Bacterial Replication Initiation as Precision Control by Protein Counting

Balanced biosynthesis is the hallmark of bacterial cell physiology, where the concentrations of stable proteins remain steady. However, this poses a conceptual challenge to modeling the cell-cycle and cell-size controls in bacteria, as prevailing concentration-based eukaryote models are not directly applicable. In this study, we revisit and significantly extend the initiator-titration model, proposed 30 years ago, and we explain how bacteria precisely and robustly control replication initiation based on the mechanism of protein copy-number sensing. Using a mean-field approach, we first derive an analytical expression of the cell size at initiation based on three biological mechanistic control parameters for an extended initiator-titration model. We also study the stability of our model analytically and show that initiation can become unstable in multifork replication conditions. Using simulations, we further show that the presence of the conversion between active and inactive initiator protein forms significantly represses initiation instability. Importantly, the two-step Poisson process set by the initiator titration step results in significantly improved initiation synchrony with $CV\sim 1/N$ scaling rather than the standard $1/\sqrt{N}$ scaling in the Poisson process, where $N$ is the total number of initiators required for initiation. Our results answer two long-standing questions in replication initiation: (i) Why do bacteria produce almost two orders of magnitude more DnaA, the master initiator proteins, than required for initiation? (ii) Why does DnaA exist in active (DnaA-ATP) and inactive (DnaA-ADP) forms if only the active form is competent for initiation? The mechanism presented in this work provides a satisfying general solution to how the cell can achieve precision control without sensing protein concentrations, with broad implications from evolution to the design of synthetic cells.

Figure 1

Protein concentration in eukaryotes vs bacteria. (a) Left: morphogen gradient in the French flag model in developmental biology. Right: Oscillation of cyclin concentration for eukaryotic cell-cycle control. (b) Balanced biosynthesis in bacteria.

Figure 2

Initiation control. (a) Hypothetical minimal cell. (b) Initiator-titration model v1 [19]. (c) Initiator-titration model v2 (this work).

Figure 3

Initiation control of the protocell by initiator protein counting. (a) Model sequence of titration and initiation. (b) Change in the copy numbers of initiators and initiator binding sites during the cell cycle under the condition of two overlapping cell cycles ($C<\tau <C+D$). The initiation condition is $I(t=t_{\mathrm{ini}})=B(t=t_{\mathrm{ini}})$. (c) Predicted initiation mass in different growth conditions ($C/\tau$) by assuming that $c_{\mathrm{I}}$ is a constant [33, 34].

Figure 4

Dynamical stability analysis for initiation of the protocell. (a) Multifork replication tracker. $i=1$ represents the group of replication forks closest to ori. (b) Linear (in time) progression of the replication forks from ori to ter on the circular chromosome. ter is on the opposite end of ori on the chromosome. (c) Stability phase diagram ($n_{\mathrm{B}}/N_{\mathrm{B}}$ vs $C/\tau$ space). Below: initiation mass vs $C/\tau$ in the condition of $n_{\mathrm{B}}/N_{\mathrm{B}}=1/30$ and constant $c_{\mathrm{I}}$ [33, 34]. Inset: stable initiation events vs unstable (oscillatory) initiation events.

Figure 5

Initiator-titration model v2 predictions (see Sec. 2d for more details). (a) External DnaA-ATP $\leftrightarrow$ DnaA-ADP conversion elements in E. coli. RIDA is the only component that depends on the active replication forks. (b) [DnaA-ATP]/[DnaA-ADP] varies during the cell cycle predicted by computer simulations in the wild-type cells. By contrast, the $\mathrm{\Delta }4$ mutant that lacks all extrinsic DnaA-ATP $\leftrightarrow$ DnaA-ADP conversion elements in (a) shows a constant ratio. (c) Predicted initiation mass in different growth conditions ($C/\tau$ with fixed $C$) by assuming a constant [DnaA] [33, 34]. RIDA, the replication-dependent DnaA-ATP $\to$ DnaA-ADP mechanism alone can restore stability as long as titration is present. None of the other extrinsic DnaA-ATP $\leftrightarrow$ DnaA-ADP conversion elements can restore the initiation stability.

Figure 6

Initiation precision and the reduction in asynchrony by $1/N$ scaling in the two-step Poisson process by titration. (a) Asynchrony (intrinsic noise), extrinsic noise, and cell-to-cell variability in initiation control. Gray dots are single-cell data of wild-type E. coli from [13]. (b) Initiation by the first-passage-time model based on a simple Poisson process with $n_{\mathrm{B}}$ the threshold at each ori. Left: $n_{\mathrm{B}}=10$ vs right: $n_{\mathrm{B}}=300$. (c) Synchronized initiation by titration in the two-step Poisson process. $N$ is the mean total number of initiator proteins required for triggering initiation at both ori's. (d) Simulation of the scaling behavior of the intrinsic noise ($C{V}_{\mathrm{int}}$), the extrinsic noise ($C{V}_{\mathrm{ext}}$), and the total CV ($C{V}_{\mathrm{tot}}$) (see Appendix pp6). Top: varying $N$ with fixed $n_{\mathrm{B}}$; bottom: varying $n_{\mathrm{B}}$ with fixed $N$. The gray dashed lines are from Eq. (17).

Figure 7

The time trajectory of the number of chromosomal initiator binding sites (titration boxes) $\tilde{B}\left(t\right)$ in different scenarios in the steady state. (a) $n=0$ and $t_{\mathrm{ini}}>t_{\mathrm{ter}}$. (b) $n=1$ and $t_{\mathrm{ini}}<t_{\mathrm{ter}}$. (c) The general case when $t_{\mathrm{ini}}\le t_{\mathrm{ter}}$. (d) The general case when $t_{\mathrm{ini}}>t_{\mathrm{ter}}$.

Figure 8

Simulation of the instability on phase diagrams. Instability is quantified by $(max{v}_{\mathrm{i}}-min{v}_{\mathrm{i}})/(max{v}_{\mathrm{i}}+min{v}_{\mathrm{i}})$ after 100 doubling times. (a) In the protocell, the unstable regimes match the theoretical predicted unstable regimes (boundaries shown by red dashed curves). (b) In the initiator-titration model v2 with a constant $\gamma$, we fixed $n_B/N_B=1/30$, and we simulate the unstable regimes on the phase diagram of $\gamma \text{−}\frac{C}{\tau }$.

Figure 9

The initiation mass behavior in different combinations of the DnaA-ATP/DnaA-ADP conversion elements, i.e., $hda$ (RIDA), $datA$ (DDAH), DARS1, and DARS2. The situations with $hda$ and without $hda$ are plotted together in each plot to show the contrast.

Figure 10

The effect of SeqA in the protocell model. (a) We consider the SeqA sequestration effect at ori binding sites, which results in no initiation events in a duration time of $t_{\mathrm{seq}}=10\min$. (b) The unstable range does not shrink with the existence of seqA. (c) The amplitude of the initiation mass oscillation, defined by $(max{v}_{\mathrm{i}}-min{v}_{\mathrm{i}})/(max{v}_{\mathrm{i}}+min{v}_{\mathrm{i}})$, is slightly reduced when $C/\tau$ is relatively large in the unstable regime. Here, we set $n_B/N_B=1/30$.