Premise

The Hill equation models hemoglobin as if it has one effective binding site. The actual molecule has four subunits that bind cooperatively — the third O₂ molecule binds differently than the first. The standard saturation curve collapses that distribution into a mean, but the occupancy PMF (the probability that exactly k of 4 sites are bound at a given pO₂) tells you more: it describes the population of hemoglobin states in tissue, not just average delivery. The question was whether the simple binomial independence assumption — four sites binding independently — produces a meaningfully different occupancy distribution than a cooperative model, and what the physiological implications are.

How it evolved

Built as a companion toolkit for a paper deriving the analytic PMF expressions. Started as a web visualization and grew to include a full Python analysis pipeline with publication-quality figures, sensitivity analysis on Adair binding constants, and RMSE validation against Winslow experimental data. The core mathematical work was upfront; the tooling expanded to make the results reproducible.

Technical crux

The Adair model uses a partition function over sequential binding constants (K₁–K₄, Winslow values) to compute the PMF analytically — P(k) = (∏ᵢKᵢ × pO₂ᵏ) / Z(pO₂). The binomial model assumes the same p at each site from the Hill equation. The divergence between them is largest at intermediate pO₂ (the physiological working range of 40–60 mmHg), which is exactly where clinical decisions about oxygen delivery are made. Sensitivity analysis on K-values showed error margins of 10–58% depending on which constant — K₃ and K₄ dominate the uncertainty.

Findings

Adair with Winslow constants closely matches experimental saturation curves; binomial independence diverges meaningfully at intermediate pO₂. The occupancy distribution shifts from predominantly k=0 at low pO₂ to predominantly k=4 at high pO₂, but the cooperative model produces a sharper sigmoidal transition — the binomial model underestimates both extremes. Published as 'Occupancy Distribution of Hemoglobin Tetramers: Analytic Probability-Mass Function and Physiological Implications.' Interactive demo live.

Open questions

Extensions to variant hemoglobins — fetal (HbF, higher O₂ affinity), sickle cell (HbS, polymerization at low occupancy), and high-altitude adapted forms — would sharpen the physiological implications. Integration with tissue oxygen diffusion models would connect the PMF to actual delivery calculations rather than stopping at the dissociation curve.