response delays and sensitivity to production fluctuations, and can reduce unnecessary accumulation of a protein. It does not suppress every possible noise source at every frequency.

A perfectly discontinuous on/off right-hand side has a threshold-switching ambiguity rather than an ordinary smooth equilibrium at $K$. Here threshold pinning means the limiting rapid-switching or steep smooth-response idealization. A continuous repression law resolves that idealization. Comparing half of the same prescribed concentration, rather than half of each model's final concentration, would be a different response-time comparison.