this means the rate of production can be computed simply by examining
OD
420
/t
. Of course, this result has to be normalized by the number of cells
that were in the reaction in the ﬁrst place since we are interested in the
number of
β
galactosidase molecules per cell. With these considerations in
hand, the convention is to deﬁne Miller units through the formula
MU = 1000
OD
420
t
×
v
×
OD
600
,
(3)
where
t
is the time until the reaction was stopped, OD
420
is a measure of the
amount of ONP in the reaction (with 0.0045 OD
420
units per
μ
M of ONP)
and OD
600
is a measure of the number of
E. coli
in the reaction (with one
OD
600
unit corresponding to about 10
9
cells/ml). Given that the absolute
activity of
β
galactosidase has been measured as
k
= 140
×
10
6
(M min)

1
(4)
determine how many
β
galactosidase molecules correspond to 1 MU. Make
a plot of the relation between Miller units and absolute number of molecules
per cell and use it to convert the Miller units for the
lac
operon into a cor
responding number of
β
galactosidase molecules .
5
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View Full Documentlac
pBAD
pBAD
+
Arabinose
lac
+ IPTG
rrnB
P
L
T7 A1
10
1
10
0
10
1
10
2
10
3
10
4
10
5
Expression level (MU)
Figure 3: Level of gene expression for diﬀerent bacterial promoters. The
expression level is reported in Miller units for various key promoters such as
the lac promoter, a promoter from bacteriophage T7, the ribosomal RNA
genes (rrnB) and the arabinose operon.
3. NotchDelta.
In class, we worked out a simple model from Julian Lewis and coworkers
that shows how NotchDelta signaling might work. In this problem, your
job is to ﬂesh out the analysis done in class.
(a) Write one paragraph summarizing the role of NotchDelta signaling in
pattern formation and email it to Rob, Hernan, Jane and YiJu (the same
people that you are mailing your referee reports to). In particular, I would
like to hear about a concrete example where this pathway is implicated in
development.
(b) Write out the four diﬀerential equations we considered in class in dimen
sionless form. Use the functions
f
(
x
) =
x
2
0
.
1 +
x
2
(5)
6
and
g
(
x
) =
1
1 + 10
x
2
,
(6)
that represent the dimensionless rates discussed in class. This notation is
consistent with what you will ﬁnd in the chapter being handed out. For the
case in which
v
= 100, numerically integrate your four equations and plot
the solutions for
N
1
(
t
),
D
1
(
t
),
N
2
(
t
) and
D
2
(
t
). Comment on the relative
time scales of the notch and the delta dynamics.
(c) Using the results from part (b) as motivation, show how the four dynam
ical equations can be reduced to two coupled equations that only feature
N
1
and
N
2
and make a plot that shows both the nullclines and phase portrait.
Make sure to explain the key features of the phase portrait and what it tells
us about how our model of the system works.
7
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 Winter '09
 DNA, Messenger RNA, miller units

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