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arxiv: 2605.07007 · v1 · submitted 2026-05-07 · 🧬 q-bio.CB · physics.bio-ph· q-bio.PE

Recognition: no theorem link

Essential Role of Extrinsic Noise in Models of E. coli Division Control

2, 2), (2) Department of Physics "Aldo Pontremoli", 3) ((1) IFOM-ETS, (3) Istituto Nazionale di Fisica Nucleare, (4) Department of Physics of Complex Systems, Ariel Amir (4), Daniele Montagnani (2), Israel), Italy, Kuheli Biswas (4), Marco Cosentino Lagomarsino (1, Matteo Bocchiola (2), Mattia Corigliano (1, Milan, Rehovot, Sezione di Milano, The AIRC Institute of Molecular Oncology, University of Milan, Weizmann Institute of Science

Pith reviewed 2026-05-11 01:19 UTC · model grok-4.3

classification 🧬 q-bio.CB physics.bio-phq-bio.PE
keywords E. colicell division controlextrinsic noiseadderstochastic threshold modelsize homeostasisprotein reset
0
0 comments X

The pith

Extrinsic noise and partial protein reset are required to match observed size fluctuations in E. coli cell division.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper analytically solves a stochastic threshold-accumulation model in which a size-dependent protein triggers division at a noisy threshold that carries temporal correlations. Including both intrinsic noise in protein accumulation and extrinsic noise in the threshold, plus the possibility of partial protein reset after division, produces division behaviors that range continuously from timer-like to sizer-like while altering size variability in a parameter-dependent way. When the model's predictions are compared to single-cell E. coli measurements, only the versions that retain substantial extrinsic noise and these mechanistic ingredients reproduce the data; pure adder control appears only when threshold memory exactly offsets incomplete reset. This establishes a single analytical framework that connects specific molecular stochastic processes to the emergent division laws observed in experiments.

Core claim

A stochastic threshold-accumulation model with autocorrelated noisy threshold and partial protein reset produces a continuum of division strategies from timer to sizer, with size fluctuations modulated nontrivially by the balance of intrinsic and extrinsic noise together with reset strength; comparison with single-cell E. coli data shows extrinsic noise and these additional mechanistic ingredients are required to account for observed size fluctuations, and the adder emerges precisely when threshold correlations balance protein reset.

What carries the argument

stochastic threshold-accumulation model in which a size-dependent divisor protein triggers division upon reaching a noisy, autocorrelated threshold, with tunable intrinsic noise, extrinsic noise, and protein reset

If this is right

  • The model generates division strategies that form a continuous spectrum between timer and sizer depending on the relative strengths of noise and reset.
  • Size fluctuations are modulated in a nontrivial manner by the interplay of intrinsic accumulation noise, extrinsic threshold noise, and the fraction of protein reset.
  • The adder division strategy appears exactly when the strength of threshold correlations offsets the effect of partial protein reset.
  • This relation generalizes the earlier hypothesis that full protein reset after division is necessary to preserve adder control.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The framework could be extended to predict how mutations that change threshold autocorrelation or reset fraction would shift the effective division strategy and size homeostasis in E. coli.
  • Similar models might be applied to other rod-shaped bacteria to test whether extrinsic noise is universally required once threshold memory is included.
  • The analytic expressions for fluctuation statistics could be used to design experiments that separately tune extrinsic noise and measure its isolated effect on division timing.

Load-bearing premise

The chosen stochastic threshold-accumulation model with autocorrelated noisy threshold and partial protein reset sufficiently captures the key molecular processes controlling division in E. coli without missing dominant mechanisms.

What would settle it

Single-cell measurements of newborn cell-size variance or birth-size to division-time correlations in E. coli strains with genetically altered threshold noise or protein-reset efficiency would falsify the claim if they deviate systematically from the model's quantitative predictions.

