## Abstract

A cell can divide only upon completion of chromosome segregation; otherwise, its daughters would lose genetic material. However, we do not know whether the partitioning of chromosomes is the key event for the decision to divide. We show how key trends in single-cell data reject the classic idea of replication-segregation as the rate-limiting process for cell division. Instead, the data agree with a model where two concurrent processes (setting replication initiation and interdivision time) set cell division on competing time scales. During each cell cycle, division is set by the slowest process (an “AND” gate). The concept of transitions between cell cycle stages as decisional processes integrating multiple inputs instead of cascading from orchestrated steps can affect the way we think of the cell cycle in general.

## INTRODUCTION

Correct chromosome replication and segregation are necessary for cell division to ensure that genetic material is propagated correctly (*1*, *2*). In bacteria such as *Escherichia coli*, a known pathway prevents cells from dividing if the chromosomes interfere with the cytokinesis machinery (*3*). However, whether completion of segregation is typically the bottleneck process for the decision to divide has never been stringently tested on single cells (*4*).

New dynamic single-cell data have revived the classic debate on the determinants of cell division, but the recent literature is fragmented into different and contrasting models (*4*). Most studies take the classic view (*5*) that after a size-regulated “B period” from cell division to initiation of replication, a fixed “C + D period,” comprising a “C period” to copy the genome and a “D period,” needed to complete segregation and running from termination to cell division (Fig. 1A), is rate limiting for cell division (Fig. 1B), but these studies do not agree on the underlying mechanisms (*6*, *7*). A recent study by Harris and Theriot (*8*) formulates the opposite hypothesis. The main assumption of this alternative view is that the rate-limiting process for cell division is instead the completion of the septum (Fig. 1B); consequently, chromosome segregation is never rate limiting for cell division.

More specifically, the authors provide evidence that surface synthesis rate is proportional to volume; they propose a model where division is set by a threshold surface to synthesize the septum (*8*). This model recapitulates the empirical size control strategy followed by these cells, whereby the added size is nearly constant, regardless of initial size (the so-called adder behavior) (*8*–*10*). With these contrasting results, the question of which process drives cell division becomes pressing (Fig. 1B). Our central finding is that both processes occur concurrently to set cell division (Fig. 1, C and D).

## RESULTS

To support this point, we started our analysis from the available experimental data. Single-cell growth in these data is well described by an exponential, with some cell-to-cell variability in the growth rate (*9*, *11*). We have focused on two underrated correlation patterns measured recently for the C + D period. Figure 2 (A and B) shows these patterns. First, we find that the growth of the cell during the C + D period, quantified by the log ratio of the division volume and the initiation volume, is anticorrelated with cell size at initiation (Fig. 2A). This pattern is very consistent across strains, measurement methods, and conditions, but is overlooked by current models, which assume that replication is the bottleneck process (*6*, *7*). Second, we find that the duration of the C + D period is anticorrelated with the growth rate of individual cells, with a near-inverse relationship (Fig. 2B), as reported by Wallden and coworkers (*6*). We note that, because of experimental limitations, the termination event has not always been recorded and, in most cases, only the cumulative C + D period is available. However, available experimental data (*12*) indicate that coupling of growth in the C + D period with initiation size mainly comes from the D period.

These patterns are not compatible with the classic assumption of replication driving division: There are no available justifications for these correlations between timing and either growth rate or cell size. If replication is always the bottleneck process, then division should occur on average at a fixed time after initiation of replication. Assuming that this fixed time is not coupled to cell size or growth rate, both plots in Fig. 2 (A and B) would show no correlation (Fig. 2, C and D, orange lines). Note that the plot in Fig. 2A is equivalent to testing the correlation of the final volume with initiation volume or of the added volume during C + D with initiation volume (*13*, *14*). For example, if replication is always the bottleneck process and division happens at a fixed average time after initiation of replication, then we expect the final volume to be maximally correlated to initiation volume, but this is not the case in the data.

