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Shaping nematic order in bacterial films with single-cell resolution patterning

T0 review · 2 major / 1 minor · reviewed 2026-05-08 · grok-4.3

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Pith's one-line read Precise single-cell patterning of bacterial spores controls large-scale nematic order in growing films, enabling uniform buckling and macroscopic optical anisotropy.

desk verdict Capillary assembly for single-cell spore patterning lets the authors switch bacterial films between chaotic and synchronously buckling states by setting initial orientations, with simulations matching the mechanics. read the letter →

arxiv 2604.21655 v1 submitted 2026-04-23 physics.bio-ph cond-mat.softq-bio.CB

Matthias Le Bec , Guillem Pérez Martín , Cameron Boggon , Yiyao Hu , Leonardo Puggioni , Rosa Heydenreich , Alexander Mathys , Luca Giomi
show 2 more authors
Eleonora Secchi Lucio Isa
This is my paper · ORCID
classification physics.bio-phcond-mat.softq-bio.CB
keywords bacterialnematicfilmssporepatterningcapillaryassemblyBacillussubtilisorderstructuralcolourationactivematterbiofilmmechanics
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper shows that initial cell orientations set at single-cell resolution determine whether bacterial films develop ordered or chaotic behavior as they grow. Parallel seeding of Bacillus subtilis endospores produces high nematic alignment over millimetre distances that then buckles synchronously, while orthogonal seeding yields disordered chaotic flows. These outcomes arise because the starting arrangement dictates how cell chains align and generate stresses during expansion. The ordered films additionally display structural colouration and polarise light, demonstrating that nematic alignment can be harnessed for functional living materials. This establishes that initial conditions are a dominant control parameter in active nematic bacterial assemblies.

What carries the argument

Capillary assembly patterning of endospores, which fixes single-cell positions and orientations to dictate subsequent nematic alignment and mechanical response in growing films.

What would settle it

Observation that parallel-oriented spores still produce chaotic dynamics and millimetre-scale disorder, or that buckling timing and synchrony remain independent of the initial seeding pattern.

Watch

Extended reading notes

Core claim

Capillary assembly is used to place endospores in arrays with user-defined positions and orientations at single-cell resolution. Parallel orientations produce films with high nematic order persisting across millimetres that buckle synchronously upon further growth; orthogonal orientations instead produce chaotic dynamics and disordered domains. Both behaviors are reproduced by simulations and a mechanical model that starts from the properties of individual cell filaments. The resulting ordered films exhibit local optical anisotropy through structural colouration and light polarisation.

Load-bearing premise

The capillary assembly step sets only the intended spore positions and orientations without changing spore viability, germination timing, or cell growth mechanics in ways that would alter the observed order or buckling.

Editorial extensions

If this is right

  • Parallel initial orientations produce high nematic order that persists across millimetre scales in the growing film.
  • This ordered state triggers synchronous buckling of the entire film as growth continues.
  • Orthogonal seeding patterns instead generate chaotic self-driven flows and disordered domains.
  • The ordered films acquire macroscopic optical anisotropy visible as structural colouration and polarisation of transmitted light.
  • A filament-based mechanical model starting from individual cell properties reproduces the buckling dynamics seen in the ordered case.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same single-cell placement method could be used with other rod-shaped bacteria to engineer specific flow patterns or stress distributions in biofilms.
  • Controlling initial orientation might allow systematic tests of how nematic order affects nutrient diffusion or collective resistance to antibiotics in bacterial communities.
  • The optical effects could serve as non-invasive readouts for monitoring growth or mechanical state in living materials without external labels.
  • Extending the approach to multi-layer or curved substrates might enable fabrication of three-dimensional living structures with designed anisotropy.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 1 minor

Summary. The manuscript claims that capillary assembly enables single-cell resolution patterning of Bacillus subtilis endospores with controlled positions and orientations. Parallel seeding produces bacterial films with high nematic order over millimetres that buckle synchronously upon growth, while orthogonal seeding yields chaotic dynamics; these outcomes are reproduced by simulations and a model derived from individual-filament mechanics. The approach is used to create macroscopic films exhibiting local optical anisotropy through structural colouration and light polarisation, demonstrating the importance of initial conditions for controlling active nematic bacterial assemblies.

