# Problem 1: Designing Experiments¶

We discussed 2 hypothesis of when activation is preferred over repression.

**a)** Devise an experimental way to discriminate among these models.

# Problem 2: Model and Simulate¶

Model the lac operon with activation and repression.

**a)** Create a CRN model and find plausible parameters in literature. Which assumptions did you make and why?

**c)** Think of simulations (and resulting figures) to demonstrate activation, repression, and combinations of both. Run simulations and discuss their outcomes for plausibility and implications for the cell.

**d)** Compare deterministic and stochastic simulations for one case. What do you observe?

# Problem 3: Revisiting Problem 1¶

Having a working lac model, let's revisit Problem 1.

**a)** Why do you think both activation and repression are used in the lac operon? Is this consistent with what you found in Problem 1?

**b)** Run a deterministic and stochastic simulation to see if the occupation theory has an impact and under which parameters (e.g., binding/unbinding rates). Discuss your observations.

# Problem 4: Michaelis-Menten and similar approximations¶

**a)** Derive the Michaelis-Menten approximation for an enzyme E, substrate S, and product P.

**b)** What is the difference to the activation / repression approximations that we used? Did you need to add an assumption?

**c)** For the approximations that we derived in the 1st lecture: Challenge the approximation (for Hill coefficient 1) and show where it holds and where it breaks in simulations. You can use deterministic or stochastic simulations to make this point. Hint: Changing $k^+$ and $k^-$ while keeping $K_d$ constant.

**d)** For the approximations that we derived in the 1st lecture: Can you modify the system for Hill coefficients larger than 1. What would be a CRN that results in such an approximation? Give a mathematical proof.