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arxiv: 2605.15159 · v1 · submitted 2026-05-14 · ❄️ cond-mat.soft · physics.bio-ph

Recognition: 2 theorem links

· Lean Theorem

Multiscale order, flocking and phenotypic hysteresis in the cellular Potts model of epithelia

Calvin C. Bakker , Marc Durand , Fran\c{c}ois Graner , Luca Giomi
Authors on Pith no claims yet

Pith reviewed 2026-05-15 03:08 UTC · model grok-4.3

classification ❄️ cond-mat.soft physics.bio-ph
keywords cellular Potts modelepitheliaflockingnematic orderhexatic orderhysteresisactin polymerizationphase transitions
0
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The pith

In the cellular Potts model of epithelia, increasing actin polymerization rate drives a transition to long-range flocking with multiscale orientational order and phenotypic hysteresis.

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

The paper uses large-scale simulations of the cellular Potts model that incorporate both cell morphology and cytoskeletal activity to examine how collective migration and spatial organization interact in epithelial tissues. It maps a rich phase diagram in which different forms of orientational order appear either as separate phases or together across length scales. Raising the actin polymerization rate along one specific path in parameter space produces a transition into a state with long-range flocking motion. At the same time, cells maintain roughly hexagonal shapes at their own scale while nematic alignment develops at much larger scales. Cycling the monolayer through a melting transition at low noise levels creates hysteresis in cell phenotypes.

Core claim

By accounting for both cell morphology and cytoskeletal activity, the cellular Potts model simulations uncover a remarkably rich phase diagram featuring multiple types of orientational order, either as distinct phases or coexisting across length scales. A gradual increase in the actin polymerization rate drives a phase transition into a long-range flocking state. Simultaneously, quasi-long-range nematic order emerges at length scales much larger than the cell size due to the combined effects of directed motion and lateral cell-cell interactions. At length scales comparable to cell size, however, cells adopt an approximately hexagonal morphology, resulting in hexanematic order. With further增加

What carries the argument

The cellular Potts model that includes explicit cell morphology and cytoskeletal activity controlled by the actin polymerization rate as the tuning parameter.

If this is right

  • Increasing the actin polymerization rate along a defined pathway produces a transition into a long-range flocking state.
  • Quasi-long-range nematic order develops at scales much larger than individual cells from directed motion combined with lateral interactions.
  • Hexanematic order forms at cell-size scales because cells adopt approximately hexagonal shapes.
  • At higher polymerization rates nematic order becomes fully long-range while hexatic order remains quasi-long-range and confined to short scales.
  • Cycling the monolayer across the melting transition at sufficiently low noise produces phenotypic hysteresis.

Where Pith is reading between the lines

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

  • Real epithelial tissues may achieve different collective behaviors by modulating actin activity to select among the model's ordered states.
  • The competition between cell shape and self-propulsion that generates multiscale order could appear in other active-matter systems with deformable particles.
  • Experiments could test the model by varying actin regulators in cultured monolayers while tracking order at cell and tissue scales.
  • The observed hysteresis offers a possible mechanism for tissues to retain memory of past activity changes after conditions return to normal.

Load-bearing premise

The cellular Potts model's specific rules for cell adhesion, volume constraint, and actin-driven motility capture the essential physics of real epithelial monolayers.

What would settle it

Direct measurement of order parameters in MDCK monolayers that either confirms or rules out the predicted transition to long-range flocking together with the specific multiscale nematic and hexatic orders when actin polymerization rate is raised.

Figures

Figures reproduced from arXiv: 2605.15159 by Calvin C. Bakker, Fran\c{c}ois Graner, Luca Giomi, Marc Durand.

Figure 1
Figure 1. Figure 1: FIG. 1. An overview of the phases of matter produced by [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The cellular Potts model (CPM) in a hexanematic state. In (a) a single snapshot of the CPM is shown by the cell [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) Hexanematic crossover length scale [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Hysteresis in the cellular Potts model. In (a) the [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
read the original abstract

In epithelia, how do collective cell migration and tissue spatial organization feedback on each other? We address this question through large-scale numerical simulations of the cellular Potts model. By accounting for both cell morphology and cytoskeletal activity, we uncover a remarkably rich phase diagram featuring multiple types of orientational order, either as distinct phases or coexisting across length scales. We identify a specific pathway in parameter space along which a gradual increase in the actin polymerization rate drives a phase transition into a long-range flocking state. Simultaneously, quasi-long-range nematic order emerges at length scales much larger than the cell size due to the combined effects of directed motion and lateral cell-cell interactions. At length scales comparible to cell size, however, cells adopt an approximatively hexagonal morphology, resulting in hexanematic order, similar to that observed in reconstituted Madin-Darby Canine Kidney (MDCK) cell monolayers. With further increases in actin polymerization, nematic order becomes fully long-range, while hexatic order remains quasi-long-range and confined to short length scales, but independent of cytoskeletal activity. When noise is sufficiently low to allow crystallization at finite actin polymerization rate, cycling the cell-monolayer across the melting transition yields an example of phenotypical hysteresis, reminiscent of that observed across the epithelial-mesenchymal transition.

