Biomedical Engineering Reference
In-Depth Information
¯
d
q
2
q
2
k
2
k
k k
1
1
1
¡
°
. [37]
¨
p
E
p
2
=
r
+
d
X
=
P
R
+
P
R
¡
°
(
)(
)
(
)
2
Q
2
2
2
2
XH
2
+
2
2
2
H H H H
XHXH
+
+
HH
¢
±
R
P
P
R
P
P
R
P
R
d
For comparison with the previous result, note that <
p
> =
k
P
k
R
/H
P
H
R
, so this can be
rewritten as
¬
-
k
/
H
E
p
2
=
p
PR
+
1
-
®
.
[38]
-
--
1
+
HH
/
PR
This is identical to the result obtained by the master equation. This method can
be readily generalized for many interacting genes when the system is fluctuating
around a steady state. As an example, we will analyze the case where one gene
represses a second gene.
Let
y
0
,
y
1
be the protein numbers of each gene, and let
f
(
y
0
) be the rate of
creation of the second protein as a function of the first. This means that the
equations describing this system are
yk
=
H
y
,
0
0
0
0
()
yf
=
y
H
y
.
[39]
1
0
1
1
Note that the equations include the entire process of producing a protein, so
mRNA levels are no longer explicitly calculated. Including the Langevin noise
term and looking at the fluctuations from steady state,
E
y
=
H E
y
+
q
F
,
0
0
0
0
0
E
yf y f
=
()
(
y
)
HE
yq cy
+
F E HE
x
yq
+
F
,
[40]
1
0
0
1
1
1
1
0
0
1
1
1
1
df
where
c
=
, and each noise term has the same conditions as before. This
0
dy
0
y
0
linearization is valid at each stable point, but not for transitions between differ-
ent stable points or for limit cycles. For very small numbers
n
of chemicals this
also breaks down, because since these processes are mostly Poissonian, the fluc-
tuations are of order
n
, so a Taylor expansion might not be valid. Fourier
transforming and taking the square and the average as before, we get
2
q
2
ˆ
()
,
yEX
=
0
0
2
2
0
XH
+
