Home > Uncategorized > Cooperation

## Cooperation

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, $[C]/([P]+[C])$

$\frac{\theta}{1-\theta}=\frac{[L]^n}{K_d}$

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

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

Rearranging:

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

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