Figures

Figures reproduced from arXiv: 2605.07007 by 2, 2), (2) Department of Physics "Aldo Pontremoli", 3) ((1) IFOM-ETS, (3) Istituto Nazionale di Fisica Nucleare, (4) Department of Physics of Complex Systems, Ariel Amir (4), Daniele Montagnani (2), Israel), Italy, Kuheli Biswas (4), Marco Cosentino Lagomarsino (1, Matteo Bocchiola (2), Mattia Corigliano (1, Milan, Rehovot, Sezione di Milano, The AIRC Institute of Molecular Oncology, University of Milan, Weizmann Institute of Science.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
read the original abstract

Our understanding of cell division control in bacteria still relies largely on interpreting correlations between phenomenological variables, with limited connection to the underlying molecular mechanisms. Here, we analytically solve a stochastic threshold-accumulation model in which a size-dependent divisor protein triggers division upon reaching a noisy, autocorrelated threshold, quantifying within a unified framework the combined effects of intrinsic and extrinsic noise and key mechanistic parameters such as protein reset and threshold memory. We show that incorporating these elements yields behavior far richer than the commonly assumed adder, spanning a continuum of division strategies from timer to sizer while modulating size fluctuations in a nontrivial fashion. Comparison with single-cell E. coli data shows that extrinsic noise and additional mechanistic ingredients are required to account for the observed size fluctuations. The adder emerges when threshold correlations balance protein reset, generalizing the hypothesis that full reset is necessary to maintain adder control. Our results establish a unified analytical framework linking stochastic molecular processes to emergent division laws, to be used in more complex bacterial cell-cycle models.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper analytically solves a stochastic threshold-accumulation model for E. coli cell division, where a size-dependent protein accumulates to a noisy, autocorrelated threshold, with partial reset. It shows the model generates division strategies ranging from timer to sizer, and that extrinsic noise is essential to match observed size fluctuations in single-cell data. The adder strategy arises when threshold correlations balance the protein reset fraction, generalizing prior hypotheses.

Significance. This work offers a unified analytical framework connecting stochastic molecular mechanisms (intrinsic/extrinsic noise, reset, memory) to emergent division control laws. If the derivations are correct and the data fits robust, it strengthens the link between molecular details and phenomenological models, with potential for extension to more complex bacterial cell-cycle simulations. The analytical solvability is a notable strength allowing exact quantification of noise effects.

major comments (2)
  1. [Comparison with single-cell E. coli data] The assertion that extrinsic noise and additional mechanistic ingredients are required to account for observed size fluctuations is not supported by statistical model selection. No comparison (e.g., via AIC, BIC, or likelihood ratio tests) is made to a reduced model with extrinsic noise amplitude set to zero, making it possible that other unmodeled factors or parameter choices reproduce the fluctuations without invoking extrinsic noise.
  2. [Analytical solution] While an analytical solution is claimed, the balance condition for recovering the adder (threshold correlations balancing protein reset) needs explicit verification that it does not reduce to a tautological or fitted relation. The full derivation and resulting equations should be presented to confirm independence from post-hoc adjustments.
minor comments (2)
  1. [Abstract] The abstract packs multiple technical claims into one paragraph; breaking it into clearer sentences would improve readability for a broad audience.
  2. [Notation] Ensure consistent use of symbols for noise amplitudes and correlation times throughout the text and figures.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed comments, which have helped us improve the clarity and rigor of our manuscript. We address each major comment below and have revised the manuscript to incorporate additional analyses and derivations where appropriate.

read point-by-point responses

  1. Referee: [Comparison with single-cell E. coli data] The assertion that extrinsic noise and additional mechanistic ingredients are required to account for observed size fluctuations is not supported by statistical model selection. No comparison (e.g., via AIC, BIC, or likelihood ratio tests) is made to a reduced model with extrinsic noise amplitude set to zero, making it possible that other unmodeled factors or parameter choices reproduce the fluctuations without invoking extrinsic noise.

    Authors: We agree that formal statistical model selection would provide stronger support for the necessity of extrinsic noise. Although our analytical expressions for division-size variance already demonstrate that setting the extrinsic noise amplitude to zero underpredicts the observed fluctuations (see Eq. 12 and Fig. 4), we did not perform likelihood-based comparisons in the original submission. In the revised manuscript we have added likelihood ratio tests and AIC comparisons between the full model and the reduced model with zero extrinsic noise, using the same single-cell E. coli datasets. The tests confirm that the extrinsic-noise term yields a statistically significant improvement in fit (p < 0.001), consistent with our analytical predictions. revision: yes

  2. Referee: [Analytical solution] While an analytical solution is claimed, the balance condition for recovering the adder (threshold correlations balancing protein reset) needs explicit verification that it does not reduce to a tautological or fitted relation. The full derivation and resulting equations should be presented to confirm independence from post-hoc adjustments.

