Gene Regulatory Network Oscillator

Feedback-Loop Parameters

Hill cooperativity n 9.0

Max transcription rate a 1.00

Decay rate (mRNA/protein/repressor) 0.20

Simulation speed 1.0×

Regime

—

Live Concentrations

mRNA X

0.00

Protein Y

0.00

Repressor Z

0.00

Period (last cycle)

—

Sim time

0.0

How it works

This is the Goodwin oscillator — the minimal mathematical model of how a single gene's own negative feedback can make it oscillate on its own, with no second gene needed. Gene X is transcribed into mRNA, translated into protein Y, which activates a repressor Z, which loops back and represses transcription of X:

dX/dt = a / (1 + (Z/K)^n)  −  b·X
dY/dt = c·X  −  b·Y
dZ/dt = e·Y  −  b·Z

The repression term is a Hill function: at low Z, transcription runs near its max rate a; as Z rises past threshold K it shuts X down sharply, with steepness set by the cooperativity exponent n. The three-step chain (X→Y→Z⊣X) injects a built-in time delay before the brake is felt.

Griffith's stability criterion: a pure negative-feedback loop like this only sustains oscillation if the repression is cooperative enough — for equal decay rates the loop needs n > 8 to escape a stable fixed point and settle onto a limit cycle; below threshold any perturbation just decays back to steady state.

Hill cooperativity n — crosses the bifurcation live: watch the regime flip from "damped, spiraling to steady state" to "sustained limit cycle" as you raise it past the threshold.
Max transcription rate a — sets the amplitude of the cycle without changing whether it oscillates.
Decay rate — the shared degradation rate b of X, Y and Z; faster decay shortens the period.
Reset — kicks the state away from equilibrium so you can see whether the trajectory relaxes back down (damped) or converges onto the same closed loop from a different starting point (a true limit cycle is attracting, so it will).

Real-world relevance: this exact three-variable delayed-negative-feedback structure underlies real biological clocks — the Drosophila PER/TIM circadian loop and the p53–Mdm2 oscillator both reduce to a Goodwin-like core, and synthetic-biology "genetic clocks" are engineered around the same principle.

mRNA X Protein Y Repressor Z Phase-space trail

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