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The Hill Equation
θ = xⁿ / (K₅₀ⁿ + xⁿ). Watch how the Hill coefficient n transforms a hyperbolic response into a sharp switch. Side-by-side comparison of all n values.
The Hill equation θ(x) = xⁿ/(K₅₀ⁿ + xⁿ) describes graded-to-switch transitions. n=1: purely hyperbolic (Michaelis-Menten); no threshold, 81× range of x gives 10→90% response. n=2: sigmoidal; range narrows to ~9×. n=4: very sharp; ~3× range. n→∞: perfect step function (Heaviside) at x=K₅₀. Sensitivity analysis: the derivative dθ/dx peaks at x=K₅₀ and sharpens with n. Biochemical ultrasensitivity (Goldbeter-Koshland): zero-order kinetics of competing kinase/phosphatase can produce effective Hill coefficients ≫ 1 even for a single site (no multiple binding needed — it's a network effect, not molecular cooperativity).