Biology Reference
In-Depth Information
mathematical form also applies to ligands other than O
2
. The second and
third terms on the right of Eq. (7-31) are identical and represent one
ligand, X, on each of the two binding sites. The last term contains the
(K
i
[X])
2
term describing the binding of two ligands to two identical
sites and the K
c
cooperativity term. If K
c
is less than 1, the system
will exhibit negative cooperativity; if K
c
is greater than 1, the system will
exhibit positive cooperativity; and if K
c
is equal to 1, there is no
cooperativity. Eq. (7-32) presents the fractional saturation function for
this system:
2
1
2
N
1
2
2K
i
½
X
þ
2K
c
ð
K
i
½
X
Þ
Y
¼
¼
:
(7-32)
2
1
þ
2K
i
½
X
þ
K
c
ð
K
i
½
X
Þ
The 2 in the 2K
i
[X] terms is included because the first ligand can go
onto either of the two identical sites. This means that the average
macroscopic Adair-binding constant for the first ligand being bound
to either of the identical binding sites is equal to twice the intrinsic
binding affinities of the individual sites.
V. APPENDIX: JUSTIFYING EQUATION (7-20)
We now use some basic probabilistic arguments to justify Eq. (7-20) for
the case of hemoglobin tetramers. Recall that we linked Eq. (7-20)
with the total number of hemoglobin species present in the solution. For
Hb
4
, it follows from Eq. (7-11) that the following oxygenated states of
hemoglobin will be present: HbO
2
, Hb(O
2
)
2
, Hb(O
2
)
3
, and Hb(O
2
)
4
.
In addition, there will also be nonoxygenated Hb
4
.Ifp(i) denotes the
concentration of Hb(O
2
)
i
, i
¼
0,1,2,3,4, we will have:
½
Hb
4
ð
O
2
Þ
i
p
ð
i
Þ¼
O
2
Þ
4
:
½
Hb
4
þ½
Hb
4
ð
O
2
Þ þ ½
Hb
4
ð
O
2
Þ
2
þ½
Hb
4
ð
O
2
Þ
3
þ½
Hb
4
ð
Using Eq. (7-12), this can be written as
i
K
4i
½
Hb
4
ð
O
2
Þ
p
ð
i
Þ¼
:
2
3
4
½
Hb
4
þ
K
41
½
Hb
4
½ð
O
2
Þþ
K
42
½
Hb
4
½ð
O
2
Þ
þ
K
43
½
Hb
4
½ð
O
2
Þ
þ
K
44
½
Hb
4
½ð
O
2
Þ
Simplifying yields
i
K
4i
½ð
O
2
Þ
p
ð
i
Þ¼
(7-33)
2
3
4
1
þ
K
41
½ð
O
2
Þþ
K
42
½ð
O
2
Þ
þ
K
43
½ð
O
2
Þ
þ
K44
½ð
O
2
Þ
:
Using Eq. (7-33) gives the probabilities for i, i
0, 1, 2, 3, or 4, O
2
molecule
s
to be bound to a hemoglobin macromolecule; the average
number N of O
2
molecules bound by a macromolecule will be given by
¼
X
4
4
X
4
1
:
N
¼
ip
ð
i
Þ¼
iK
4i
½
O
2
i
2
3
1
þ
K
41
½ð
O
2
Þþ
K
42
½ð
O
2
Þ
þ
K
43
½ð
O
2
Þ
þ
K
44
½ð
O
2
Þ
i
¼
0
i
¼
0
(7-34)
