Biomedical Engineering Reference
In-Depth Information
where
κ
il
>
0 is a rate parameter,
b
il
:
R
n
+
→{
0
,
1
}
is a boolean-valued regulation
function, and
I
is an index set. The regulation functions
b
il
capture the conditions
under which the protein encoded by gene
i
is synthesized at a rate
κ
il
.These
conditions are written down as combinations (sums of products) of step functions
s
+
,s
−
:
R
+
×
R
+
→{
0
,
1
}
,where
s
+
(
x
j
,θ
j
)=1if
x
j
>θ
j
,and
s
+
(
x
j
,θ
j
)=0
if
x
j
<θ
j
,and
s
−
(
x
j
,θ
j
)=1
−
s
+
(
x
j
,θ
j
). The parameters
θ
j
are threshold
concentrations.
This class of PWA systems was first introduced by Glass and Kauffman [
22
],
and is widely used for modeling genetic regulatory networks [
9
,
17
,
22
,
34
]. Step
functions are not defined at threshold points, but solutions of the system “across” or
“along” a threshold can still be defined in the sense of Filippov, as the solutions of
differential inclusions, as shown in Sect.
2.2.2.4
and Fig.
2.3
.
In the PWA formalism, the bistable system in Eq. (
2.17
) is defined inside the
(invariant) set
Ω
=[0
,κ
1
/γ
1
]
×
[0
,κ
2
/γ
2
]. Assuming for the sake of simplicity that
κ
10
=
κ
20
=0, one gets the equations:
x
1
=
κ
1
s
−
(
x
2
,θ
2
)
−
γ
1
x
1
,
(2.22)
=
κ
2
s
−
(
x
1
,θ
1
)
−
x
2
γ
2
x
2
.
The space of state variables
Ω
is now divided into four boxes, or
regular domains
,
where the vector field is uniquely defined:
∈
R
2
+
:0
<x
1
<θ
1
,
0
<x
2
<θ
2
}
B
00
=
{
x
∈
R
2
+
:0
<x
1
<θ
1
,θ
2
<x
2
<κ
2
/γ
2
}
B
01
=
{
x
∈
R
2
+
:
θ
1
<x
1
<κ
1
/γ
1
,
0
<x
2
<θ
2
}
B
10
=
{
x
∈
R
2
+
:
θ
1
<x
1
<κ
1
/γ
1
,θ
2
<x
2
<κ
2
/γ
2
}
B
11
=
{
x
.
In addition, there are also
switching domains
, where the system is defined only as a
differential inclusion, corresponding to the segments where each of the variables
is at a threshold (
x
i
=
θ
i
and
x
j
∈
[0
,κ
j
/γ
j
]). In each of the four regular
domains, the differential system is affine, and simple to study. In
B
00
for instance
x
1
=
κ
1
−
γ
2
x
2
,
and the solution can easily be written and
converges exponentially towards a steady state (
κ
1
/γ
1
,κ
2
/γ
2
). If we suppose that
θ
i
<
κ
i
γ
i
γ
1
x
1
, x
2
=
κ
2
−
, then this steady state is
outside
B
00
, and the solution will switch to another
system when it crosses one of the thresholds. This succession of possible transitions
will result in a
transition graph
, describing the possible sequences of boxes.
For the bistable switch, there are two classical stable steady states,
P
1
and
P
2
,
and an unstable Filippov equilibrium point,
P
3
, analogous to a saddle point (see
Fig.
2.6
):
P
1
=
κ
1
γ
1
,
0
, P
2
=
0
,
κ
2
γ
2
, P
3
=(
θ
1
,θ
2
)
.
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