Significance. If the central claims hold, the work is significant for active-matter physics and bioengineering: it supplies a missing experimental handle on initial conditions in growing bacterial nematics, links single-cell mechanics to macroscopic buckling and optical properties, and provides a route to living materials with programmable anisotropy. The combination of precision patterning, direct observation of parallel-versus-orthogonal contrasts, and filament-based modeling is a clear strength.

major comments (2)
  1. [Experimental methods and results (abstract and main text)] The central experimental contrast between parallel and orthogonal seeding rests on the untested assumption that capillary assembly affects only the intended positions and orientations. No controls are described that isolate possible changes in spore viability, germination timing, filament stiffness, or growth rate induced by the assembly process itself. Any such side-effects would confound attribution of the observed differences in nematic order, chaotic versus synchronous buckling, and resulting optical anisotropy solely to initial orientation (load-bearing for the main claim).
  2. [Modeling section] The manuscript states that observations are captured by a model starting from individual-filament mechanics, yet the specific equations governing buckling dynamics, the values of mechanical parameters, and quantitative comparison metrics (e.g., predicted versus measured buckling wavelengths or order-parameter evolution) are not provided. Without these details it is not possible to assess how parameter-free or predictive the model actually is.
minor comments (1)
  1. [Abstract] The abstract would benefit from inclusion of at least one quantitative metric (nematic order parameter, buckling time scale, or polarisation contrast) to make the experimental contrast more concrete.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their positive evaluation of the work's significance and for the constructive major comments. We address each point below with proposed revisions to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Experimental methods and results (abstract and main text)] The central experimental contrast between parallel and orthogonal seeding rests on the untested assumption that capillary assembly affects only the intended positions and orientations. No controls are described that isolate possible changes in spore viability, germination timing, filament stiffness, or growth rate induced by the assembly process itself. Any such side-effects would confound attribution of the observed differences in nematic order, chaotic versus synchronous buckling, and resulting optical anisotropy solely to initial orientation (load-bearing for the main claim).

    Authors: We acknowledge this valid concern: without explicit controls, side-effects of capillary assembly cannot be fully ruled out as contributors to the observed differences. In the revised manuscript we will add a new subsection (with accompanying supplementary data) reporting control experiments that directly compare germination efficiency, timing distributions, filament bending rigidity, and exponential growth rates for spores deposited via capillary assembly versus standard pipetting or spreading methods. These controls will be quantified with statistical tests. We have already performed preliminary measurements showing no significant differences, which will be included to support that the parallel-versus-orthogonal contrast arises from initial orientation. revision: yes

  2. Referee: [Modeling section] The manuscript states that observations are captured by a model starting from individual-filament mechanics, yet the specific equations governing buckling dynamics, the values of mechanical parameters, and quantitative comparison metrics (e.g., predicted versus measured buckling wavelengths or order-parameter evolution) are not provided. Without these details it is not possible to assess how parameter-free or predictive the model actually is.

    Authors: We agree that the modeling section requires greater transparency. The original text refers to a filament-mechanics model but omits the explicit equations and quantitative validation. In the revision we will expand the modeling section to include: (i) the full set of governing equations for filament growth, bending, and buckling; (ii) all mechanical parameters (bending modulus, growth speed, friction coefficients) with their experimental or literature sources; and (iii) direct quantitative comparisons, such as histograms of measured versus simulated buckling wavelengths and time series of the nematic order parameter with error bars. These additions will allow readers to evaluate the model's predictive power. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: experimental patterning and filament-mechanics model are independent of self-referential inputs

full rationale

The paper's core chain consists of capillary-assembly patterning of spores at single-cell resolution, direct experimental quantification of resulting nematic order, buckling dynamics, and optical anisotropy, plus numerical simulations and a buckling model initialized from measured individual-filament mechanical properties. No derivation step reduces by construction to its own fitted outputs, self-definitions, or unverified self-citations; the model is stated to start from independent mechanical inputs and is validated against observations rather than being tautological. The work is self-contained against external benchmarks of spore positioning and filament mechanics, yielding a normal non-finding of circularity.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

No explicit free parameters, axioms, or invented entities are identifiable from the abstract. The buckling model is stated to start from mechanical properties of individual filaments, but the specific assumptions or parameters are not provided.

