Biology Reference
In-Depth Information
fractional saturation curve shifts towards the higher affinity dimeric
species (see Figure 7-4). As a consequence of the dynamic linked
equilibrium between subunit association and O
2
binding, the fraction
of dimeric species approaches zero as the fractional saturation
approaches zero and monotonically increases with increasing fractional
saturation (i.e., oxygen concentration).
The solid lines in Figure 7-4 were calculated using a simultaneous least-
squares fit of Eq. (7-18) to the data. (The data-fitting procedures are
presented in the next chapter.) The D curve represents the dimer binding
curve (i.e., the limiting form at zero hemoglobin concentration), and
the T curve represents the tetramer binding curve (i.e., the limiting form
at an infinite hemoglobin concentration). The intermediate curves
correspond to the binding curve at intermediate hemoglobin
concentrations as predicted by Eqs. (7-18) and (7-19).
The observation that O
2
binding by hemoglobin is dependent upon
hemoglobin concentration might appear to be a serious complication
requiring a much more complex model for data analysis. However, it
also provides another means to probe the structure and function of
hemoglobin. The properties of hemoglobin can now be studied as
a function of O
2
concentration, or as a function of hemoglobin
concentration, or as a simultaneous function of both. Ackers and
coworkers studied the simultaneous function of both O
2
and hemoglobin
concentrations, giving us a better understanding of how hemoglobin
transports O
2
in our bodies.
IV. DERIVING FRACTIONAL SATURATION FUNCTIONS
WITH BINDING POLYNOMIALS
There are multiple ways to approach the derivation of the fractional
saturation equations. The easiest and the most generally applicable
approach is based on binding polynomials. A binding polynomial,
X
,
is simply a mathematical description of the sum of the concentrations
(i.e., probabilities) of each of the hemoglobin species in the solution. Given
the mathematical form of the binding polynomial and a little calculus,
it is easy to derive any desired fractional saturation function.
The binding polynomial approach for modeling cooperativity and
oxygen-binding problems is based on Eqs. 1 through 67 from Hill's book
(1960). The following equatio
n
relates the mean number of O
2
molecules
bound by a macromolecule, N, to the natural logarithm of the binding
polynomial, ln
X
, and the O
2
concentration, [O
2
]:
O
2
@
ln
O
2
¼
@
X
ln
X
N
¼½
O
2
:
(7-20)
@½
@
ln
½
Although Eq. (7-20) may not appear intuitive, its derivation is
elementary. To illustrate its importance, we shall use it to justify
