Hemoglobin grabs oxygen better once it’s already holding some — its four sites COOPERATE, so its binding curve is a switch-like S, not a gentle rise. The Hill equation captures that: the exponent n says how cooperative. n=1 is a lazy curve (like myoglobin); n=4 snaps between empty and full. That switch is how your blood loads O₂ in the lungs and dumps it in the tissues. Slide the cooperativity.
The Hill equation θ = Lⁿ/(K + Lⁿ) gives the fraction of binding sites occupied at ligand concentration L, with Hill coefficient n encoding cooperativity: n=1 is independent binding (a hyperbola, like myoglobin), n>1 is positive cooperativity (a sigmoid, hemoglobin ≈ 2.8–3), and larger n approaches an all-or-nothing switch. All curves cross θ=½ at Lⁿ=K. The steepness at the midpoint is what makes cooperative proteins act as sharp molecular switches. A fail-loud self-check throws unless n>1 is steeper around K than n=1 and both pass through θ=½ at L=K. ◆ real biochemistry, node-verified.
The Hill coefficient is a PHENOMENOLOGICAL summary (real hemoglobin isn’t exactly n=4; it’s ~2.8), not a literal count of independent sites. The sigmoid-vs-hyperbola contrast and the switch behaviour are the exact content.