Biomedical Engineering Reference
In-Depth Information
+
+
1
2
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Fig. 2.9
An interaction graph
x ω
1
(
x
)
ω
2
(
x
)
(0
,
0)
{
2
}∅
(0
,
1)
∅ ∅
(1
,
0)
{
2
}
1
}
(1
,
1)
∅
1
}
(2
,
0)
{
1
,
2
}{
1
}
(2
,
1)
a
b
+1
+2
1
2
0..2
0..1
-1
{
1
}
1
}
Fig. 2.10
Logical thresholds and resources. (
a
) The interaction graph of Fig.
2.9
together with the
bounds
b
1
=2
and
b
2
=1
, and the logical thresholds
t
11
=2
,
t
12
=1
and
t
21
=1
.(
b
) The table
gives the set or resources
ω
i
(
x
)
of
i
=1
,
2
according to the state
x
of the system
The
third step
consists in associating with every vertex
i
and every set of
regulators
Ω
⊆
G
i
a logical parameter
K
i,Ω
∈
X
i
, in such a way that: for all
i
, and for all subsets
Ω
and
Ω
Ω
K
i,Ω
(condition
C2). Intuitively,
K
i,Ω
is the level toward which
i
evolves (focal level) when
Ω
is
the set of resources of
i
. In other words, at state
x
,thelevelof
i
is: increasing if
x
i
<K
i,ω
i
(
x
)
;stableif
x
i
=
K
i,ω
i
(
x
)
; and decreasing if
x
i
>K
i,ω
i
(
x
)
. The signs
of the interactions of
G
are taken into account through the condition C2, which
states that the focal level of
i
increases (
K
i,Ω
≤
of
G
i
,if
Ω
⊆
then
K
i,Ω
≤
K
i,Ω
) when its resources increase
Ω
), that is, when there are more activators and less inhibitors (so that the
resources of
i
favor effectively the synthesis of the protein encoded by gene
i
).
More precisely, once logical parameters have been given, the behavior of the
system is described by a directed graph, called
asynchronous state graph
,and
defined by: the set of vertices is
X
; for every state
x
and every vertex
i
such that
x
i
=
K
i,ω
i
(
x
)
, there is an arc (or
transition
) from
x
to the state
x
(
Ω
⊆
defined by:
x
i
=
x
i
+1if
x
i
<K
i,ω
i
(
x
)
x
i
−
1 if
x
i
>K
i,ω
i
(
x
)
and
x
j
=
x
j
∀
j
=
i.
See Fig.
2.11
for an illustration.
If every variable is stable at state
x
(that is if
x
i
=
K
i,ω
i
(
x
)
for every
i
), then
x
has
no outgoing transition in the asynchronous state graph, and it corresponds to a
stable
state
of the system. More generally, the
attractors
of the system are the smallest
non-empty subsets of states
A
⊆
X
that we cannot leave, that is, such that for every
transition
x
→
y
of the state graph, if
x
∈
A
then
y
∈
A
.So
{
x
}
is an attractor if
and only if
x
is a stable state. Attractors that are not stable states (attractors of size
at least two) are called
cyclic attractors
, because once the system is inside such an
attractor, it cannot reach a stable state, and thus, it necessarily describes sustained
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