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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.

Use the logic approximation for gene regulation, zero basal production and initially vanishing X, Y and Z. Let $K=K_{xy}=K_{xz}=K_{yz}=K_{xx}/2$, $Y_{\infty }=Z_{\infty }={\beta }_y/\alpha$, and assume the activated Q level exceeds the X activation threshold. The circuit combines a coherent feed-forward loop with negative autoregulation. While X is below its self-repression threshold it obeys $\dot{X}={\beta }_x-\alpha X$, hence X first crosses $K$ atY production begins then. While X remains above $K$, $Y=Y_{\infty }[1-e^{-\alpha (t-t_X)}]$. The AND gate for Z first becomes active when Y reaches $K$, soThe approximation drops the short X build-up time because ${\beta }_x\gg {\beta }_y$. Strictly the input must last longer than this onset time to produce a nonzero amount of Z. This requires ${\beta }_y/\alpha >K$: at equality the threshold is approached only asymptotically, and below it Z never switches on. No numerical delay follows without the threshold-to-production ratio.

In the strong negative autoregulation limit, X is held close to $K_{xx}=2K$; a smooth repression law regularizes the rapid switching of the ideal Boolean rule. After the input disappears at $t_0$, production of X stops and $X\simeq 2K{e}^{-\alpha (t-t_0)}$. It takes $(\ln 2)/\alpha$ for X to fall below $K$. Until then Y remains at $Y_{\infty }>K$ and the Z AND gate stays on, so Z stays at $Z_{\infty }$. After that Z decays by dilution, taking another $(\ln 2)/\alpha$ to halve. ThereforeMore generally replace the first $\ln 2$ by $\ln [X(t_0)/K]$. This distinguishes disappearance of the external ligand from disappearance of the persistent X transcription factor.