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  1. Consider a system with a transcription factor I that inhibits the expression of a protein X, as shown on slide 16 of lecture 7. In the cell, X and I dilute and degrade, but the cell maintains goff + gon to be one. In addition, the transcription factor I degrades whether it is bound to the gene or not. (a) Develop a model of the gene network I --| X that includes the reactions shown on slide 16 of lecture 7, plus degradation/dilution reactions for X and I, plus a degradation reaction that degrades I when it is bound up in goff, thereby producing gon. Invent symbols for all of the rate constants. (b) Show that goff + gon is conserved by finding a mass vector. (c) Assume that I is constitutively expressed. Choose values for the rates, and then simulate the ODEs for the system for different expression levels of I.
  2. Consider the AND gate and the NOR gates in Figure 2 of this paper. For each circuit: (a) Devise a network model, like that shown on the top of slide 18 of Lecture 7. (b) Produce a chemical reaction model. (c) Develop a Hill function model, choose rates, and simulate the system showing the steady state that results for each of the four different possible input combinations.
  3. Model the Repressilator.
(a) Develop a Boolean Network model Repressilator using the synchronous model and only modeling protein. Draw the update rules and the entire network. Implement the model in gro, in chemostat mode, making sure there is some randomness in how often the rules update, and plot the trajectories of a few characteristic cells.
(b) Develop an asynchronous model of the system, this time including RNA. You do not need to draw the entire network (unless you want to). Implement this model in gro as well. Plot the trajectories of a few characteristic cells.
(c) Implement the model in the paper using copy numbers of RNA and protein in gro. Plot the trajectories of a few characteristic cells.

Turn in A3 here: