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Hill equation

The Hill equation fits cooperative binding as y = Vmax · xⁿ / (Kⁿ + xⁿ) — Michaelis-Menten generalised to a variable Hill coefficient.

What is Hill equation?

The Hill coefficient n captures cooperativity. n = 1 reduces to Michaelis-Menten (single binding site, no cooperativity); n > 1 = positive cooperativity (haemoglobin binds 4 O₂ with n ≈ 2.8 — classical example); n < 1 = negative cooperativity.

K is the half-saturation concentration (analogous to Km in M-M). For Hill the curve approaches Vmax with steeper slope than M-M when n > 1.

When should I use Hill equation?

Receptor-binding studies with suspected cooperativity.
Allosteric enzymes.
When the M-M fit leaves systematic curvature in the residuals.

What data does it need?

Numeric X (concentration) + Y (response).

What does it report?

Vmax, K, Hill n with 95% CIs.

What does it assume?

Single binding-site cluster (no mixed populations).
Y normally distributed given X.

How do I interpret the result?

n far from 1 with tight CI ⇒ real cooperativity. n CI containing 1 ⇒ no evidence for cooperativity; M-M is the simpler fit.

References

Hill (1910). The possible effects of the aggregation of the molecules of haemoglobin. J. Physiol. 40.