The repressilator is the founding synthetic-biology circuit: three genes, each repressing the next in a ring, produce sustained protein oscillations with no external clock:
dp_i/dt = α / (1 + p_(i-1)^n) − β·p_i for i = 1,2,3 (cyclic)
- Promoter strength α — maximum production rate of each protein when its repressor is absent.
- Degradation rate β — how fast each protein is broken down; higher β damps and speeds up oscillation.
- Hill coefficient n — cooperativity of repression; higher n makes the on/off switching sharper, favouring stronger oscillations.
- Simulation speed — time-scaling only, doesn't change the underlying dynamics.
Real repressilators (built from LacI, TetR and λ CI repressors in E. coli) are used as a proof-of-concept "biological clock" and as a building block for programmable cells in synthetic biology.