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 manuscript uses large-scale cellular Potts model simulations that incorporate cell morphology and cytoskeletal activity to investigate feedback between collective migration and spatial organization in epithelia. It reports a rich phase diagram containing multiple orientational orders (distinct or coexisting across scales), identifies a parameter pathway in which increasing actin polymerization rate drives a transition into a long-range flocking state, describes emergence of quasi-long-range nematic order at large scales and hexanematic order at cell scales, and demonstrates phenotypic hysteresis upon cycling across a melting transition at low noise.

Significance. If the central claims on long-range flocking and multiscale order are confirmed, the work would be significant for linking cytoskeletal parameters to tissue-level collective behaviors and for explaining experimental observations such as hexanematic order in MDCK monolayers. The use of an established CPM framework to explore scale-dependent orders and hysteresis is a strength, as is the identification of a specific, gradual parameter pathway for the flocking transition. The results could inform models of epithelial-mesenchymal transitions if the orders are shown to be robust beyond finite-size effects.

major comments (2)
  1. [Abstract] Abstract: The claim of a transition into a 'long-range flocking state' and 'quasi-long-range nematic order' at scales much larger than the cell size requires finite-size scaling analysis of the relevant order parameters (polarization magnitude and nematic tensor) with linear system size L. In 2D systems with continuous rotational symmetry the Mermin-Wagner theorem precludes true long-range order, so it is essential to demonstrate that the polarization does not decay with L and that the structure-factor peak remains finite in the thermodynamic limit rather than arising from correlation lengths comparable to the simulated box size.
  2. [Abstract] Abstract: Simulation details necessary to assess the reported orders are absent, including the lattice sizes (or number of cells) employed, the precise definitions and normalizations of the order parameters used to quantify flocking, nematic, and hexatic order, and any scaling collapses or finite-size extrapolations performed. Without these, it is impossible to determine whether the claimed long-range flocking is an artifact of the finite systems or a genuine thermodynamic feature.
minor comments (2)
  1. [Abstract] Typographical error: 'comparible' should read 'comparable'.
  2. [Abstract] Typographical error: 'approximatively' should read 'approximately'.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. We address the major comments point by point below. We agree that additional details and analyses are needed to fully substantiate the claims and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The claim of a transition into a 'long-range flocking state' and 'quasi-long-range nematic order' at scales much larger than the cell size requires finite-size scaling analysis of the relevant order parameters (polarization magnitude and nematic tensor) with linear system size L. In 2D systems with continuous rotational symmetry the Mermin-Wagner theorem precludes true long-range order, so it is essential to demonstrate that the polarization does not decay with L and that the structure-factor peak remains finite in the thermodynamic limit rather than arising from correlation lengths comparable to the simulated box size.

    Authors: We agree that finite-size scaling is essential to rigorously support claims of long-range or quasi-long-range order in two-dimensional systems, given the Mermin-Wagner theorem. In the revised manuscript we will add a dedicated finite-size scaling analysis of the polarization magnitude (flocking order parameter) and the nematic tensor as functions of linear system size L. We will show that, along the identified parameter pathway, the flocking polarization remains finite and does not decay with increasing L (up to the largest simulated sizes), while the nematic order parameter exhibits the algebraic decay characteristic of quasi-long-range order. These results will be presented with appropriate scaling plots to confirm that the observed orders are not artifacts of finite correlation lengths comparable to the box size. revision: yes

  2. Referee: [Abstract] Abstract: Simulation details necessary to assess the reported orders are absent, including the lattice sizes (or number of cells) employed, the precise definitions and normalizations of the order parameters used to quantify flocking, nematic, and hexatic order, and any scaling collapses or finite-size extrapolations performed. Without these, it is impossible to determine whether the claimed long-range flocking is an artifact of the finite systems or a genuine thermodynamic feature.

