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A2 : Due by 11:30 AM, Oct. 8 via Catalyst

  1. Consider the reaction network 2X + Y → X + 2Y. Assume the rate constant for the reaction is 1.
    1. Find the stoichiometric matrix.
    2. Assume the system is mass action, and find the kinetics vector.
    3. Write out the ODE model.
    4. Simulate the ODEs from a couple of initial conditions and show (nicely formatted) plots of the resulting trajectories for each species.
    5. Determine a mass vector or prove that the system is not conservative.
    6. Find the equilibria of the system. Which of the equilibra are stable. Which are not stable? Why?
    7. Extra credit: Linearize the ODE model around the equilibra and find the eigenvalues to determine stability. Note that if the system is conservative, you will need to reduce the dimension of the system by using the conservation law before you analyze stability.
  2. Repeat (1) with the model of transcription and translation from the lecture notes (Lec. 4).
  3. Repeat (1) with the following network
    X → 2X
    X+Y → 2Y
    Y → ∅
    where the first two reactions have rate constant 1, and the third reaction has rate constant 2.
  4. Work through the gro tutorial number 1 (available on the main page of the gro website). Then work out a new example in gro that models both RNA and GFP with the following reactions:
    ∅ → RNA
    RNA → RNA + GFP
    Assume the first reaction has kinetics αRV, where V is the volume of the cell, and the second reaction has kinetics αP RNA. Suppose that the GFP dilutes but is not degraded, while the RNA dilutes and is also degraded, as in
    RNA → ∅
    with kinetics βR. Save data on the time, RNA, GFP, and volume to a .csv file and plot it in your favorite math software. Use αR = 2.0, αP = 3.0, and βR = 0.1.
  5. Derive the ODEs that model the rate of change of the concentration of RNA and the concentration of GFP, simulate them, and overlay them with the data obtained from gro, as was done with the simpler system in the tutorial.
  6. Extra credit: Save the trajectories of all the cells, and make a plot like that in the gro documentation under Tutorial->Molecules or in Fig. 3 of the gro paper. This plot should give you a better idea of the variance in the behaviors you can expect to see.

Turn in A2 here: