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Synthetic Gene Network
book

Synthetic Gene Network

by Bor-Sen Chen, Yu-Chao Wang
May 2014
Intermediate to advanced content levelIntermediate to advanced
216 pages
6h 47m
English
CRC Press
Content preview from Synthetic Gene Network
If V
()
(
)
00x =
, then (3.30) will be reduced to (3.2).
If the synthetic gene network is free of the external disturbances, i.e.,
0v =
, then we have
()
()
0
<0
T
ExQxdtVx
 
(3.31)
For some positive constant V
()
(
)
0x
, it means
0x
in probability as
t →∞
.
Appendix 3.2: Proof of proposition 3.2
Let us choose a Lyapunov function V
()
0x >
for the stochastic gene network
in (3.7). By the Ito formula (Chen and Hsu 1995), we get
() ()
()
()
()
()
()
()
()
()
()
2
2
1
2
2
1
1
2
1
2
T
d
m
TT
idi iid
i
T
d
m
TT
idi iid
i
dV x V x
EE Nfxxv
dt x
Vx
gxxM Mgxx
x
Vx
Nfx x
x
Vx
gxxM Mgxx
x
=
=
⎛⎞
=++
⎡⎤
⎜⎟
⎣⎦
⎝⎠
++ +
⎛⎞
+Δ+
⎜⎟
⎝⎠
+Δ+ Δ+



(3.32)
By the fact that
()
()
() ()
()()
() ()
()()
2
1
4
1
4
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Publisher Resources

ISBN: 9781466592698