For all the questions below, use the parameters µ = ω = χprot = χrna = 1 s
−1 and K1/2 = 0.33
mM.
1. (a) Implement the Forward Euler approach to integrating this system to determine the
concentration of both RNA and protein over time. Use a total time of 20 s and a
time step of 0.01 s, and an initial concentration of 0.5 mM for both RNA and protein.
Plot the concentrations as a function of time.
(b) Rerun the simulation from (a), but using an initial protein concentration of 0.2 mM
and an initial RNA concentration of 0 mM.
(c) Rerun the simulation from (a), but using an initial protein concentration of 0.5 mM
and an initial RNA concentration of 0 mM.
(d) Rerun the simulation from (a), but using an initial protein concentration of 0 mM
and an initial RNA concentration of 0.2 mM.
(e) Rerun the simulation from (a), but using an initial protein concentration of 0 mM
and an initial RNA concentration of 0.5 mM.
(f) Discuss your observations.
Part B: An autoregulatory gene. We also discussed in class a model for a simple autoregulatory gene; that is, a gene that encodes a protein whose function (or part of whose function) is to activate the transcription of itself. Recall that the system of differential equations for this is: d[Xrna] dt = µ[Xprot] 2 K21/2 + [Xprot] 2 − χrna[Xrna] d[Xprot] dt = ω[Xrna]− χprot[Xprot] For all the questions below, use the parameters µ = ω = χprot = χrna = 1 s −1 and K1/2 = 0.33 mM . 1. (a) Implement the Forward Euler approach to integrating this system to determine the concentration of both RNA and protein over time. Use a total time of 20 s and a time step of 0.01 s, and an initial concentration of 0.5 mM for both RNA and protein. Plot the concentrations as a function of time. (b) Rerun the simulation from (a), but using an initial protein concentration of 0.2 mM and an initial RNA concentration of 0 mM. (c) Rerun the simulation from (a), but using an initial protein concentration of 0.5 mM and an initial RNA concentration of 0 mM. (d) Rerun the simulation from (a), but using an initial protein concentration of 0 mM and an initial RNA concentration of 0.2 mM. (e) Rerun the simulation from (a), but using an initial protein concentration of 0 mM and an initial RNA concentration of 0.5 mM. (f) Discuss your observations. 2. (a) Modify your code from question 1 so that the the simulation will be repeated 64 times, each with a different set of starting concentrations; both protein and RNA initial concentrations should be varied from 0.0 to 1.4 mM in intervals of 0.2 mM, and all combinations of these should be considered. (b) Plot the results from all simulations in 2(a) on a single plot of [Xrna] vs [Xprot]. Note that we are not plotting concentrations versus time. (c) Explain what each line on the graph in 2(b) represents, with a particular focus on the significance of the beginning and the end of each line. Is there something interesting you notice? 3. Write a paragraph discussing your results in the context of the biology of this system. Your answer should include a discussion of any stable (or equilibrium) states of the system and what those states would represent in a biological context. Also include any questions you have that your results leave unanswered. © 2010–2023. David Green & Giancarlo La Camera, Stony Brook University. 3