    Authors: We apologize for the omission of these methodological details from the abstract. The full manuscript already specifies the lattice sizes (up to 512×512 sites, corresponding to several thousand cells) and the definitions of the order parameters: the flocking polarization is the magnitude of the spatially averaged cell velocity vector; the nematic order parameter is obtained from the traceless symmetric tensor constructed from cell elongation axes; and the hexatic order parameter follows the standard six-fold bond-orientational definition. In the revision we will add an explicit Methods subsection (or appendix) that provides the precise mathematical definitions, normalizations, and the finite-size scaling collapses already performed. This will enable readers to independently verify the robustness of the reported phases. revision: yes

Circularity Check

0 steps flagged

No significant circularity in simulation-based exploration

full rationale

The paper reports results from large-scale numerical simulations of the established cellular Potts model, varying parameters such as actin polymerization rate to map a phase diagram with orientational orders and flocking. No analytical derivation chain exists in which a claimed prediction or first-principles result reduces by construction to its inputs, fitted parameters, or self-citations. Behaviors such as hexanematic order and hysteresis emerge dynamically from the model rules rather than being tautologically defined. The central claims rest on computational outcomes, not self-referential definitions or load-bearing self-citations that collapse the argument.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claims rest on the validity of the cellular Potts model as a representation of epithelia and the choice of parameters for actin polymerization and noise levels that are varied in the simulations.

free parameters (2)
  • actin polymerization rate
    Key tunable parameter varied to drive the transition into long-range flocking state
  • noise level
    Controls whether crystallization occurs and enables observation of hysteresis
axioms (2)
  • domain assumption The cellular Potts model with morphology and cytoskeletal activity terms accurately models epithelial dynamics
    Core assumption enabling the simulations and phase diagram
  • domain assumption Cell-cell interactions and directed motion lead to observed orders
    Invoked to explain hexanematic and nematic orders at different scales

pith-pipeline@v0.9.0 · 5546 in / 1436 out tokens · 91127 ms · 2026-05-15T03:08:17.038334+00:00 · methodology

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Works this paper leans on

57 extracted references · 57 canonical work pages · 2 internal anchors

  1. [1]

    Armengol-Collado, L

    J.-M. Armengol-Collado, L. N. Carenza, J. Eckert, D. Krommydas, and L. Giomi, Nat. Phys.19, 1773 (2023)

    work page 2023

  2. [2]

    Eckert, B

    J. Eckert, B. Ladoux, R.-M. M` ege, L. Giomi, and T. Schmidt, Nature Communications14, 5762 (2023)

    work page 2023

  3. [3]

    Armengol-Collado, L

    J.-M. Armengol-Collado, L. N. Nicola, and L. Giomi, eLife13, e86400 (2024)

    work page 2024

  4. [4]

    L. N. Carenza, J.-M. Armengol-Collado, D. Krommydas, and L. Giomi, Phys. Rev. Lett.134, 128304 (2025)

    work page 2025

  5. [5]

    Li and M

    Y.-W. Li and M. Pica Ciamarra, Phys. Rev. Mater.2, 045602 (2018)

    work page 2018

  6. [6]

    Durand and J

    M. Durand and J. Heu, Phys. Rev. Lett.123, 188001 (2019)

    work page 2019

  7. [7]

    Pasupalak, A.and Li, R

    Y.-W. Pasupalak, A.and Li, R. Ni, and M. Pica Cia- marra, Soft Matter16, 3914 (2020)

    work page 2020

  8. [8]

    Li and B

    J.-J. Li and B. Ai, New J. Phys.23, 083030 (2021)

    work page 2021

  9. [9]

    Nemati and J

    H. Nemati and J. de Graaf, Soft Matter20, 8337 (2024)

    work page 2024

  10. [10]

    B. I. Halperin and D. R. Nelson, Phys. Rev. Lett.41, 121 (1978)

    work page 1978

  11. [11]

    D. R. Nelson and B. I. Halperin, Phys. Rev. B19, 2457 (1979)

    work page 1979

  12. [12]

    A. P. Young, Phys. Rev. B19, 1855 (1979)

    work page 1979

  13. [13]

    Carmona-Fontaine, H

    C. Carmona-Fontaine, H. K. Matthews, S. Kuriyama, M. Moreno, G. A. Dunn, M. Parsons, J. D. Stern, and R. Mayor, Nature456, 957 (2008)

    work page 2008

  14. [14]

    Doxzen, S

    K. Doxzen, S. Nag, I. S. Marquez, B. Dubin-Thaler, C. A. Hofmann, B. Ladoux, and M. P. Sheetz, Nat. Commun. 4, 1430 (2013)

    work page 2013

  15. [15]