In contrast, the C + D patterns emerge naturally from a model in which the division event is set by two concurrent processes. In a concurrent processes framework, the C + D period is the juxtaposition of two processes: the time by which the replication-related process is ready to divide, defined as C + D′, and the time by which the cell interdivision process (e.g., the septum) is completed. The measurable C + D is the time for the cell to really divide and is set at the single-cell level by the slowest between these two processes. Whenever division is set by the cell-related interdivision process, the cells that initiate replication at a larger size will grow less during the C + D period (because, in this case, the final size ignores the initiation size), which gives the pattern of Fig. 2A. Equally, each time the interdivision process is slowest, the D period will tend to be shorter for cells growing faster than average due to the size-based interdivision control. For example, if the interdivision process adds an average constant size (*8*, *9*), this target size will be reached faster by single cells growing faster than average, which will decrease the duration of their C + D period. This yields the pattern shown in Fig. 2B.

To go beyond these qualitative considerations, and to produce testable quantitative predictions on the assumption of concurrent processes setting cell division, we formulated and solved a mathematical model based on this principle (see Materials and methods). In this model, two size-dependent processes exist. A chromosome process setting initiation and a cell-related process trying to set division run in parallel, and division follows the slowest process (as in an AND gate) between completion of the interdivision process and completion of a C + D′ period after initiation (Fig. 1D). We note that the available data do not allow us to make a clear distinction between the different chromosome-related processes (e.g., DNA replication and segregation). Biologically, cell division is limited by nucleoid occlusion (*3*, *15*); hence, a chromosome-aware cell division process should be limited at least by completion of segregation. Our minimal description of the chromosome cycle complies with the classic view and posits that a size-uncoupled C + D period is minimally required to divide after initiation. However, things could be more complex (*16*). Future, more detailed, data on cell cycle transitions in single cells may help to develop more mechanistic models of the decisional process leading to cell division.

We solve the model both analytically and by numerical simulation. Figure 2 (C and D) shows the predicted patterns of the C + D period (light blue lines). The model reproduces both correlation patterns shown in Fig. 2 (A and B). Figure 2C also shows that when replication is limiting, both slopes are flat (Fig. 2, C and D, orange lines), indicating that the classic framework is too restrictive. Instead, models, assuming that replication is never limiting (*8*), may reproduce the trends in Fig. 2 (A and C), but the slope is quantitatively too strong for the data in Fig. 2A. Specifically, if replication is never limiting, then the size at division *V*_{f} is independent from the size at initiation *V*_{I}. The growth between initiation and division (*y* axis of Fig. 2A) is log(*V*_{f}/*V*_{I}) = log(*V*_{f}) − log(*V*_{I}). If log(*V*_{f}) is independent of *V*_{I} (because replication is never limiting), then the slope is equal to −1. The data reside in the intermediate regime, where competition between the two processes setting cell division is relevant. While one cannot rule out all possible scenarios, it is possible to show that available models, assuming that replication-segregation is always limiting for cell division, fail to reproduce these patterns [see (*17*), which provides an extensive technical analysis comparing different models]. Wallden and coworkers (*6*) take the growth rate dependence of the C + D period in Fig. 2B as a basic assumption. However, their model still fails to reproduce Fig. 2A. Overall, the concurrent processes assumption explains the patterns for the C + D period naturally from the conceptually simple assumption of competing time scales between the different processes that set cell division.

The fact that the model captures the data does not depend on a specific parameter set but emerges from the hypothesis of concurrent processes. This point can be shown by direct analytical estimates of the patterns in Fig. 2 (see Materials and methods for details). Our model (see Fig. 1D and Materials and methods) is specified by the two size control parameters and [ranging from 0 (strong control) to 1 (no control)], by the characteristic initiation and division sizes encoded by the two concurrent processes, and by the intrinsic cell-to-cell stochasticity of their duration (see Materials and methods). The values of these parameters affect the probability *p*_{H} that the interdivision process is rate limiting, which is the only relevant emergent parameter of the model. For any parameter set, the plots in Fig. 2 depend on *p*_{H} only: They deviate from constancy if replication is not always limiting, i.e., when *p*_{H} deviates from 0, and the case of replication never limiting (*p*_{H} = 1) appears quantitatively too extreme to fit the data. In the data, we estimate from the slopes that *p*_{H} is between 0.25 and 0.75 (see Materials and methods), well within the parameter region in which there is actual competition between the two processes. Thus, regardless of parameter values and details, only competing time scales efficiently reproduce the data.

