Essays. Essay Number 2. Equations of State for Fractional Saturation of Human Hemoglobin with Oxygen

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1 Essays Essay Number 2 Equations of State for Fractional Saturation of Human Hemoglobin with Oxygen Formulation of the Adair Equation with Equivalent O 2 Binding Sites Francis Knowles

2 Equations of State for Fractional Saturation of Human Hemoglobin with Oxygen Formulation of the Adair Equation with Equivalent O 2 Binding Sites Updated: 20 August :39 PM The Adair equation was the first attempt to model the properties of tetrameric hemoglobin. Adair s experimental skills resulted in an accurate determination of the molecular weight of hemoglobin. That measurement, in concert with the known equivalent weight of hemoglobin, based on iron analyses, conclusively established that the hemoglobin molecule possessed four sites for binding of O 2. One could only conclude that there were four successive reactions in which a molecule of O 2 reacted with one hemoglobin molecule. The simplest model consistent with this view is represented in Equation (2.1). ± O, K ± O, K ± O, K Hb Hb (HbO ) Hb (HbO ) I II III Hb(HbO ) ± O, K (HbO ) IV V (2.1) There are four sequential O 2 binding reactions resulting in an ordered sequence of five species. For convenience in formulating an equation of state, the five species of Adair s reaction sequence are assigned roman numerals, I thru V in Eq. (2.1), species I being deoxy-hemoglobin and species V being a fully oxygenated hemoglobin. The next step is to formulate an equation of state which defines fractional saturation of the sample of hemoglobin tetramers with O 2. This is presented in Eq. (2.2), where F is fractional saturation of hemoglobin with O 2 : terms within brackets represent the molar

3 concentration of the molecular species within the bracket; the integer before each term in a bracket represents the number of O 2 molecules bound by that species. F, a dimensionless number, is the ratio of two molar concentrations of O 2. The polynomial in the numerator is the actual molar concentration of O 2, bound to hemoglobin, in solution. The polynomial in the denominator is the maximum molar concentration of O 2 which could be supported by hemoglobin in solution. The range of values available to F is between zero and one: 0 F 1. [II] + 2 [III] + 3 [V] + 4 [V] F = 4 [I] + 4 [II] + 4 [III] + 4 [IV] + 4 [V] (2.2) Eq. (2.2) is a precise mathematic description of Adair s model. Analytical expressions relating the concentration of each of the species II thru V to the equilibrium constants K1, K2, K3, and K4; [O 2 ]; and [I] are readily derived. These analytical expressions, listed below, are substituted into Eq. (2.2), resulting in the Adair equation, Eq. (2.3). The term for DHb, [I], appearing in each of the analytical expressions, cancels out of Eq. (2.3). [I] = [I] [II] = K1 [O2] [I] [III] = K1 K2 [O2] 2 [I] [IV] = K1 K2 K3 [O2] 3 [I] [V] = K1 K2 K3 K4 [O2] 4 [I] K [O ] + 2 K K [O ] + 3 K K K [O ] + 4 K K K K [O ] F = (2.3) 4 ( 1 + K [O ] + K K [O ] + K K K [O ] + K K K K [O ] )

4 The Adair equation, Eq. (2.3), is subject to compression by factoring the numerator and denominator terms. (K [O ] ( 1 + K [O ] (2 + K [O ] (3 + 4 K [O ] )))) F = (2.4) ( 4 ( 1 +K [O ] ( 1 + K [O ] (1 + K [O ] (1 + K [O ] ))))) Adair s analysis of the properties of hemoglobin, which appeared almost 90 years ago, also appeared at a time when numerical calculations were particularly challenging, especially when it comes to non-linear least-squares curve fitting procedures on 4 th order polynomials. Further though on this subject by Adair, suggested a justified simplification to the solution of Eq. (2.4). All of the O 2 binding sites are equivalent. The concept of equivalence can be presented in many ways. In its simplest form, one can say that the all O 2 binding sites behave in an identical fashion, regardless of the presence of other O 2 molecules on other O 2 binding sites in the hemoglobin tetramer. Equivalent binding sites, however, result in different equilibrium constants as the hemoglobin tetramer successively binds four O 2 molecules. One can carry out a thought experiment in which one binding site is separated from its three siblings in the hemoglobin tetramer without changing any chemical property of the O 2 binding site. Consider the equilibrium binding properties of such an O 2 binding site, Eq. (2.5). k forward k Binding Site + O forward 2 Binding Site-O 2; K (2.5) k k reverse reverse

