Biology Reference
In-Depth Information
Here, we shall assume the distribution of experimental uncertainties
meets the basic assumptions of the least-squares fitting procedure.
Assume we attempt to extrapolate the values presented in Figure 8-3 to
zero and to an infinite concentration of the drug. The experimentally
measured value at zero in Figure 8-3 cannot simply be used as the zero
extrapolation because it contains experimental measurement error.
These two limits could then be used in a linear transform of the data,
such that the values range from 0 to 1, as in Eq. (8-48). The data would
then be in a form that could potentially be fit to one (or more) of the
fractional saturation functions above:
Original data-Zero limit
Infinite limit-Zero limit
Fractional data
¼
:
(8-48)
This approach is not optimal, because the extrapolated values of the data
at zero and at infinite drug concentrations both contain uncertainties.
They have not been determined to infinite precision and, as a
consequence, will introduce an unknown systematic uncertainty into the
transformed fractional data. A better approach is to perform the
inverse transform of the fitting equation, as in Eq. (8-49), and then fit the
original untransformed data to the transformed fitting equation.
Incorporating these limits in the fitting equations will introduce two
additional fitting parameters into the fitting process:
Transformed function
¼ð
Infinite Limit-Zero Limit
Þ
Y
þ
Zero Limit
:
(8-49)
For example, when Eq. (8-15) is modified by Eq. (8-49) and least-squares
fit to the data in Figure 8-3, there are three parameters estimated
simultaneously: the zero limit, the infinite limit, and the K
d
. The
resulting values of the zero and infinite drug concentration limits are
1.140 and 1.578, respectively, which are clearly not equal to the first and
last data point values. The reasons for this are that the data values
contain experimental uncertainties and the last data point is not at an
infinite concentration. The estimated value of the dissociation constant,
K
d
, is 1.073.
B. Cross-Correlation of the Estimated Parameters
Usually, there will appear to be a correlation between estimated
parameters. For example, when Eq. (8-7) is fit to a data set, the two
estimated parameters, K
21
and K
22
, will appear to be correlated (i.e., the
estimated value of K
21
is linearly dependent upon the value of K
22
and vice versa). This correlation is not caused by the molecular
mechanism of hemoglobin action but is, rather, a consequence of fitting
a complex equation to a small number of data points spanning
a limited range of oxygen concentrations. It is important to be aware of
