This is a genuine, independently-computed 2D counterpart to the 3D double-well landscape simulator: instead of reducing the circuit to a single order parameter rolling on a Landau potential, it simulates the actual discrete chemistry — two mutually-repressing genes A and B with real integer protein/mRNA molecule counts, evolved by a tau-leaping approximation of the Gillespie stochastic simulation algorithm (birth-death chemical master equation):
Synthesis A: rate = Ωβ/(1+(B/Ω)²) A += 1
Degradation A: rate = A A -= 1
Synthesis B: rate = Ωβ/(1+(A/Ω)²) B += 1
Degradation B: rate = B B -= 1
Each of the N cells is an independent Markov jump process. Ω (molecule copy number) sets how many actual molecules realize the same mean concentration — a real biological knob: small cells with few regulatory molecules are shot-noise-dominated (Poisson fluctuations scale as 1/√Ω), exactly the mechanism synthetic-biology papers cite for "leaky" genetic memory, distinct from the free noise-amplitude dial the 3D version uses. β is promoter strength (cooperative dimeric repression, Hill exponent 2, fixed — the standard Gardner–Collins toggle-switch form); larger β makes the two attractor branches more separated and harder to escape.
The phase plane plots each cell's live (A,B) molecule counts. The two curves are the deterministic nullclines A = Ωβ/(1+(B/Ω)²) and B = Ωβ/(1+(A/Ω)²); their three intersections are the two stable fixed points (filled) and the unstable saddle between them (open) — this IS the actual bifurcation structure of the circuit, not a rendering trick. The predicted escape time reduces the 2D flow to an effective 1D potential along x = A−B on a fixed-composition slice through the saddle, then applies a Kramers-style rate using the exact linear instability rate at the saddle and the shot-noise diffusion coefficient there. Caveat, stated plainly: unlike the 3D version's genuine gradient (Landau) potential, this circuit's real 2D flow is not conservative away from the symmetric line, so no exact global potential exists — the predicted τ is therefore only an order-of-magnitude WKB-style estimate (it reproduces the correct exponential dependence on Ω and β but can be off from the observed average by a large constant factor, worse the further β sits above its bistability threshold); the "Observed avg. interval" readout is the trustworthy number.
- Ω (copy number) — higher values mean more molecules realize the same concentration, so relative Poisson noise shrinks and spontaneous flips become exponentially rarer.
- β (promoter strength) — deepens and separates the two attractor basins; stronger cooperative repression suppresses spontaneous switching.
- Population N — more independent cells give a cleaner statistical read of the state-A/state-B split.
- Stress pulse — adds a transient zero-mean "futile cycle" of extra synthesis+degradation events (extra molecular turnover), raising shot noise without shifting the mean — a real biochemical noise-injection mechanism, not just a slider hack.