Blood Oxygen Dissociation (Hill Equation)
The Hill equation models how hemoglobin binds oxygen.
Learn the oxygen-hemoglobin dissociation curve formula with worked examples.
The Formula
The Hill equation describes the sigmoidal (S-shaped) relationship between the partial pressure of oxygen (pO₂) and the fraction of hemoglobin saturated with oxygen (Y). It was proposed by Archibald Hill in 1910 to model the cooperative binding of oxygen to hemoglobin.
Hemoglobin is a protein in red blood cells that carries oxygen from the lungs to the tissues. Each hemoglobin molecule can bind up to four oxygen molecules. The binding is cooperative, meaning that once one oxygen binds, it becomes easier for subsequent oxygen molecules to attach. This cooperative behavior creates the characteristic S-shaped dissociation curve.
The P₅₀ value is the partial pressure of oxygen at which hemoglobin is 50% saturated. For normal human blood at 37°C and pH 7.4, P₅₀ is approximately 26.6 mmHg. Factors that shift the curve include pH (Bohr effect), temperature, and 2,3-DPG concentration.
Variables
| Symbol | Meaning |
|---|---|
| Y | Fractional saturation of hemoglobin (0 to 1, or 0% to 100%) |
| pO₂ | Partial pressure of oxygen (in mmHg) |
| P₅₀ | pO₂ at 50% saturation (≈ 26.6 mmHg for normal blood) |
| n | Hill coefficient (≈ 2.8 for hemoglobin, measures cooperativity) |
Example 1
What is the hemoglobin saturation at pO₂ = 40 mmHg (typical venous blood)? Use P₅₀ = 26.6 mmHg and n = 2.8.
Apply the formula: Y = 40²·⁸ / (26.6²·⁸ + 40²·⁸)
40²·⁸ ≈ 28,198 and 26.6²·⁸ ≈ 9,001
Y = 28,198 / (9,001 + 28,198) = 28,198 / 37,199
Y ≈ 0.758 or about 75.8% saturated
Example 2
What is the hemoglobin saturation at pO₂ = 100 mmHg (typical arterial blood)?
Apply the formula: Y = 100²·⁸ / (26.6²·⁸ + 100²·⁸)
100²·⁸ ≈ 398,107 and 26.6²·⁸ ≈ 9,001
Y = 398,107 / (9,001 + 398,107) = 398,107 / 407,108
Y ≈ 0.978 or about 97.8% saturated
When to Use It
The Hill equation is fundamental in respiratory physiology and clinical medicine.
- Predicting oxygen delivery to tissues at different oxygen pressures
- Understanding the effects of altitude on blood oxygen levels
- Analyzing the Bohr effect (pH changes shifting the dissociation curve)
- Clinical assessment of patients with respiratory conditions