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

Abstract

Growth laws emerging from studies of cell populations provide essential constraints on the global mechanisms that coordinate cell growth^1-3. 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 consequences^5-14. 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.

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 growth^,, where the total cell mass growth rate λ_m is identical to cell number growth rate λ_c. 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 λ_m=λ_c. 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. 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 OD₆₀₀·ml per 10⁹ cells. All three measures are linearly correlated with the OD₆₀₀·ml per 10⁹ cells. The cell volume is expected to be a precise quantitative measure of cell size. However, data sets from different published studies^,,, 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 OD₆₀₀·ml per 10⁹ cells as the population-averaged cellular mass $(\overline{m})$ 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. c, Comparison between the data in this study and those extracted from Allman et al. 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 $\overline{o}$ and growth rate. d, Comparison between the data in this study and those extracted from Si et al. Their $\overline{o}$ were characterized by using run-out protocol followed by Hoechst 33342 staining and microscopic image analysis. e, Comparison between the data in this study and those extracted from Zhu et al. Their $\overline{o}$ 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′). 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′) is defined by its relative location between oriC (m′ = 0) and terC (m′ = ±1). b, Dependence of relative gene copy number (X^c) 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 $lo{g}_2^{X_c}$ and m′. 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 $\overline{m}$ scales linearly (R² > 0.94) with λC, a measure of average number of replication positions per chromosome, with best-fit slope $m_0^{\prime }=1.62\pm 0.03$ OD₆₀₀·ml/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 ($\overline{m}$), population-averaged cellular oriC No. ($\overline{o}$), 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.

Fig. 1 |. SMK growth law does not describe the steady-state growth of E. coli.

The SMK growth law states that $\overline{m}\infty e^{\lambda T}$, where T is a growth-rate-independent constant of ~1 h, but this is an incorrect description, even in the fast growth regime, where λ > 0.7 h⁻¹. a-d, Population-averaged DW per cell (a), OD₆₀₀ ml per 10⁹ cells (b), relative FSC (normalized to the FSC of cells grown in M18) (c) or cell volume from microscopy images (d) plotted against growth rate on semi-log axes. All of the experimental cultures were ensured to be in steady state (Extended Data Fig. 2; see Methods). The different symbols indicate different growth media (Extended Data Fig. 1). For a–c and the x axis of d, data are mean ± s.d.; for the y axis of d, data are mean with the 95% confidence intervals (CIs) of population-averaged cell volume. Sample sizes and the mean values for each symbol are provided in Extended Data Fig. 8. Many of the error bars are smaller than the size of the symbols. The straight dashed lines were semi-log fit to the data in fast-growth regime (λ > 0.7 h⁻¹); the slopes of the lines for a–d are 0.82, 0.83, 0.95 and 0.67, respectively, which—in most cases—differs substantially from the value 1 asserted by the SMK growth law and also required by Donachie’s cell-mass relation (see the main text). Data from the slow-growth regime deviated from the lines in all cases. e, Comparison between the optical density data from b (grey circles) and the original data from Schaechter et al.. The error bars of the grey circles are presented in the same manner as in b; the green squares are the data of each individual experiment, with no statistical methods applied. f, Comparison between the cell volume data from d (grey circles) and the datasets extracted from recent reports, Gray et al. (red triangles, first right y axis) or Si et al. (blue dots, second right y axis). The data from Gray et al. are quite similar to ours in the region of overlap, including an apparent transition at λ ≈ 0.7 h⁻¹ in this semi-log plot. The behaviour of the data from Si et al. near to growth rates of ~0.7 h⁻¹ is difficult to discern owing to the sparseness of the data. We note that, even though all three studies (Gray et al., Si et al. and the present study) characterized the cell volume on the basis of microscopy images, the absolute cell volumes reported by the different studies differ by about twofold. Such systematic differences probably reflect the different image-processing methods used, as reported previously. Gray et al. and the present study used two closely related software packages, Oufti and MicrobeTracker, respectively, to characterize cell volume, and the resulting values are considerably more comparable with each other than with those of Si et al.. The error bars of the grey circles in e and f are as described for those in b and d, respectively. Data points other than the grey circles are presented as their original value in the publications, with no further statistics applied; the CIs or s.d. of the population-averaged cell volume have been omitted for simplicity.

