Fig. 3.

Non-Poisson transcription dynamics and TCF of the transcription rate of an unrepressed gene. a A simple model of the non-Poisson transcription process: (Step 1: Initial binding) initial binding of RNAP to promoter to form the closed complex; (Step 2: Successful initiation) transition of the RNAP-promoter complex into the elongation complex; and (Step 3: Elongation) synthesis of mRNA. Step 1 is modeled as a Poisson process with the mean reaction time, ${\tau }_1$, for which the reaction waiting time is distributed according to ${\psi }_{}^{\left(1\right)}\left(t_1\right)={\tau }_1^{-1}{\mathrm{e}}^{-t_1∕{\tau }_1}$. The reaction waiting time of Step 2 is modeled as a gamma distribution, ${\psi }^{\left(2\right)}\left(t_2\right)=t_2^{a-1}{\mathrm{e}}^{-t_2\mathrm{∕}b}\mathrm{∕}\left(\Gamma \left(a\right)b^a\right)$, with $⟨t_2⟩=ab\left(\equiv {\tau }_2\right)$ and $⟨\delta t_2^2⟩\mathrm{∕}⟨t_2⟩\equiv b$. The elongation process (Step 3) is a highly sub-Poisson process for which the reaction time is modeled as ${\psi }_{}^{\left(3\right)}\left(t_3\right)=\delta \left(t_3-{\tau }_3\right)$. The next round of RNAP binding to the promoter is not allowed before the preceding RNAP completes the second step to leave the promoter. During transcriptional elongation by RNAPs, other RNAP can associate with the promoter and proceed to the next step. Multiple RNAPs can simultaneously perform the elongation process. b Distribution of transcription waiting times or the times between successive transcription events of the single-active gene copy. This distribution is given by the convolution of ${\psi }^{\left(1\right)}\left(t\right)$ and ${\psi }^{\left(2\right)}\left(t\right)$. The shape of the transcription waiting time distribution is shown for three different sets of parameter values for ${\tau }_1$, ${\tau }_2$, and $b$. The mean value, ${\tau }_1+{\tau }_2$, is fixed at 4 s. c TCF, ${\varphi }_{\kappa }\left(t\right)$, of the active gene transcription rate corresponding to each transcription waiting time distribution (see Supplementary Note 18). When the coefficient of variation (CV) in the transcription waiting time is small enough, ${\varphi }_{\kappa }\left(t\right)$ exhibits an oscillatory feature. For the model shown in a every cell has the same transcription dynamics, in which case the oscillation period in ${\varphi }_{\kappa }\left(t\right)$ is constant in time and approaches the mean transcription waiting time, ${\tau }_1+{\tau }_2$, as the CV of the transcription waiting time decreases. In the presence of coupling to a disordered environment, the period of the oscillation in ${\varphi }_{\kappa }\left(t\right)$ gradually increases over time, as shown by the blue line in Fig. 2b (Supplementary Note 6 and Supplementary Figure 7). Further details on the stochastic simulation method used in Fig. 3b, c can be found in Supplementary Note 19