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General quantitative relations linking cell growth and the cell cycle in Escherichia coli


Growth laws emerging from studies of cell populations provide essential constraints on the global mechanisms that coordinate cell growth1,2,3. The foundation of bacterial cell cycle studies relies on two interconnected dogmas that were proposed more than 50 years ago—the Schaechter–Maaloe–Kjeldgaard growth law that relates cell mass to growth rate1 and Donachie’s hypothesis of a growth-rate-independent initiation mass4. These dogmas spurred many efforts to understand their molecular bases and physiological consequences5,6,7,8,9,10,11,12,13,14. Although they are generally accepted in the fast-growth regime, that is, for doubling times below 1 h, extension of these dogmas to the slow-growth regime has not been consistently achieved. Here, through a quantitative physiological study of Escherichia coli cell cycles over an extensive range of growth rates, we report that neither dogma holds in either the slow- or fast-growth regime. In their stead, linear relations between the cell mass and the rate of chromosome replication–segregation were found across the range of growth rates. These relations led us to propose an integral-threshold model in which the cell cycle is controlled by a licensing process, the rate of which is related in a simple way to chromosomal dynamics. These results provide a quantitative basis for predictive understanding of cell growth–cell cycle relationships.

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Fig. 1: SMK growth law does not describe the steady-state growth of E. coli.
Fig. 2: Donachie’s cell-mass relation \(\bar m\propto {\mathrm{e}}^{\lambda \left( {C + D} \right)}\) does not hold.
Fig. 3: Donachie’s constant-initiation-mass hypothesis breaks down.
Fig. 4: The linear relation given by equation (2) unifies the slow- and fast-growth regimes.

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Data availability

The data that support the findings of this study are available from the corresponding author on reasonable request. The deep-sequencing data used to characterize the λC and C period have been deposited at NCBI BioProject with the accession code PRJNA615952. Source data for Figs. 1a–d, 2a,b, 3a,d and 4a,c and Extended Data Figs. 3b–d, 5 and 7a,b are provided with the paper.

Code availability

Simulation data can be generated using the custom-made code and the parameter sets provided. The code that was used for the simulations in this study are available at


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We thank numerous colleagues for discussions. This research was financially supported by the National Key R&D Program of China (2018YFA0902701), the Strategic Priority Research Program (XDPB0305) and the Key Research Program (KFZD-SW-216) of the Chinese Academy of Sciences, Shenzhen Grants (JCYJ20170818164139781, KQTD2015033117210153 and Engineering Laboratory [2016]1194) to C.L., the National Natural Science Foundation of China (number 31700045) to H.Z., the National Natural Science Foundation of China (nos. 11804355 and 31770111), the Guangdong Nature Science Foundation (2018A030310010) and Shenzhen Grants (JCYJ20170413153329565 and KQTD2016112915000294) to Y.B, the National Key R&D Program of China (2018YFA0903400) to X.F., grant no. NIH R01 GM025326 to N.K. and grant no. NIH R01 GM095903 to T.H.

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Authors and Affiliations



C.L. initiated and directed the research. H.Z. set up the experimental system to evaluate the steady-state growth status with contributions from C.L.; H.Z. and M.J. quantified the growth rate and cell-cycle-related parameters with contributions from C.L., X.H., F.Z., and Y.W.; Y.B., X.F. and TH conceptualized the integral-threshold model and performed the numerical simulations and mathematical analysis with contributions from T.T. All of the authors analysed the results and wrote the manuscript.

Corresponding author

Correspondence to Chenli Liu.

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The authors declare no competing interests.

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Extended data

Extended Data Fig. 1 Composition of growth media used in the study.

The composition of the 32 growth media used in this study, detailed chemical information on the buffer and supplement (supp.) are available in Supplementary Table 1. Corresponding mass growth rate under steady-state growth status and the symbols used in figures for each growth medium are presented as well.

Extended Data Fig. 2 Steady-state growth is validated by monitoring mass and cell-number growth simultaneously.

A fundamental but often ignored point in bacterial physiology studies is the establishment of steady state growth15,16, where the total cell mass growth rate λm is identical to cell number growth rate λc. In an exploratory experiment on early growth of cells in medium M1 (Extended Data Fig. 1) after inoculation from seed culture, overnight cultured cells were inoculated into pre-warmed medium with starting OD600 at 0.02, then grown without further dilution. The OD600, cell number, and population-averaged cellular origin number were characterized at different time points (Methods). We found that the total cell mass growth (a) quickly entered exponential phase (the grey area, from 30 to 140 minutes), but cell number growth (b) showed a considerable lag. As a result, the average cellular mass (c) and origin number (d) varied throughout the exponential growth phase, which clearly indicated that the cells were not in true steady state. By employing serial dilutions (see Methods), we found that cells grown for more than 10 mass doublings after inoculation from seed-culture were safely in steady state. This was a key step in ensuring the validity of the findings presented in this study. Following this protocol, we show in (e) that experimental cultures in 14 representative growth media covering the entire range of growth rates examined in this study lie on the steady state line λm=λc. Representative growth curves in the steady state are shown in (f): After 10 mass doublings after inoculation from seed-culture, OD600 and the cell number concentration are plotted versus time, taking the dilution ratio into account to plot the ‘sawtooth’ behavior as a single smooth curve. The cell number (red lines) and cell mass (blue lines) growth curves formed two parallel lines in semi-log plots, indicating the steady-state growth had been achieved.

