Consider a standard model of transcription and translation with probabilistic creation and degradation of discrete mRNA and protein molecules. The propensity functions for each reaction are as follows:
• Probability of transcribing 1 mRNA molecule: 0.2dt

• Probability of degrading 1 mRNA molecule: 0.5dt and is proportional to the number of mRNA molecules.

• Probability of translating 1 protein: 5dt and is proportional to the number of mRNA molecules.

• Probability of degrading 1 protein molecule: 0.5dt and is proportional to the number of protein molecules.
In each case, dt will be the time step chosen for your simulation, which we take as dt = 0.05 sec.

(i) Simulate the stochastic system above until time T = 100. Plot the resulting number of mRNA and protein over time.

(ii) Now assume that the proteins are degraded much more slowly than mRNA and the propensity function of protein degradation is now 0.05dt. To maintain similar protein levels, the translation probability is now 0.5dt (and still proportional to the number of mRNA molecules). Simulate this system as above. What difference do you see in protein level? Comment on the effect of protein degradation rates on noise.