Page 126 - Thermodynamics of Biochemical Reactions

7.1 The Binding of Oxygen by Hemoglobin Tetramers 123

Substituting these equilibrium conditions in equation 7.1-1 yields
dG‘ = -S’dT+ VdP + p’(TotT)dn;(TotT) + p’(O,)dnb(O,)
+ RT In( 10)nc(H)dpH (7.1-1 1)

where p’ (TotT) is the transformed chemical potential of the T component and
nl(TotT) is the amount of the T component, namely

n’,(TotT) = d(T) + n’(T(0,)) + d(T(O,),) + n’(T(O,),) + n’(T(O,),) (7.1-12)
The amount nh(0,) of the molecular oxygen component is given by
nX02) = n’(O,> + n’(T(0,)) + 2n‘(T(O2),)
+ 3n’(T(OJJ + 4WIO2),) (7.1 -1 3)

Equation 7.1-12 shows that the natural variables for G‘ are 7; P. nL(T), n:(O,),
and pH, and so the criterion for spontaneous change and equilibrium is dC’ < 0
at constant 7; P, n;(TotT), nh(O,), and pH. The number of natural variables is
five, D’ = 5. The number of independent intensive properties is F’ = 4, and they
can be taken to be 7; P, pH, and [O,].
It is convenient to use the fundamental equation in matrix forin (see
Chapter 5), The stoichiometric number matrix v‘ for reactions 7.1-3 to 7.1-6 is
rx 7.1-3 rx 7.1-4 rx 7.1-5 rx 7.1-6
T -1 0 0 0
T(O2) 1 -1 0 0
vf = (7.1 - 1 4)
T(O,), 0 1 -1 0
T(OJ3 0 0 1 -1
T(O,), 0 0 0 1
0, -1 -1 -1 -1
Substituting this matrix in equation 5.5-3 yields
4
dG‘ = -S’dT+ VdP + 1 ArC:d(: + RTln(lO)n,(H)dpH (7.1-15)
I= 1
where

A,G: = ~’(T(02)~) ~’(T(02)~- - p’(Oz), i = 1, 2, 3, 4 (7.1-16)
-
1)
Since at equilibrium ArG: = 0, these four equations can be used to derive the
expressions for the apparent equilibrium constants K‘ for the four reactions that
are given in equations 7.1-3 to 7.1-6.
In the absence of experimental methods for distinguishing experimentally
between the five forms of the tetramer, the fractional saturation of hemoglobin is
measured. The fractional saturation of tetramer YT is defined by

Substituting the equilibrium expressions defined in equations 7.1-13 to 7.1-16
yields

Y, =
K&iC021 + 2Kk1Kk2C0212 + 3Kk1K&2Kk3[0213 + 4KklKk2K&3Kk4[0214
+
4(1 + Kk11021 + Kk1Kk,[0212 + G~&,~k3[0,13 ~k,~k,~&,~k,~0,1~)
(7.1 - 18)
This is often referred to as the Adair equation. A plot of the fractional saturation
for tetramer, which shows the cooperative effect, is given in Fig. 7.1.