Fig. 3.

The RNA polymerase growth law. (A) Illustration of the RNA polymerase autocatalytic cycle. Among the active RNA polymerases, a fraction ${\alpha }_{rpoA-D/Z}$ is allocated to transcribe (transc.) the rpoA-D and rpoZ genes (rpoD not shown). The transcribed mRNAs produce the RpoA-D and RpoZ protein subunits during their lifetime. The rate of protein synthesis by mRNA is equal to $R_{m_i}\times {\tau }_{transl.}^{-1}$, where $R_{m_i}$ is the average number of ribosomes on an mRNA of type $i$ and ${\tau }_{transl.}=L_i{\tau }_{aa}$ is the duration for a single ribosome to translate (transl.) mRNA of type $i$, whose length is $L_i$ base-pair triplets, and ${\tau }_{aa}^{-1}$ is the ribosome elongation rate. The $Rp{o}_j$ proteins, $j\in \left\{A,B,C,D,Z\right\}$ self-assemble to form new RNA polymerases, which join the collective pool of RNA polymerases. The fraction of active RNA polymerases ${\varphi }_b$ is taken from refs. 18 and 19. (B) The RNA polymerase growth law. ${\tau }_{lifetime}$ is the rpoB mRNA life time, taken to be 3 min (17), ${\tau }_{transc.}$ is the duration of transcription, calculated based on the transcription rate from ref. 18 and the length of the rpoB gene. The fraction of active RNA polymerases and the RNA polymerase assembly duration are modulated by using separate Hill functions, in accordance with the experiment under consideration (see C and D for details). (C) The growth rate as a function of the fluorescent reporter protein, induced subsequently to the induction of the rpoB and rpoC genes in the experiment detailed in ref. 20. Since RpoB precedes RpoC in the assembly of RNA polymerase, upon the depletion of the RpoB pool, the synthesis rate of RpoB governs the RNA polymerase assembly duration. The theoretical fits were produced by using the RNA polymerase growth law (B), by fitting only once the $K$ and $h$ parameters (curly brackets in B). To facilitate the comparison between the theoretical fits and the data, we artificially shifted the M9+glucose by 500 arbitrary (arb.) units of fluorescence and the M9+casamino acids by 1,000 arbitrary units of fluorescence. (D) The effect of rifampicin on the growth rate of E. coli; data were taken from ref. 21. As rifampicin levels increase, the fraction of actively translating RNA polymerases, ${\widehat{\varphi }}_b$, decreases. We find that, when the concentration of rifampicin $c$ is $c\approx 17\hspace{0.17em}\mu g/mL$, the growth rate is reduced by half compared with the nominal ($c=0$) case. The measured RNA-to-protein ratio (which is proportional to the ribosomal protein mass fraction) remains constant, indicating that ribosomes do not limit the growth rate in this experiment. This is because RNA transcription becomes limiting, which equally attenuates both ribogenesis and protein synthesis due to a global shortage in all forms of mRNA, as explained in The RNA Polymerase Autocatalytic Cycle.