1.Considering that the half-life for the mRNA and the protein is respectively 3min and 1h. Workout the values of (k_1, d_1, k_2, d_2) so that, at steady state, [mRNA] = 3 and [protein] = 500.
2.Using the previously found parameter values, plot on the same graph [mRNA](t) and [protein](t) for t=[0, 5h].
3. Parameter scanning: Consider parameter d_2 to vary [-20%, 20%] of its nominal value (10 uniformly spread values). Generate on the same graph the 10 [protein](t) trajectories (t=[0, 5h]).
Exercise 3 (Non linear regression)
Some synthetic data has been generated to mimic the characterisation of a repressed gene construct. The experiment consisted in measuring the reporter concentration at steady-state, expressed by the repressed construct, as a function of different repressor concentration.
The measurements are as follow:
Plot the measurements: Reporter = f(Repressor)
In order to analyse the data, you need to build an hypothetical underlying model of the repressed gene construct. Using ODEs, write down the equations governing the dynamics of the system. In order to model the repressor effect on the transcription, you will use a Hill function.
From the previous ODEs, define the steady-state equilibrium of the system. What are the parameters that characterise the repressed gene construct ? Use the nls(...) function in R.
Using a non linear regression analysis, fit your model to the measurements. Plot on top of the measurements, the resulting fit.
Convert your system of ODEs so that it becomes representative of a PoPS device. Plot its transfer function.