= 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.
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