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 = (Sx − Y) / τY
dZ/dt = (gate − 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 relaxes toward Sx with time constant τY (so it rises toward 1 while X is active and decays toward 0 when X turns off). Z relaxes toward 1 while the chosen logic gate is open and toward 0 otherwise, always with time constant τZ. This 2D version renders the same three-node circuit as the original 3D simulator, plus a live strip chart of X(t), Y(t) and Z(t) so the sign-sensitive delay is visible directly on the time axis — scroll to zoom the time window, drag to scrub through the last ~40 seconds of history.
- Coherent type-1, AND logic — Z can only turn ON after Y has climbed past KY, 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 KY, 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. Watch the strip chart: the Y trace crossing the dashed KY line is exactly the moment the gate can open.