Design limitations:-
This problem illustrates possible limitations that are involved in any realistic design question. Here, we examine this through the open loop and negative feedback transcriptional component. Specifically, we want to compare the robustness of these two topologies to perturbations. We model these perturbations as a time-varying disturbance affecting the production rate of mRNA m and protein P. To slightly simplify the problem, we focus only on disturbances affecting the production of protein. The open loop model becomes
Answer the following questions:
(i) After performing linearization about the equilibrium point, determine analytically the frequency response of P to d for both systems.
(ii) Sketch the magnitude plot of this response for both systems, compare them, and determine what happens as κ and α increase (note: if your calculations are correct, you should find that what really matters for the negative feedback system is the product ακ, which we can view as the feedback gain). Is increasing the feedback gain the best strategy to decrease the sensitivity of the system to the disturbance?
(iii) Pick parameter values and use MATLAB to draw plots of the frequency response magnitude and phase as the feedback gain increases and validate your predictions in (b). (Suggested parameters: δ = 1 hrs-1, γ = 1 hrs-1, K = 1 nM, n = 1, ακ = {1,10,100,1000,...}.)
(iv) Investigate the answer to (c) when you have δ = 20 hrs-1, that is, the timescale of the mRNA dynamics becomes faster than that of the protein dynamics. What changes with respect to what you found in (c)?
(v) When δ is at least 10 times larger than γ, you can approximate the m dynamics to the quasi-steady state. So, the two above systems can be reduced to one differential equation. For these two reduced systems, determine analytically the frequency response to d and use it to determine whether arbitrarily increasing the feedback gain is a good strategy to decrease the sensitivity of response to the disturbance.