This is the Elowitz & Leibler (2000) repressilator: three repressor genes wired in a ring, each one transcriptionally repressing the next (LacI ⊣ TetR ⊣ cI ⊣ LacI). Because the loop has an odd number of inhibitory links and enough delay from mRNA to protein, the network has no stable fixed point and instead relaxes into sustained oscillations — a synthetic clock built from ordinary gene regulation.
for each gene i, repressed by protein p_(i-1):
dm_i/dt = -m_i + alpha_eff/(1 + p_(i-1)^n) + alpha0
dp_i/dt = -beta * (p_i - m_i)
alpha_eff = alpha / (1 + burden * mean(m1,m2,m3))
n = 2 (cooperative Hill repression)
Concentrations are in dimensionless promoter units. The host burden term models finite ribosomes/RNA polymerase: as total circuit transcription rises, every gene's effective α drops, exactly as measured in real burden-monitor strains. IPTG mimics a real inducer that inactivates the LacI repressor, so gene C's promoter is freed and locked ON — turning the oscillator into a fixed logic output, the same trick used to build genetic toggle switches and memory elements.
- α (transcription rate) — raises oscillation amplitude and speeds up the clock.
- β (protein decay) — the ratio of protein-to-mRNA lifetimes; tuning it can dampen or destabilize the oscillation.
- Host burden — throttles all three genes together as total expression rises, modelling limited cellular resources.
- Leaky expression α₀ — baseline "off-state" transcription; too much leak can suppress oscillation.
- Add IPTG — chemically induces the circuit, freezing gene A's repression of gene B and gene C's ring position into a fixed high/low logic state (memory).
Circuits like this underpin real synthetic-biology tools: genetic clocks for timed drug release, toggle switches for cellular memory, and burden-aware designs that keep engineered circuits from crashing their host cell.