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# Assignment 3 : Due Oct. 19 at 11:30 am

1. Make a simulation of E. coli in gro that includes transcription mRNA, translation of gfp, degradation of mRNA, and degradation gfp. Choose the rates to be reasonable and justify your choices. Do the following:

- a) Run the simulation in chemostat mode and save data so that you can plot the gfp/volume of each cell as a function of time from initial conditions with one cell with no mRNA or GFP. Determine how long you need to run the simulation in order to have the mean and variance of gfp/volume taken over all cells settle to 95% of their steady state values. Call this time the settling time T
_{s}. Make a histogram of gfp/volume at time T_{s}and report the mean and variance. - b) Now make plots of T
_{s}the mean and variance (after the system has settled) versus the transcription rate, as you vary the transcription rate (in separate simulations) from 10% to 1000% of the value you chose for the initial simulation. The idea is to show how changes in the transcription rate lead to changes in the overall behavior, as characterized by T_{s}, the mean, and the variance. - c) Repeat (b) for the translation rate, and both of the degradation rates.
- d) Summarize the "tunability" of this system.

2. Consider the three reactions

- A → B + C
- C + D → E
- E → C + D

starting with two As and two Ds and no Bs, Cs or Es.

- a) Enumerate the states and draw the Markov process the results from this system (i.e. draw the reachability graph with the rates of each transition noted).
- b) Determine the rate matrix.
- c) Find the probabilities of being in each state as a function of time by solving the master equation. You can do this analytically and/or numerically using an ODE solver.
- d) Find the mean and variance for the number of Es as a function of time and plot the mean and the one standard deviation window versus time Note: To simplify things, you may set the rates of the three reactions 1.0.

3. Download and install the ApE plasmid editor. Download this ApE file describing the DNA for the bistable switch we have built in our lab. Two of the ribosome binding sites are highlighted. Find and highlight the other two. Then, go to the Salis Lab's RBS Calculator and design four new ribosome binding sites for the rfp, gfp, lacI and tetR coding regions to have strengths 20000, 40000, 80000 and 100000 respectively. Create a new DNA design in a new ApE file with these custom RBSs inserted.

4. Describe in a few sentences each of your five favorite iGEM 2011 projects found by perusing the iGEM 2011 team pages.

5. **Extra credit**: Show that the Euler approximation of the reaction A → B starting with 1 A molecule approaches the actual master equation as dt → 0.