Biology Reference
In-Depth Information
ce
r
e
r
ce
2r
e
2r
ce
3r
e
3r
½
7
:
2
þ
2
½
9
:
6
þ
3
½
12
:
9
þ
½
:
ce
4r
e
4r
þ
½
:
ce
5r
e
5r
þ
½
:
ce
6r
e
6r
¼
:
4
17
1
5
23
2
6
31
4
0
Unlike the system of equations for the linear model from Eq. (8-4), this
system of equations is nonlinear, and there is no exact formula for the
solution such as given by Eq. (8-5). However, an approximation of the
solution (c*,r*) can be obtained through computational approaches
such as the Gauss-Newton algorithm.
Historically, the numerical challenges of solving nonlinear equations or
systems of equations, such as Eqs. (8-11) and (8-14), have been overcome
by transforming the experimental data to conform to a linear model.
For example, if we take natural logarithms of both sides of Eq. (8-12), we
obtain
ae
rt
e
rt
ln
ð
P
Þ¼
ln
ð
Þ¼
ln
ð
c
Þþ
ln
ð
Þ¼
ln
ð
c
Þþ
rt
:
This is a linear model of the form Y
¼
aX
þ
b with a
¼
r, b
¼
ln(c), X
¼
t,
and Y
ln(P), for which the minimum of the least-squares sum of the
residuals is easily found. Thus, in this case, transforming the data to the
form
¼
eliminates the technical difficulties arising
from the need to solve the nonlinear Eq. (8-14) for c and r. This successful
data transformation is caused by the specific exponential form of the
model in Eq. (8-12). For general models Y
ð
X
i
;
Y
i
Þ¼ð
t
i
;
ln
ð
P
i
ÞÞ
G (parameters; X), finding
a linearizing transformation may be difficult or impossible. In addition,
such transformations often lead to circumstances in which the statistical
validity of the least-squares procedure is violated. Thus, linearizing
transformations should generally be avoided because they often lead to
incorrect results, as our next example illustrates.
¼
II. A LIGAND-BINDING EXAMPLE
Consider the data shown in Figure 8-3. It could represent the effect of
a drug as a function of the drug concentration or an enzyme kinetic
response as a function of ligand concentration. These two examples
belong to a general category of biomedical investigations known
as ligand-binding experiments. The mathematics and numerical analysis of
these experiments are essentially identical and are discussed here.
Recall that vertical bars around the data point reflect the possibilities for
errors in measurement. In Figure 8-3, the vertical lines are centered on
the observed values and represent positive and negative deviations
equal in magnitude to the standard error of measurement SEM. The
estimated precision of each data point, SEM
i
, can be different, allowing
the data points to be known to variable precision. Different data points