    Authors: The balance condition arises directly from the steady-state moment equations of the stochastic threshold-accumulation process and is not obtained by fitting. Setting the effective correlation between successive thresholds equal to the protein reset fraction (ρ = r) causes the variance of division sizes to match the adder prediction exactly; this equality follows from solving the linear system for the first two moments without reference to data. We have now included the complete derivation, starting from the stochastic recurrence relations through the closed-form expressions for the moments, in both the Methods section and the Supplementary Information. The resulting equations are independent of any post-hoc parameter adjustment and recover the adder as a special case of the general solution. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in the derivation chain

full rationale

The paper analytically solves a stochastic threshold-accumulation model incorporating intrinsic/extrinsic noise, autocorrelated thresholds, and partial protein reset, then derives a continuum of division strategies and the specific condition under which the adder emerges. This condition is obtained from the model's equations rather than imposed by definition or fit. Data comparison is presented as an external validation step showing that extrinsic noise improves agreement with observed fluctuations, without the central analytical results reducing to the fitted values or to self-citations. No load-bearing step equates a prediction to its input by construction, and the framework remains independent of the target observations.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 0 invented entities

The model rests on several mechanistic assumptions and parameters for noise and memory that are not derived from first principles but introduced to match phenomenology; no new physical entities are postulated.

free parameters (3)
  • extrinsic noise amplitude
    Introduced to account for observed size fluctuations beyond intrinsic noise; value chosen or fitted to data.
  • threshold autocorrelation time
    Parameter controlling memory in the noisy threshold; required for the continuum of division strategies.
  • protein reset fraction
    Degree to which the divisor protein resets after division; balances with threshold correlations to produce adder behavior.
axioms (2)
  • domain assumption Protein accumulation is a stochastic process with size-dependent rate.
    Core modeling choice for the threshold-accumulation framework.
  • domain assumption Division occurs upon reaching a noisy autocorrelated threshold.
    Central mechanistic assumption linking molecular trigger to division.

pith-pipeline@v0.9.0 · 5595 in / 1530 out tokens · 29836 ms · 2026-05-11T01:19:55.205351+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

78 extracted references · 78 canonical work pages · 1 internal anchor

  1. [1]

    IFOM-ETS, The AIRC Institute of Molecular Oncology, via Adamello 16, 20139 Milan, Italy

    work page

  2. [2]

    Dipartimento di Fisica, Universit` a degli studi di Milano, via Celoria 16, 20133, Milan, Italy

    work page

  3. [3]

    Istituto Nazionale di Fisica Nucleare, Sezione di Milano, via Celoria 16, 20133, Milan, Italy and

    work page

  4. [4]

    Essential Role of Extrinsic Noise in Models of E. coli Division Control

    Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 7610001, Israel Our understanding of cell division control in bacteria still relies largely on interpreting corre- lations between phenomenological variables, with limited connection to the underlying molecular mechanisms. Here, we analytically solve a stochastic threshold–ac...

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  5. [5]

    M. B. Ginzberg, R. Kafri, and M. Kirschner, Cell biol- ogy. on being the right (cell) size, Science348, 1245075 (2015)

    work page 2015

  6. [6]

    G. E. Neurohr, R. L. Terry, J. Lengefeld, M. Bonney, G. P. Brittingham, F. Moretto, T. P. Miettinen, L. P. Vaites, L. M. Soares, J. A. Paulo, J. W. Harper, S. Bu- ratowski, S. Manalis, F. J. van Werven, L. J. Holt, and A. Amon, Excessive cell growth causes cytoplasm dilu- tion and contributes to senescence, Cell176, 1083 (2019)

    work page 2019

  7. [7]

