The 3D twin of this simulator recomputes the lac operon's binding curves from scratch every frame — an instantaneous equilibrium with no memory. Real E. coli is not like that: because the lac permease it makes is itself the protein that imports more inducer, the operon has genuine positive feedback with a time constant, and Novick & Weiner showed in 1957 that induced cells stay induced for many generations even after external inducer is removed. This 2D model integrates that actual feedback loop as a single ODE for lacZYA enzyme level E, with the effective intracellular signal S folding in both the external inducer and E's own contribution:
S = Iext + c·E (permease imports more inducer as E rises)
f(S) = S^n / (K^n + S^n) n=3, K=8 (cooperative promoter response)
capGain = 0.15 + 0.85·fCap fCap from cAMP = Kg/(Kg+glucose), same
AND-logic as the 3D twin's CAP activator
dE/dt = α0 + α1·capGain·f(S) − γ·E α0=0.02 (basal leak), α1=1, γ=0.1
Integrated live with 4th-order Runge–Kutta. For a wide range of external inducer — including exactly zero — this equation has two stable steady states (OFF near E≈0.2–0.7 and INDUCED near E≈6–9.7) separated by an unstable threshold, confirmed numerically by scanning dE/dt for sign changes: three real roots exist in the bistable window, collapsing to one above it. That is a genuine saddle-node bifurcation, not a scripted state machine.
- Lactose analog — sets Iext, the external inducer concentration (like IPTG/TMG in the classic experiments).
- Glucose — lowers capGain via the same cAMP/CAP saturation curve as the 3D twin; push it high enough and the bistable window closes entirely — try the sweep at glucose=0 vs glucose=80 to see it shrink.
- Raise the inducer slider until the enzyme trace jumps to the induced branch, then drag it back to zero — the cell usually stays induced, exactly the 1957 "maintenance effect." Only "Dilute by growth" (repeated halving of E across generations, the real mechanism — cell volume and its contents dilute at every division) can cure it back to OFF once inducer is also removed.
- The sweep panel below finds the rising threshold directly from the live equation; the falling sweep typically finds none in range — the switch's memory is real, not a display artifact.
Real biology: Ozbudak et al., Nature 2004, "Multistability in the lactose utilization network of Escherichia coli," measured exactly this bistable, hysteretic induction curve in single cells — the founding experimental demonstration that a natural gene circuit can be a true bistable switch.