Hill Equation & Cooperative Binding

Cooperative Binding

Many biological receptors and proteins do not bind ligands independently — binding at one site influences affinity at other sites. This is called cooperative binding and is described by the Hill equation.

Fractional Saturation

The fraction of binding sites occupied (θ) at ligand concentration [L]:

$\theta = \frac{[L]^n}{K_d^n + [L]^n}$

Where:

• [L] — free ligand concentration
• K_d — dissociation constant (= EC50, the concentration at half-maximal binding)
• n — Hill coefficient

Hill Coefficient Interpretation

EC50

At EC50, θ = 0.5. From the Hill equation: EC50 = K_d.

Hill Plot Linearization

Taking the logit transform:

$\log\left(\frac{\theta}{1-\theta}\right) = n \cdot \log([L]) - n \cdot \log(K_d)$

A plot of log(θ/(1-θ)) vs log([L]) gives a straight line with slope = n.

Oxygen Binding to Hemoglobin

Hemoglobin is a classic example of cooperative binding:

• n ≈ 2.8 (4 subunits, strong cooperativity)
• P50 ≈ 26 mmHg (half-saturation at pO2 = 26 mmHg)
• Arterial blood (pO2 ≈ 100 mmHg): ~97% saturated
• Venous blood (pO2 ≈ 40 mmHg): ~75% saturated

This sigmoidal oxygen-dissociation curve enables efficient O2 loading in the lungs and release in tissues.

Functions to Implement

• hill_saturation(L, K_d, n=1) — fractional saturation θ
• hill_plot_y(L, K_d, n) — Hill plot y-axis: log(θ/(1-θ))
• ec50(K_d) — returns K_d (EC50 equals K_d by definition)
• oxygen_saturation(pO2_mmHg, n=2.8, P50=26) — hemoglobin saturation