A noisy linear map underlies oscillations in cell size and gene expression in bacteria

Abstract

During bacterial growth, a cell approximately doubles in size before division, after which it splits into two daughter cells. This process is subjected to the inherent perturbations of cellular noise1,2 and thus requires regulation for cell-size homeostasis. The mechanisms underlying the control and dynamics of cell size remain poorly understood owing to the difficulty in sizing individual bacteria over long periods of time in a high-throughput manner. Here we measure and analyse long-term, single-cell growth and division across different Escherichia coli strains and growth conditions3. We show that a subset of cells in a population exhibit transient oscillations in cell size with periods that stretch across several (more than ten) generations. Our analysis reveals that a simple law governing cell-size control—a noisy linear map—explains the origins of these cell-size oscillations across all strains. This noisy linear map implements a negative feedback on cell-size control: a cell with a larger initial size tends to divide earlier, whereas one with a smaller initial size tends to divide later. Combining simulations of cell growth and division with experimental data, we demonstrate that this noisy linear map generates transient oscillations, not just in cell size, but also in constitutive gene expression. Our work provides new insights into the dynamics of bacterial cell-size regulation with implications for the physiological processes involved.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.

from$8.99

All prices are NET prices.

Figure 1: Transient oscillations in cell size.
Figure 2: Simulated transient cell-size oscillations using the noisy linear map (equation (1)).
Figure 3: Dependence of cell size oscillation on a, and oscillations in gene expression.
Figure 4: a in different data sets.

Change history

  • 05 June 2015

    The wrong equation was mistakenly inserted in the online version of the Methods, this has now been corrected.

References

  1. 1

    Raj, A. & van Oudenaarden, A. Nature, nurture, or chance: stochastic gene expression and its consequences. Cell 135, 216–226 (2008).

    CAS  Article  Google Scholar 

  2. 2

    Huh, D. & Paulsson, J. Random partitioning of molecules at cell division. Proc. Natl Acad. Sci. USA 108, 15004–15009 (2011).

    ADS  CAS  Article  Google Scholar 

  3. 3

    Wang, P. et al. Robust growth of Escherichia coli. Curr. Biol. 20, 1099–1103 (2010).

    CAS  Article  Google Scholar 

  4. 4

    Osella, M., Nugent, E. & Cosentino Lagomarsino, M. Concerted control of Escherichia coli cell division. Proc. Natl Acad. Sci. USA 111, 3431–3435 (2014).

    ADS  CAS  Article  Google Scholar 

  5. 5

    Taheri-Araghi, S. et al. Cell-size control and homeostasis in bacteria. Curr. Biol. 25, 385–391 (2015).

    CAS  Article  Google Scholar 

  6. 6

    Campos, M. et al. A constant size extension drives bacterial cell size homeostasis. Cell 159, 1433–1446 (2014).

    CAS  Article  Google Scholar 

  7. 7

    Kitagawa, G. Introduction to Time Series Modeling 31–48 (Chapman and Hall/CRC, 2010).

    Google Scholar 

  8. 8

    Donachie, W. D. & Blakely, G. W. Coupling the initiation of chromosome replication to cell size in Escherichia coli. Curr. Opin. Microbiol. 6, 146–150 (2003).

    CAS  Article  Google Scholar 

  9. 9

    Reyes-Lamothe, R., Nicolas, E. & Sherratt, D. J. Chromosome replication and segregation in bacteria. Annu. Rev. Genet. 46, 121–143 (2012).

    CAS  Article  Google Scholar 

  10. 10

    Erickson, H. P., Anderson, D. E. & Osawa, M. FtsZ in bacterial cytokinesis: cytoskeleton and force generator all in one. Microbiol. Mol. Biol. Rev. 74, 504–528 (2010).

    CAS  Article  Google Scholar 

  11. 11

    de Boer, P. A. J. Advances in understanding E. coli cell fission. Curr. Opin. Microbiol. 13, 730–737 (2010).

    CAS  Article  Google Scholar 

  12. 12

    Moffitt, J. R., Lee, J. B. & Cluzel, P. The single-cell chemostat: an agarose-based, microfluidic device for high-throughput, single-cell studies of bacteria and bacterial communities. Lab Chip 12, 1487–1494 (2012).

    CAS  Article  Google Scholar 

  13. 13

    Nobs, J.-B. & Maerkl, S. J. Long-term single cell analysis of S. pombe on a microfluidic microchemostat array. PLoS ONE 9, e93466 (2014).

    ADS  Article  Google Scholar 

  14. 14

    Amir, A. Cell size regulation in bacteria. Phys. Rev. Lett. 112, 208102 (2014).

    ADS  Article  Google Scholar 

  15. 15

    Levine, J. H., Lin, Y. & Elowitz, M. B. Functional roles of pulsing in genetic circuits. Science 342, 1193–1200 (2013).

    ADS  CAS  Article  Google Scholar 

  16. 16

    Süel, G. M., Kulkarni, R. P., Dworkin, J., Garcia-Ojalvo, J. & Elowitz, M. B. Tunability and noise dependence in differentiation dynamics. Science 315, 1716–1719 (2007).