We found that similar conclusions apply to other testable predictions of the model, beyond the C + D period. First, the model predicts that competition between concurrent processes should affect the relation between the growth in the B and C + D periods. To test this prediction, we use the method introduced by Chandler-Brown and coworkers in yeast (*18*), comparing growth (measured here by log ratio of final to initial volume) in the two periods. If replication is never limiting, then fluctuations in the initiation size make the B period longer and the C + D period shorter, affecting, in opposite directions, growth in the B and C + D periods, which become anticorrelated. If replication is always limiting, then growth in the C + D period should instead be independent from growth in the B period. Our results show that the concurrent processes model predicts an intermediate situation between these two, which is where all the data are found (Fig. 3, A and B). The argument to understand the slopes in Fig. 3B is similar to the one that explains the slope of Fig. 2A. The growth during C + D, log(*V*_{f}/*V*_{I}) (plotted on the *y* axis), can be expressed as the difference of the growth during the whole cycle and the growth during the B period, log(*V*_{f}/*V*_{I}) = log(*V*_{f}/*V*_{0}) − log(*V*_{I}/*V*_{0}). If replication is never limiting, then log(*V*_{f}/*V*_{0}) is independent of log(*V*_{f}/*V*_{I}), while log(*V*_{I}/*V*_{0}) and log(*V*_{f}/*V*_{I}) are in a linear relation with slope one. Equivalently, if the size at initiation is irrelevant for the size at division, then the growth in the B period must be anticorrelated (with slope −1) to the growth in the C + D period. If instead replication is always limiting, then the growth during the C + D period is uncoupled to both size at initiation and initial size, hence independent on log(*V*_{I}/*V*_{0}). Second, competition between concurrent processes should also affect measurable patterns for the interdivision cycle, where abundant data are available. Motivated from the trend in Fig. 2B, we focused on the anticorrelation reported between interdivision time and individual cell growth rates (*19*). Our results (Fig. 3, C and D) show that the model captures the decreasing trend of this correlation with average growth rate. Once again, rather than the specific numerical values of the parameters, we find that the data take intermediate values between the two extreme cases of replication always or never limiting division.

## DISCUSSION

In *E. coli*, the average cell size varies across conditions, but the average size per replication origin at initiation remains nearly constant (*20*). This old observation was recently proven to be very robust to perturbations (*21*, *22*). Classically, this observation was interpreted as evidence for a replication-limited cell division circuit (*5*, *20*). However, since we have shown that single-cell behavior challenges this interpretation, the classic view needs to be revisited. In the hypothesis of concurrent processes, different processes (e.g., a chromosome cycle and an interdivision cycle) are sometimes the slowest to be completed and set cell division in a given condition. This also translates into a concurrence of size scales and can happen if the characteristic division sizes encoded by the different processes are similar. However, since mean cell size per origin is roughly constant across conditions, we can surmise that the division size encoded by the chromosome process has to grow with population growth rate, along with the number of origins. If this hypothesis applies, then the size scale of the cell-related interdivision process also has to grow, or concurrence would be lost by varying the growth media. In addition, different analyses indicate that a single size scale governs the whole size distribution across conditions [a behavior called “scaling” (*10*, *13*, *19*)], suggesting either a matching or a perfect dominance of a single encoded size.

