Biology Reference
In-Depth Information
2
2
be a PDS with transition functions
Example 3.2.
Let
F = Z
3
and
F
: F
→ F
f
1
=
f
1
(
x
1
,
x
2
)
=
2
x
2
,
x
1
.
f
2
=
f
2
(
x
1
,
x
2
)
=
x
1
+
The dimension of
F
is 2, where each of the two variables
x
1
and
x
2
can be in one of
three states, i.e., 0, 1, or 2.
A key issue in the study of GRNs is understanding the relationships among
genes, proteins, metabolites, etc. in the network. Thus an important characteristic
of a PDS is the way its variables are “connected." To visualize the network topol-
ogy, a graph called a wiring diagram is used, where the vertices are labeled by the
PDS variables and the directed edges signify the direction of interaction between two
variables.
Definition 3.2.
Let
F
be an
n
-dimensional PDS on variables
x
1
,...,
x
n
.The
wiring
diagram
(also known as a
dependency graph
)of
F
is a directed graph with vertex set
V
. A directed edge is
d
rawn from
x
i
to
x
j
if, and only if,
x
i
is present
in at least one polynomial term (with a nonzero coefficient) of the local function
f
j
.
Figure
3.1
(a) shows the wiring diagram of the PDS from Example
3.2
.
While the wiring diagram of a PDS provides only a
static
snapshot of the network
topology with no reference to strength and timing of the interactions, such a graph
carries very useful information about the biological system and several methods
have been developed for its inference. Sections
3.5.1
and
3.5.2
present two such
methods.
Just as a phase portrait is used to plot the simultaneous change of two or three
variables in a system of differential equations [
13
], the
dynamic
properties of a PDS
are captured in a graph called a
state space graph
. In fact, the dynamics of a PDS
F
={
x
1
,...,
x
n
}
n
n
is uniquely represented by its state space graph, which has
n
vertices.
: F
→ F
|F|
n
n
Definition 3.3.
The
state space graph
of a PDS
F
: F
→ F
on variables
n
and contains an edge directed
x
1
,...,
x
n
is a directed graph whose vertex set is
F
X
n
.
from
u
to
v
if, and only if,
F
(
u
)
=
v
, where
u
,
v
∈
0 1
1 2
2 1
1 1
x
2
2 0
0 2
2 2
x
1
0 0
1 0
(a)
(b)
FIGURE 3.1
Wiring diagram (a) and state space graph (b) of the PDS from Example
3.2
.
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