How it Works
Three genes — LacI, TetR and cI — are wired into a closed repression ring: cI represses lacI, LacI represses tetR, and TetR represses cI. Each gene's mRNA level rises toward a maximum transcription rate α whenever its repressor protein is absent, and is pushed toward zero by a Hill function as that repressor accumulates, with a small constant leak α0 that never fully switches off. Protein is then translated from mRNA and degraded at a rate β relative to mRNA turnover. Because the three negative feedback loops are chained around a ring rather than acting alone, a rise in one protein a moment later suppresses the next gene, which releases the gene after it, and so on — producing a self-sustaining relay race of gene expression instead of a stable equilibrium.
The simulation integrates these coupled ordinary differential equations every frame with a small-step midpoint (RK2) method so the curves stay numerically stable even at high transcription rates. Whether the system actually oscillates depends on the balance of the four sliders: a steep enough Hill coefficient, a high enough α relative to α0, and a β large enough that protein responds quickly to its own mRNA are all needed to destabilise the symmetric fixed point and produce the sustained, phase-shifted oscillation seen in the chart and pulsing ring diagram.
dmTetR/dt = -mTetR + α/(1+pLacI^n) + α0
dmcI/dt = -mcI + α/(1+pTetR^n) + α0
dpi/dt = β·(mi − pi), i ∈ {LacI, TetR, cI}
Frequently Asked Questions
What is a repressilator?
The repressilator is a synthetic genetic circuit built by Elowitz and Leibler in 2000 from three transcriptional repressor proteins — LacI, TetR and cI — wired into a ring where each one represses the next. Because there is no direct positive feedback, the three genes push each other into a sustained oscillation rather than settling down.
Why does the repressilator oscillate instead of settling to a fixed point?
With three repressors arranged in a ring, any rise in one protein suppresses the next gene a fraction of a cycle later, which in turn releases the gene after it, and so on around the loop. This delayed negative feedback destabilises the symmetric steady state whenever the Hill coefficient, transcription rate and protein-to-mRNA degradation ratio are large enough, producing a stable limit-cycle oscillation instead.
What do LacI, TetR and cI actually do?
LacI, TetR and cI are natural bacterial and phage repressor proteins that normally bind their own specific operator DNA sequences to switch off a target gene. In the repressilator they are repurposed so that cI represses the lacI gene, LacI represses the tetR gene, and TetR represses the cI gene, forming a closed repression ring.
What does the Hill coefficient control?
The Hill coefficient n sets how sharply a repressor switches its target gene off as its concentration rises — a higher n produces a steeper, more switch-like response. Cooperative repressor binding (n greater than roughly 1-2) is required for the ring to oscillate; with n too close to 1 the feedback is too gradual and the system relaxes to a steady state.
What happens if leakiness is too high?
Leakiness (α0) is the baseline transcription rate that continues even when a gene is fully repressed. If α0 is pushed too high relative to the maximum transcription rate α, every gene stays partially active all the time, the repression signal is no longer strong enough to suppress its target, and the oscillation collapses into a steady, non-oscillating state.