Biomedical Engineering Reference
In-Depth Information
Signal
cAMP
·
CRP
CRP
activation
Cya
GyrAB
gyrAB
cya
P
Fis
P2 P1/P'1
P
fis
CRP
Supercoiling
crp
P1 P2
TopA
Stable RNAs
topA
P1
P1
P2
rrn
Legend
Fis
Synthesis of protein Fis
from gene
fis
Concise description
of interactions
P
fis
Activation
Inhibition
Fig. 2.7
Genetic network, including proteins and regulations that come into play during a
nutritional stress response in
E.coli
: CRP activation module (Cya, CRP, Fis), DNA topology
module (GyrAB, TopA, Fis), stable RNA output module (Rrn) (Adapted from [
34
])
2.2.4.2
The Carbon Starvation Response in
Escherichia coli
One of the successful applications of the PWA formalism is the study of the genetic
network that regulates nutritional stress response in
Escherichia Coli
. The model
was developed by Ropers et al. [
34
] to describe the dynamics of a family of genes
that regulate the carbon starvation response in
E.coli
(Fig.
2.7
):
crp
(
x
c
),
cya
(
x
y
),
fis
(
x
f
),
gyrAB
(
x
g
),
topA
(
x
t
), and
rrn
(
x
r
). Nutritional stress is represented by an
input
u
:
u
=0if carbon is present (no stress), and
u
=1in the absence of
carbon. The PWA equations are shown in Table
2.1
, and their mathematical study
can be found in [
24
].
For the case
u
=1, the asymptotic dynamics of the system in Table
2.1
satisfies:
∈{
0
,
1
}
κ
c
+
κ
c
+
κ
c
γ
c
>θ
c
>θ
c
;
x
y
(
t
)=
θ
y
x
c
(
t
)
→
(in finite time);
x
g
(
t
)=
θ
g
(in finite time);
,
x
f
(
t
)
→
0;
t
(
t
)
→
0
.
(2.24)
Therefore, solutions converge to an equilibrium point in the sense of Filippov. In
practice, there are sliding modes along the planes
x
g
=
θ
g
and
x
y
=
θ
y
.
For the case
u
=0, the asymptotic dynamics of the system in Table
2.1
can be
reduced to the equations on
x
g
and
x
f
with:
κ
y
+
κ
y
γ
y
κ
c
γ
c
1.
x
c
(
t
)
→
, after some finite time;
2. Sliding mode along the plane
x
t
=
θ
t
with the solution eventually jumping down
to the region
x
t
<θ
t
, and staying there;
and
x
y
(
t
)
→
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