If the two size scales encoded by a chromosome cycle and a cell-related interdivision cycle are coupled, causation could act in either (or both) directions, i.e., the characteristic cell size could be caused by the encoded initiation size per origin, or vice versa (or be the result of a mutual coupling) (*23*, *24*). Notably, deletion of the nucleoid occlusion SlmA, which prevents division in the presence of unsegregated chromosomes, leaves mean cell size unaffected (*3*, *15*), in line with the idea that the replication-associated process does not always drive cell division. The exact causal (and temporal) chain of events might be detectable by examining single cells under perturbations and during nutrient shift experiments. Last, if we abandon the long-standing assumption that replication initiation always drives cell division in *E. coli* favoring a scenario of concurrent processes, then we are forced to rethink the regulatory process linking replication to cell division (*23*), and alternative hypotheses should be revisited in data (*16*, *25*).

More generally, following the pioneering views of Boye and Nordström (*23*), our findings support a change of perspective on the cell cycle. We can draw a comparison with the way the cell cycle is described in animal cells. A prevalent narrative of the cell cycle, particularly for (higher) eukaryotes, is a succession of well-determined transitions under a common master clock. However, this paradigm struggles to find the “final trigger” of cell cycle intervals (*26*). Instead, in line with our concurrent processes view, emerging evidence supports the idea that a key aspect could be the integration of multiple decisions (*26*, *27*). For example, cell division in higher eukaryotes requires coordination of the actin cortex and the spindle (*27*), with required cross-talk between these two systems that is analogous to the one addressed here. The concept of concurrent processes on competing time scales can be important to characterization of cell cycle stages other than division, as well as the cell cycles of other organisms. We expect that this different perspective will affect future analyses of the coordination of cell division and other cell cycle stages with metabolic and physiological cues.

## MATERIALS AND METHODS

### Data analysis

Figures 2 and 3 (A and C) use data on SeqA foci formation from (*6*, *12*). The data contain information on cell division, size versus time, and replication initiation time of single tracked cells. Figure 3D also uses datasets from (*6*, *10*, *19*, *28*). The data contain information on cell division and size versus time of single tracked cells for a total of about 10^{6} cells. The dataset in (*6*) contains about 400 cells in the slow growth condition and about 90 pairs of mother-daughter cells in the intermediate growth condition. The dataset in (*12*) contains 80 cells. For the slow growth data in (*6*) and for (*12*), cells typically have a single DNA replication round during a cell division cycle, and mother-daughter tracking is not necessary. For the intermediate growth conditions of (*6*), cells typically have two overlapping rounds of DNA replication, and mother-daughter tracking is needed to define the C + D period, because it spans two cell cycles. The plots in Figs. 2 and 3 show binned averages (where the *x*-axis variable was subdivided into bins of equal size); the averages of the *y*-axis variables were considered only for bins containing more than five data points.

### Mathematical model and analytical predictions

We describe here the mathematical framework for the concurrent processes model and the main analytical results (see Fig. 4). We point the reader with modeling expertise to related work, where a similar framework was established to study interdivision correlations (*13*, *14*), and to a related manuscript addressed to a technical audience, where the same formalism is used for a detailed comparison of the models available in the literature (*17*).

*Control parameters and their interpretation*. Figure 4A shows a list of the variables and control parameters of the model. To quantify the coupling between size and cell cycle progression (and growth) for each cell cycle interval *X*, the model uses the control parameters . These parameters range from 0 (size threshold) to 1 (timer, no size control). The value of can be estimated as the slope of the plot of final versus initial logarithmic size during the cell cycle interval. For example, is the slope of the plot of the logarithmic size at division versus the logarithmic size at initiation. Equivalently, is one minus the slope of the so-called size-growth plot (*29*) of relative growth during a cell cycle interval versus the cell size at the entry of the interval (see Fig. 2, A and C).

In the model, initiation of replication is set by a size-coupled process acting between consecutive initiations, with a control parameter , which describes the strength of the size correction. When , initiation follows an adder (*7*), while for , initiation is triggered by a critical size (*6*) (a sizer).The case of overlapping rounds of DNA replication (*30*) is described by this model by the assumption that the initiation circuit encodes a size in units of replication origins (*7*). Similarly, division size is set by a process with control parameter , which is an adder when .

