Biology Reference
In-Depth Information
Hint: Use the fact that if f and g are solutions to the differential
equations f
0
¼
and g
0
¼
kf
þ
F
ð
t
Þ
kg
þ
G
ð
t
Þ
(with k
>
0 and initial
conditions f
ð
t
0
Þ¼
g
ð
t
0
Þ >
0) and F
ð
t
Þ
G
ð
t
Þ
0, we have f
ð
t
Þ
g
ð
t
Þ
for
all t
>
t
0
.
Exercise 10-9 establishes that the evolving solution of the system
ð
C
A
;
C
B
Þ
will approach the square
even for those
initial conditions that are outside of this square. On the other hand, if
there is no external input in the system (no infusion of A or B), then
C
A
<a
ð
0
C
A
a
=a;
0
C
B
b
=bÞ
after some time, and we get from Eq. (10-14) that the actual
endogenous peak concentration of B will never reach b
=a þ e
=b
. In particular,
with time its upper limit will approach
b
b
1
n
A
(10-18)
T
A
þ
1
a
=a þ e
which is less than b
. To get the above inequality, substitute the
estimate for C
A
in the term controlling the secretion in the second
Eq. (10-14), and estimate the maximal concentration of B following the
Hint in Exercise 10-9. We may work in a similar way to estimate the
concentration of A using the second inequality in Eq. (10-11). (How?)
Therefore, the solution of the unperturbed system
=b
ð
C
A
;
C
B
Þ
will be inside
the square
and the concentration of one
hormone stimulated by an infusion of the other hormone will remain
bounded in this square. (Why?) The latter justifies the previously used
term maximal attainable amplitude.
ð
0
C
A
a
=a;
0
C
B
b
=bÞ
All estimates may be further refined through a recurrent procedure
inherent in the core system (Eq. (10-14)). For example, one can combine
the two inequalities from Eq. (10-17) to get an explicit lower bound for
C
B
of
b
b
1
0
1
C
B
ð
t
Þ:
(10-19)
n
B
n
A
b
a
T
A
þ
1
@
A
b
T
B
þ
1
a
Accordingly, we can use this to write an explicit upper bound for C
A
:
a
a
1
a
a
1
b
b
1
C
A
T
B
n
B
1
T
B
n
B
1
;
where M
¼
:
0
n
B
1
n
A
C
B
;
min
þ
M
þ
b
a
T
A
þ
1
@
A
b
T
B
þ
1
a
The inequalities derived above can assist in determining reasonable
(initial) values for the model parameters. As we see next, they can also
be used in examining changes in sensitivity.