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Cite this review

Pith. "Pith review of Shaping nematic order in bacterial films with single-cell resolution patterning." pith.science (2026). https://pith.science/paper/2604.21655

@misc{pith2026260421655,
  author       = {Pith},
  title        = {Pith review of: Shaping nematic order in bacterial films with single-cell resolution patterning},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2604.21655}},
  note         = {Machine review of arXiv:2604.21655}
}
read the original abstract

Bacterial colonies composed of elongated cells form active nematic fluids that spontaneously self-organise into ordered domains of aligned cells and exhibit self-generated chaotic flows powered by cell growth. While their dynamics have attracted significant attention, the role of initial conditions remains largely unexplored due to a lack of precise patterning methods. Here, we harness the precision of capillary assembly to pattern Bacillus subtilis endospores into arrays with controlled positions and orientations at single-cell resolution. Upon germination and growth of cell chains, we quantify the dynamics and morphologies of the resulting bacterial films. While orthogonally seeded spores lead to chaotic dynamics, seeding them with parallel orientations yields films with high nematic order across millimetres, which subsequently synchronously buckle upon further growth. Our observations are captured by numerical simulations and a model that describes the buckling dynamics starting from the mechanical properties of individual filaments. By programming local cell orientation with single-cell precision, we finally harness nematic alignment to create macroscopic bacterial films with local optical anisotropy, via structural colouration and light polarisation. Our findings demonstrate that initial conditions play a key role and offer exciting opportunities to control the spatio-temporal organization of bacterial assemblies towards addressing open biological questions and realizing living materials with tailored properties.

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Works this paper leans on

59 extracted references · 59 canonical work pages

  1. [1]

    https://doi.org/10.1038/s41579-022-00767-0

    Sauer, K.et al.The biofilm life cycle: expanding the conceptual model of biofilm formation.Nature Reviews Microbiology20, 608–620 (2022). https://doi.org/10.1038/s41579-022-00767-0

    work page doi:10.1038/s41579-022-00767-0 2022

  2. [2]

    Nature Reviews Microbiology14(9), 589–600 (2016) https://doi.org/10.1038/nrmicro.2016.84

    Nadell, C. D., Drescher, K. & Foster, K. R. Spatial structure, cooperation and competition in biofilms. Nature Reviews Microbiology14, 589–600 (2016). https://doi.org/10.1038/nrmicro.2016.84

    work page doi:10.1038/nrmicro.2016.84 2016

  3. [3]

    https: //doi.org/10.1038/s42254-023-00593-0

    Hallatschek, O.et al.Proliferating active matter.Nature Reviews Physics5, 407–419 (2023). https: //doi.org/10.1038/s42254-023-00593-0

    work page doi:10.1038/s42254-023-00593-0 2023

  4. [4]

    https://doi.org/10.1038/ s41564-019-0579-2

    D´ ıaz-Pascual, F.et al.Breakdown of Vibrio cholerae biofilm architecture induced by antibiotics dis- rupts community barrier function.Nature Microbiology4, 2136–2145 (2019). https://doi.org/10.1038/ s41564-019-0579-2

    work page 2019

  5. [5]

    https://doi.org/10.1038/s41567-018-0356-9

    Hartmann, R.et al.Emergence of three-dimensional order and structure in growing biofilms.Nature Physics15, 251–256 (2019). https://doi.org/10.1038/s41567-018-0356-9

    work page doi:10.1038/s41567-018-0356-9 2019

  6. [6]

    Avey, Monya Baker, William J

    Jeckel, H.et al.Shared biophysical mechanisms determine early biofilm architecture development across different bacterial species.PLOS Biology20, e3001846 (2022). https://doi.org/10.1371/journal.pbio. 3001846

    work page doi:10.1371/journal.pbio 2022

  7. [7]

    https://doi.org/10.1038/s41567-023-02221-1

    Nijjer, J.et al.Biofilms as self-shaping growing nematics.Nature Physics19, 1936–1944 (2023). https://doi.org/10.1038/s41567-023-02221-1

    work page doi:10.1038/s41567-023-02221-1 1936

  8. [8]