The qualitative predictions of the model are very robust to variations in the actual mechanisms of cell cycle progression, e.g., whether initiation is controlled by a sizer per origin or an adder per origin, or something else, specified by the parameters and in the model. This point can be shown by direct analytical estimates.

*Stochasticity*. Stochasticity parameters describe size-independent cell-to-cell variability in duration of the interdivision and interinitiation processes. The “C + D′” period duration is assumed to be a size-independent Gaussian random variable with assigned mean and variance (a “timer”). Last, cells are assumed to grow exponentially, and the growth rate is a random variable with prescribed distribution (e.g., Gaussian).

*Formulation of the model*. The model (see Fig. 4B) is composed of two concurrent stochastic processes: An interdivision process sets a possible division (log) size *q*_{H} (or equivalently a division time) as a function of the size at birth, and a chromosome cycle defines the initiation size *q*_{I}, and thus also indirectly a concurrent possible division time (and size *q*_{R}) compatible with the termination of replication and segregation.

The defining equations of the model are(1)where *i* is a cell cycle index (the cell with index *i* is a daughter of the cell with index *i* − 1), *q*_{X} = log(*V*_{X}) are logarithmic sizes, and δ*q*_{X} = *q*_{X} − 〈*q*_{X}〉 are fluctuations. The two parameters and set the characteristic division and initiation size scales of the two concurrent processes. Last, η_{X} are noises describing cell-to-cell variability (the growth rate α is a random variable in our simulations but, for simplicity, will be considered constant in the following calculations). Equation 1 states that size at division and size at initiation are corrected by the interdivision and interinitiation processes, respectively, with control parameters and (*13*).

The logarithm of the final size encoded by the replication-related process in a given cell cycle *i*, *q*_{R}, can be expressed as(2)which can be rewritten as(3)where(4)

and is the time for the replication-related process to be ready for cell division.

The concurrent processes condition states that the final size is the one dictated by the slowest process; hence(5)

*Mean-field approximation*. To compute the effective final size, we rewrite Eq. 5 as(6)where *s*^{i} is a random variable with values in {0,1}. The probability that *s*^{i} is 1 depends on *q*_{0} and *q*_{I}. We consider the approximation where *s*^{i} = 1 with probability *p*_{H} independently of *q*_{0} and *q*_{I} (but dependently on their averages 〈*q*_{0}〉 and 〈*q*_{I}〉 and noises, see below). We verified that this approximation (which we refer to as “mean-field” approximation) works very well with simulations in the noise range of empirical data. *p*_{H} is the probability that the interdivision process is rate limiting, a relevant outcome of the model (depending on the parameters), and is estimated below.

*Predictions of the model*. In this model, the value parameter is not an input but the result of the interplay between the concurrent processes. It is measured by the slope of Fig. 2 (A and C) or equivalently by the covariance(7)where is the variance of the logarithmic initiation size. The equivalence between these two definitions is explained as follows. Figure 2 (A and C) implies the condition (where *ξ*_{C+D} is an effective noise term characterizing the cell-to-cell variability). Computing the covariance between the two fluctuations leads to Eq. 7.

Some algebra on the combination of Eqs. 1, 6, and 7, leads to the equation(8)

The ratio in the right-hand side of Eq. 8 is strictly 1 when *p*_{H} = 0 (i.e., replication is always rate limiting) and strictly 0 when *p*_{H} = 1 (i.e., replication is never limiting). Instead, intermediate values of this ratio can only be attained for intermediate values of *p*_{H} for many combinations of the other parameters. Note that Eq. 8, given and , allows us to directly estimate *p*_{H}. In the data, since empirically , we estimate that *p*_{H} ≈ 0.24 to 0.69 for a sizer at initiation () and *p*_{H} ≈ 0.3 to 0.75 for an adder between initiations ().

