Biological Clock — Goodwin Oscillator & Repressilator

Genetic feedback loops generating circadian and synthetic oscillations

n = 8.0
γ = 1.00
α = 4.00
η = 0.020

Goodwin (1965) Oscillator & Repressilator (Elowitz 2000)

Goodwin oscillator: single negative feedback loop X→Y→Z⊣X:

Ẋ = α/(1+Zⁿ) − γX , Ẏ = X − γY , Ż = Y − γZ

Oscillations require Hill coefficient n > 8 (for 3 nodes). The Repressilator (Elowitz & Leibler 2000) uses 3 genes in cyclic repression (LacI→TetR→CI⊣LacI), the first synthetic genetic oscillator:

ṁᵢ = −m + α/(1+pⱼⁿ) + α₀ , ṗᵢ = −β(pᵢ−mᵢ)

Both are limit cycle oscillators. Period T ∝ 1/γ. Noise causes phase diffusion; cells in a population desynchronize over time (phase noise ∝ √t).