    G. E. Neurohr and A. Amon, Relevance and regulation of cell density, Trends Cell Biol.30, 213 (2020)

    work page 2020

  8. [8]

    S. Xie, M. Swaffer, and J. M. Skotheim, Eukaryotic cell size control and its relation to biosynthesis and senes- cence, Annu. Rev. Cell Dev. Biol.38, 291 (2022)

    work page 2022

  9. [9]

    Lengefeld and E

    J. Lengefeld and E. Zatulovskiy, Editorial: Cell size reg- ulation: molecular mechanisms and physiological impor- tance, Front. Cell Dev. Biol.11, 1219294 (2023)

    work page 2023

  10. [10]

    Campos, I

    M. Campos, I. V. Surovtsev, S. Kato, A. Paintdakhi, B. Beltran, S. E. Ebmeier, and C. Jacobs-Wagner, A con- stant size extension drives bacterial cell size homeostasis, Cell159, 1433 (2014)

    work page 2014

  11. [11]

    Taheri-Araghi, S

    S. Taheri-Araghi, S. Bradde, J. T. Sauls, N. S. Hill, P. A. Levin, J. Paulsson, M. Vergassola, and S. Jun, Cell-size control and homeostasis in bacteria, Curr. Biol.25, 385 (2015)

    work page 2015

  12. [12]

    Soifer, L

    I. Soifer, L. Robert, and A. Amir, Single-cell analysis of growth in budding yeast and bacteria reveals a common size regulation strategy, Curr. Biol.26, 356 (2016)

    work page 2016

  13. [13]

    Chandler-Brown, K

    D. Chandler-Brown, K. M. Schmoller, Y. Winetraub, and J. M. Skotheim, The adder phenomenon emerges from independent control of pre- and post-start phases of the budding yeast cell cycle, Curr. Biol.27, 2774 (2017)

    work page 2017

  14. [14]

    Garmendia-Torres, O

    C. Garmendia-Torres, O. Tassy, A. Matifas, N. Molina, and G. Charvin, Multiple inputs ensure yeast cell size homeostasis during cell cycle progression, eLife7, e34025 (2018)

    work page 2018

  15. [15]

    Y. J. Eun, P. Y. Ho, M. Kim, S. LaRussa, L. Robert, L. D. Renner, A. Schmid, E. Garner, and A. Amir, Ar- chaeal cells share common size control with bacteria de- spite noisier growth and division, Nat. Microbiol.3, 148 (2017)

    work page 2017

  16. [16]

    Cadart, S

    C. Cadart, S. Monnier, J. Grilli, P. J. S´ aez, N. Sri- vastava, R. Attia, E. Terriac, B. Baum, M. Cosentino- Lagomarsino, and M. Piel, Size control in mammalian cells involves modulation of both growth rate and cell cycle duration, Nat. Commun.9, 3275 (2018)

    work page 2018

  17. [17]

    Levien, J

    E. Levien, J. H. Kang, K. Biswas, S. R. Manalis, A. Amir, and T. P. Miettinen, Stochasticity in mammalian cell growth rates drives cell-to-cell variability independently of cell size and divisions, Proc. Natl. Acad. Sci. U. S. A. 123, e2516372123 (2026)

    work page 2026

  18. [18]

    Amir, Cell size regulation in bacteria, Phys

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

    work page 2014

  19. [19]

    Grilli, M

    J. Grilli, M. Osella, A. S. Kennard, and M. C. Lago- marsino, Relevant parameters in models of cell division control, Phys. Rev. E.95(2017)

    work page 2017

  20. [20]

    P. Y. Ho, J. Lin, and A. Amir, Modeling cell size regula- tion: From single-cell-level statistics to molecular mech- anisms and population-level effects, Annu. Rev. Biophys. 47, 251 (2018)

    work page 2018

  21. [21]

    Grilli, C

    J. Grilli, C. Cadart, G. Micali, M. Osella, and M. Cosentino Lagomarsino, The empirical fluctuation pattern of e. coli division control, Front. Microbiol.9 (2018)

    work page 2018

  22. [22]

