A symmetric synthetic toggle switch (two genes that mutually repress each other via Hill-function kinetics) has two stable steady states. Reducing the two gene levels to a single order parameter x = (protein A level − protein B level), the deterministic dynamics near the two attractors are well approximated by overdamped motion in a double-well (Landau) potential:
U(x) = b·(x⁴/4 − x²/2) minima at x = ±1, barrier at x = 0
dx/dt = −dU/dx + √(2D)·ξ(t) (Langevin / chemical-Langevin noise)
Here b plays the role of promoter strength / cooperativity (a stronger, more switch-like Hill response deepens and separates the two wells) and D is the intrinsic biochemical noise set by molecule copy number — small copy numbers (low D on this dial would actually mean high real noise, but here D is noise amplitude directly) mean noisier, more frequent spontaneous flips. Each of the N marbles is one cell, independently integrated by Euler–Maruyama with a clamped time step.
The mean residence time in a well before a noise-driven escape follows Kramers' rate theory for a cubic-derivative double well:
ΔV = b/2 (barrier height above a well minimum)
τ ≈ (2π / (b·√2)) · exp(ΔV / D)
- Noise D — the diffusion strength of the Langevin term; higher D means shorter, more frequent switching events.
- Barrier b — reshapes the landscape live: deeper wells (larger b) exponentially suppress spontaneous switching, exactly as stronger cooperative repression stabilizes a real toggle switch.
- Population N — more independent cells give a cleaner statistical read on the state-A / state-B split and the switching statistics.
- Stress pulse — a transient 4× noise burst mimics a real cellular stress event that can kick a cell out of its current basin.
This landscape-and-noise framing (an "epigenetic"/Waddington-style potential for a gene circuit) is the same qualitative picture used in synthetic-biology bistability analysis to reason about how robust a genetic memory element is to molecular noise, distinct from the deterministic, externally-induced flipping shown by a plain Hill-kinetics toggle-switch model.