HW_7_final_2011

A a simplified version of the enzymatic reaction used

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(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 one-step 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 fold-change is given by fold-change = 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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a 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 fold-change 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, 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
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