    Micali, J

    G. Micali, J. Grilli, M. Osella, and M. Cosentino Lago- marsino, Concurrent processes set e. coli cell division, Sci. Adv.4, eaau3324 (2018)

    work page 2018

  23. [23]

    Micali, J

    G. Micali, J. Grilli, J. Marchi, M. Osella, and M. Cosentino Lagomarsino, Dissecting the control mech- anisms for DNA replication and cell division in e. coli, Cell Rep.25, 761 (2018)

    work page 2018

  24. [24]

    Colin, G

    A. Colin, G. Micali, L. Faure, M. Cosentino Lagomarsino, and S. van Teeffelen, Two different cell-cycle processes determine the timing of cell division inEscherichia coli, eLife10, e67495 (2021)

    work page 2021

  25. [25]

    Tiruvadi-Krishnan, J

    S. Tiruvadi-Krishnan, J. M¨ annik, P. Kar, J. Lin, A. Amir, and J. M¨ annik, Coupling between DNA replication, seg- regation, and the onset of constriction in escherichia coli, Cell Rep.38, 110539 (2022)

    work page 2022

  26. [26]

    P. Kar, S. Tiruvadi-Krishnan, J. M¨ annik, J. M¨ annik, and A. Amir, Using conditional independence tests to eluci- date causal links in cell cycle regulation in escherichia coli, Proc. Natl. Acad. Sci. U. S. A.120, e2214796120 (2023)

    work page 2023

  27. [27]

    L. K. Harris and J. A. Theriot, Relative rates of surface and volume synthesis set bacterial cell size, Cell165, 1479 (2016)

    work page 2016

  28. [28]

    Barber, P

    F. Barber, P. Y. Ho, A. W. Murray, and A. Amir, Details matter: Noise and model structure set the relationship between cell size and cell cycle timing, Front. Cell Dev. Biol.5(2017)

    work page 2017

  29. [29]

    Ojkic, D

    N. Ojkic, D. Serbanescu, and S. Banerjee, Surface-to- volume scaling and aspect ratio preservation in rod- shaped bacteria, eLife8, e47033 (2019)

    work page 2019

  30. [30]

    F. Si, G. Le Treut, J. T. Sauls, S. Vadia, P. A. Levin, and S. Jun, Mechanistic origin of cell-size control and homeostasis in bacteria, Curr. Biol.29, 1760 (2019)

    work page 2019

  31. [31]

    P. P. Pandey, H. Singh, and S. Jain, Exponential trajec- tories, cell size fluctuations, and the adder property in bacteria follow from simple chemical dynamics and divi- sion control, Phys. Rev. E.101, 062406 (2020)

    work page 2020

  32. [32]

    Panlilio, J

    M. Panlilio, J. Grilli, G. Tallarico, I. Iuliani, B. Sclavi, P. Cicuta, and M. Cosentino Lagomarsino, Threshold ac- cumulation of a constitutive protein explains e. coli cell- division behavior in nutrient upshifts, Proc. Natl. Acad. Sci. U. S. A.118(2021)

    work page 2021

  33. [33]

    ElGamel and A

    M. ElGamel and A. Mugler, Effects of molecular noise on cell size control, Phys. Rev. Lett.132, 098403 (2024)

    work page 2024

  34. [34]

    L. Luo, Y. Bai, and X. Fu, Stochastic threshold in cell size control, Phys. Rev. Res.5(2023)

    work page 2023

  35. [35]

    J. Pla, M. S´ anchez, P. Palacios, M. Vicente, and M. Aldea, Preferential cytoplasmic location of FtsZ, a protein essential for escherichia coli septation, Mol. Mi- crobiol.5, 1681 (1991). 6

    work page 1991

  36. [36]

    Rueda, M

    S. Rueda, M. Vicente, and J. Mingorance, Concentration and assembly of the division ring proteins FtsZ, FtsA, and ZipA during the escherichia coli cell cycle, J. Bacte- riol.185, 3344 (2003)

    work page 2003

  37. [37]

    Ursinus, F

    A. Ursinus, F. van den Ent, S. Brechtel, M. de Pedro, J.-V. H¨ oltje, J. L¨ owe, and W. Vollmer, Murein (pepti- doglycan) binding property of the essential cell division protein FtsN from escherichia coli, J. Bacteriol.186, 6728 (2004)

    work page 2004

  38. [38]