5 The rate constants for the forward and reverse reactions are as defined above. And, the equilibrium constant for this isolated O 2 binding site is defined as K. What, then, is the equilibrium constant for binding of the first O 2 molecule by DHb, species I? The answer is apparent from Eq. (2.6). Since there are four equivalent O 2 -binding sites in DHb, DHb, binds one O 2 molecule 4 times as fast as an isolated equivalent binding site. Both species II and the isolated O 2 binding site, on the other hand, release O 2 at the same rate. 4k 4 k I + O forward forward 2 IO 2; K 1 = = 4 K k k reverse reverse (2.6) Therefore, K 1 = 4 K. Similarly: K 2 = 1.5 K, K 3 = 0.67 K, and K 4 = 0.25 K. Substitution of these values for K 1 thru K 4 into Equation (2.4) results in a ratio of polynomials with a single unknown quantity, K. Still, not a trivial calculation in 1925, but one is guaranteed an answer for their effort. Adair s effort was, nonetheless, notable insofar as he formulated the equation of state and realized the means of obtaining a solution to the unknown quantity in the instance where all binding sites are equivalent. The numerator and denominator terms of the Adair equation for the special case in which all O 2 binding sites are equivalent, is presented in Eq. (2.7). (K [O ] (4 + K [O ] (12 + K [O ] ( K[O ])))) F = (2.7) (4 (1 + K [O ] (4 + K [O ] (6 + K [O ] (4 + K [O ])))))

6 Does the Adair Equation with Equivalent Binding at all O 2 Binding Sites Describe O 2 Equilibrium Binding Curves? HbA, M IHP, 0.05 M BisTris, ph 7.00, 20.0 degrees C A non-linear least-squares curve fitting procedure was carried out on O 2 equilibrium binding data 1 for human hemoglobin in a supporting electrolyte comprised of M IHP, 0.05 M BisTris, ph 7.00 with HCl, 20.0 degrees C. The Adair equation, constrained to equivalent binding sites in all intermediate species, does not accurately describe the O 2 equilibrium binding data. The only unknown quantity in Eq. (2.7) is the equilibrium binding constant, K. The best-fitting value of K is 17 L/ mmol. The predicted and observed O 2 equilibrium binding curves are illustrated in Figure (2.1). At low values of [O 2 ] the predicted values of F exceed the observed values of F. And, at high values of [O 2 ], the observed values of F exceed the predicted values. A more precise way of viewing these results is to plot the difference between the observed results and the results predicted by Eq. (2.7) using the best fitting value of K, 17 L/mmol together with the corresponding values for [O 2 ]. This result is illustrated in Figure (2.2). If the Adair equation with equivalent O 2 binding at all sites did, in fact describe the observed O 2 equilibrium binding curve, the difference between observed and predicted results versus [O 2 ] would be a straight line with the equation F PRE F OBS = [O 2 ]. Experimental data will always possess some variance and the best fitting straight line would be considered a good fit if the intercept and the slope were on the order of 0.001, or less and R 2 was approximately The best-fitting straight line has the formula: y = F PRE , R 2 = Another graphical way to illustrate the difference between observed and predicted values of F for the equilibrium binding curve is to graph F PRE versus F OBS over the range of experimental values of [O 2 ]. In this instance, the values of F PRE will closely match those for F OBS if the equation of state is an accurate mathematical model of the process which accompanies the conversion of DHb to oxyhemoglobin. The expected

7 F; Observed and Predicted result is a straight line with the equation F PRE = F OBS Again, Figure (2.3) clearly demonstrates the dissimilarity between F PRE and F OBS. 1 Adair Equation with Equivalent Binding Sites M IHP, 0.05 M BisTris, ph 7, 20 o C [O2], mmol / L Figure (2.1). Comparison of Observed ( ) and Predicted ( ) O 2 Equilibrium Binding Curves. The best-fitting values of F with O 2 are predicted by Eq. (2.7) using the best fitting value of K, 17 L/mmol and the concentration of O 2 corresponding to the observed values of F.

8 F: Predicted - Observed Adair Equation with Equivalent Binding Sites M IHP, 0.05 M BisTris, ph 7, 20 o C y = x R² = [O2], mmol/l Figure (2.2). Differences Between Predicted and Observed O 2 Equilibrium Binding Curves. Predicted values of F with O 2 were calculated using Eq. (2.7) with the best-fitting value of K, 17 L/mmol and the concentration of O 2 corresponding to observed values of F.

