Chemistry Reference
In-Depth Information
described as steady-state models, because the effect
approaches a steady state with increasing dose.
Curves 7A and 7B have three factors in common; they
both originate from the origin (i.e., the zero dose-zero
effect point) they both approach a maximum effect of
four units asymptotically, and they both contain a single
constant,
K
, with a value of 0.5. The difference between
the two curves is obvious from their mathematical
expressions. The derivation of the mathematical expres-
sions is discussed in Section 3 dealing with possible bio-
logical bases for the relationships. It is of interest that
curve 7A can be constructed as a straight line by plot-
ting 1/D against 1/E. This is because E = E
max
[D/(D +
K)] may be rewritten as 1/E = l/E
max
+ [K/E
max
][1/D].
This plot allows the experimental determination of
E
max
from the y-intercept and subsequently also the value for
K
from the slope. This expression and procedure is iden-
tical to the Lineweaver-Burk plot used in the study of
enzyme-substrate kinetics. Curve 7B may also be plot-
ted as a straight line by plotting dose on a linear scale
against the value of E
max
/[E
max
− E] on a logarithmic
scale. However, to do this,
E
max
must be known, and the
value is often diffi cult to determine experimentally.
An additional important feature of curves 7A and
7B is that if dose is plotted on a logarithmic scale, the
curves become sigmoid in shape, being asymptotic to
zero effect and to the maximum effect. Additionally,
the effect axis may be presented as a percent value.
Thus, in Figure 7, effect values of 0, 1, 2, 3, and 4 would
become 0, 25, 50, 75, and 100% effect values.
4
6A
3
2
1
0
0
1
2
3
4
Dose (D)
FIGURE 6
Example of a power-function model when 0<n<1.
E = KD
n
.
point cannot be plotted on log-log paper but neverthe-
less exists as a condition of the model. When = 0.5 as in
curve 6A, the expression becomes E = K(D)
0.5
.
Both the logarithmic and the power-function model
share the feature that as the dose becomes infi nitely
large, the effect also becomes infi nitely large. Many, if
not all, biological systems have some maximum effect
that they are capable of demonstrating and the maxi-
mum response in any population can be no more than
100%. Figure 7 illustrates two curvilinear relationships
in which a maximum effect is approached asymptoti-
cally with increasing dose. These relationships may be
2.1 The Shape of Dose-Response Curves:
S, Hormesis, U-Shaped
When successive equal increments of dose cause
greater and greater increment in effect, the curvilin-
ear relationship swings upward instead of downward.
The exponential model and the power-function model
are examples of such relationships.
The exponential model is illustrated in Figure 8. As
with the logarithmic curves in Figure 5, the value for
b must be greater than 1. At zero dose, the effect will
always have a value of 1 for the simple exponential
model described by E = b
a(D)
or log
b
E = a(D) as seen
with curve 8A. Curve 8A, but not curve 8B, will be a
straight line when plotted on semilogarithmic graph
paper with effect on the log scale.
The power-function model represented in Figure 9
requires that the value of
n
be greater than 1, whereas
the previous power-function model illustrated in
Figure 6 requires that the value of
n
be less than 1
but greater than 0. Curve 9, as well as curve 6A, will
become a straight line when the data for dose and effect
are plotted on log-log scale.
4
7A
7B
3
2
1
0
0
1
2
3
4
Dose (D)
FIGURE 7
Examples of two equilibrium models. Curve 7A,
E = E
max
[D/(D + K)]. Curve 7B, E = E
max
[1−exp(−KD)].