    ADS  Article  Google Scholar 

  17. 17

    Wakamoto, Y. et al. Dynamic persistence of antibiotic-stressed mycobacteria. Science 339, 91–95 (2013).

    ADS  CAS  Article  Google Scholar 

  18. 18

    Scott, M., Gunderson, C. W., Mateescu, E. M., Zhang, Z. & Hwa, T. Interdependence of cell growth and gene expression: origins and consequences. Science 330, 1099–1102 (2010).

    ADS  CAS  Article  Google Scholar 

  19. 19

    You, C. et al. Coordination of bacterial proteome with metabolism by cyclic AMP signalling. Nature 500, 301–306 (2013).

    ADS  CAS  Article  Google Scholar 

  20. 20

    Hui, S. et al. Quantitative proteomic analysis reveals a simple strategy of global resource allocation in bacteria. Mol. Syst. Biol. 11, 784 (2015).

    Article  Google Scholar 

  21. 21

    Kiviet, D. J. et al. Stochasticity of metabolism and growth at the single-cell level. Nature 514, 376–379 (2014).

    ADS  CAS  Article  Google Scholar 

  22. 22

    Tan, C., Marguet, P. & You, L. Emergent bistability by a growth-modulating positive feedback circuit. Nature Chem. Biol. 5, 842–848 (2009).

    CAS  Article  Google Scholar 

  23. 23

    Marguet, P., Tanouchi, Y., Spitz, E., Smith, C. & You, L. Oscillations by minimal bacterial suicide circuits reveal hidden facets of host-circuit physiology. PLoS ONE 5, e11909 (2010).

    ADS  Article  Google Scholar 

  24. 24

    Cookson, N. A. et al. Queueing up for enzymatic processing: correlated signaling through coupled degradation. Mol. Syst. Biol. 7, 561 (2011).

    Article  Google Scholar 

  25. 25

    Elowitz, M. B. & Leibler, S. A synthetic oscillatory network of transcriptional regulators. Nature 403, 335–338 (2000).

    ADS  CAS  Article  Google Scholar 

  26. 26

    Stricker, J. et al. A fast, robust and tunable synthetic gene oscillator. Nature 456, 516–519 (2008).

    ADS  CAS  Article  Google Scholar 

  27. 27

    Mondragón-Palomino, O., Danino, T., Selimkhanov, J., Tsimring, L. & Hasty, J. Entrainment of a population of synthetic genetic oscillators. Science 333, 1315–1319 (2011).

    ADS  MathSciNet  Article  Google Scholar 

  28. 28

    Hegreness, M., Shoresh, N., Hartl, D. & Kishony, R. An equivalence principle for the incorporation of favorable mutations in asexual populations. Science 311, 1615–1617 (2006).

    ADS  CAS  Article  Google Scholar 

  29. 29

    Kvarnström, M., Logg, K., Diez, A., Bodvard, K. & Käll, M. Image analysis algorithms for cell contour recognition in budding yeast. Opt. Express 16, 12943–12957 (2008).

    ADS  Article  Google Scholar 

  30. 30

    Godin, M. et al. Using buoyant mass to measure the growth of single cells. Nature Methods 7, 387–390 (2010).

    CAS  Article  Google Scholar 

Download references

Acknowledgements

We thank S. Jun for providing the original mother machine and P. Cluzel for providing their data set on E. coli and B. subtilis growth. We also thank the Light Microscopy Core Facility at Duke University for their help in conducting microscopy experiments. This work was partially supported by a National Science Foundation Career Award (L.Y.), the National Institutes of Health (NIH) (L.Y., R01GM098642, R01GM110494), a DuPont Young Professorship (L.Y.), a David and Lucile Packard Fellowship (L.Y.), a DARPA Biochronicity Grant (DARPA-BAA-11-66, N.E.B.), a NIH Director's New Innovator Award (DP2 OD008654-01, N.E.B.), and a Burroughs Wellcome Fund CASI Award (BWF 1005769.01, N.E.B.).

Author information

Affiliations

Authors

Contributions

Y.T. conceived the research, designed and performed both modelling and experimental analyses, interpreted the results, and wrote the manuscript. A.P. conceived the research, designed and performed experimental analyses, interpreted the results, and wrote the manuscript. H.P. designed and performed experimental analyses and interpreted the results. S.H. fabricated the microfluidic device and performed experiments. R.S. developed the software for image analysis. N.E.B. assisted in data interpretation and manuscript revisions. L.Y. conceived the research, assisted in research design and data interpretation, and wrote the manuscript. All authors approved the manuscript.

Corresponding author

Correspondence to Lingchong You.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Extended data figures and tables

Extended Data Figure 1 Analysis of cell growth parameters.

ac, Division ratio (a), growth rate (b) and doubling time (c) are plotted against initial cell length (n = 11,168). Black and red lines show binned average and trend line, respectively. In a and b, the trend lines are the linear regression line, whereas equation (4) was used in c. The error bars indicate standard deviation of each bin.

