Oxygen binding to hemoglobin at 4 cooperative sites. alp > 1 for pos cooperativity, alp < 1 for neg coop.

## Description

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.

## Equations

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Keener J and Sneyd J. Mathematical Physiology. New York, NY: Springer-Verlag, 1998, 766 pp. Dash RK and Bassingthwaighte JB. Erratum to: Blood HbO2 and HbCO2 dissociation curves at varied O2, CO2, pH, 2,3-DPG and Temperature Levels. Ann Biomed Eng 38(4): 1683-1701, 2010. Hill AV. The diffusion of oxygen and lactic acid through tissues. Proc R Soc Lond (Biol) 104: 39-96, 1928. Adair GS. The hemoglobin system. VI. The oxygen dissociation curve of hemoglobin. J Biol Chem 63: 529-545, 1925. Hill AV. The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii, 1910 Hill R. Oxygen dissociation curves of muscle hemoglobin. Proc Roy Soc Lond B 120: 472-480, 1936. Roughton FJW, Deland EC, Kernohan JC, and Severinghaus JW. Some recent studies of the oxyhemoglobin dissociation curve of human blood under physiological conditions and the fitting of the Adair equation to the standard curve. In: Oxygen Affinity of Hemoglobin and Red Cell Acid Base Status. Proceedings of the Alfred Benzon Symposium IV Held at the Premises of the Royal Danish Academy of Sciences and Letters, Copenhagen 17-22 May, 1971, edited by Rorth M and Astrup P. Copenhagen: Munksgaard, 1972, p. 73-81. Winslow RM, Swenberg M-L, Berger RL, Shrager RI, Luzzana M, Samaja M,and Rossi-Bernardi L. Oxygen equilibrium curve of normal human blood and its evaluation by Adair's equation. J Biol Chem 252: 2331-2337, 1977.

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**Model development and archiving support at https://www.imagwiki.nibib.nih.gov/physiome provided by the following grants:** NIH U01HL122199 Analyzing the Cardiac Power Grid, 09/15/2015 - 05/31/2020, NIH/NIBIB BE08407 Software Integration, JSim and SBW 6/1/09-5/31/13; NIH/NHLBI T15 HL88516-01 Modeling for Heart, Lung and Blood: From Cell to Organ, 4/1/07-3/31/11; NSF BES-0506477 Adaptive Multi-Scale Model Simulation, 8/15/05-7/31/08; NIH/NHLBI R01 HL073598 Core 3: 3D Imaging and Computer Modeling of the Respiratory Tract, 9/1/04-8/31/09; as well as prior support from NIH/NCRR P41 RR01243 Simulation Resource in Circulatory Mass Transport and Exchange, 12/1/1980-11/30/01 and NIH/NIBIB R01 EB001973 JSim: A Simulation Analysis Platform, 3/1/02-2/28/07.