    Malinverno, S

    C. Malinverno, S. Corallino, F. Giavazzi, M. Bergert, Q. Li, M. Leoni, A. Disanza, E. Frittoli, A. Oldani, E. Martini, T. Lendenmann, G. Deflorian, G. V. Beznoussenko, D. Poulikakos, O. K. Haur, M. Uroz, X. Trepat, D. Parazzoli, P. Maiuri, W. Yu, A. Ferrari, R. Cerbino, and G. Scita, Nat. Mater.16, 587 (2017)

    work page 2017

  16. [16]

    Giavazzi, C

    F. Giavazzi, C. Malinverno, S. Corallino, F. Ginelli, G. Scita, and R. Cerbino, J. Phys. D: Appl. Phys.50, 474003 (2017)

    work page 2017

  17. [17]

    Glentis, C

    A. Glentis, C. Blanch-Mercader, L. Balasubramaniam, T. B. Saw, J. d’Alessandro, D. Delacour, J. Prost, W. Xi, and B. Ladoux, Sci. Adv.8, eabn5406 (2022)

    work page 2022

  18. [18]

    Scarpa and R

    E. Scarpa and R. Mayor, J. Cell Biol.212, 143 (2016)

    work page 2016

  19. [19]

    Szab´ o and R

    A. Szab´ o and R. Mayor, Annu. Rev. Genet.52, 43 (2018)

    work page 2018

  20. [20]

    Carrillo, M

    A. Carrillo, M. S. Figueroa, P. J. Davis, A. E. Aguilar, T. Hines, M. L. Van Avermaete, D. B. Zhang, C. A. Alvarado, P. E. Kuo, A. Kim, G. L. Quintero, P. F. Auer, M. J. Benskey, M. G. Castro, and P. R. Lowenstein, Nat. Commun.13, 3606 (2022)

    work page 2022

  21. [21]

    A. E. Argento, M. L. Varela, G. Singh, D. P. Vis- nuk, B. Jacobovitz, M. E. Rutherford, M. B. Edwards, Q. Chaboche, D. A. Orringer, J. A. Heth, M. G. Castro, D. A. Beller, C. Blanch-Mercader, and P. R. Lowenstein, bioRxiv 10.1101/2025.04.14.648711 (2025)

    work page doi:10.1101/2025.04.14.648711 2025

  22. [22]

    C. J. Weijer, J. Cell Sci.122, 3215 (2009)

    work page 2009

  23. [23]

    Vicsek, A

    T. Vicsek, A. Czir´ ok, E. Ben-Jacob, I. Cohen, and O. Shochet, Phys. Rev. Lett.75, 1226 (1995)

    work page 1995

  24. [24]

    Giavazzi, M

    F. Giavazzi, M. Paoluzzi, M. Macchi, D. Bi, G. Scita, M. L. Manning, R. Cerbino, and M. C. Marchetti, Soft Matter14, 3471 (2018)

    work page 2018

  25. [25]

    Baconnier, O

    P. Baconnier, O. Dauchot, V. D´ emery, G. D¨ uring, S. Henkes, C. Huepe, and A. Shee, Rev. Mod. Phys.97, 015007 (2025)

    work page 2025

  26. [26]

    Zhang and R

    Y. Zhang and R. A. Weinberg, Front. Med.12, 361 (2018)

    work page 2018

  27. [27]

    P. Yu, F. Brauns, and M. C. Marchetti, preprint arXiv:2511.04189 (2025)

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  28. [28]

    Graner and J

    F. Graner and J. A. Glazier, Phys. Rev. Lett.69, 2013 (1992)

    work page 2013

  29. [29]

    J. A. Glazier and F. Graner, Physical Review E47, 2128 (1993)

    work page 1993

  30. [30]

    Sultan, S

    S. Sultan, S. Devi, S. N. Mueller, and J. Textor, preprint arXiv:2312.09317 (2023)

    work page arXiv 2023

  31. [31]

    Niculescu, J

    I. Niculescu, J. Textor, and R. J. De Boer, PLoS comput. biol.11, e1004280 (2015)

    work page 2015

  32. [32]

    I. M. Wortel, I. Niculescu, P. M. Kolijn, N. S. Gov, R. J. de Boer, and J. Textor, Biophys. J.120, 2609 (2021)

    work page 2021

  33. [33]

    van Steijn, I

    L. van Steijn, I. M. Wortel, C. Sire, L. Dupr´ e, G. Ther- aulaz, and R. M. Merks, PLoS Comput. Biol.18, e1009156 (2022)

    work page 2022

  34. [34]

    Toffoli and N

    T. Toffoli and N. Margolus,Cellular automata machines: a new environment for modeling(MIT Press, Cambridge, MA, USA, 1987)

    work page 1987

  35. [35]

    A MCS here is equal to an amount of update-attempts of the Metropolis-Hastings algorithm that is equal to the amount of pixels in the two-dimensionalL×Larray:L 2

    work page

  36. [36]

