
14 Synthetic Gene Network: Modeling, Analysis, and Robust Design Methods
fl uctuations. The covariance of stochastic intrinsic parameter fl uctuation
&
M
a
n
a
is given as
(
)
(
)
(
)
Cov ,
aa aa
nt nkktDD
2
att
kd=D
, where
F
t
V
denotes the
delta function, i.e.,
F
t
V
= 1 if t =
V
and
F
t
V
= 0 if t
V
, i.e., &
M
i
denotes the
standard deviation
U
i
of the stochastic parameter variation &
M
i
n
i
.
Suppose the synthetic gene network also suffers from environmental
disturbances due to changing extra-cellular environments and interactions
with the cellular context in its host cell. Then the stochastic gene network
can be represented as (Chen and Wu 2009).
() ()
()
() () ...