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Hill (Hill, A. V. (1910-01-22). “The possible effects of the aggregation of the molecules of hæmoglobin on its dissociation curves”J. Physiol. 40 (Suppl): iv-vii.)  provided an equation to quantify cooperation of protein and ligand binding, i.e. enhanced binding when ligand is already bound. Hemoglobin -oxygen interaction was Hill’s focus.

Let P be the protein, L the ligand and C the protein ligand complex and K_d the dissociation constant for the reaction.

P +nL \rightarrow C

At equilibrium with cooperation, the Hill coefficient n >1,

K_d = \frac{ [P] [L]^n}{[C]}


Let \theta be the fraction of protein molecules with ligands attached, [C]/([P]+[C])


Let [L_{50}] represent the concentration when 50% of the protein is bound by ligand. Therefore, K_d=[L_{50}]^n


\frac{\theta}{1-\theta}= \frac{[L]^n}{[L_{50}]^n}


\theta = \frac{[L]^n}{[L_{50}]^n+[L]^n}


For hemoglobin-oxygen,  oxygen concentration is usually expressed as the partial pressure of oxygen (pO_2)  and \theta is the oxygen saturation (SaO_2) of the hemoglobin molecule. The p_{50}0_2 is 26.6 mm Hg (3.55 kPa) and leads to

SaO2 =\frac{pO_2^n}{p_{50} O_2^n +pO_2^n}

This is the familiar form of the hemoglobin oxygen dissociation curve, where  n= 2.4 to 2.8, yielding the familiar sigmoid curve.

Hemoglobin oxygen dissociation curve with n=2.8


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