Equations
Toggle switch (Gardner 2000):
du/dt = α/(1+v^n) − γu
dv/dt = α/(1+u^n) − γv
Nullclines intersect at:
1 steady state (monostable)
3 steady states (bistable, n>1)
Repressilator (Elowitz 2000):
ẋ = α/(1+z^n) − x
ẏ = α/(1+x^n) − y
ż = α/(1+y^n) − z → OSCILLATION
Bistability requires positive feedback (or mutual inhibition). The toggle switch (Gardner et al., Nature 2000) was the first synthetic bistable gene circuit. The repressilator (Elowitz, Nature 2000) — three mutually repressing genes — produces sustained oscillations: the first synthetic genetic clock.