This is the Gardner–Cantor–Collins genetic toggle switch (Nature, 2000): two genes whose protein products mutually repress each other's promoter. With protein concentrations u (repressor A) and v (repressor B):
du/dt = α₁ / (1 + v^n) − u
dv/dt = α₂ / (1 + u^n) − v
Each term is a Hill-function repression curve (protein decays at rate 1 in these dimensionless units, and is synthesized at a rate that falls off as the *other* repressor's concentration rises). When the cooperativity n > 1 and the promoters are strong enough, the two nullclines u = α₁/(1+v^n) and v = α₂/(1+u^n) cross three times: two stable fixed points (Gene A high / Gene B low, or vice-versa) separated by an unstable saddle. That is bistability — the circuit acts as a biological flip-flop, and which state it lands in depends only on where it started, not on any memory circuit built from more genes.
- α₁, α₂ sliders — maximum promoter output; raising both moves the fixed points further apart.
- n slider — repression cooperativity; below n≈1.5–2 (depending on α) the two wells merge into one and bistability is lost — try dragging it down.
- Background dots — a grid of "cells" starting from every combination of initial u,v, integrated live by 4th-order Runge–Kutta; each settles into whichever basin it started in, colored blue (A-dominant) or red (B-dominant). Their split shows the two basins of attraction.
- Probe cell — click anywhere on the floor to drop one cell there and watch its trajectory (white trail) converge to a fixed point.
- IPTG / aTc pulses — mimic the real experiment: a chemical inducer transiently blocks one repressor's DNA-binding, letting the other gene win the race and flipping every cell currently on that side toward the opposite stable state.
Real-world relevance: this exact circuit (built from the lac and tet repressors in E. coli) was the first synthetic gene circuit engineered from scratch to show a designed, predictable dynamical behavior — the founding demonstration of synthetic biology as a discipline, alongside the repressilator oscillator published the same year.