Biology Reference
In-Depth Information
To summarize, we use up- or down-regulatory Hill functions to write
the term controlling the secretion of A in the form:
S
A
ð
C
B
Þ¼
aF
up
;ð
down
Þ
ð
C
B
Þþ
S
A
;
basal
;
(10-8)
where S
A
;
basal
0 is independent of B and controls the basal secretion of
A. The quantities
represent C
A
;
max
and
C
A
;
min
, respectively. In case the basal secretion is negligible, it might be
ignored and considered zero. This is the assumption in our next
example, which illustrates how control functions can be used to express
the schematic diagrams describing the system in terms of coupled
ordinary differential equations.
ð
a
þ
S
A
;
basal
Þ=a
and S
A
;
basal
=a
Example 10-6
.........................
Assuming that S
A
;
basal
0, write a system of
differential equations describing the schematic hormone network in
Figure 10-13:
¼
0 and S
B
;
basal
¼
S
OLUTION
:
We begin with the basic differential equations describing the rate of
change of the concentrations C
A
and C
B
:
dC
A
dt
¼a
(
+
)
B
Elimination
S
A
and
dC
B
1
C
A
þ
dt
¼a
2
C
B
þ
S
B
;
D
(
−
)
where
0 are the elimination constants of hormones
A and B, and S
A
and S
B
are the respective control functions for the
secretion rates. Because Figure 10-13 indicates that the increase of the
concentration of A is inhibited by hormone B, we use a down-regulatory
Hill function with parameters T
1
and n
1
to express
a
1
>
0 and
a
2
>
A
Elimination
FIGURE 10-13.
Schematic hormone network for Example 10-6.
T
n
1
1
S
A
ð
C
B
Þ¼
a
1
F
down
ð
C
B
Þ¼
a
1
T
n
1
:
n
1
ð
C
B
Þ
þ
As the increase of the concentration of B is stimulated by hormone A (as
evident from the positive conduit indicated in Figure 10-13), we use an
up-regulatory Hill function to express
n
2
ð
C
A
Þ
S
B
ð
C
A
Þ¼
a
2
F
up
ð
C
A
Þ¼
T
n
2
:
n
2
ð
C
A
Þ
þ
We need to account for the presence of delay in the way hormone A
affects the secretion of hormone B. Because the delay D reflects the fact
that secretion at time t is affected by the hormone action in a past
moment, t
D, the control function S
B
can be expressed as S
B
ð
t
Þ¼
S
B
ð
. These considerations give the following system of
differential equations representing the diagram from Figure 10-13:
C
A
ð
t
D
ÞÞ