Fig. 2 |. Donachie’s cell-mass relation m¯∞eλ(C+D) does not hold.

Standard run-out experiments^, followed by flow cytometry analysis were applied to quantify the population-averaged number of DNA replication origins per cell $\overline{o}$ (see Methods). a, Semi-log plot of $\overline{o}$ versus growth rate λ. The straight line is a semi-log fit $\overline{o}=e^{a+\beta \lambda }$, with best fit values α = 0.28 and β = 0.99 h (R² > 0.97). b, Population-averaged cellular mass (in units of OD₆₀₀ ml per 10⁹ cells) is not proportional to e^{λ(C + D)}. The straight line is linear fit to the data in the fast-growth regime (λ > 0.7 h⁻¹). The linear fit does not go through the origin, clearly showing that $\overline{m}$ is not proportional to e^{λ(C + D)}, even for fast-growth rates. For a and b, data are mean ± s.d.; many of the error bars are smaller than the size of the symbols. Sample sizes and mean values for each symbol are provided in Extended Data Fig. 8.

Fig. 3 |. Donachie’s constant-initiation-mass hypothesis breaks down.

During steady-state growth, wild-type bacterial cells initiate DNA replication at all available origins synchronously once per cell cycle. The cell mass per origin at initiation, termed the initiation mass, was hypothesized by Donachie to be a constant independent of growth rate. a, The initiation mass m_i (in units of 10⁻¹⁰ OD₆₀₀ ml, characterized as described in the main text) is plotted against growth rate λ. In contrast to the Donachie hypothesis, it varied by ~50% across different growth rates, reaching a peak value ~5.5 × 10⁻¹⁰ OD₆₀₀ ml at a growth rate 0.7 h⁻¹. The growth rate dependence of the initiation mass is well described by the black curve from $m_i=\frac{m_0}{\ln 2}(\alpha +\beta \lambda )e^{-(\alpha +\beta \lambda )}$. b, Comparison between the data from a (grey circles) and those extracted from Wallden et al. (MG1655, red squares, right y axis). For the data from Wallden et al., the error bars for initiation volume are set to 10% as they described; the error bars for growth rate represent the s.d. at the single-cell level as they described. The culture temperature for the middle data point (arrow) was 30 °C, while the other two data points and ours were obtained at 37 °C. c, Comparison between the data from a (grey circles) and those extracted from Si et al. (MG1655, green diamonds, right y axis). The error bars for the data from Si et al. represent the s.d. of biological replicates. d, Relative DnaA protein concentration exhibits a non-monotonic dependence on growth rate. Relative DnaA protein concentration varied by ~50%, reaching a minimum value at ~0.7 h⁻¹, approximately where the initiation mass reaches a peak. It was quantified by quantitative proteomics and was normalized to the population average of the total protein concentration across growth rates (see Methods). Except for the data points that have already been published elsewhere and described above (the red squares in b and the green diamonds in c), in a–d, data are mean ± s.d.; many of the error bars are smaller than the size of the symbols. Sample sizes and mean values for each symbol are provided in Extended Data Fig. 8.

Fig. 4 |. The linear relation given by equation (2) unifies the slow- and fast-growth regimes.

a, The population-averaged cellular mass $\overline{m}$ scales linearly with λ(C + D), a measure of the generation number during the time from initiation of a round of DNA replication to the cell division at which the corresponding sister chromosomes segregate. b, Hypothetical mechanistic basis of the integral threshold model. The key feature of our model is that cell division depends on the synthesis of a licensing product, which could be the PCC (i), divisome^, (ii), septum^, (iii) or their combinations. c, Quantitative proteomics showed that the concentrations of ZapA, FtsK and FtsZ, which are proposed to be involved in PCC formation or septum formation, were roughly constant across different growth rates, suggesting that they are candidate proteins involved in the synthesis (or construction) of the licensing product. By contrast, the concentration of RpsA, a ribosomal protein, increased linearly with growth rate, suggesting that it is not directly involved in such synthesis. For a and c, data are mean ± s.d.; many of the error bars are smaller than the size of the symbols. Sample sizes and mean values for each symbol are provided in Extended Data Fig. 8.