Let us now consider the pattern in Fig. 2B. When replication sets division (with probability 1 − *p*_{H}), the mean initiation volume 〈*V*_{I}〉 dictates the duration of the C + D period (which corresponds to the C + D′ period) independently on the growth rate; when replication is not the bottleneck (with probability *p*_{H}), the final size will be initiation independent and encoded by the interdivision process 〈*V*_{H}〉, and for a fixed growth rate α, the duration of the C + D period will be proportional to (1/α)(〈*q*_{H}〉 − 〈*q*_{I}〉) =: χ/α, where χ is roughly the log ratio of the sizes encoded by the two processes χ ≈ log(〈*V*_{H}〉/〈*V*_{I}〉) [assuming that log(〈*V*_{X}〉) ≈ 〈*q*_{X}〉, which is valid for small noise (*7*, *13*)]. The effective duration of the C + D periods is estimated as(9)

Equation 9 is approximate because, at the fixed single-cell growth rate, the probability that the interdivision sets division varies compared to *p*_{H} (which is its average value).

Equation 9 predicts that the duration of the period is a sum between a constant and an inverse relationship with the single-cell growth rate. The assumption that replication limits division can only predict the constant part, at odds with the data. Conversely, Fig. 2 shows how the data are in line with the predictions of the concurrent processes model.

Equation 9 contains 〈*q*_{H}〉 and , neither of which is directly accessible from experiments. We could turn it in an expression containing the final (log) size 〈*q*_{f}〉, which is measured directly in experiments. Equation 6, under the mean-field approximation described above, gives(10)where we have used independence of C + D′ duration. Substituting this expression in Eq. 9, yields(11)which can be solved for the only unknown parameter, the product to estimate its value from the data, leaving no adjustable parameters for the predictions shown in Fig. 2D.

### Model parameters

We discuss here the values of the model parameters used in the plots and how they were fixed. The cyan lines in Fig. 2C represent theoretical predictions and have a negative slope corresponding to Eq. 8 with *p*_{H} = 0.48, 0.3, and 0.75 for an adder between initiations (see below), respectively, for the Wallden *et al*. slow growth, Adiciptaningrum *et al*., and Wallden *et al*. intermediate growth datasets (*6*, *12*). These values were fixed from Fig. 2A and Eq. 8. The same values of *p*_{H} are used in all other plots. In Fig. 2D, the orange lines are the best fit for the case where replication is limiting [corresponding to the model of (*7*)]. The average duration of the C + D period is set from the empirical averages to 160, 102, and 75 min for the Wallden *et al*. slow growth, Adiciptaningrum *et al*., and Wallden *et al*. intermediate growth datasets, respectively. Cyan curves follow Eq. 9 with *p*_{H} fixed as above, and was fixed from Eq. 11 to the following values: 49 min (Wallden *et al*. slow growth), 52 min (Adiciptaningrum *et al*.), and 41 min (Wallden *et al*. intermediate growth).

