Biology Reference
In-Depth Information
freely-available web application
Discrete Visualizer of Dynamics
(
DVD
) (available at
http://dvd.vbi.vt.edu
)
. More details about DVD are provided in Section
1.3.5
below.
Mathematically, the diagram in Figure
1.5
b depicts another directed graph. This
time, however, the directed graph represents the state-to-state transitions of the system.
The edges of the directed graph correspond to transitions between the states as time
evolves. Fixed points are recognized as the states that transition to themselves (with
edges pointing back to the same state in the diagram). The state space diagram in
Figure
1.5
b indicates that the three-variable system with wiring diagram given in
Figure
1.5
a and transition functions described by Eqs. (
1.2
) has two fixed points:
(
and a limit cycle of length two. It also shows that the state space
graph has three
disconnected components
: The first is {(0, 0, 0)} containing only the
fixed point
0
,
0
,
0
)
and
(
1
,
1
,
1
)
(
0
,
0
,
0
)
, the second is
{
(
0
,
0
,
1
), (
1
,
0
,
1
), (
0
,
1
,
1
), (
1
,
1
,
0
), (
1
,
1
,
1
)
}
,
and the third one is
. Any trajectories originating from points
in the second component will terminate at the fixed point
{
(
0
,
1
,
0
), (
1
,
0
,
0
)
}
. Any trajectory
originating from a point in the third component will oscillate, alternating between the
two states of the cycle.
Example 1.3.
Consider a Boolean network defined by the following set of transition
functions describing the interaction dynamics of the three variables
x
1
,
(
1
,
1
,
1
)
x
2
,
and
x
3
:
x
1
(
+
)
=
f
x
1
(
x
1
(
),
x
2
(
),
x
3
(
))
=
x
3
(
)
t
1
t
t
t
t
x
2
(
t
+
1
)
=
f
x
2
(
x
1
(
t
),
x
2
(
t
),
x
3
(
t
))
=
x
1
(
t
)
x
3
(
t
+
1
)
=
f
x
3
(
x
1
(
t
),
x
2
(
t
),
x
3
(
t
))
=
x
2
(
t
).
(1.3)
The wiring diagram for this system and its state space transition diagram are presented
in Figure
1.6
. The verification is left as an exercise (Exercise
1.7
). In addition to
the fixed points
(
0
,
0
,
0
)
and
(
1
,
1
,
1
)
, the state space diagram contains two cycles
of length three:
(
0
,
0
,
1
)
→
(
1
,
0
,
0
)
→
(
0
,
1
,
0
)
→
(
0
,
0
,
1
)
and
(
0
,
1
,
1
)
→
(
. We say that the state space diagram contains four
components, two fixed points, and two cycles of length three.
When the number of variables is small, it is common to use capital letters to
denote them instead of using
x
1
,
1
,
0
,
1
)
→
(
1
,
1
,
0
)
→
(
0
,
1
,
1
)
x
2
,...,
x
n
. With such notation, for the example in
x1
0 0 0
0 0 1
0 1 1
1 1 1
x2
1 0 0
1 0 1
x3
0 1 0
1 1 0
FIGURE 1.6
Wiring diagram and the state space transition diagram for the Boolean dynamical system
in Example
1.3
given by Eqs. (
1.3
). Graphs produced with DVD [
12
].
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