Regulatory Network Toggle Switch
Two genes, mutual repression — bistability and oscillation
Two genes that repress each other can hold a memory: high-A/low-B or high-B/low-A. Crank the Hill coefficient and the two stable states appear; flatten it and they merge. Swap one repression for an activation and the same two parts become an oscillator. The same alphabet writes both memories and clocks.
Phase portrait — a vs b
ȧ = 0ḃ = 0
Concentrations over time
a(t)b(t)
Bistable — two stable cell-memory states
Toggle: ȧ = α/(1+bⁿ) − γa, ḃ = α/(1+aⁿ) − γb. Cooperativity n bends the nullclines until they intersect three times — two stable nodes and a saddle. Each stable node is a cellular memory state. Switch to activator–repressor and the same parts produce a limit cycle: an oscillator from the same alphabet.