Page 133 - Thermodynamics of Biochemical Reactions

130 Chapter 7 Thermodynamics of the Binding of Ligands by Proteins

quadratic formula shows that the equilibrium concentration of TotD is given by
-2 + (4 + 16K”[heme])”2
[TotD] = ~ (7.4-4)
8K“

The fraction of heme in the dimer is given by
2[To t D] - 1 (7.4-5)
’’ 2[TotD] + 4[TotT] - 1 + 2K”[TotD]
=
Substituting equation 7.4-4 into equation 7.4-5 yields
2
.fD = 1 + (1 + 4K”[heme])i71 (7.4-6)
Substituting this equation into equation 7.4-2 yields

Y= Y, + 2(YD - Y,) (7.4-7)
1 + (I + 4K”[heme])’

At specified [O,], Y is a function only of [heme] in a way that is described by
three parameters, Y,, Y,, and K“. This equation is the same as that derived by
Ackers and Halvorson (1974), although it has a rather different form.
If it were possible to titrate hemoglobin with oxygen at sufficiently high
[heme], Y, could be obtained directly. However, for the values of the seven
equilibrium constants obtained by Mills. Johnson, and Ackers (1976), the tet-
ramer is partially dissociated at the highest practical heme concentrations of
about 5 mM. Equation 7.4-7 indicates that if Y can be determined at several high
[heme], a linear extrapolation is possible (see equation 7.4-12). As is evident from
equation 7.4-7, the question as to whether [heme] is high or low depends on
whether [heme] > bK” or [heme] < $K“. Of course, this criterion depends on
[O,]. In considering plots of Y versus some function of [heme], [heme] = +K“
can be used to divide the dependence of Y on [heme] into high-heme and
low-heme regions. If 4K”[heme] >> 1, equation 7.4-7 reduces to

Y= Y, + - ‘T) (7.4-8)
(K”)1’2[heme]1

Thus a plot of Y versus [heme]-”2 at a specified [O,] must approach linearity
as [heme] is increased. The intercept of the limiting slope of a plot of Y versus
at [hemel- ‘I2 = 0 is Y,, and the limiting slope is (Y, - Y,)/(K”)1i2.
This slope is determined by two factors, (Y, - YT) and K ‘I. The slope will be low
at high [O,] because Y, - YT is small. The slope will be low at low [O,] because
K“ is large. Once Y, has been determined at a series of [O,] by use of
extrapolations of this type, Kk,, K&,, Kk3, and Kk4 can be calculated by the
method of nonlinear least squares.
The values of Y, at various [O,] can be determined by extrapolations at low
[heme]. If 4K“[heme] << 1, the square root term in equation 7.4-7 can be
rewritten using
x
(1 + x)lr2 % 1 + 7 (x << 1) (7.4-9)

to obtain

Y= Y, + (‘D - ‘T) (7.4- 10)
1 + K”[heme]

Since 4K”[heme] << 1 is satisfied, we can use

(7.4-1 1)