Biology Reference
In-Depth Information
Figure 7-3. It was soon observed, however, that the value of n required to
describe the actual experimental observations was approximately 2.5,
when the theoretical assumption implied that n should be an integer.
Hill rationalized the value of 2.5 as the statistical average of a mixture of
different-sized aggregates (i.e., a mixture of monomers, dimers, trimers,
tetramers, and so on).
In 1913, Hill noted that the limiting slope of the Hill equation is zero
when the O
2
concentration approaches zero and n
1. He showed that
the following mathematical fact follows from Eq. (7-9):
>
dY
lim
¼
0
;
n
>
1
;
k
>
0
(7-10)
d
½
O
2
½
O
2
!
0
(see Exercise 7-4 below). In 1913, accurate experimental measurement of
this limiting slope was impossible, and the limited experimental data
that existed indicated the limiting slope might be zero. Hill noted that if
the limiting slope was actually not equal to zero, then models of the
form of Eq. (7-9) were incorrect and should not be used. It is now known
the limiting slope of the hemoglobin-oxygen binding curves is,
in fact, not equal to zero. While this experimental observation
demonstrated the Hill Equation should not be used for the study of
binding phenomena, the Hill Equation is still commonly used because
it gives a reasonable approximation of binding behavior.
E
XERCISE
7-2
Use Eq. (7-8) and the law of mass action to derive Hill's model for the
fractional saturation given by Eq. (7-9).
E
XERCISE
7-3
Plot the fractional saturation Y from Eq. (7-9) as a function of the O
2
concentration using different values for the number of binding sites
n
1 and different values of the association constant k. Then answer the
following questions:
>
(a) If the valu
e
of k is kept fixed, what is the effect on the fractional
saturation Y when the number of binding sites n increases? Is this
to be expected in the context of the problem? Explain why or
why not.
(b) If the valu
e o
f n is kept fixed, what is the effect on the fractional
saturation Y when the association constant k increases? Is this
to be expected in the context of the problem? Explain why or
why not.
