## Competition for sites CO Vs O on hemoglobin

Binding of oxygen in the blood to hemoglobin is affected by a number of molecules that compete with O2 for the heme sites. In particular, carbon monoxide binding to the heme sites is one to two orders of magnitude greater than oxygen binding. The following analysis of a hypothetical two-site hemoglobin with both O2 and CO dissolved in the blood is given below.

1 If the liquid (blood) is in saturated with O2 in air while the binding equilibria are attained, L, not Lo is constant.

However, if the liquid is initially saturated with O2 and subsequently equilibrates with the fixed initial O2

concentration, then [Lo] is constant. The latter case is assumed here

02 partial pressure, mmHg 0 5 10 15 20

Fig. 11.15 Adair plot for hemoglobin and myoglobin. The single equilibrium constant for myoglobin is the first-site equibrium constant for hemoglobin. Conversion from oxygen partial pressure to concentration uses the Henry's law constant given in Table 8.1

O, concentration,

Fig. 11.15 Adair plot for hemoglobin and myoglobin. The single equilibrium constant for myoglobin is the first-site equibrium constant for hemoglobin. Conversion from oxygen partial pressure to concentration uses the Henry's law constant given in Table 8.1

To simplify the notation, hemoglobin is designated as Hb, O2 by L and CO by M. The bound-ligand dissociation reactions, their mass-action laws and their equilibrium constants are:

Hb-L = Hb + L Hb-L2 = Hb-L + L Hb-M = Hb + M HbM = Hb-M + M Hb-ML = Hb-M + L Hb-ML = Hb-L + M

Kil = [Hb][L]/[Hb-L] = 25 mM K2L = [Hb-L][L]/[Hb-L2] = 15 mM Kim = [Hb][M]/[Hb-M] = 0.25 mM K2M = [Hb-M][M]/[Hb-M2] = 0.15 mM Kml = [Hb-M] [L]/Hb-ML] = 1.5 mM Klm = [Hb-L][M]/Hb-ML] = ... mM

Except for Kil and the two-order-of-magnitude factor between it and Kim, these equilibrium constants have been chosen arbitrarily. The only restriction was that the dissociation constants of the di-liganded hemoglobin be smaller than the mono-ligand value in order to represent cooperative binding. The mixed di-ligand equilibrium constant, Klm, was assumed to be the geometric mean of K2M and K2L. As shown below the Klm is related to the preceding equilibrium constants.

Possible combinations of bound ligands are shown below. The sites on the hemoglobin molecule are drawn as vertical line segments.

Conservation equations for hemoglobin and the ligands O2 and CO are: [Hbtot] = [Hb] + [Hb.L] + [Hb.M] + [Hb-L2] + [Hb.M2] + [Hb.ML] [Lo] = [L] + [Hb.L] + 2[Hb-L2] + [Hb.ML] [Mo] = [M] + [Hb.M] + 2[Hb-M2] + [Hb.ML]

To start, the bound-ligand concentrations are expressed in terms of the free ligand concentrations: [Hb.L] = K-L [Hb][L]

[Hb.L2] = K -L [Hb.L][L] = K-L K -L [Hb][L]2 [Hb.M] = K-M [Hb][M]

[Hb.M2] = K -M [Hb.M][M] = K-M K - M [Hb][M]2 [Hb.LM] = K mMl K-M [Hb][L][M] = K lM K-L [Hb][L][M]

The last of these equations shows that KLMK1L = KMLK1M, so that KLM is not an independent equilibrium constant.

Substituting the above equations into the conservation equations reduces the problem from one with nine unknowns to one with three unknowns:

[Hbtot] = [Hb] + K-L [Hb][L] + K-J [Hb][M] + K-L K -L [Hb][L]2 + K-M K - M [Hb][M]2 + K mMl K-M [Hb][L][M] (11.36a)

[Lo] = [L] + K-L [Hb][L] + 2 K-L K - L [Hb][L]2 + K ML K-J [Hb][L][M] (11.36b)

[Mo] = [M] + K-M [Hb][M] +2 K-jM K -M [Hb][M]2 + K M\, K-M [Hb][L][M] (11.36c)

Solution method To further simplify the calculation, the above three equations are expressed in terms of the following dimensionless quantities:

A = K-L [Lo]; B = K-M [Mo]; C = K-L K- L [L0]2; D = K-M K -M [M0]2; E = K MlK-M [L o ][M o ] ; X = [L]/[Lo]; Y = [M]/[Mo]; V = [Hb^/PM,]; W = [Hbtot]/[Lo]

The first step in the solution is to factor [Hb] from the terms on the right-hand side of the Eq (11.36a) and use the result to eliminate this variable from Eqs (11.36b) and (11.36c). This results in two equations with two unknowns, X and Y:

These two equations are solved by a Monte Carlo search: X and Y are guessed as random numbers between 0 and 1 (which they must be) and substituted into the above equations. When the right-hand sides of these equations are simultaneously between 1 and 1.001, the solution has been obtained

A modicum of reality is injected into the calculation using realistic values of [Hbtot], [Lo] and [Mo]. The concentration of hemoglobin in human blood is ~ 15 g/dl, which, with a molecular weight of 68,000, corresponds to [Hbtot] = 2200 |mM (micromoles per liter). The concentrations of O2 and CO are taken to be those in water determined by the partial pressures of these gases (in atm) and their Henry's law constants. The latter are obtained from Table 8.1:

The calculation assumes that these concentrations are established by equilibration of the liquid (blood) with air with a variable partial pressure of carbon monoxide. Equilibrium with hemoglobin takes place out of contact with air, so that [Lo] and [Mo] are the fixed total concentrations of the ligands, bound and free.

The results are displayed in the two graphs in Fig. 11.16. The top graph shows the fraction of the sites on the hypothetical hemoglobin molecule (two sites per molecule) that are occupied by each of the ligands. These are given by:

The horizontal line in this graph shows that oxygen binds to Hb as if CO were absent, irrespective of the CO partial pressure. The constant value of fQ = 0.068 means that X = [L]/[Lo] << 1, a result shown in more detail in the O2 curve in the bottom graph.

CO binding does not start in earnest until its partial pressure is ~ 0.2 atm, where the concentration in solution is ~200 mM (top). Thereafter, binding increases until at 4000 mM (4 atm partial pressure) ~ 95% of the total sites on hemoglobin are occupied by CO molecules. At any CO partial pressure, less than 0.1% of the available CO remains in solution (bottom).

CO partial pressure, atm

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