Extended Data Fig. 3 Tight correlation of different measures for cellular mass or size.

a, Density distributions (in Probability Density Function, PDF) for cell volume normalized by average cell size, as quantified by automated image analysis (Methods), for cells taken from the conditions described in Fig. 1d. Distributions for cells at comparable growth rates from Gray et al.18 were taken for comparison. When normalized by mean cell size, the different distributions appear very similar. b-d, The dry weight (DW) per cell (b), relative FSC (forward scatter) (c), and cell volume (d) plotted against the OD600ml per 109 cells. All three measures are linearly correlated with the OD600ml per 109 cells. The cell volume is expected to be a precise quantitative measure of cell size. However, data sets from different published studies5,18,40,43 show an approximately two-fold difference for the same strain or closely related stains under similar growth conditions, possibly due to the difficulty in quantifying the actual cell diameter based on microscopic images. Given the variability in the measured FSC or cell volume, and the convenience and robustness in quantifying the cell number concentration and OD600, we used the OD600ml per 109 cells as the population-averaged cellular mass (\(\bar m\)) for the rest of the current study. Symbols and error bars in panels b-d (except for the y axis of panel d) represent the mean±SDs of the; many of the error bars were smaller than the size of the symbols. Symbols and error bars on y axis of panel d represent mean with 95% CI of population-averaged cell volume. Sample size and mean value for each symbol are provided in Extended Data Fig. 8.

Extended Data Fig. 4 Linear relation between λ(C+D) and growth rate λ.

Comparison between the data in this study and those extracted from Table 3 (a) or 4 (b) in Michelsen et al.22 c, Comparison between the data in this study and those extracted from Allman et al21. The C and D periods were characterized by resolving the DNA histogram of cells in batch culture (a, c) or continuous culture (b). d-e, The semi-log relationship between \(\bar o\) and growth rate. d, Comparison between the data in this study and those extracted from Si et al5. Their \(\bar o\) were characterized by using run-out protocol followed by Hoechst 33342 staining and microscopic image analysis5. e, Comparison between the data in this study and those extracted from Zhu et al23. Their \(\bar o\) data were derived from replication origin per genome, and genome equivalents per cell for cells in batch culture. The straight lines represent Eq. 1 (d,e) or its derived form (a-c). The relationship between C+D period and growth rate is also presented in the inset to each panel. The solid line in the inserted plots represents Eq. 1’s derivative. The error bars for the gray filled circles in panels a-e represent the mean±SDs of the data; many of the error bars were smaller than the size of the symbols. Sample size and mean value for each symbol are provided in Extended Data Fig. 8. Data points other than the gray filled circles are presented as their original value in the publications, with no further statistics applied.

Extended Data Fig. 5 Negative correlation of initiation mass with relative DnaA protein concentration.

Shown are the initiation mass from Fig. 3a and the corresponding relative DnaA concentrations from Fig. 3d. Symbols and error bars represent the mean±SDs of the data. Sample size and mean value for each symbol are provided in Extended Data Fig. 8.

Extended Data Fig. 6 Measurement of the C period by deep sequencing.

a, Definition of the relative chromosomal location (\(m^\prime\)). To characterize the C period, the genome was binned into over 900 fragments of size 5,000bp. The relative chromosomal location for each fragment (\(m^\prime\)) is defined by its relative location between oriC (\(m^\prime = 0\)) and terC (\(m^\prime = \pm 1\)). b, Dependence of relative gene copy number (Xc) on chromosome location for cells grown in M1 (Extended Data Fig. 1). The relative gene copy number was obtained by normalizing the deep sequencing counts for each fragment to the count number for the fragment containing terC (Methods). c, Linear correlation between the logarithm of the relative copy number of the fragment \(log_2^{X_C}\) and \(m^\prime\). Representative plots of 4 biologically independent samples have been presented. Colors represent growth media: blue, green, navy, and red correspond to M1, M3, M13, and M23, respectively (Extended Data Fig. 1).

Extended Data Fig. 7 The Linear relation between C and C + D.

a, The logarithm of population-averaged cellular origin number, which equals to λ(C+D), is proportional to λC, which was determined by deep sequencing as described in Methods and Extended Data Fig. 6. The direct proportionality between the independently measured λ(C+D) and λC suggests that the C period is proportional to the sum of the C and D periods. b, The population-averaged cellular mass \(\bar m\) scales linearly (R2 > 0.94) with λC, a measure of average number of replication positions per chromosome, with best-fit slope \(m_0^\prime\)=1.62 ± 0.03 OD600ml/109 cells or 0.89 ± 0.04 pg DW/cell. Symbols and error bars represent the mean±SDs of the data. Sample size and mean value for each symbol are provided in Extended Data Fig. 8.

Extended Data Fig. 8 Sample size and mean value of the experiments in this study.

Thirty-two different kinds of growth media were used in this study. Here we summarize the number of biologically independent experiments for each quantity we characterized. Note that the growth rate (λ), population-averaged cellular mass OD600ml per 109 cells (\(\bar m\)), population-averaged cellular oriC No. (\(\bar o\)), and the derived λ(C+D), C+D, initiation mass (\(m_i\), in units of 10–9 OD600 ml) were examined simultaneously for a same experimental culture, so these parameters have the same replicate number.

Supplementary information

Supplementary Information

Supplementary Notes 1–5, Figs. 1–7, Tables 1 and 2, and references.

Reporting Summary

Supplementary Data 1

Summarized dataset for Figs. 1a–d, 2a,b, 3a,d and 4a,c, and Extended Data Figs. 3b–d, 5 and 7a,b.

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Zheng, H., Bai, Y., Jiang, M. et al. General quantitative relations linking cell growth and the cell cycle in Escherichia coli. Nat Microbiol 5, 995–1001 (2020).

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