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# Assignment 4 : Due Oct. 26 at 11:30 am

1. Starting with the same ApE file (for the bistable switch) as last week, replace the tetR gene and the promoter it acts on with the cI gene and the appropriate promoter. You can get the sequences at the biobrick registry of standard biological parts. Extra credit: Balance the translation strength of CI with LacI by matching the RBS strength of the new CI you add with the existing LacI.

2. Using the extended generator, determine the rate of change of the expected number of B molecules and the variance in the number of B molecules in the reaction

- A ↔ B

where the rate of the forward reaction is s/(k_{1}+s) and of the reverse reaction is k_{2} B. Here, s is the concentration of a signaling molecule and k_{1} and k_{2} are constants. Note that the total number of molecules is A+B = n, a constant. So you would use n-B wherever A shows up in your equations. Show how changes in s and n affect the mean and coefficient of variation of B (AT STEADY STATE) in a sensitivity plot with B on the horizontal axis and the coefficient of variation on the vertical axis.

3. Many repressors are inactivated by signaling molecules, which we assume bind quickly to the transcription factors. The reactions are

- s + R ↔ R*
- g
_{on}+ n R ↔ g_{off}

Say the rate constants are k_{1}, ..., k_{4}. Determine a Hill function describing the fraction of gene that is on as a function of s only. Plot the function for various values of n.

4. Simulate the bistable switch in gro using the architecture in the ApE file we've been using -- but make up rates however you want to. Show that you get a bimodal distribution of gene expression.

5. Grad students: Describe two possible project ideas. For each, explain the high level motivation and describe what low-level genetic construct would be made and tested. What kind of modeling could you do? How can you tell if the system is working experimentally?

6. Extra credit: Find me a quote to use for Lecture 8 on Engineering Gene Expression (10/17).