Biomedical Engineering Reference
In-Depth Information
Figure 1
. Basic model for constitutive expression of a single gene. Only four individual reac-
tions are considered: creation of mRNA from a DNA template, creation of proteins from indi-
vidual mRNA molecules, and the degradation/dilution of both species.
r k r
,
H
=
H
R
R
p kr p
.
=
[19]
P
P
The fluctuations in mRNA and proteins level are found by differentiating the
above equations again with respect to
z
1
and
z
2
and evaluating at
z
1
=
z
2
= 1:
2
2
r
r
=
r
,
p
rp
r
p
=
,
[20]
1
+
RP
HH
/
¬
-
k
H
HH
/
2
2
-
p
p
=
p
PR
+
1
.
-
--
1
+
/
®
PR
Further moments can be obtained sequentially in this manner. Note that for a
random variable with Poissonian distribution all moments are equal, so the vari-
ance over the mean equals one, as is the case for the mRNA in this model. The
protein number fluctuates with a higher than Poissonian noise, the correction
determined primarily by the term
P
k
(the "burst size") that corresponds to
the average number of proteins produced per mRNA (9).
In simple cases like this, the moments can also be obtained as a function of
time. For a single gene, the noise out of equilibrium can be 40% larger than its
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