This is the same Gardner–Cantor–Collins genetic toggle switch as the 3D version, but simulated with the physics real cells actually run on: a finite, integer number of repressor-protein molecules, born and degraded one at a time, not a smooth concentration obeying a deterministic ODE. Each cell tracks discrete counts nA, nB and four birth/death reaction propensities (chemical master equation):
prod(A) = Ω·α₁ / (1 + (n_B/Ω)^n) deg(A) = n_A
prod(B) = Ω·α₂ / (1 + (n_A/Ω)^n) deg(B) = n_B
Every animation substep uses tau-leaping (Gillespie, 2001): the number of molecules produced or degraded in that instant is drawn from a Poisson distribution with mean = propensity × dt, so each cell's trajectory is a genuinely random walk through molecule-count space, not an integrated curve. Averaged over many cells this converges exactly to the same du/dt, dv/dt Hill-function equations the 3D deterministic model integrates — but any single cell can, purely from molecular shot noise, wander far enough from its stable point to cross the unstable ridge and land in the other basin, with no inducer pulse at all. That is a real, well-documented phenomenon (noise-induced switching) that a continuous ODE — which has no concept of "how many molecules" — cannot produce even in principle.
- α₁, α₂, n sliders — same promoter-strength and cooperativity meaning as the deterministic model; they set the mean-field nullclines (drawn as guide curves) and fixed points.
- System size Ω — the molecule-count scale (roughly the mean copy number at a fixed point). Noise amplitude scales as 1/√Ω: turn Ω down and cells flip spontaneously far more often; turn it up and the population "freezes" into whichever basin it started in, approaching the deterministic limit.
- Scatter plot — 49 independent cells (7×7 seed grid across state space), each an integer (n_A, n_B) point, colored blue (A-dominant) or red (B-dominant); the thin curves are the deterministic mean-field nullclines for reference only — nothing here is computed from them.
- Probe cell — click anywhere on the plot to drop one cell there and follow its molecule-count trail (white).
- Spontaneous flips — counts every population-wide basin change that happened without a pulse button being pressed, using a 15% hysteresis margin so shot-noise jitter near the ridge isn't miscounted.
Real-world relevance: Gardner et al.'s original 2000 paper on this exact circuit already flagged molecular noise as the practical limit on how reliably a bistable gene circuit holds its state — this is why synthetic-biology memory circuits in real cells are engineered with high copy number and steep cooperativity (large Ω, large n) specifically to suppress the spontaneous switching this simulator makes visible.