Consider the block diagram in Fig. E24.2 of the multiple feedback loops involved in the Central Dogma schematic from Fig. 24.3, namely, genetic regulation C₁, transnational regulation C₂, and enzyme inhibition C₃. Assume that the processes P₁, P₂, and P₃obey first-order dynamics, with corresponding gains and time constants K_i, Ï„_i.
(i) Derive the transfer function from the external input (u) to the output (y) for each of the three cases shown in Fig. E24.2 (a), (b), and (c)
(ii) Assume that the feedback mechanisms operate via proportional control with corresponding controller gains K_ci. Derive the closed-loop transfer function from the external input u to the output y in block diagram (b).
(iii) Consider a simplified biological circuit in which only genetic regulation is active (C₁). Derive the closed-loop transfer function and comment on the key differences between this transfer function and the one from part (b).

(iv) Give several reasons why the natural feedback architecture with all three controllers operating is more effective than the control architecture in part (c).

P1 P2 P3 (a) P1 P2 P3 C2 C3 C1 (b) P1 P2 P3 C1 Figure E24.2 (c) Primary transcript Protein Nuclear transport MRNA Transcription Translation Protein activation Active protein Nucleus Cytoplasm Cell membrane Figure 24.3 The layers of feedback control in the Central Dogma (Copyright 2002 from Molecular Biology of the Cell, Fourth Edition by Alberts et al. Reproduced by permission of Garland Science/Taylor & Francis LLC).