Fig. 4: The incoherent feedback between amino acid pool and ribosome production gives rise to damped oscillatory response regardless of delays due to ribosome transcription.

Simulations were performed with two different models: with instantaneous transcription (A, C) and with gradual mRNA and rRNA pool production (B, D). In the instantaneous transcription model, the regulatory functions χ_i follow ppGpp levels without delay (as in ref. ¹³), while they are given by Eq. (2) in the gradual transcription model, where here we set τ_χ = 10min to emphasize the effect of the delay. In all panels, circles indicate simulation results color-coded by the time from the shift, and X-shaped symbols represent the pre- and post-shift steady states. A, B Three phases of post-shift adaptation, which are present in both models. The x axis displays the ribosomal sector mass fraction, and the y axis shows the growth rate. The dashed line in both panels connects (0,0) and (\({\phi }_{{\rm{R}}}^{{\rm{final}}}\),λ^final), and highlights the second phase (as in Fig. 9 of ref. ¹³). C, D Ribosome allocation through the shift. The x axis displays the relative translation rate, and the y axis shows the ribosomal regulatory function χ_R. The plots show that oscillations occur even when the regulatory function instantaneously follows the translation elongation rate along the steady-state relation. Panel E provides a sketch of the incoherent feedback loop between the amino acid pool ψ_A and the ribosomal fraction ϕ_R, which is explicitly described in this study and is responsible for the observed oscillatory behavior.