Biomedical Engineering Reference
In-Depth Information
¯
0
0
0
0
0
¡
°
¯
00000
0 0 0 0
00 00
000 0
0000
¡
°
¡
°
dk
0
0
0
0
¡
°
11
R
¡
°
H
¡
°
¡
°
0
k
0
0
0
R
1
¡
°
¡
°
P
1
,
,
[22]
A
=
(
=
H
¡
°
¡
°
P
1
s
s
f
¡
°
¡
°
dk
0
d
0
0
H
¡
°
¡
°
2
R
2
2
p
R
2
¡
°
¡
°
1
p
¡
°
¡
°
H
¢
±
1
¡
°
P
2
¡
°
0
0
0
k
0
¢
±
P
2
s
f
(
)
where
.
k
=
f
p
s
p
R
2
1
1
p
1
p
1
Written in terms of these matrices, the master equation in generating func-
tion form would be
¬
s
F
s
-
F
(
)
-
F
=
1
(
z
A z
.
[23]
-
-
i
ii
ij
j
s
z
s
z
-
®
i
j
i
j
At steady state,
, and taking the derivative with respect to
z
l
we obtain
F
=
0
¬
¬
2
2
s
F
s
F
s
F
-
s
F
s
F
-
-
-
(
)
.
[24]
0
=
1
(
z
A z
(
A
A z
-
-
-
-
i
ii
ij
j
il
-
ll
lj
j
-
ss
zz
ss
z z
s
z
s
z
s
z
®
®
i
j
j
i
l
j
l
l
l
j
Setting all
z
i
= 1, we have for each
i
s
F
s
F
(
)
(
)
,
[25]
0
=
(
Az
º
0
=
(
A
F
=
(
A x
ii
ij
j
1
s
z
s
z
1
j
i
j
1
1
corresponding to the macroscopic result. Similarly, differentiating again and
evaluating at
z
i
= 1,
¬
¬
s
2
F
s
2
F
s
F
-
s
2
F
s
2
F
s
F
-
-
-
,
0
=
(
Az
A
-
+
(
Az
A
-
-
-
-
-
ii
ij
j
il
®
ll
lj
j
li
®
ss
zz
ss
z z
s
z
ss
zz
ss
zz
s
z
j
j
i
l
1
j
l
l
1
l
i
1
j
i
i
1
1
1
(
)
(
)
T
( 2
,
[26]
(
)
(
)
=
( 2
A
T
F
A
F
+
A
T
F
A
F
1
1
1
1
E
s
where
=
s
. These linear equations can be solved for the means, variances,
and correlations.
This approach is very general and the resulting matrix equations can be
solved directly. However, even for the case of just two interacting genes this
2
ij
ij
z
i