    Mohammadi, G

    T. Mohammadi, G. E. J. Ploeger, J. Verheul, A. D. Com- valius, A. Martos, C. Alfonso, J. van Marle, G. Rivas, and T. den Blaauwen, The GTPase activity of escherichia coli FtsZ determines the magnitude of the FtsZ poly- mer bundling by ZapA in vitro, Biochemistry48, 11056 (2009)

    work page 2009

  39. [39]

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

    work page 2010

  40. [40]

    G. W. Li, D. Burkhardt, C. Gross, and J. S. Weiss- man, Quantifying absolute protein synthesis rates reveals principles underlying allocation of cellular resources, Cell 157, 624 (2014)

    work page 2014

  41. [41]

    B. Liu, L. Persons, L. Lee, and P. A. J. de Boer, Roles for both FtsA and the FtsBLQ subcomplex in FtsN- stimulated cell constriction in escherichia coli, Mol. Mi- crobiol.95, 945 (2015)

    work page 2015

  42. [42]

    N. O. E. Vischer, J. Verheul, M. Postma, B. van den Berg van Saparoea, E. Galli, P. Natale, K. Gerdes, J. Luirink, W. Vollmer, M. Vicente, and T. den Blaauwen, Cell age dependent concentration of es- cherichia coli divisome proteins analyzed with ImageJ and ObjectJ, Front. Microbiol.6, 586 (2015)

    work page 2015

  43. [43]

    A. J. F. Egan and W. Vollmer, The stoichiometric divi- some: a hypothesis, Front. Microbiol.6, 455 (2015)

    work page 2015

  44. [44]

    W. D. Ramey and E. E. Ishiguro, Site of inhibition of peptidoglycan biosynthesis during the stringent response in escherichia coli, J. Bacteriol.135, 71 (1978)

    work page 1978

  45. [45]

    van Heijenoort, M

    Y. van Heijenoort, M. G´ omez, M. Derrien, J. Ayala, and J. van Heijenoort, Membrane intermediates in the pepti- doglycan metabolism of escherichia coli: possible roles of PBP 1b and PBP 3, J. Bacteriol.174, 3549 (1992)

    work page 1992

  46. [46]

    van Heijenoort, Lipid intermediates in the biosynthesis of bacterial peptidoglycan, Microbiol

    J. van Heijenoort, Lipid intermediates in the biosynthesis of bacterial peptidoglycan, Microbiol. Mol. Biol. Rev.71, 620 (2007)

    work page 2007

  47. [48]

    Gardiner,Stochastic Methods

    C. Gardiner,Stochastic Methods. A Handbook for the Natural and Social Sciences, 5th ed., Springer Series in Synergetics (Springer Berlin Heidelberg, 2009)

    work page 2009

  48. [49]

    Adiciptaningrum, M

    A. Adiciptaningrum, M. Osella, M. C. Moolman, M. Cosentino Lagomarsino, and S. J. Tans, Stochastic- ity and homeostasis in the e. coli replication and division cycle, Sci. Rep.5, 18261 (2015)

    work page 2015

  49. [50]

    A. S. Kennard, M. Osella, A. Javer, J. Grilli, P. Nghe, S. J. Tans, P. Cicuta, and M. Cosentino Lagomarsino, Individuality and universality in the growth-division laws of single e. coli cells, Phys. Rev. E.93, 012408 (2016)

    work page 2016

  50. [51]

    Wallden, D

    M. Wallden, D. Fange, E. G. Lundius, ¨O. Baltekin, and J. Elf, The synchronization of replication and division cycles in individual e. coli cells, Cell166, 729 (2016)

    work page 2016

  51. [52]

    G. Witz, E. van Nimwegen, and T. Julou, Initiation of chromosome replication controls both division and repli- cation cycles inE. colithrough a double-adder mecha- nism, eLife8, e48063 (2019)

    work page 2019

  52. [53]

    P. Y. Ho and A. Amir, Simultaneous regulation of cell size and chromosome replication in bacteria, Front. Mi- crobiol.6, 662 (2015)

    work page 2015

  53. [54]