Figure 3 (B and D) shows the result of numerical simulations, where the parameters were constrained from the data. The plots in Fig. 2 fix the values of *p*_{H} and for the three datasets in Fig. 3B, while Fig. 3D corresponds to a wide range of growth conditions and required more general choices. Specifically, the numerical simulations use the following input parameters, directly fixed from the data: (i and ii) The average growth rate 〈α〉 and its variance . In Fig. 3D, the average growth rates range from 0.002 to 0.04 min^{−1} to match the experimental values. The variance of the growth rate was chosen by keeping a constant coefficient of variation (CV) of 0.15. The growth rate was a normal or log-normal random variable (the two choices do not affect the results, see below), extracted independently for every cell cycle. (iii and iv) The average added volume per origin between consecutive initiations 〈*ν*_{I}〉 and its variance . The average added volume per origin was fixed to 0.45 μm^{3} as measured in (*6*), and the variance was fixed to a constant CV of 0.15. The added volume between consecutive initiations was assumed to be log-normally distributed (and extracted independently for every cell cycle). (v and vi) The average time needed for replication and segregation (the C + D′ period) and its variance . In Fig. 3D, the average duration of the C + D′ period was set to 45 min independently from the average growth rate, and its CV was maintained constant and equal to 0.2 for the simulations corresponding to replication always limiting (orange lines) and concurrent processes (cyan lines). The average duration of the C + D′ period was set to 0 for the simulations where replication is never limiting (purple lines). was assumed to be a normal random variable (extracted independently for every cell cycle). Note that the average added volume per origin between consecutive initiations, the average growth rate, and the average C + D′ duration set the characteristic division size of the interinitiation process, . (vii and viii) The average interdivision added volume 〈Δ_{H}〉 and its variance . Note that 〈Δ_{H}〉 sets the characteristic size of the interdivision process . In Fig. 3B, 〈Δ_{H}〉 was fixed to achieve the same values of *p*_{H} as in Fig. 2 (B and D). In Fig. 3D, the interdivision added volume was fixed so that for the case of concurrent cycles, the replication-related process and the interdivision process would compete to set division in the same way across all conditions. Specifically, we chose 〈Δ_{H}〉 = 〈*ν*_{I}〉 exp(45 min ⋅ 〈α〉) for the case of concurrent processes (Fig. 3D, cyan line). The case where replication is never limiting (purple line) was simulated by assuming the same 〈Δ_{H}〉 but setting to zero. Last, we set 〈Δ_{H}〉 = 0 to simulate the case of replication always limiting (orange line). With these choices, *p*_{H} ≈ 0 for the replication always limiting case (Fig. 3D, orange line), *p*_{H} ≈ 0.6 for concurrent cycles (cyan line), and *p*_{H} ≈ 1 for the replication never limiting models (purple line). The variance was set to a constant CV of 0.15 over the different growth conditions. The interdivision volume was assumed to be a log-normally distributed random variable as above.

We chose to present the data with (adder) simulations, but we explored other values of these parameters, and in particular and , initiation triggered at fixed size per origin, finding very robust results. In addition, while Fig. 3 follows a specific set of parameters, we explored systematically in simulations different characteristic sizes and noise levels for the two processes. For instance, as mentioned above, the simulations for Gaussian-distributed growth rate, interdivision volume, and added volume per origin between consecutive initiations show no substantial differences in the trends shown in Fig. 3 (data not shown).

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## REFERENCES AND NOTES

**Acknowledgments:**We are very grateful to N. Kleckner, C. Cadart, S. van Teeffelen, A. Amir, B. Sclavi, S. Tans, and I. Iuliani for their feedback.

**Funding:**This work was supported by the International Human Frontier Science Program Organization (grant HFSP RGY0070/2014). M.O. was supported by the “Departments of Excellence 2018–2022” grant awarded by the Italian Ministry of Education, University and Research (MIUR) (L. 232/2016). J.G. was supported by an Omidyar Postdoctoral Fellowship at the Santa Fe Institute. G.M. was supported by grant nr. 31003A_169978 from the Swiss National Science Foundation to Martin Ackermann.

**Author contributions:**M.C.L., M.O., G.M., and J.G. designed the research and contributed with key ideas and analyses at different stages. G.M. relentlessly performed data analysis, model simulations, and analytical calculations. J.G. provided a forceful drive for the analytical calculations. M.C.L. conceived the project and wrote the paper, with constant help and feedback from the other authors.

**Competing interests:**The authors declare that they have no competing interests.

**Data and materials availability:**The code used to simulate the concurrent cycles model is available on Mendeley Data and can be accessed at https://data.mendeley.com/datasets/tgvykrrph4/. All other data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.

- Copyright © 2018 The Authors, some rights reserved; exclusive licensee American Association for the Advancement of Science. No claim to original U.S. Government Works. Distributed under a Creative Commons Attribution NonCommercial License 4.0 (CC BY-NC).