Incoherent Feed-Forward Loop: Genetic Pulse Generator (2D)

Inducer Input (X)

Input level X 1.0

Circuit Kinetics

Y production β_Y 1.20

Y decay γ_Y 0.60

Z production β_Z 2.50

Z decay γ_Z 0.80

Repression threshold K 1.50

Hill coefficient n 2.0

Live Concentrations

X (input) Y (repressor) Z (output pulse)

0.00

Peak / plateau

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How it works

This is the same incoherent type-1 feed-forward loop (I1-FFL) circuit as the 3D version, rendered as a 2D gene-network diagram instead of an animated molecule swarm. X directly activates Y and Z; Y then represses Z, so the two regulatory arms compete:

dY/dt = β_Y·X − γ_Y·Y
dZ/dt = β_Z·X / (1 + (Y/K)ⁿ) − γ_Z·Z

Both equations are integrated numerically from scratch every frame (RK4, fixed sub-steps), never precomputed. When X steps to a constant "on" value, Z rises immediately through the fast direct X→Z path, while Y — with no repression acting on its own production — climbs more slowly toward its own steady state β_Y·X/γ_Y. As Y crosses the Hill threshold K, the term 1/(1+(Y/K)ⁿ) collapses and clamps Z's production down, so Z's trace rises then falls back to a lower plateau even though X never turns off — the defining transient-pulse signature of this motif.

Step ON / OFF — jumps X between 0 and its slider level instantly.
β_Y, γ_Y — how fast the repressor Y is made and cleared; a slower Y (low γ_Y) widens the pulse because the "off" signal takes longer to build.
β_Z, γ_Z — Z's own production/decay, setting the initial rise rate and final plateau height.
K — the Y level that half-represses Z. Lower K means Y clamps Z down sooner and harder, giving a sharper, more transient pulse (higher peak-to-plateau ratio); higher K lets more of the pulse persist into a flatter, more sustained response.
Hill coefficient n — cooperativity of the Y⊣Z repression; higher n gives a steeper, more switch-like shutoff.

This is the same well-documented motif found in E. coli galactose/arabinose regulation and rebuilt synthetically by Basu et al. and Entus et al. to test fold-change detection and pulse generation from a sustained inducer step.

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