Biomedical Engineering Reference
In-Depth Information
2.4. Noise Properties of a Constitutively Expressed Gene
Based on the results for these elementary reactions the equation for the
moment-generating functions of more complex networks can be easily deduced.
First let us consider a constitutive expressed gene in a single copy in the chro-
mosome of a bacterium. In this case the state of this system at any time is de-
fined by the number of mRNA molecules
r
and number of proteins
p
for that
gene. mRNA molecules are synthesized off the template DNA strand at a rate
k
R
and are translated at a rate
k
P
. The mRNA and protein degradation are described
by the destruction rates H
R
and H
P
, respectively (Figure 1).
Based on the results in Table 1, the moment-generating function can be
deduced directly:
s
F
s
F
s
F
. [15]
Fzz t k z F kzz
(
,
, )
=
( )
+
( )
s
H
( )
z
H
( )
z
1
2
R
1
P
1
2
R
1
P
2
z
s
z
s
z
1
1
2
The first two terms are the transcription and translation reactions (Table 1, type
I) and the last two terms model degradation of mRNA and proteins, respectively
(Table 1, type II). Below the equation will be solved for the moments in the
steady state
F
( )
=
. In this case:
s
F
s
F
s
F
k
(1
z Fk z z
)
=
(
s
1)
H
(
z
1)
H
(
z
1)
.
[16]
R
1
P
1
2
R
1
P
2
z
s
z
s
z
1
1
2
The mean mRNA level
r
and protein level
p
are found by taking the de-
rivative with respect to
z
1
and
z
2
, respectively:
s
F
s
2
F
[
]
k
(1
z
)
k Fk z z
=
(
1)
H
(
z
1)
R
1
R
P
1
2
R
1
s
z
s
z
2
1
1
2
s
F
s
F
[
]
+
kz
( )
s
H
H
( )
z
zz
.
[17]
P
2
R
P
2
z
s s
1
2
1
Evaluating both expressions at
z
1
=
z
2
= 1 gives
k
r
=
R
,
H
R
kk
RP
p
=
.
[18]
HH
RP
These results are consistent with the equivalent deterministic system:
