Abstract
A master equation approach to the kinetics of co-operative ligand binding reactions in proteins is developed. As a specific example, equations based on the Perutz description are applied to several reactions of hemoglobin, both in the absence and in the presence of organic phosphates.
The observed time dependence of O
2 binding to stripped hemoglobin is fitted with a total of four parameters, of which two are held fixed at values determined from the experimentally measured rate constants for the last step of ligand binding. An analysis of Gibson's data (1970a) for oxygen-binding kinetics of unstripped hemoglobin shows that phosphate binding not only stabilizes the deoxy quaternary conformation of hemoglobin but also increases
both the strength of the quaternary conformational constraints and the kinetic time scaling parameter for the deoxy quaternary conformation.
A single set of five parameters is used to fit the deoxygenation data of Salhany
et al. (1970) in the presence of varying concentrations of P
2Glyc
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Abbreviations used: P
in2Glyc, 2,3-diphosphoglycerate; IHP, inositol hexaphosphate; HPT, 8-hydroxy-l,3,6-pyrenetrisulfonate; Hb, hemoglobin.
. The five parameters are determined by fitting the theory to the data in the absence of P
2Glyc and at a single concentration of P
2Glyc. The predictive ability of the model is then tested by using these same parameter values to calculate the curves for intermediate P
2Glyc concentrations, and then comparing the predicted curves with experimental data. Whereas deoxygenation data in the presence of P
2Glyc can be fitted with the subunits considered as equivalent, we find that α, β subunit inequivalence is particularly important in interpreting deoxygenation data in the presence of the inhibitor, IHP. Since the phosphate dependence of the reaction is explicitly included, the model can also describe biphasic deoxygenation, as observed by Gray & Gibson (1971) for the deoxygenation reaction in the presence of IHP.
In the deoxygenation reaction, the quaternary conformation is found to change from oxy to deoxy just prior to the release of the second oxygen molecule; the switch occurs somewhat earlier in the presence of organic phosphates. As a consequence of this switch, the release of oxygen lags behind the binding of organic phosphates; the lag is more pronounced for IHP than for P
2Glyc. The analogous result of MacQuarrie & Gibson (1972) concerning the lag between the release of HPT and binding of CO is well reproduced; the fractional saturation of hemoglobin with CO and the fraction of HPT released are both fitted with a single set of five parameters.
The dependence of the Adair (1925a,b) rate constants (for α = β) and Olson-Gibson (1972) rate constants (for α ≠ β) on the concentration of organic phosphate is calculated. It is found that (i) the experimentally observed increase in the overall deoxygenation rate with increasing organic phosphate concentration is primarily due to the increase in the rate of dissociation of the second oxygen molecule, (ii) the “on” rate constants most affected by organic phosphate are
k
1′ and
k
2′ the rate constants for the binding of the first and second oxygen molecules, (iii) the “off” rate constants,
k
j
, are affected more than the “on” rate constants,
k
j
′ (iv) the rate constants for the reactions of the β subunits are affected more by IHP than those of the α subunits, and (v) the rate constants involving the intermediate species with two β subunits and one α subunit liganded are influenced the most by the binding of phosphates.