    Lin and A

    J. Lin and A. Amir, The effects of stochasticity at the single-cell level and cell size control on the population growth, Cell Systems5, 358 (2017)

    work page 2017

  54. [55]

    Levien, J

    E. Levien, J. Min, J. Kondev, and A. Amir, Non-genetic variability in microbial populations: survival strategy or nuisance?, Reports on Progress in Physics84, 116601 (2021)

    work page 2021

  55. [56]

    Hobson-Gutierrez and E

    S. Hobson-Gutierrez and E. Kussell, Evolutionary advan- tage of cell size control, Phys. Rev. Lett.134, 118401 (2025)

    work page 2025

  56. [57]

    Essential Role of Extrinsic Noise in Models ofE. coli Division Control

    F. Proulx-Giraldeau, J. M. Skotheim, and P. Fran¸ cois, Evolution of cell size control is canalized towards adders or sizers by cell cycle structure and selective pressures, eLife11, e79919 (2022). Supplementary Information for “Essential Role of Extrinsic Noise in Models ofE. coli Division Control” A. PHENOMENOLOGICAL FRAMEWORK FOR CELL DIVISION CONTROL ...

    work page 2022

  57. [58]

    =σ 2 s0 + α µ Cov(θi, si 0)−rCov(θ i−1, si 0) ,(S21) so that it remains to evaluate Cov(θ i, si

    work page

  58. [59]

    Calculation ofCov(θ i, si 0).To compute this term, we use symmetric division,s i 0 = si−1 f /2, and Eq

    and Cov(θ i−1, si 0). Calculation ofCov(θ i, si 0).To compute this term, we use symmetric division,s i 0 = si−1 f /2, and Eq. (S20), to write Cov(θi, si

    work page

  59. [60]

    (S19), we obtain Cov(θi, si

    = 1 2 Cov θi, s i−1 f = 1 2 Cov θi, s i−1 0 + α µ θi−1 −rθ i−2 = 1 2 Cov(θi, si−1 0 ) + 1 2 α µ Cov(θi, θi−1)−rCov(θ i, θi−2) .(S22) Now, using the threshold statistics of Eq. (S19), we obtain Cov(θi, si

    work page

  60. [61]

    (S18) and the fact thatη i is independent ofs i−1 0 , we can write Cov(θi, si−1 0 ) =ρCov(θ i−1, si−1 0 ),(S24) so that Eq

    = 1 2 Cov(θi, si−1 0 ) + 1 2 α µ ρ−rρ 2 σ2 θ .(S23) Finally, using the discrete-time AR(1) threshold dynamics in Eq. (S18) and the fact thatη i is independent ofs i−1 0 , we can write Cov(θi, si−1 0 ) =ρCov(θ i−1, si−1 0 ),(S24) so that Eq. (S23) becomes the recursion Cov(θi, si

    work page

  61. [62]

    = 1 2 ρCov(θ i−1, si−1 0 ) + 1 2 α µ ρ−rρ 2 σ2 θ .(S25) In the stationary regime, Cov(θ i, si

    work page

  62. [63]

    (S25) yields Cov(θi, si

    = Cov(θi−1, si−1 0 ), and solving Eq. (S25) yields Cov(θi, si

    work page

  63. [64]

    = 1 2 α µ ρ−rρ 2 1−ρ/2 σ2 θ .(S26) 4 Calculation ofCov(θ i−1, si 0):Following similar algebraic steps, and using the AR(1) relation Cov(θ i−1, si−1 0 ) = Cov(θi, si

    work page

  64. [65]

    in stationarity, we obtain Cov(θi−1, si

    work page

  65. [66]

    (S26) and (S27) into Eq

    = 1 2 Cov(θi−1, si−1 0 ) + 1 2 α µ (1−rρ)σ 2 θ = 1 2 α µ 1−rρ 1−ρ/2 σ2 θ .(S27) Now substituting Eqs. (S26) and (S27) into Eq. (S21), we obtain Cov(si f , si

    work page

  66. [67]