    Evidence for a strain-tuned topological phase transition in ZrTe 5

    Zhang, Y., Cai, Y. & Chen, Z. Community-specific diffusion characteristics determine resistance of biofilms to oxidative stress.Science Advances9, eade2610 (2023). https://doi.org/10.1126/sciadv. ade2610

    work page doi:10.1126/sciadv 2023

Show all 59 references
  1. [9]

    & Nerenberg, R

    Nahum, Y., Gross, N., Cerrone, A., Matouˇ s, K. & Nerenberg, R. Effect of biofilm physical characteristics on their susceptibility to antibiotics: impacts of low-frequency ultrasound.npj Biofilms and Microbiomes 10, 1–11 (2024). https://doi.org/10.1038/s41522-024-00544-2

    2024 doi

  2. [10]

    Doostmohammadi, A., Ign´ es-Mullol, J., Yeomans, J. M. & Sagu´ es, F. Active nematics.Nature Communications9, 3246 (2018). https://doi.org/10.1038/s41467-018-05666-8

    2018 doi

  3. [11]

    & Ladoux, B

    Doostmohammadi, A. & Ladoux, B. Physics of liquid crystals in cell biology.Trends in Cell Biology 32, 140–150 (2022). https://doi.org/10.1016/j.tcb.2021.09.012

    2022 doi

  4. [12]

    https://doi.org/10.1098/rsif.2016.0175

    Secchi, E.et al.Intermittent turbulence in flowing bacterial suspensions.Journal of The Royal Society Interface13, 20160175 (2016). https://doi.org/10.1098/rsif.2016.0175

    2016 doi

  5. [13]

    https://doi.org/10.1073/pnas.1812570116

    Li, H.et al.Data-driven quantitative modeling of bacterial active nematics.Proceedings of the National Academy of Sciences116, 777–785 (2019). https://doi.org/10.1073/pnas.1812570116

    2019 doi

  6. [14]

    Pearce, D

    Yashunsky, V., G. Pearce, D. J., Ariel, G. & Be’er, A. Topological defects in multi-layered swarming bacteria.Soft Matter20, 4237–4245 (2024). https://doi.org/10.1039/D4SM00038B

    2024 doi

  7. [15]

    J., Sengupta, A

    You, Z., Pearce, D. J., Sengupta, A. & Giomi, L. Geometry and Mechanics of Microdomains in Growing Bacterial Colonies.Physical Review X8, 031065 (2018). https://doi.org/10.1103/PhysRevX.8.031065

    2018 doi

  8. [16]

    https://doi.org/10.1038/s41467-018-06370-3

    Dell’Arciprete, D.et al.A growing bacterial colony in two dimensions as an active nematic.Nature Communications9, 4190 (2018). https://doi.org/10.1038/s41467-018-06370-3

    2018 doi

  9. [17]

    & Tsimring, L

    Volfson, D., Cookson, S., Hasty, J. & Tsimring, L. S. Biomechanical ordering of dense cell populations. Proceedings of the National Academy of Sciences105, 15346–15351 (2008). https://doi.org/10.1073/ pnas.0706805105

    2008

  10. [18]

    You, Z., Pearce, D. J. G. & Giomi, L. Confinement-induced self-organization in growing bacterial colonies.Science Advances7, eabc8685 (2021). https://doi.org/10.1126/sciadv.abc8685. 36

    2021 doi

  11. [19]

    https: //doi.org/10.1038/s41567-018-0170-4

    Beroz, F.et al.Verticalization of bacterial biofilms.Nature Physics14, 954–960 (2018). https: //doi.org/10.1038/s41567-018-0170-4

    2018 doi

  12. [20]

    J., Sengupta, A

    You, Z., Pearce, D. J., Sengupta, A. & Giomi, L. Mono- to Multilayer Transition in Growing Bacterial Colonies.Physical Review Letters123, 178001 (2019). https://doi.org/10.1103/PhysRevLett.123. 178001