9 F, Predicted 1 Adair Equation with Equivalent Binding Sites M IHP, 0.05 M BisTris, ph 7, 20 o C 0.5 y = x R² = F, Observed Figure (2.3). Comparison of observed and predicted values of F of hemoglobin with O 2.

10 The results described above do not permit the conclusion that Eq. (2.7), the Adair equation for equivalent binding for the four O 2 -binding sites, describes the O 2 equilibrium binding curve for purified human hemoglobin in M IHP, M BisTris, ph 7.00, 20.0 o C. The results demonstrated above and the conclusion that may be reached concerning the inability of Eq. (2.7) to describe the mechanism whereby DHb is converted to oxyhb is based on only one set of data. The ability of IHP to depress the affinity of hemoglobin for O 2 either approaches or sets the upper limit of the strength of an E-molecule if measurement of the concentration of O 2 at the point at which F approaches 0.5 is taken as a standard point of reference. It would be best to carry out the curve fitting operation described above for HbA under a variety of conditions to determine if the results obtained with IHP at ph 7.00 and 20.0 o C are universal with regard to the conditions of the experiment. Several examples are described below. Whole Blood, Under Standard Conditions The experiment described above was carried out with purified hemoglobin in a supporting electrolyte of known composition. Experiments with whole blood, on the other hand, employ hemoglobin contained in red blood cells, the red blood cells being supported in an isotonic solution at the physiological ph-value of 7.4 and in equilibrium with a gas phase in which the partial pressure of CO 2 at 37 o C is 40 torr. The purpose of such studies has been to determine the O 2 -equilibrium binding properties of hemoglobin under conditions that approximate, as closely as possible, the natural physiological environment, a subject of wide interest in the fields of physiology, medicine and respiratory physiology, interests which predate those of biochemists, protein chemists and x-ray crystallographers. Physical chemists, on the other hand, were at the forefront of efforts to characterize the properties of hemoglobin. Considerable effort has been devoted to this subject. We compare the results initially established by Roughton 2, using the demanding methods of gas analysis, with the more recent results of Winslow and collaborators 3 using modern methods of spectroscopy for

11 determination of F and oxygen-sensitive electrodes to for determination of oxygen concentrations [O 2 ]. A non-linear least-squares curve fitting operation was carried out on the data collected for the standard O 2 -equilibrium binding curve, the results being fitted to the Adair equation subjected to the constraint of equivalent O 2 binding sites, Eq. (2.7). Equation (2.7) was not found to provide a satisfactory fit to O 2 equilibrium binding data. Winslow s data and the best-fitting data are presented in Figure (2.4). The results show a strikingly similar pattern to that obtained with purified human hemoglobin in the presence of IHP, described above. At values of [O 2 ] less than mmol/l, Eq. (2.7) over-estimates observed values of F. For values of [O 2 ] greater than mmol/l, Eq. (2.7) underestimates observed values of F. Examination of the O 2 -equilibrium binding curves in whole blood, established by Roughton, using methods of gas analysis to establish values of F yield similar results (results not illustrated). The results illustrated in Figure (2.4) establish the inability of Eq. (2.7) to account for the standard O 2 equilibrium binding curve in whole blood. The standard curve of Roughton and Severinghaus is quite similar to that of Winslow and collaborators, being shifted slightly to the left (Roughton s results are not illustrated). A thoughtful discussion of these small differences was presented in Winslow s paper. The difference was attributed to aging changes in concentration of the E-molecule, diphosphoglycerate, inside the red blood cell in the time required by the methods of gas analysis, which exceeded that required for Winslow s simultaneous spectroscopic and electrometric procedures. The concentration of O 2 required for F = 0.5 was mmol/l for Roughton s standard curve in contrast to mmol/l for Winslow s standard curve. These results are indicative of the similarity in their results rather than being a significant discrepancy between two sets of data..

12 F: Observed and Predicted 1 Adair Equation with Equivalent Binding Sites Whole Blood (Winslow et al. (1977) [O2], mmol / L Figure (2.4). Comparison of observed ( ) and predicted ( ) values of fraction saturation of whole blood with O 2 under standard conditions. Predicted values were obtained using the best-fitting value of K, 24.6 L/mmol, in the Adair equation with equivalent O 2 binding sites, Eq. (2.7), and the concentration of O 2 corresponding to observed values of F.