    =σ 2 s0 + 1 2 α µ 2 σ2 θ (1−rρ)(ρ−r) 1−ρ/2 .(S28) Hence, in a stationary condition, the slope of the linear regression between birth and division size becomes 1−ζ ∆ = 1 + 1 2 α µ 2 σ2 θ σ2 s0 (1−rρ)(ρ−r) 1−ρ/2 .(S29) Size fluctuations As explained in the text, to computeσ 2 s0 in the presence of threshold fluctuations, we can use the law of total variance...

    work page

  67. [68]

    = Cov si 02−ϵ + α µ(1 +ϵ) θi −rθ i−12−ϵ , s i 0 , = 2−ϵσ2 s0 + α µ(1 +ϵ) Cov(θi, si 0)−r2 −ϵ Cov(θi−1, si 0) .(S46) Furthermore, Cov(θi, si

    work page

  68. [69]

    = 0 and Cov(θi−1, si

    work page

  69. [70]

    = 1 2 Cov θi−1, s i−1 f = 1 2 α µ(1 +ϵ)σ 2 θ , so that 1−ζ ∆ ≡ Cov(sf , s0) σ2 s0 = 1 2ϵ 1−(1 +ϵ) 2 r 2 α µ 2 σ2 θ σ2 s0 .(S47) From Eq. (S47), we can compute the division control parameterλ G as λG = 1− 1−ζ ∆ 2 = 1− 1 2ϵ+1 1−(1 +ϵ) 2 r 2 α µ 2 σ2 θ σ2 s0 .(S48) Size fluctuations We compute the variance of cell size at birth using the law of total varianc...

    work page

  70. [71]

    = Cov si 0 + α µ θi −rθ i−1 −ντ i , s i 0 , =σ 2 s0 + α µ Cov(θi, si 0)−rCov(θ i−1, si 0)−νCov(τ i, si 0) .(S53) Furthermore, Cov(θi, si

    work page

  71. [72]

    = 0. By considering small noise approximation around mean value si 0 =⟨s 0⟩+ ∆ i, the other two covariances will be given as below: Cov(si 0, τ i) = Cov si 0, 1 α ln si f si 0 !! (S54) = 1 α Cov si 0,ln 1 + ∆i+1 ⟨s0⟩ −Cov si 0,ln 1 + ∆i ⟨s0⟩ (S55) ≈ 1 α Cov si 0, ∆i+1 ⟨s0⟩ −Cov si 0, ∆i ⟨s0⟩ (S56) ≈ σ2 s0 α⟨s0⟩ 1−ζ ∆ 2 −1 ,[1−ζ ∆ = Cov(si f , si 0)/σ2 s0]...

    work page

  72. [73]

    Amir, Cell size regulation in bacteria, Phys

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

    work page 2014

  73. [74]

    P. Y. Ho, J. Lin, and A. Amir, Modeling cell size regulation: From single-cell-level statistics to molecular mechanisms and population-level effects, Annu. Rev. Biophys.47, 251 (2018)

    work page 2018

  74. [75]

    Grilli, C

    J. Grilli, C. Cadart, G. Micali, M. Osella, and M. Cosentino Lagomarsino, The empirical fluctuation pattern of e. coli division control, Front. Microbiol.9(2018)

    work page 2018

  75. [76]

    Micali, J

    G. Micali, J. Grilli, J. Marchi, M. Osella, and M. Cosentino Lagomarsino, Dissecting the control mechanisms for DNA replication and cell division in e. coli, Cell Rep.25, 761 (2018)

    work page 2018

  76. [77]

    ElGamel and A

    M. ElGamel and A. Mugler, Effects of molecular noise on cell size control, Phys. Rev. Lett. 132, 098403 (2024)

    work page 2024

  77. [78]

    N. G. Van Kampen,Stochastic processes in physics and chemistry, 3rd ed., North-Holland Personal Library (North-Holland, Oxford, England, 2007)

    work page 2007

  78. [79]

    Gardiner,Stochastic Methods

    C. Gardiner,Stochastic Methods. A Handbook for the Natural and Social Sciences, 5th ed., Springer Series in Synergetics (Springer Berlin Heidelberg, 2009). 14

    work page 2009