Lac Operon Bistable Switch (2D)

This 2D counterpart integrates the real dynamical feedback the 3D twin's per-frame equilibrium formula cannot show: because the enzyme the lac operon makes is the very protein that imports more inducer, the system has genuine positive feedback with a time constant. A single ODE for lacZYA enzyme level is integrated live with 4th-order Runge–Kutta, driving a particle-cloud view of the repressor and enzyme populations, a live enzyme-vs-time trace, and a dedicated bifurcation panel that sweeps external inducer forward and backward through the actual equation to trace out a genuine hysteresis loop. Push the inducer up, then back down to zero — the switch usually stays induced, reproducing the 1957 Novick–Weiner "maintenance effect" numerically rather than by a scripted state.