Biology Reference
In-Depth Information
Thus, when these transformations are applied to more complex models,
the resulting transformed data will not be a straight line. Figure 8-4
presents a Scatchard transformation of a ligand-binding model that has
two different, noninteracting binding sites: A low-affinity site and
a high-affinity site.
Clearly, the curve in Figure 8-4 cannot be analyzed as a straight line.
However, even if the data corresponded to a simple model where the
transformation yields a straight line, a simple linear fit of the
transformed data may be of questionable statistical validity. The
problem lies in the very nature of the least-squares procedure. Because
model parameters are estimated to minimize the sum of the squares of
the residuals, defined as the vertical distances between the data points
and the model, this assumes all of the uncertainties in the data can be
attributed to the y-axis. With the transformed data, however, this
is not always the case. For example, in Figure 8-4, the uncertainties at the
left side of the graph are in the y-axis, while on the right the
uncertainties are mostly in the x-axis.
So why were these linearizing transformations developed? Better
methods have been available for a long time but require a lot of
computer power. At the time the linearizing methods were developed,
computers were not available, and calculations had to be performed by
hand. The data transformations required in the past are no longer
needed or desired. Biology is not linear, and our methods of analysis
should not be linear either.
1.5
1.0
0.5
5
10
15
20
Y
FIGURE 8-4.
A typical Scatchard plot of data containing both low- and high-affinity binding sites. The precision of
the individual data points is represented by lines (error bars) that radiate from the origin. These
error bars were generated by assuming that all of the uncertainties within the data are in the
measured fractional saturation. When expressed as fractional saturation versus free concentration,
all of these error bars are vertical. However, with the Scatchard transformation, both axes contain
errors because the Scatchard transformation includes the fractional saturation on both the
y-axis and the x-axis.
(Adapted from Johnson, M. L. and Frasier, S. G. [1985]. Nonlinear least-squares analysis. Methods in
Enzymology, 117, 301-342, with permission from Elsevier.)
