Oxygen binding to hemoglobin at 4 cooperative sites. alp > 1 for pos cooperativity, alp < 1 for neg coop.
Hemoglobin, a protein with 4 interdependent binding sites, can become saturated with oxygen, i.e all of its binding sites can be occupied at high concentrations. The fractional saturation is calculated here by a cooperative scheme by which there is a constant ratio of increases in affinity as each site is filled in succession. The cooperativity factor "alp" is >1 for positive cooperativity, and < 1 for anticooperativity. The results are compared with the result using a Hill equation with a Hill coefficient of 2.7. The value is chosen because the Hill equation with nH (Hill coefficient with nH = 2.7 fits oxyhemoglobin saturation curves well). The math is straightforward, based on the equation for single site binding, modified in recognition that there are, at varying concentrations, 4 sites available to fill. When one is filled, only 3 remain, reducing the odds from 4 to 3, and so on. The actual O2 carriage depends on the relative abundances of HbO, HbO2, etc, and the fact that there is twice as much O2 on HbO2 as on HbO, etc. The sum of the products of the relative concentrations times the O2s being carried in each form is divided by 4*HbO4, the maximum that can be carried. This model serves as a basis of other cooperativity models wherein the filling of the first and successive sites causes (by cooperativity = positive feedback through molecular conformational rearrangement) successively higher affinities. The ratio, "alp" is not necessarily constant. For example the Adair eqautions are equivalent to having "alp as a variable.
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