Feed-Forward Loop Network Motif

Input Signal (Sx)

Pulse duration 2.0 s

Promoter Logic for Z

Y activation threshold K_Y 0.50

Y response time τ_Y 1.5 s

Z response time τ_Z 1.0 s

Live State

X (input TF)off

Y (intermediate)0.00

Z (output)0.00

Signal elapsed

0.0 s

Z gate state

CLOSED

How it works

The feed-forward loop (FFL) is the single most over-represented three-gene wiring pattern in real transcriptional networks (E. coli, yeast, human — Shen-Orr & Alon, 2002): a master regulator X activates an intermediate Y, and both X and Y jointly control a target Z.

dY/dt = βY·Sx(t) − Y/τY
dZ/dt = βZ·gate(Sx,Y) − Z/τZ

AND gate:  gate = 1  iff  Sx=1  AND  Y > KY
OR  gate:  gate = 1  iff  Sx=1  OR   Y > KY

Sx(t) is the external step input (1 while the signal is on, 0 otherwise). Y rises toward 1 with time constant τY only while X is active, and decays exponentially when X turns off. Z is produced only while the chosen logic gate is open, and always decays with time constant τZ.

Coherent type-1, AND logic — Z can only turn ON after Y has climbed past K_Y, so short pulses of X (shorter than it takes Y to cross the threshold) never switch Z on at all. This is a persistence detector / low-pass filter: real biology uses it (e.g. the E. coli arabinose and flagella systems) to ignore transient noise and respond only to sustained signals. Turning X off instantly closes the gate, so Z's OFF response is fast.
Coherent type-1, OR logic — Z turns ON immediately with X (fast response), but stays ON after X switches off for as long as Y remains above K_Y, since either input alone keeps the gate open. This gives a delayed OFF response — the circuit keeps signalling for a while after the trigger disappears.
This asymmetry between ON-delay and OFF-delay depending only on the promoter's Boolean logic is called sign-sensitive delay, a hallmark result of network-motif theory.

Use Fire Pulse with a short duration and AND logic to see Z fail to respond; lengthen the pulse past the Y time constant and Z switches on. Switch to OR logic to see Z persist after the pulse ends.

AND

Gene X (input) Gene Y (intermediate) Gene Z (output)

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