Why is the oxygen dissociation curve shaped like it is?

At first, the shape of the Hb makes it hard for oxygen to bind to one of the polypeptide subunits so at low concentrations of oxygen, the saturation of oxygen is low too.
Once the first molecule of oxygen has bound to the Hb, the shape changes making it easier for more O2 to bind. This is called positive cooperativity. Therefore rapid saturation of Hb occurs when there is not much change in partial pressure of O2 so the gradient is very steep.
To become fully saturated, a fourth O2 molecule has to bind to the Hb. This is hard because of the chance that O2 will find the single empty site when the majority of binding sites are occupied so the saturation does not increase despite there sometimes being large increases in partial pressure of O2 so the curve flattens.

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