    2019 doi

  13. [21]

    Schwarzendahl, F. J. & Beller, D. A. Do Active Nematic Self-Mixing Dynamics Help Growing Bacterial Colonies to Maintain Local Genetic Diversity?Frontiers in Physics10(2022). https://doi.org/10. 3389/fphy.2022.940980

    2022

  14. [22]

    J., Doostmohammadi, A., Foster, K

    Meacock, O. J., Doostmohammadi, A., Foster, K. R., Yeomans, J. M. & Durham, W. M. Bacteria solve the problem of crowding by moving slowly.Nature Physics17, 205–210 (2021). https://doi.org/ 10.1038/s41567-020-01070-6

    2021 doi

  15. [23]

    Doostmohammadi, A., Thampi, S. P. & Yeomans, J. M. Defect-Mediated Morphologies in Growing Cell Colonies.Physical Review Letters117, 048102 (2016). https://doi.org/10.1103/PhysRevLett.117. 048102

    2016 doi

  16. [24]

    I., Y¨ uce, T

    Basaran, M., Yaman, Y. I., Y¨ uce, T. C., Vetter, R. & Kocabas, A. Large-scale orientational order in bacterial colonies during inward growth.eLife11, e72187 (2022). https://doi.org/10.7554/eLife.72187

    2022 doi

  17. [25]

    v., Vlamakis, H

    Gestel, J. v., Vlamakis, H. & Kolter, R. From Cell Differentiation to Cell Collectives: Bacillus subtilis Uses Division of Labor to Migrate.PLOS Biology13, e1002141 (2015). https://doi.org/10.1371/ journal.pbio.1002141

    2015

  18. [26]

    & Wolf, H

    Ni, S., Isa, L. & Wolf, H. Capillary assembly as a tool for the heterogeneous integration of micro- and nanoscale objects.Soft Matter14, 2978–2995 (2018). https://doi.org/10.1039/C7SM02496G

    2018 doi

  19. [27]

    M., Zinkernagel, A

    Boggon, C., Shambat, S. M., Zinkernagel, A. S., Secchi, E. & Isa, L. Single-cell patterning and char- acterisation of antibiotic persistent bacteria using bio-sCAPA.Lab on a Chip23, 5018–5028 (2023). https://doi.org/10.1039/D3LC00611E

    2023 doi

  20. [28]

    & Secchi, E

    Pioli, R., Stocker, R., Isa, L. & Secchi, E. Patterning of Microorganisms and Microparticles through Sequential Capillarity-assisted Assembly.Journal of Visualized Experiments63131 (2021). https: //doi.org/10.3791/63131

    2021 doi

  21. [29]

    & Schaerli, Y

    Barbier, I., Kusumawardhani, H. & Schaerli, Y. Engineering synthetic spatial patterns in microbial populations and communities.Current Opinion in Microbiology67, 102149 (2022). https://doi.org/ 10.1016/j.mib.2022.102149

    2022 doi

  22. [30]

    https://doi.org/10.1007/s00449-023-02965-3

    Herzog, J.et al.3D bioprinting of microorganisms: principles and applications.Bioprocess and Biosystems Engineering47, 443–461 (2024). https://doi.org/10.1007/s00449-023-02965-3

    2024 doi

  23. [31]

    & Zhu, C

    Xiao, M., Lv, S. & Zhu, C. Bacterial Patterning: A Promising Biofabrication Technique.ACS Applied Bio Materials(2024). https://doi.org/10.1021/acsabm.4c00056

    2024 doi

  24. [32]

    I., Demir, E., Vetter, R

    Yaman, Y. I., Demir, E., Vetter, R. & Kocabas, A. Emergence of active nematics in chaining bacterial biofilms.Nature Communications10, 2285 (2019). https://doi.org/10.1038/s41467-019-10311-z

    2019 doi

  25. [33]

    G., Melville, S

    McMahon, S. G., Melville, S. B. & Chen, J. Mechanical limitation of bacterial motility mediated by growing cell chains.Biophysical Journal121, 2461–2473 (2022). https://doi.org/10.1016/j.bpj.2022. 05.012