13 F: Predicted and Observed HbA, M KPi, ph 7.00, 20.0 o C 1 HbA, 0.05 M KPi, ph 7.00, 20.0 deg C [O2], mmol / L Figure (2.5). Comparison of observed ( ) and predicted ( ) values of fractional saturation of purified hemoglobin in a supporting electrolyte comprised of M KPi, ph 7.00, 20.0 o C. Predicted values were obtained using the best-fitting value of K, 60.8 L/mmol, together with corresponding O 2 -values in the Adair equation with equivalent O 2 binding sites, Eq. (2.7).

14 Accurate O 2 equilibrium binding data is available in a supporting electrolyte of M KPi, ph 7.00, 20.0 o C (Knowles, unpublished data). Under these conditions, HbA demonstrates F = at [O 2 ] = mmol/l. This value for [O 2 ] in the liquid phase corresponds to a partial pressure of O 2 in the gas phase of 8.38 torr which is close to 25% of the corresponding values of F observed with both HbA-IHP at neutral ph and 20.0 o C and whole blood under standard conditions. The best fitting value of K in Eq. (2.7) is L/mmol. The results illustrated in Figure (2.5) establish the inability of Eq. (2.7) to account for the O 2 equilibrium binding curve in M KPi, ph 7, 20 o C. HbA, M NaCl, 0.05 M BisTris, ph 7, 20 o C Two of the examples cited above are characterized by very low affinity for O 2. Whole blood has F equal to 0.5 at mmol O 2 /L, corresponding to a partial pressure of O 2 in the gas phase of about 29 torr at 37 o C. Purified hemoglobin has F equal to mmol O 2 /L, corresponding to a partial pressure of O 2 in the gas phase of 35.9 torr in M IHP, 0.05 M Bistris. ph 7 at 20 o C. Purified hemoglobin in M NaCl, 0.05 M BisTris, ph 7.0 at 20 o C has a much higher affinity for O 2, showing F = 0.5 at an O 2 concentration of 8.1 μmol/l corresponding to a partial pressure of O 2 in the gas phase of 4.7 torr. These results and the best fitting curve are illustrated in Figure (2.5). Again, the Adair equation with equivalent O 2 binding sites fails to describe the O 2 equilibrium binding curve. The divergence between observed and predicted values of F follows the same pattern illustrated above for HbA-IHP, whole blood and HbA-Pi: F PRE over-estimates F OBS at values of F below 0.57 and underestimates F OBS at values of F greater than 0.57.

15 F with O2, Observed and Predicted 1 Adair Equation with Equivalent Binding Sites M NaCl, 0.05 M BisTris, ph 7, 20 o C [O2], mmol/ L Figure 2.6. Comparison of observed ( ) and predicted ( ) values of fractional saturation of hemoglobin in M NaCl, 0.05 M BisTris, ph 7.0, 20.0 o C. Predicted values were obtained using the best-fitting value of K, L/mmol, and corresponding O 2 values in the Adair equation with equivalent O 2 binding sites, Eq. (2.7).

16 F: Predicted and Observed HbA, 0.05 M BisTris, ph 7.00, 20.0 o C M BisTris, ph 7.00, 20 deg C [O2], mmol / L Figure 2.7. Comparison of observed ( ) and predicted ( ) values of fractional saturation of hemoglobin in 0.05 M BisTris, ph 7.0, 20.0 o C. Predicted values were obtained using the best-fitting value of K, L/mmol together with corresponding O 2 values, in the Adair equation with equivalent O 2 binding sites, Eq. (2.7).

17 The supporting electrolyte comprised of M BisTris, ph 7.00 with HCl, 20.0 o C is observed to be E-free 4. At neutral values of ph, purified HbA demonstrates its highest affinity for O 2 in this electrolyte system. The best fitting value of K in Eq. (2.7) is L / mmol. Comparison of F PRE and F OBS, Figure (2.6), yields results similar to those described above: F PRE over-estimates F OBS at low values of F and underestimates F OBS at higher values of F. Eq. (2.7) fails to predict the O 2 equilibrium binding curve in the presence of M BisTris, ph 7.0, 20.0 o C. The characteristic pattern of misfit of observed and predicted value of F, observed with IHP, whole blood, KPi, and M NaCl is repeated in M BisTris. The O 2 equilibrium binding curves described cover, almost, the entire range of conditions available to HbA at neutral ph-values. The ratio of values for the concentration of O 2 at observed values of F = 0.5 for M IHP and M Bistris, mmol / L and 1.9 μmol / L, respectively stand in the ratio of It is possible to raise the affinity of HbA for O 2 if the ph value is lowered below 7 or the temperature is lowered below 20.0 o. Such information is available (unpublished results, F Knowles 5 ). More interesting, however, is the tabulated data for the CO equilibrium binding curve for whole blood. This can be fitted to Eq. (2.7) and the observed and predicted CO equilibrium binding curves compared. These results are presented below.

