Persistence correction to a gene-circuit pulse threshold

Suppose an input is produced for time $T$, then decays, and activates two identical mutually activating OR-gated outputs. Write $S=\beta /\alpha$ and $h=K/S\in (0,1)$. The input remains above threshold until ${\alpha }^{-1}\ln [(e^{\alpha T}-1)/h]$. Each output reaches threshold at $-2\ln (1-h)/\alpha$ if input remains available. Equating these times gives the limiting pulse duration $T_{\min }={\alpha }^{-1}\ln [1+h/(1-h{)}^2]$. Exact equality depends on the ideal gate's boundary convention; a robust latch requires a longer pulse. Treating cessation of production as immediate removal incorrectly discards the input's residual lifetime.