Biology Reference
In-Depth Information
Substituting Eq. (
2.28
) and Eq. (
2.29
) into Eq. (
2.31
) and making necessary simpli-
fications gives us
=−
μ
+
β
[
d
[
x
]
x
]
.
(2.32)
dt
As seen in this equation, an increase in the volume over time can successfully be
modeled by adding the growth rate to the protein degradation rate.
Modeling of the lactose repressor dynamics:
The repressor protein plays a major
role in the regulation of the
lac
operon. In both absence and presence of the external
lactose, the repressor protein is produced at a constant rate. A binary complex between
allolactose and the repressor protein is formed. This complex cannot bind to the
operator site (
O
). Yagil and Yagil [
11
] have modeled the dynamics of the repressor
protein (
R
) as follows:
nA
K
1
R
+
RA
n
.
(2.33)
Here
n
represents the effective number of allolactose molecules required to form this
complex. If we assume this reaction is at equilibrium, we have
[
RA
n
]
K
1
=
n
,
(2.34)
[
R
][
A
]
where
K
1
is the association constant for this reaction. The repressormolecules can also
bind to the operator region (
O
) in the absence of allolactose and block the transcription
process. We assume this interaction is of the following form:
R
K
2
O
+
OR
.
(2.35)
At equilibrium, we have
[
OR
]
K
2
=
]
.
(2.36)
[
R
][
O
Here
K
2
is also an association constant of this reaction. Let
O
tot
be the total operator
concentration. It is plausible to take the total concentration of the operator as constant.
Therefore,
O
tot
=[
O
]+[
OR
]
.
(2.37)
From Eq. (
2.36
) and (
2.37
), we can write
[
]
O
tot
=
O
1
]
.
(2.38)
1
+
K
2
[
R
Now, let
R
tot
be the total concentration of the repressor protein. Since the repressor
protein is not regulated by the extracellular lactose, we can assume that the total
concentration of this protein also stays unchanged. Hence,
R
tot
=[
R
]+[
OR
]+[
RA
n
]
.
(2.39)
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