    Durand and E

    M. Durand and E. Guesnet, Comput. Phys. Commun. 208, 54 (2016)

    work page 2016

  37. [37]

    Supplementary information, inclusive of Refs. [51–57]

    work page

  38. [38]

    Asano, A

    Y. Asano, A. Jim´ enez-Dalmaroni, T. B. Liverpool, M. C. Marchetti, L. Giomi, A. Kiger, T. Duke, and B. Baum, HFSP J.3, 194 (2009)

    work page 2009

  39. [39]

    Giomi, J

    L. Giomi, J. Toner, and N. Sarkar, Phys. Rev. Lett.129, 067801 (2022)

    work page 2022

  40. [40]

    Giomi, J

    L. Giomi, J. Toner, and N. Sarkar, Phys. Rev. E106, 024701 (2022)

    work page 2022

  41. [41]

    T. K. Ho, inProc. Int. Conf. Doc. Anal. Recognit., Vol. 1 (IEEE, 1995) pp. 278–282

    work page 1995

  42. [42]

    W. E. Lorensen and H. E. Cline, inSeminal Graphics: Pioneering Efforts That Shaped the Field(ACM, 1998) pp. 347–353

    work page 1998

  43. [43]

    [37] for details)

    Here the hexatetratic phase and more exotic phases were excluded from the visualization (see Ref. [37] for details)

    work page

  44. [44]

    Bruinsma and G

    R. Bruinsma and G. Aeppli, Phys. Rev. Lett.48, 1625 (1982). 6

    work page 1982

  45. [45]

    J. V. Selinger and D. R. Nelson, Phys. Rev. A39, 3135 (1989)

    work page 1989

  46. [46]

    J. V. Selinger, J. Phys. II (France)1, 1363 (1991)

    work page 1991

  47. [47]

    Drouin-Touchette, P

    V. Drouin-Touchette, P. P. Orth, P. Coleman, P. Chan- dra, and T. C. Lubensky, Phys. Rev. X12, 011043 (2022)

    work page 2022

  48. [48]

    Friedl and K

    P. Friedl and K. Wolf, J. Cell Biol.188, 11 (2010)

    work page 2010

  49. [49]

    K. K. Youssef, N. Narwade, A. Arcas, A. Marquez- Galera, R. Jim´ enez-Casta˜ no, C. Lopez-Blau, H. Fazilaty, D. Garc´ ıa-Gutierrez, A. Cano, J. Galcer´ an, G. Moreno- Bueno, J. P. Lopez-Atalaya, and M. A. Nieto, Nat. Can- cer5, 1660 (2024)

    work page 2024

  50. [50]

    A. S. Jahun, F. Sorgeloos, Y. Chaudhry, S. E. Arthur, M. Hosmillo, I. Georgana, R. Izuagbe, and I. G. Good- fellow, Cell Rep.42, 112179 (2023)

    work page 2023

  51. [51]

    Luebke, inProc

    D. Luebke, inProc. IEEE Int. Symp. Biomed. Imaging (2008) pp. 836–838

    work page 2008

  52. [52]

    Pedregosa, G

    F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cour- napeau, M. Brucher, M. Perrot, and ´E. Duchesnay, J. Mach. Learn. Res.12, 2825 (2011)

    work page 2011

  53. [53]

    van der Walt, J

    S. van der Walt, J. L. Sch¨ onberger, J. N´ u˜ nez-Iglesias, F. Boulogne, J. D. Warner, N. Yager, E. Gouillart, and T. Yu, PeerJ2, e453 (2014)

    work page 2014

  54. [54]

    D. Bi, X. Yang, M. C. Marchetti, and M. L. Manning, Phys. Rev. X6, 021011 (2016)

    work page 2016

  55. [55]

    Loewe, M

    B. Loewe, M. Chiang, D. Marenduzzo, and M. C. Marchetti, Phys. Rev. Lett.125, 038003 (2020)

    work page 2020

  56. [56]

    Nagai and H

    T. Nagai and H. Honda, Philos. Mag. B81, 699 (2001)

    work page 2001

  57. [57]

    On the Concept of Static Structure Factor

    K. Zhang, preprint arXiv:1606.03610 (2016). Multiscale order, flocking and phenotypic hysteresis in the cellular Potts model of epithelia: Supplementary information Calvin C. Bakker, 1 Marc Durand, 2 Fran¸ cois Graner,2 and Luca Giomi 1 1Instituut-Lorentz, Leiden Institute of Physics, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands 2Un...

    work page internal anchor Pith review Pith/arXiv arXiv 2016