(a) A simpliﬁed version of the enzymatic reaction used to measure the
level of gene expression in the cell is given by
ONPG + LacZ
k
→
ONP + LacZ
.
(1)
By considering this simple onestep and irreversible reaction, write diﬀer
ential equations for the amount of ONPG and ONP as a function of time
and solve the equation for [
ONPG
](
t
). Then, substitute this result into
the equation for ONP and solve it and examine the dynamics of [
ONP
](
t
)
in the small time limit. Use this to evaluate under what set of conditions
the rate of increase in ONP concentration is linear in the concentration of
β
galactosidase molecules. If you have 1 ml of cells at an OD600 of 1 (about
10
9
cells) each with 1000 LacZ molecules ﬁnd out how long you can watch
the reaction before the linearity condition no longer holds. The rate of hy
drolysis (our constant
k
above in the chemical equation) has been measured
to be about 140
×
10
6
(M min)

1
.
(b) Experimentally, we are interested in measuring the level of gene
expression of a promoter that is subject to repression. Recall from the ther
modynamic models that we characterize the level of expression through the
quantity
p
bound
which tells us the probability that the promoter is occupied
by the polymerase. Also, remember that the foldchange is given by
foldchange =
p
bound
(
R
6
= 0)
p
bound
(
R
= 0)
,
(2)
where we compare the level of expression in two strains, one of which harbors
the repressor of interest (
R
6
= 0) and the other of which does not (
R
= 0).
Find an expression for the steady state concentration of
β
galactosidase.
In particular, you need to write a diﬀerential equation that gives the rate
of change in the
β
galactosidase concentration. This equation should have
4
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View Full Documenta production term that depends upon
p
bound
and a decay term. Given the
long lifetimes of proteins, what is the appropriate degradation time of inter
est? Use this diﬀerential equation to ﬁnd the steady state concentration of
β
galactosidase. Then, use this steady state level of
β
galactosidase in the
context of your diﬀerential equation for [
ONP
] production in part (a). Show
how in the linear regime described in (a) the foldchange in gene expression
can be obtained by measuring the rate of increase in yellow substrate for
two diﬀerent strains, one harboring the repressor and the other not. Make
sure to state all assumptions that go into your answer.
(c) A popular way to report the level of gene expression measured using
this assay is through Miller Units (MU). For example, in ﬁg. 3, we show
the level of expression of various viral and bacterial promoters reported in
Miller units. To get these results, a volume
v
of cells that have been lysed
is combined with ONPG at a concentration of 1.86 mM. The reaction is
stopped at some time and the MU are calculated by measuring the OD420
which is a proxy for the resulting concentration of ONP. Since we earlier
showed that in the small time limit that ONP increases linearly in time,
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 Winter '09
 DNA, Messenger RNA, miller units

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