Initiation of chromosome replication controls both division and replication cycles in E. coli through a double-adder mechanism

Living cells proliferate by completing and coordinating two cycles, a division cycle controlling cell size and a DNA replication cycle controlling the number of chromosomal copies. It remains unclear how bacteria such as Escherichia coli tightly coordinate those two cycles across a wide range of growth conditions. Here, we used time-lapse microscopy in combination with microfluidics to measure growth, division and replication in single E. coli cells in both slow and fast growth conditions. To compare different phenomenological cell cycle models, we introduce a statistical framework assessing their ability to capture the correlation structure observed in the data. In combination with stochastic simulations, our data indicate that the cell cycle is driven from one initiation event to the next rather than from birth to division and is controlled by two adder mechanisms: the added volume since the last initiation event determines the timing of both the next division and replication initiation events.

Single-cell measurements of E. coli cell cycles can be described using either a division-centric or a replication-centric framework. Time-lapse of E. coli cells growing in microfluidic channels. Fluorescence signal from FROS labeling is visible as red spots in each cell. The green dotted line is an aid to the eye, illustrating the replication of a single origin. B. Consistent with an adder behavior, the added length between birth and division is uncorrelated with length at birth. C. The classical cell cycle is defined between consecutive division events, shown here with replication and division for slow growth conditions (i.e. without overlapping rounds of replication). D. We introduce an alternative description framework where the cell cycle is defined between consecutive replication initiation events. The observables that are relevant to characterize the cell cycle in these two frameworks are indicated (see also

division-centric replication-centric measured variables
Size at birth* Λ Size per origin at initial replication initiation* Size at division* Λ Size per origin at final replication initiation* Duration between birth and division Duration between consecutive replication initiations Size at replication initiation* Λ Size per origin at birth* Duration between birth and replication initiation Duration between replication initiation and birth derived variables = 1 log Cell growth rate* (between birth and division) = 1 log Λ Λ Cell growth rate* (between consecutive replication initiations) = − Division "adder" Λ = Λ − Λ Replication "adder" = − Birth-to-initiation "adder" Λ = Λ − Λ Initiation-to-birth "adder" = ∕ Growth ratio between birth and division = Λ ∕Λ Growth ratio between consecutive initiations = ∕ Growth ratio between birth and initiation = Λ ∕Λ Growth ratio between initiation and birth * variables indicated by a star are measured from a linear fit of exponential elongation. 138 A popular idea dating back to the 1960's and still often used today to explain the coupling of 139 division and replication cycles is the initiation mass model. The observations that cell volume grows 140 exponentially with growth rate (Schaechter et al., 1958) and that, across a range of conditions, the 141 time between replication initiation and division is roughly constant (Helmstetter et al., 1968) led 142 Donachie to propose that the volume per origin of replication is held constant (Donachie, 1968). In 143 particular, the model proposes that initiation occurs when a cell reaches a critical volume. A simple 144 prediction of this model is that, for a given cell, the cell length at which initiation occurs should 145 be independent of other cell cycle variables such as the length at birth . However, as can be 146 seen in Figure 2A, we observe that the initiation length and birth length are clearly correlated 147 in all conditions, rejecting the initiation mass model.  Figure 2. Models for initiation control. A. The initiation mass model predicts that the length at initiation should be independent of the length at birth . However, we observe clear positive correlations between and in all growth conditions. B. In contrast, the length accumulated between two rounds of replication Λ is independent of the initiation size Λ , suggesting that replication initiation may be controlled by an adder mechanism.  However, within each growth condition, that period is clearly dependent on fluctuations in growth rate. B. The length accumulated from initiation to division is constant for each growth condition, suggesting an adder behavior for that period. In A and B, the Pearson correlation coefficient R and p values are indicated for each condition.

Replication initiation mass
at each origin until it reaches a critical amount, triggering replication, after which it is degraded and 160 starts a new accumulation cycle. Given that, for a molecule at constant concentration, the added 161 volume over some time period is proportional to the amount produced of the molecule, the result 162 of this process is that the cell adds a constant volume per origin Λ between initiation events 163 (with Λ = Λ − Λ where indexes stand for "initial" and "final" respectively, see Figure 1D and 164 Table 1 for more details). If replication is indeed triggered by such an adder mechanism, then one 165 would expect the observed added lengths Λ to be independent of the length Λ at the previous 166 initiation. As shown in Figure 2B, our data support this prediction.

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Connecting replication and division cycles 168 Having validated the multiple origins accumulation model for replication control, we now investigate 169 its relation to the division cycle. A common assumption is that the period from initiation to 170 division (classically split into C (replication) and D (end of replication to division) periods) is constant 171 and independent of growth rate (Cooper and Helmstetter, 1968;Ho and Amir, 2015). As visible in  The double-adder model postulates that E. coli cell cycle is orchestrated by two independent "adders", one for replication and one for division, reset at replication initiation. Both adders (shown as coloured bars) start one copy per origin at replication initiation and accumulate in parallel for some time. After the division adder (green) has reached its threshold, the cell divides, and the initiation adder (orange) splits between the daughters. It keeps accumulating until it reaches its own threshold and initiates a new round of division and replication adders. Note that the double-adder model is illustrated here for the simpler case of slow growth. The double-adder model 180 These observations motivated us to formulate a model in which the cell cycle does not run from 181 one division to the next, but rather starts at initiation of replication, and that both the next initiation 182 of replication and the intervening division event, are controlled by two distinct adder mechanisms. 183 In this replication-centric view, the cell cycles are controlled in a given condition by three variables: 184 an average growth rate , an average added length per origin Λ , and an average added length 185 Λ between replication initiation and division. In particular, we assume that these three variables 186 fluctuate independently around these averages for each individual cell cycle, and that all other 187 parameters such as the sizes at birth, initiation, and the times between birth and division or between 188 initiation and division, are all a function of these three fundamental variables. This double-adder 189 model is sketched in Figure 4 for the case of slow growth conditions: a cell growing at a rate and 190 of length initiates replication and thereby starts two adder processes. First, the cell will divide 191 when reaching a size Λ = + Λ = (Λ + Λ ) where = 2 is the number of replication origins. 192 Second, the next replication round will be initiated at a given origin after the corresponding Λ has 193 increased by Λ .

