HW_7_final_2011

# This means the rate of production can be computed

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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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lac 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. Notch-Delta. In class, we worked out a simple model from Julian Lewis and coworkers that shows how Notch-Delta 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 Notch-Delta signaling in pattern formation and email it to Rob, Hernan, Jane and Yi-Ju (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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