Biology Reference
In-Depth Information
output depicted in Figure 10-20 uses this system of differential equations
to simulate exogenous infusion with the same assumptions used
earlier for the simulations shown in Figure 10-17. Note that in
Figure 10-17 we exploited Eq. (10-20), where the endogenous and the
exogenous component were not separated. As expected, the model
output for hormone A remains unchanged, and the sum of endogeneous
and exogeneous B in Figure 10-20 equates to the profile shown in black
in Figure 10-17, illustrating the equivalence of the two approaches.
Another example is shown below in Section IV, Part E.
Changes in the profiles of the control functions can be used to model
alterations in system sensitivity. The analysis shows that if a model has
stable periodic behavior, the increase in one of the Hill coefficients
would not change the system performance (see, for example, Glass and
Kauffman [1973]). On the other hand, a decrease in the same parameter
may transform the steady state from a repellor into an attractor and
affect the periodic behavior. Changes in the action thresholds may also
affect the periodicity. Exogenous hormone delivery can be simulated
by a simple increase in the basal secretion, or by introducing a third
node, if we would like to distinguish between the exogenous and
endogenous components of one and the same substance, as we did in
Eq. (10-21).
FIGURE 10-20.
Simulated bolus infusion (at t ¼ 96:00) of the
system hormone B (dashed line) in the model
outlined in Eq. (10-15). The exogenous
hormone is shown with a dotted line, whereas
hormone A is plotted in black.
D. Oscillations Generated by a Perturbation
In the reference models in the previous section, the pulsatility was
generated by a system having a unique periodic solution and a unique
fixed repelling point. In this section, we demonstrate how oscillations
appear as a result of disrupting a system that does not have a periodic
solution and its fixed point is an asymptotically stable focus
(Figure 10-15, left panel). We illustrate this concept with the following
model:
dC
A
dt
¼
1
ð
Þþ
3C
A
t
60
2
½
C
B
ð
t
0
:
25
Þ=
5
þ
1
(10-22)
2
dC
B
dt
¼
½
C
A
ð
t
Þ=
4
3C
B
ð
t
Þþ
40
1
:
2
½
C
A
ð
t
Þ=
4
þ
In this example, formalizing again the network in Figure 10-14 (left
panel), the parameters are chosen so that there is no periodic solution [in
contrast, for example, with Eq. (10-15)] and the unique fixed point
attracts all trajectories in the phase space. Therefore, this system by itself
cannot generate stable oscillations. However, if it is externally
stimulated, it can be removed from its steady state, and oscillations will
be generated. We can show this by simulating two brief (10-minute),
unequal suppressions of the secretion of B at t
¼
94 and t
¼
104