194
Simulations of the double-adder model 195 To assess to what extent our double-adder model can recover our quantitative observations, 196 we resorted to numerical simulations. We first obtained from experimental data the empirical correlated ( ≈ 0.3) between mother and daughter. Accounting for this mother-daughter correlation 205 in growth rate was found not to be critical for capturing features of E. coli cell cycle, but was included 206 in the model to reproduce simulation conditions of of previous studies. 207 As can be seen in Figure 5, the double-adder model accurately reproduces measured distri-208 butions and correlations at all growth rates. In particular, the global adder behavior for cell size 209 regulation naturally emerges from it ( Figure 5A). Similarly, the specific relation between length 210 at initiation and length at birth , which prompted us to reject the initiation mass model, is 211 reproduced by the model as well ( Figure 5B). Finally, the distribution of the number of origins at 212 birth, which reflects the presence of overlapping replication cycles is reproduced as well ( Figure 5D). 213 An exhaustive comparisons between experiments and simulations can be found in remarkably ≈ 1 indicates almost full independence in this case. 247 We can now systematically explore which set of variables best explains the correlation structure 248 in the data, by searching for the set of variables that maximizes independence . For example, 249 while a model that assumes a timer between initiation and division would treat the time as            to set the timing of division (Micali et al., 2018a) Amir, 2015), an adder for the regulation of replication initiation can be easily implemented at 313 the molecular level by having a "sensor" protein that builds up at each origin, and that triggers 314 replication initiation whenever a critical mass is reached at a given origin. If this sensor protein 315 is additionally homeostatically controlled such that its production relative to the overall protein 316 production is kept constant, than the average volume per origin will also be kept constant across 317 conditions.

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It is more challenging to define a molecular system that can implement the second adder that 319 controls division. The main challenge is that this adder does not run throughout the entire cell This observation not only suggests that, at replication initiation some local molecular event occurs 333 that will eventually trigger division at the same position, but it is also remarkably consistent with 334 the idea of an adder running only between replication initiation and division. One long-standing 335 idea that is consistent with these observations is that some molecular event that occurs during 336 replication initiation triggers the start of FtsZ ring formation, and that the timing from initiation to 337 division is controlled by the polymerization dynamics of the FtsZ ring (Weart and Levin, 2003). 338 Moreover, it has been shown that the constriction rate of the ring is strongly correlated with the 339 cell elongation rate (Coltharp et al., 2016). Therefore fast growing cells tend to divide faster than 340 slow growing ones, a compensation mechanism which might at least partially explain the adder 341 nature of the regulation.  Table 1. Correlations between length at birth of mother and daughter.

523
Other models 525 In this article we have shown that models relying on the concept of initiation mass, as well as those involving a constant timer from initiation to division are incompatible with measurements. Still, those models are able to reproduce a wide range of experimental measurements, and we wanted to understand where they would break. We give here two examples of such an analysis. In the first case we tried to reproduce the model proposed in Wallden et al. (2016). This model assumes that cells initiate replication around a specific initiation mass length and then grow for an amount of time depending on growth rate ( ) before dividing (1A). In panels B and C of 1 we show that we are successfully reproducing the model used e.g. in Figure 6 of Wallden et al. (2016). The histogram of the number of origins at birth shown in 1D shows a clear failure of the model where cells in slow growth conditions are all born with an ongoing round of replication in contradiction with experimental data (see e.g. Figure 3 of Wallden et al. (2016)). The second model we are investigating here has been recently proposed by Micali et al. (2018b). It uses an inter-initiation adder for replication regulation, and combines it with a classical adder (birth to division) without coupling those two regulation systems together explicitly. We simulated such a model with the added constraint that division can only occur if at least two origins are present in the cell. The results are shown in Fig.2. The model surprisingly reproduces most of the features of the experimental data with one exception: the initiation to division variable Λ is clearly not anymore an adder. This can be trivially explained: as the two mechanisms are uncoupled, an initiation at a large size automatically leads to a small Λ on average while an early initiation at small size leads to a large Λ on average.   Decomposition: the classic adder model. 559 In order to illustrate the functioning of the our decomposition approach, we apply it to the familiar case of the classic division adder model. By considering all possible combinations of standard cell cycle variables and estimating their independence, we find that the decomposition offering the most independent set of variables corresponds to the classic adder model defined by , and , as can be seen in 1.            which we have the most accurate measures, we note that shows a slight deviation from adder behavior. As shown here, we found that this could be corrected by slightly reducing the variance level of the division adder distribution (to 70% of its original value). As the initiation measurement is made imprecise for experimental (e.g. acquisition rate) and biological (variable cohesion of origins, it is reasonable to assume that we overestimate the variance of that parameter.