Biology Reference
In-Depth Information
Because B inhibits the secretion of A, we used a down-regulatory Hill
function in the first equation. This equation also accounts for the delay
(via the parameter D B ) in the B
A system interaction depicted in
the schematic diagram. In contrast, because A stimulates the secretion
of B, an up-regulatory Hill function is used to represent the control
function in the second Eq. (10-14).
!
We should note that, because of the presence of delay, solving these
equations for t
t 0 requires preset initial conditions for C B on the entire
interval
½
t 0
D B ;
t 0
.
E XERCISE 10-7
Give the system of ODEs corresponding to the schematic diagram of the
right panel in Figure 10-14.
Attractor
A. Limit Cycles and Steady States
We now demonstrate that a nonzero delay and large nonlinearity in the
control functions (sufficiently high Hill coefficients) can guarantee
steady periodic behavior, because of the existence of a nontrivial limit
cycle.
Repellor
Limit cycle
As the next exercise shows, the system defined by Eq. (10-14) has a
unique equilibrium point (steady state). When there is no delay (that is,
when D B ¼
FIGURE 10-15.
Illustrative trajectories in the phase space
(C A , C B ) if the steady state is an attractor
(top) or a repellor (bottom). In the latter
case, a unique asymptotically stable
periodic solution acts as limit cycle and
attracts all system trajectories (except for
the fixed point). (Reprinted from Farhy, L.
S. [2004]. Modeling of oscillations in
endocrine networks with feedback,
Methods in Enzymology, 384, 54-81.
Copyright 2004, with permission from
Elseveri.)
0), this equilibrium point is asymptotically stable and
attracts all trajectories in the phase space (see Figure 10-15, left panel).
E XERCISE 10-8
For the model of Eq. (10-14) show that:
(a) The system has a unique equilibrium state.
(b) When D B
¼
0, the equilibrium state of Eq. (10-14) is asymptotically
stable.
Hint: Apply the theory presented in Chapter 2 by completing the
following steps.
1. Represent the system defined by Eq. (8) in the form
x 0 ¼a
þ
ð
Þ
x
F
y
y 0 ¼b
y
þ
G
ð
x
Þ;
where x = x(t) and y = y(t),
0, and F and G are monotonic
decreasing and increasing, respectively.
a; b;
F
;
G
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