Biomedical Engineering Reference
In-Depth Information
F
EX
XH
q
ˆ
,
[31]
r
ˆ()
=
rr
i
+
R
so after multiplying by the complex conjugate and taking the average,
q
2
2
,
[32]
rEX
ˆ()
=
r
XH
2
+
2
R
The steady-state fluctuations are given by the inverse Fourier transform with
t
=
0:
4
d
d
q
2
q
2
q
2
1
2
dx
¨
¨
.
[33]
E
r
2
=
X
Q XH H
r
d
=
r
=
r
2
+
2
2
x
2
+
1
2
H
R
R
R
d
d
But since the production of mRNA is in this model a single-step independent
random process, it has a Poisson distribution, so the variance equals the mean,
which implies
q
2
k
.
[34]
H
=
r
R
º
q
2
=
2
k
r
R
2
H
R
R
For the number of proteins, we have
,
EHEE F
p
+=+
pr
q
P
p
p
E
rw
ˆ()
X
+
q
F
ˆ
pp
,
[35]
EX
p
ˆ ()
=
i
+
P
but in this case we also need to notice that
, since these
are two independent random processes with zero mean. So in this case,
E F
rw
ˆ
()
ˆ
*
=
E
rw
ˆ
()
F
ˆ
*
=
0
p
p
2
EX
r
ˆ()
+
q
2
2
2
q
q
p
2
p
ˆ
()
r
.
[36]
EX
p
=
=
+
(
)(
)
2
2
2
2
XH
+
XHXH
2
+
2
2
+
2
XH
+
P
P
R
P
k
We use
2
and
2
R
(since this represents the internal noise and
q
=
2
k
q
=
2
k
H
r
R
p
p
R
for a fixed number of mRNAs the production of proteins is also a Poissonian
process). Performing the inverse transform,
5
