Two mutually repressing genes with Hill cooperativity: bistable A-high/B-high attractors, phase-plane nullclines, hysteresis under inducer sweeps, and noise-driven switching.
The phase portrait shows why memory appears: two attractors coexist when nullclines intersect on stable outer branches. The hysteresis panel keeps the previous state while bias is swept.
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About this model
The genetic toggle switch is a minimal synthetic biology motif: two genes encode repressors A and B, and each repressor inhibits expression of the other gene. This simulator uses nondimensional Hill equations dA/dt = β + α_A/(1+Bⁿ) − A and dB/dt = β + α_B/(1+Aⁿ) − B. When repression is cooperative enough, the phase-plane nullclines intersect so that two stable attractors coexist: A-high/B-low and B-high/A-low. The current inducer bias changes the relative expression strengths α_A and α_B. The time-series panel shows deterministic relaxation plus optional expression noise, the phase portrait overlays the trajectory with both nullclines, and the hysteresis panel sweeps the bias upward and downward to show memory: the switch flips at different thresholds depending on its previous state.
Who it's for: Students in systems biology, synthetic biology, nonlinear dynamics, or biophysics learning gene circuits, bistability, and stochastic switching.
Key terms
Genetic toggle switch
Mutual repression
Hill function
Bistability
Hysteresis
Nullcline
Attractor
Noise-driven switching
How it works
A mutual-repression genetic toggle switch: cooperative Hill repression creates bistability, hysteresis under inducer sweeps, and noise-driven switching between expression states.
Key equations
dA/dt = β + α_A/(1+Bⁿ) − A, dB/dt = β + α_B/(1+Aⁿ) − B, with inducer bias setting α_A = αeˢ, α_B = αe⁻ˢ. Noise is additive Langevin-style expression noise.
Frequently asked questions
Why is Hill cooperativity important?
Without a sufficiently steep repression curve, the nullclines usually intersect in only one stable state. Cooperative binding makes the response switch-like, allowing two stable expression states to coexist.
What does the hysteresis loop mean biologically?
It means the circuit has memory. The inducer level needed to flip from B-high to A-high differs from the level needed to flip back, so cells can remember a past stimulus after the bias is partially removed.
How does noise cause switching?
Expression noise randomly perturbs A and B. If a fluctuation is large enough to cross the separatrix between basins of attraction, the deterministic dynamics carry the circuit into the other stable state.