Biology Reference
In-Depth Information
FIGURE 10-12.
Effect of manipulating the model. Left panel: Changes in the response curve caused by varying the
potency and/or efficacy of the interaction. Solid line: control; upper dashed: decrease in potency of
the regulating hormone (threefold increase in ED
50
, that is, of the parameter T ); dotted: twofold
decrease in efficacy (that is, twofold increase in the value of a); lower dashed: combined twofold
decrease in efficacy (that is, of the parameter a) and threefold increase in ED
50
(that is, of the
parameter T ). Right panel: Alterations in the control response curve (black) associated with 10-fold
increase (dashed) or twofold decrease (dotted) of the Hill coefficient n. Remark: Note the difference
in the time scales in the left and right panels; the control function is the same in both plots.
(the parameter a) change the maximal value of the control function. The
Hill coefficient n controls the slope of the control function. As n increases
(values as large as 100 exist in biology; see Vrzheshch et al. [1994];
Mikawa et al. [1998]), the slope of the control function also increases as
illustrated in Figure 10-12 (right panel). For large n, the control function
acts as an on/off switch at the concentration value C
T. Plots
similar to those in Figure 10-12 (left panel) are often seen in textbooks to
illustrate the anticipated effect on percent of maximal response
caused by decreased responsiveness and/or sensitivity.
¼
The parameter a in Eq. (10-7) depends upon, and is determined from, the
maximal possible attainable concentration of hormone A, C
A
;
max
. The
latter is the maximal physiologically possible endogenous concentration
of A under a variety of conditions, including extremes such as responses
to external high pharmacological stimulations. This maximal value may
be known from experiments or hypothesized in case of mathematical
simulations, in which case it could be considered a parameter of the
model.
E
XERCISE
10-6
Show that if a is the control coefficient from Eq. (10-7), then the
quantities
ð
a
þ
S
A
;
basal
Þ=a
and S
A
;
basal
=a
represent C
A
;
max
and C
A
;
min
,
respectively. Then, show that a
¼ a
C
A
;
max
S
A
;
basal
¼ að
C
A
;
max
C
A
;
min
Þ:
Hint: Use Eq. (10-1), the fact that the maximal and minimal
concentrations of A are achieved when dC
0, and the fact that
F
up
ð
down
Þ
has maximal value 1 and minimal value 0.
=
dt
¼