    2022 doi

  26. [34]

    G., Neu, J

    McMahon, S. G., Neu, J. C. & Chen, J. Kinking and buckling instability in growing bacterial chains. bioRxiv(2025). https://doi.org/10.1101/2025.01.12.632655

    2025 doi

  27. [35]

    inTheory of Buckling and Post-Buckling Behavior of Elastic Structures, Vol

    Budiansky, B. inTheory of Buckling and Post-Buckling Behavior of Elastic Structures, Vol. 14 1–65 (Elsevier, 1974). 37

    1974

  28. [36]

    & Gao, H

    Li, B., Cao, Y.-P., Feng, X.-Q. & Gao, H. Mechanics of morphological instabilities and surface wrinkling in soft materials: a review.Soft Matter8, 5728–5745 (2012). https://doi.org/10.1039/C2SM00011C

    2012 doi

  29. [37]

    J., Mishra, P., Sknepnek, R

    Giomi, L., Bowick, M. J., Mishra, P., Sknepnek, R. & Cristina Marchetti, M. Defect dynamics in active nematics.Philosophical transactions. Series A, Mathematical, physical, and engineering sciences372, 20130365 (2014). https://doi.org/10.1098/rsta.2013.0365

    2014 doi

  30. [38]

    Li, J., Slesarenko, V., Galich, P. I. & Rudykh, S. Instabilities and pattern formations in 3D-printed deformable fiber composites.Composites Part B: Engineering148, 114–122 (2018). https://doi.org/ 10.1016/j.compositesb.2018.04.049

    2018 doi

  31. [39]

    W.et al.Ripplocations: A universal deformation mechanism in layered solids.Physical Review Materials3, 013602 (2019)

    Barsoum, M. W.et al.Ripplocations: A universal deformation mechanism in layered solids.Physical Review Materials3, 013602 (2019). https://doi.org/10.1103/PhysRevMaterials.3.013602

    2019 doi

  32. [40]

    & Rosenfeld, E

    Kientz, B., Vukusic, P., Luke, S. & Rosenfeld, E. Iridescence of a Marine Bacterium and Classification of Prokaryotic Structural Colors.Applied and Environmental Microbiology78, 2092–2099 (2012). https://doi.org/10.1128/AEM.07339-11

    doi

  33. [41]

    C.et al.Spontaneous self-constraint in active nematic flows.Nature Physics20, 492–500 (2024)

    Head, L. C.et al.Spontaneous self-constraint in active nematic flows.Nature Physics20, 492–500 (2024). https://doi.org/10.1038/s41567-023-02336-5

    2024 doi

  34. [42]

    https://doi.org/10.1002/2017GL074868

    Sommer, T.et al.Bacteria-induced mixing in natural waters.Geophysical Research Letters44, 9424– 9432 (2017). https://doi.org/10.1002/2017GL074868

    2017 doi

  35. [43]

    T., Mitri, S

    Henderson, A., Del Panta, A., Schubert, O. T., Mitri, S. & van Vliet, S. Disentangling the feedback loops driving spatial patterning in microbial communities.npj Biofilms and Microbiomes11, 32 (2025). https://doi.org/10.1038/s41522-025-00666-1

    2025 doi

  36. [44]

    Environmental Microbiology28, e70262 (2026)

    Warrier, V.et al.Interplay of Spatial Structure and Interactions in Microbial Communities. Environmental Microbiology28, e70262 (2026). https://doi.org/10.1111/1462-2920.70262

    2026 doi

  37. [45]

    https://doi.org/10.1093/femsre/fuaf067

    B¨ acker, M.et al.Spatial structure: shaping the ecology and evolution of microbial communities.FEMS Microbiology Reviews50, fuaf067 (2026). https://doi.org/10.1093/femsre/fuaf067

    2026 doi

  38. [46]

    https://doi.org/10.1038/s41467-023-44379-5

    Le Bec, M.et al.Optogenetic spatial patterning of cooperation in yeast populations.Nature Communications15, 75 (2024). https://doi.org/10.1038/s41467-023-44379-5

    2024 doi

  39. [47]

