A beautiful set of quantitative experiments on the lac operon were done by the Müller-Hill group in the 1990 s. where repression of expression of the lacZ gene was measured in a population of different mutant $E .$ coli cells. The mutant cells differed in the number, sequence, and position of the operator sites that bound the Lac repressor. In this problem, we explore how, using thermodynamic models of gene expression, these data can be used to obtain a number of quantities characterizing the Lac repressor-DNA interaction as well as DNA looping.
(a) Using the data from Oehler et al. (1994) shown in Figure 19.22 determine the in vivo binding energy of Lac repressor to each one of its operators and reproduce Figure 19.23
(b) Use your results from (a), and the repression measured by Oehler et al. (1994) in cells with two operators present, which leads to DNA looping, in order to determine the looping energy and to reproduce Figure 19.27
(c) As mentioned many times throughout the book, Müller et al. (1996) performed an experiment where the repression level was measured as a function of the distance between operators. The experiment and its results are shown in Figure 1.11 (p. 19 ). Based on their repression data and the thermodynamic models from the chapter, make a plot of the looping energy as a function of the interoperator distance. Show analytically that a maximum in repression corresponds to a minimum in looping energy. At what interoperator distance is the inferred looping free energy at a minimum? Is this consistent with the measured persistence length of DNA in vitro, which is $50 mathrm{nm} ?$
(d) Fit the looping energy obtained in (c) to the functional form $Delta F_{text {loop }}=a / N_{mathrm{bp}}+b ln N_{mathrm{bp}}+c N_{mathrm{bp}}+e .$ Use this looping
energy to make predictions about the outcome of a hypothetical experiment similar to the one performed by Müller et al. $(1996),$ but now using cells bearing $10,200,$ and 900 Lac repressor molecules per cell.
Relevant data for this problem are provided on the book’s website.
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