Extended Data Figure 2 A noisy linear map in E. coli cell size control across different growth temperatures and strains.

The same analysis as in Fig. 1e (LI versus LF) was performed for MC4100 grown at 27 °C (a, n = 3,772), MC4100 grown at 25 °C (b, n = 4,539), MG1655 grown at 37 °C (c, n = 10,964), and B/r strain grown at 37 °C (d, n = 5,541). The data sets for MG1655 and B/r strain were from a previous study3. Black and red lines show binned average and linear regression line, respectively. In ad, a = 1.02, 1.08, 1.03 and 1.14, respectively. The error bars indicate standard deviation of each bin.

Extended Data Figure 3 Frequency analysis of noisy linear map.

a, Time-frequency analysis of the noisy linear map model for LI. The top panel shows a spectrogram of 700-generation simulation constructed using 70-generation segments with 50% overlap. The bottom panel shows the temporal dynamics of LI. The same parameters as in Fig. 2a were used. b, The power spectrum of noisy linear map (X(f)). Equation (3) is plotted for different a (0 ≤ a ≤ 1.8 with 0.2 interval). X(f) = 1/(2πf)2 for a = 2 is the straight line in this log–log plot. Note that the maximum value of f is 0.5 as the time resolution is 1 generation. c, Dependence of oscillation frequencies on a simulated using the rescaled linear map (equation (2)). The noisy linear map model was simulated (Fig. 3a–c) and the distributions of oscillation frequencies are shown for four different values of a (from top to bottom, n = 24, 76, 117 and 79, respectively).

Extended Data Figure 4 Comparison of LI oscillation characteristics between all lineages (blue) and lineages without aberrant cell cycles (red).

ag, Probability of oscillation (a), average oscillation frequency (b), and distributions of oscillation frequencies (cg) are shown. In a and b, filled symbols represent experimental data (data shown in red are the same as Fig. 3a, b): circles are MC4100 grown at three different temperatures; squares and triangles are MG1655 and B/r strain, respectively. The unfilled circles were generated from simulations using the rescaled linear map (equation (2), the same plot as Fig. 3a, b). In a, the data shown in blue include 160 (37 °C), 54 (27 °C) and 65 (25 °C) lineages for MC4100, 158 lineages for MG1655, and 80 lineages for B/r strain. The data shown in red include 143 (37 °C), 48 (27 °C), and 57 (25 °C) lineages for MC4100, 97 lineages for MG1655, and 60 lineages for B/r strain. In bg, only lineages that were considered oscillatory were used. For the data set in blue, n = 58 (37 °C), 21 (27 °C), and 39 (25 °C) for MC4100, 46 for MG1655, and 26 for B/r strain. For the data set in red, n = 51 (37 °C), 18 (27 °C) and 36 (25 °C) for MC4100, 34 for MG1655, and 23 for B/r strain.

Extended Data Figure 5 A noisy linear map in total per-cell YFP.

Total YFP before division is plotted against total YFP at birth (n = 11,168), revealing a linear map with a = 1.05. Black and red lines show binned average and the linear regression line, respectively. The error bars indicate standard deviation of each bin.

Extended Data Figure 6 Oscillation in [Y] observed in MC4100 at 27 °C.

a, Oscillation scores of YFP concentration are shown in an ascending order. The dashed line indicates a threshold for oscillation. b, For four different oscillation frequencies, ACFs with the highest oscillation scores are shown. c, The distribution of oscillation frequencies (n = 47).

Extended Data Figure 7 Oscillation in [Y] observed in MC4100 at 25 °C.

The same plots as in Extended Data Fig. 6 but for MC4100 grown at 25 °C. n = 61 in c.

Extended Data Figure 8 Oscillation in [Y] observed in MG1655.

The same plots as in Extended Data Fig. 6 but for MG1655 (ref. 3). n = 60 in c.

Extended Data Figure 9 Oscillation in [Y] observed in B/r strain.

The same plot as in Extended Data Fig. 6 but for B/r strain3. n = 20 in c.

Extended Data Figure 10 A noisy linear map in cell size control in the data sets from ref. 12 and our experimental data of S. pombe

The same analysis as in Fig. 1e (LI versus LF) was performed. These data sets were obtained using a microfluidic device different from the mother machine. ac, E. coli growth under three different media as indicated (n = 8,795, 4,637 and 684, respectively). d, e, Growth of B. subtilis (n = 1,592) (d) and of S. pombe (n = 85) (e). Black and red lines show binned average and the linear regression line, respectively. In ae, a = 1.20, 0.864, 0.684, 1.41 and 0.645, respectively. Cell length (µm) (ad) and cell area (µm2) (e) are shown. The error bars indicate standard deviation of each bin.

Supplementary information

Supplementary Information

This file contains a Supplementary Discussion and Supplementary References. (PDF 134 kb)

PowerPoint slides

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Tanouchi, Y., Pai, A., Park, H. et al. A noisy linear map underlies oscillations in cell size and gene expression in bacteria. Nature 523, 357–360 (2015). https://doi.org/10.1038/nature14562

Download citation

Further reading

Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.