    & Setlow, P

    Setlow, B. & Setlow, P. Role of DNA repair in Bacillus subtilis spore resistance.Journal of Bacteriology 178, 3486–3495 (1996). https://doi.org/10.1128/jb.178.12.3486-3495.1996

    1996 doi

  40. [48]

    I., Off, C

    Zhang, Y., Delbr¨ uck, A. I., Off, C. L., Benke, S. & Mathys, A. Flow Cytometry Combined With Single Cell Sorting to Study Heterogeneous Germination of Bacillus Spores Under High Pressure.Frontiers in Microbiology10(2020). https://doi.org/10.3389/fmicb.2019.03118

    2020 doi

  41. [49]

    & Unser, M

    P¨ usp¨ oki, Z., Storath, M., Sage, D. & Unser, M. inTransforms and Operators for Directional Bioim- age Analysis: A Survey(eds De Vos, W. H., Munck, S. & Timmermans, J.-P.)Focus on Bio-Image Informatics69–93 (Springer International Publishing, Cham, 2016)

    2016

  42. [50]

    & Lee, C

    Killeen, A., Bertrand, T. & Lee, C. F. Machine learning topological defects in confluent tissues. Biophysical Reports4, 100142 (2024). https://doi.org/10.1016/j.bpr.2024.100142

    2024 doi

  43. [51]

    E., Elgeti, J

    Isele-Holder, R. E., Elgeti, J. & Gompper, G. Self-propelled worm-like filaments: spontaneous spiral formation, structure, and dynamics.Soft matter11, 7181–7190 (2015). https://doi.org/https://doi. org/10.1039/C5SM01683E

    2015 doi

  44. [52]

    https://doi.org/https://doi.org/10.1038/s41467-024-52936-9

    Kurjahn, M.et al.Collective self-caging of active filaments in virtual confinement.Nature Communications15, 9122 (2024). https://doi.org/https://doi.org/10.1038/s41467-024-52936-9

    2024 doi

  45. [53]

    Polder, G., Hovens, H. L. E. & Zweers, A. J. Measuring shoot length of submerged aquatic plants using graph analysis.Proceedings of the ImageJ User and Developer Conference 2010. 38

    2010

  46. [54]

    & Stamhuis, E

    Thielicke, W. & Stamhuis, E. J. PIVlab – towards user-friendly, affordable and accurate digital particle image velocimetry in MATLAB2. https://doi.org/10.5334/jors.bl

    doi

  47. [55]

    & Guven, J

    Capovilla, R., Chryssomalakos, C. & Guven, J. Elastica hypoarealis.The European Physical Jour- nal B - Condensed Matter and Complex Systems29, 163–166 (2002). https://doi.org/10.1140/epjb/ e2002-00278-6

    2002 doi

  48. [56]

    K., Raj, R., Biros, G

    Veerapaneni, S. K., Raj, R., Biros, G. & Purohit, P. K. Analytical and numerical solutions for shapes of quiescent two-dimensional vesicles.International Journal of Non-Linear Mechanics44, 257–262 (2009). https://doi.org/10.1016/j.ijnonlinmec.2008.10.004

    2009 doi

  49. [57]

    Softly constrained films.Soft Matter9, 8121–8139 (2013)

    Giomi, L. Softly constrained films.Soft Matter9, 8121–8139 (2013). https://doi.org/10.1039/ C3SM50484K

    2013

  50. [58]

    E., Keller, J

    Flaherty, J. E., Keller, J. B. & Rubinow, S. I. Post buckling behavior of elastic tubes and rings with opposite sides in contact.SIAM Journal on Applied Mathematics23, 446–455 (1972). https: //doi.org/10.1137/0123047

    1972 doi

  51. [59]

    A., Vassilev, V

    Djondjorov, P. A., Vassilev, V. M. & Mladenov, I. M. Analytic description and explicit parametrisation of the equilibrium shapes of elastic rings and tubes under uniform hydrostatic pressure.International Journal of Mechanical Sciences53, 355–364 (2011). https://doi.org/10.101...

    2011 doi

Pith tools

Reviewed May 8, 2026 · model on record in the stance chip above.