Biology Reference
In-Depth Information
Since there are at most a few molecules of the operator site per cell, it is reasonable
to assume
[
OR
]
max
{[
R
]
,
[
RA
n
]}
. Then Eq. (
2.39
) simplifies to
R
tot
=[
R
]+[
RA
n
]
.
(2.40)
Putting Eq. (
2.34
) into Eq. (
2.40
) and solving the resultant equation for
[
R
]
, we obtain
R
tot
[
R
]=
n
.
(2.41)
1
+
K
1
[
A
]
By plugging the repressor protein concentration from Eq. (
2.41
) into Eq. (
2.38
),
we obtain the portion of free operator site concentration in terms of the allolactose
concentration as
n
[
]
O
tot
=
O
1
+
K
1
[
A
]
n
,
(2.42)
K
+
K
1
[
A
]
where
K
K
2
R
tot
. This equation describes the available operator concentration
for RNA polymerase binding and for initiating the transcription of the structural
genes. The right-hand side of the function in Eq. (
2.42
) is a sum of two functions: (i)
a decreasing Hill function,
=
1
+
n
1
K
1
[
A
]
n
, and (ii) an increasing Hill function
n
K
+
K
1
[
A
]
K
+
K
1
[
A
]
n
1
+
K
1
[
A
]
of the form given in Eq. (
2.25
). Denote
f
(
[
A
]
)
=
n
and notice that
f
takes
K
+
K
1
[
A
]
a positive value 1
and
the concentration of
A
is always non-negative, this is the lowest value
f
can take.
Biologically this can be interpreted as the process of ongoing transcription of the
structural genes occurring at a constant rate in the absence of allolactose, and providing
basal concentrations of the proteins encoded by the structural genes. On the other
hand, when allolactose is abundant, lim
/
K
when
[
A
]=
0. Since
f
is an increasing function of
[
A
]
1. This can be interpreted
as having all of the operator sites available to mRNA polymerase for transcription of
the structural genes, with transcription occurring at the maximal possible rate. Yagil
and Yagil [
11
] have experimentally shown that Eq. (
2.42
) can accurately fit their
observations and estimated the parameters
K
,
K
1
, and
n
from the data.
The Yildirim-Mackey models of the lac operon:
The dynamics of mRNA is modeled
with the following equation
f
(
[
A
]
)
=
[
A
]→∞
K
1
e
−
μτ
M
A
M
n
1
+
dM
dt
=
α
M
τ
K
1
e
−
μτ
M
A
τ
M
n
−
γ
M
M
,
(2.43)
K
+
where
A
. The rate of change in mRNA con-
centration is the difference between its gain and loss terms. We assume that the
production rate of mRNA is proportional to the fraction of free operators as described
in Eq. (
2.42
). Here
=
A
(
t
−
τ
M
)
and
γ
M
=
γ
M
+
μ
τ
M
α
M
represents such a proportionality constant.
A
M
represents
the allolactose concentration at time
t
−
τ
M
. The production of mRNA from DNA
via transcription is not an instantaneous process. In fact, it requires a period of time
τ
M
for RNA polymerase to transcribe the first ribosomes binding site. Hence, the
production rate of mRNA is not a function of the available allolactose concentration
at time
t
, but a function of the available concentration of allolactose at time
t
τ
−
τ
M
.
The exponential prefactor
e
−
μτ
M
accounts for the dilution of the allolactose through
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