Fig. 5: Stochastic entrainment of gene expression by the repressilator.

A Schematic diagram of the repressilator driving a gene expression network. The cI protein from the repressilator acts as an activating transcription factor for mRNA (X₁) which translates into output protein (X₂). The red arrow from X₂ to X₁ indicates negative transcriptional feedback from the protein molecules. When the repressilator is connected to the gene expression network, for linearised feedback the PSD can be estimated with the composite Padé PSD method which is based on Theorem 2.1. In B, these PSD estimates (after normalisation by the total area under the PSD curve) are plotted for six values of θ and compared for \(\theta =0.4\,{\min }^{-1}\) to the PSD obtained with the DFT method. One can observe the stochastic entrainment phenomenon as θ increases. C The heat-map for the entrainment score (see (44)) as a function of θ and the feedback strength parameter k_fb. Observe that the entrainment score is monotonically increasing in both variables k_fb and θ, but it is more sensitive to k_fb. D For nonlinear transcriptional feedback PSD estimates obtained with Padé PSD are plotted and compared with the DFT method for \(\theta =1\,{\min }^{-1}\) and \(\theta =5\,{\min }^{-1}\). All the PSDs were estimated with Q = 10 simulated trajectories.