160 Synthetic Gene Network: Modeling, Analysis, and Robust Design Methods
and
F
t,
V
= 0 if
t ≠
V
, i.e., &
C
j
, &
D
j
, &
I
j
and &
J
s
denote the corresponding
standard deviations of the stochastic parameter fl uctuations &
C
j
()
i
nt
,
&
D
()
ji
nt
, &
I
()
ji
nt
and &
J
s
()
i
nt
, respectively. Thus, the nonlinear stochastic
coupled synthetic genetic oscillators under intrinsic kinetic parameter
fl uctuations in the host cell i can be represented by (Chen and Hsu 2012)
11
( ) ( ( )) ( ( )) ( ( )) ( ( )) ( )
==
⎛⎞⎛ ⎞
=+ + +
⎜⎟⎜ ⎟
⎝⎠⎝ ⎠
∑∑
NN
iiijjWiijWji
jj
xt fxt cgx t f xt cg x t nt
(8.6)
where
1
( ( )) ( ( )) ( )
N
Wi ijW j i
j
fxt cgxt nt
=
⎛⎞
+
⎜⎟
⎝⎠
∑
denotes the intrinsic kinetic
parameter fl uctuations of synthetic genetic oscillator in the host cell i. For
the convenience of analysis ...