18 Whole Blood, ph 7.4, P (CO 2 ) 40 torr, 37 o C CO Equilibrium Binding Curve A non-linear least-squares curve fitting operation was carried out on the data of Roughton and coworkers 4 for the standard CO-equilibrium binding curve, the results being fitted to the Adair equation subjected to the constraint of equivalent CO binding sites, Eq. (2.7). The best-fitting value of K was 9,775 L/mmol CO. Equation (2.7) was not found to provide a satisfactory fit to CO equilibrium binding data. Roughton s data and the best-fitting data are presented in Figure (2.8). The results show a strikingly similar pattern to that obtained with O 2 and whole blood under standard conditions. At values of [CO] less than μmol / L, Eq. (2.7) over-estimates observed values of F. For values of [CO] greater than μmol / L, Eq. (2.7) under-estimates observed values of F. The ratio of values for the concentration of the sixth axial ligand, O 2 and CO at observed values of F = 0.5, mmol / L and μmol / L, respectively stand in the ratio of The use of Eq. 2.7 as a probe of the O 2 and CO equilibrium curves described above reveals a striking similarity in the pattern of residuals. While the mechanism underlying the equilibrium binding curves is not significantly revealed by the use of the Adair equation with equivalent binding sites, the series of experiments reveal a common underlying mechanism. In the presence of E-molecules, whatever happens to HbA as it binds oxygen parallels whatever happens to HbA as it binds CO.

19 F: Predicted and Observed 1 Whole Blood, ph 7.4, P(CO2) 40 torr [CO], μmol / L Figure (2.8). Comparison of observed ( ) and predicted ( ) values of fraction saturation of whole blood with O 2 under standard conditions. Predicted values were obtained using the best-fitting value of K, 9,775 L/mmol, in the Adair equation with equivalent O 2 binding sites, Eq. (2.7), and the concentration of CO corresponding to observed values of F.

20 Conclusion The results described above could be subjected to extensive analysis. Beyond that presented in the final paragraph of the discussion of results, concerning the universal commonality of well chosen equilibrium binding curves with O 2 and CO, such analysis can be deferred. The succeeding essays develop simple variations of the Adair equation wherein an ordered sequence of reactions is revealed based on: (i) the stereochemical mechanism revealed by crystal structures 5 ; (ii) chain heterogeneity; (iii) equivalent binding for two pairs of equivalent subunits; and (iv) the full development of the form of the Adair equation by inclusion of conformational equilibria not concerted with binding of a sixth axial ligand molecule.

21 References 1. F. Knowles (1985). Determination of Equlibrium Binding Constants for a Sequential Model of Dioxygen Binding by Hemoglobin-Inositol Hexaphoshate Complexes: The Structural Pathway from Deoxy- to Oxy-hemoglobin. Arch. Biochem. Biophys. 240, F.J.W. Roughton, E.C. DeLand, J.C. Kernohan, J.W. Severinghaus (1972). Recent Studies of the Oxyhaemoglobin Dissociation Curve of Human Blood under Physiological Conditions and the Fitting of the Adair Equation to the Standard Curve. In Oxygen Affinity of Haemoglobin and Red Cell Acid Base Status (edited by M RØrth and P Astrup), R.M. Winslow, M-L Swenberg, R. Berger, R.L. Shrager, M. Luzzana, M. Samaja, L Rossi-Bernardi (1977). Oxygen Equilibrium Curve of Normal Human Blood and its Evaluation by Adair s Equation. J. Biol. Chem. 252(7), F.J.W. Roughton (1970). Equilibrium of Carbon Monoxide with Human Haemoglobin in Whole Blood. Ann. N.Y. Acad. Sci. 174, Perutz, M. F., Femi, G., Luisi, B., Shaanan,,B., and Liddington, R. C. (1987). Stereochemistry of Cooperative Mechanisms in Hemoglobin, Accts Chem. Res. 20,