# 16.E: Transcription regulation via effects on RNA polymerases (Exercises)

**16.1** The ratio [RDs]/[Ds] is the concentration of a hypothetical repressor (R) bound to its specific site on DNA divided by the concentration of unbound DNA, i.e. it is the ratio of bound DNA to free DNA. When the measured [RDs]/[Ds] is plotted versus the concentration of free repressor [R], the slope of the plot showed that the ratio [RDs]/[Ds] increased linearly by 60 for every increase of 1x10-11 M in [R]. What is the binding constant Ks for association of the repressor with its specific site?

**16.2** The binding of the protein TBP to a labeled short duplex oligonucleotide containing a TATA box (the probe) was investigated quantitatively. The following table gives the fraction of total probe bound (column 2) and the ratio of bound to free probe (column 3) as a function of [TBP]. These data are provided courtesy of Rob Coleman and Frank Pugh.

| [TBP] nM | ||
---|---|---|---|

| 0.10 | 0.040 | 0.042 |

| 0.20 | 0.16 | 0.19 |

| 0.30 | 0.33 | 0.5 |

| 0.40 | 0.44 | 0.78 |

| 0.50 | 0.52 | 1.1 |

| 0.70 | 0.62 | 1.6 |

| 1.0 | 0.71 | 2.45 |

| 2.0 | 0.83 | 4.88 |

| 3.0 | 0.87 | 6.69 |

| 5.0 | 0.93 | 14 |

| 10 | 0.97 | 32.3 |

| 20 | 0.99 | 99 |

Plot the data for the two different measures of bound probe. Note that since the denominator for column 2 is a constant, the ratio of bound to total probe will level off, whereas the amount of free probe can continue to decrease with increasing [TBP], and thereby getting a continuing increase in the ration of bound to free probe.

What is the equilibrium constant for TBP binding to the TATA box?

**16.3** What is the fate of the *lac* repressor after it binds the inducer?

**16.4** How does the *lac* repressor prevent transcription of the *lac*operon?

For the next two questions, let's imagine that you mixed increasing amounts of the DNA binding protein called AP1 with a constant amount of a labeled duplex oligonucleotide containing the binding site (TGACTCA). After measuring the fraction of DNA bound by AP1 (i.e. the fractional occupancy) as a function of [AP1], the data were analyzed by nonlinear, least squares regression analysis at a wide range of possible values for DG. The error associated with the fit of each of those values to experimental data is shown below; the higher the variance of fit, the larger the error.

**16.5** What is the most accurate value of DG for binding of AP1 to this duplex oligonucleotide?

**16.6** What is the most accurate measure of the equilibrium constant, Ks, for binding of AP1 to this duplex oligonucleotide?

For the next two problems, consider a hypothetical eubacterial operon in which the operator overlaps the -10 region of the promoter. Measurement of the lag time before production of abortive transcripts (in an abortive initiation assay) as a function of the inverse of the RNA polymerase concentration (1/[RNAP]) gave the results shown below. The filled circles are the results of the assay in the absence of repressor, and the open circles are the results in the presence of repressor bound to the operator.

**16.7** What is the value of the forward rate constant (kf ) for closed to open complex formation under the two different conditions?

**16.8** What is the value of the equilibrium constant (KB ) for binding of the RNA polymerase to the promoter under the 2 conditions?