Consider the simple block diagram for the feedback control of circadian rhythms (i.e., 24 h) in Fig. 24.5.To simplify things considerably, assume that transcription obeys a first order process with gain = 5 [mRNA/Protein] and time constant = 1.0 h. Furthermore, assume that translation and degradation act together like a simple delay, with unknown magnitude.
(a) Calculate the frequency response characteristics for all the elements in this feedback loop. Be careful with your time units. 

(b) In order to generate sustained circadian rhythms, how much time delay is required in the translation/degradation step? (requires a numerical answer)

(c) At this oscillation condition, what is the gain of the translation/degradation process?

Light Phosphorylation & dimerization of PER, TIM TIM Degradation PER Translation TIM Translation tim per Transcription Transcription Nucleus Cytoplasm -Cell membrane Light Translation and MRNA(M) Transcription process Protein (P) degradation processes Figure 24.5 Schematic of negative feedback control of Drosophila circadian clock (adapted from Tyson et al., 1999): detailed system (top), and simplified model (bottom).