Biology Reference
In-Depth Information
dC
A
dt
¼a
1
C
A
ð
t
Þþ
a
n
B
ð
C
B
ð
t
D
B
Þ=
T
B
Þ
þ
1
n
A
;
1
n
P
dC
B
dt
¼b
ð
C
A
ð
t
Þ=
T
A
;
1
Þ
ð
P
B
ð
t
Þ=
T
P
Þ
C
B
ð
t
Þþ
b
n
A
;
1
n
C
n
P
ð
C
A
ð
t
Þ=
T
A
;
1
Þ
þð
C
C
ð
t
Þ=
T
C
Þ
þ
1
ð
P
B
ð
t
Þ=
T
P
Þ
þ
1
n
A
;
2
dP
B
dt
¼
ð
C
A
ð
t
Þ=
T
A
;
2
Þ
c
ð
P
max
P
B
Þ
n
A
;
2
ð
C
A
ð
t
Þ=
T
A
;
2
Þ
þ
1
n
A
;
1
n
P
ð
C
A
ð
t
Þ=
T
A
;
1
Þ
ð
P
B
ð
t
Þ=
T
P
Þ
b
y
1
:
n
A
;
1
n
C
n
P
ð
C
A
ð
t
Þ=
T
A
;
1
Þ
þð
C
C
ð
t
Þ=
T
C
Þ
þ
1
ð
P
B
ð
t
Þ=
T
P
Þ
þ
(10-23)
Here, for simplicity, we assumed that circulating levels of B do not
feedback on the secretion. This corresponds to a model with a much
higher concentration in the storage pool than in the circulation. The
parameter c controls the rate of A-stimulated synthesis of B. The
parameter
represents the ratio between the volumes of the storage
pool and the pool into which B is secreted. Typically, the second pool is
larger and
y
1. It is assumed that the control functions that correspond
to the A-driven synthesis and release are different with distinct
thresholds T
A
;
1
and T
A
;
2
and corresponding Hill coefficients n
A
;
1
and
n
A
;
2
. The control, exerted on the secretion by the current concentrations
of B in the storage pool, is presented by the up-regulatory function
ð
y >
n
P
ð
Þ=
Þ
P
B
t
T
P
1
. The following values were assigned to the parameters
n
P
ð
P
B
ð
t
Þ=
T
P
Þ
þ
in Eq. (10-23):
a ¼
1
;
b ¼
2
;
y ¼
6
a
¼
4
b
¼
4000
c
¼
2
P
max
¼
900
;
;
;
;
;
D
B
¼
2
;
T
A
;
1
¼
4
;
T
A
;
2
¼
3
;
T
B
¼
40
;
T
C
¼
10
;
T
P
¼
500
;
¼
¼
¼
¼
¼
:
n
A
;
1
2
;
n
A
;
2
2
;
n
B
3
;
n
C
2
;
n
P
2
The infusion of C was approximated with a separate model equation
(see Section IV, Part C.3) with an elimination constant equal to 5 and an
infusion rate assumed to be a nonzero constant only during the time of
infusion. The infusion is determined by
8
<
0if
t<60
inf
ð
t
Þ¼
3000
if
60 <t<70
:
:
0if
t
>
70
The model output is shown in Figure 10-25.
FIGURE 10-25.
Simulated rebound response following withdrawal of
continuous C-infusion (timeline 60-70). Legend: A:
black; B: dotted; B in the storage pool: dashed.
The plot depicts a reboundlike increase in the secretion of B following
withdrawal of the inhibitor C. From a mechanistic point of view,
