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arxiv: 2606.30401 · v1 · pith:QL6IOTTZnew · submitted 2026-06-29 · ❄️ cond-mat.soft · physics.bio-ph

Stress tensor field and mesoscopic stresses in the vertex model for tissues

Paulo C. Godolphim , Leonardo G. Brunnet , Rodrigo Soto This is my paper

Pith reviewed 2026-06-30 03:35 UTC · model grok-4.3

classification ❄️ cond-mat.soft physics.bio-ph
keywords vertex modelstress tensormesoscopic stresstissue mechanicscell forcestension distributionforce inference
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The pith

The vertex model admits a family of stress tensor fields linking microscopic forces to mesoscopic stresses, with prior expressions as special cases.

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

The paper derives a microscopic expression for the stress tensor in the vertex model of tissues. This connects mesoscopic stresses directly to the forces acting in the model. It shows that the stress field allows different ways of distributing tensions across cells. Earlier expressions for the stress tensor arise as particular choices within this general field. The freedom points to possible connections with how cytoskeletal elements transmit forces inside real cells.

Core claim

We provide a microscopic derivation of the stress tensor for the VM, linking mesoscopic stresses to the VM forces. The stress field presents a freedom on how tensions are distributed across cells, which allows previous expressions to emerge as particular realizations of the field and suggests a link between mesoscopic stresses and cytoskeletal force-transmission architectures in real cells.

What carries the argument

A microscopically derived stress tensor field for the vertex model that is parameterized by the choice of how tensions are assigned among adjacent cells.

If this is right

  • Mesoscopic stresses become computable directly from the vertex-model forces without additional assumptions.
  • All previously published stress expressions for the vertex model appear as particular tension-distribution choices.
  • The freedom in tension distribution offers a way to represent different cytoskeletal force-transmission patterns inside cells.
  • Force-inference procedures can be extended by selecting the tension-distribution parameter that matches observed cell mechanics.

Where Pith is reading between the lines

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

  • Tissue simulations could vary the tension-distribution parameter to test whether it alters predicted large-scale flows or rearrangements in measurable ways.
  • The same construction might be applied to other discrete cell models to obtain comparable stress fields.
  • Live imaging of stress-sensitive probes in real tissues could be used to back out an effective tension-distribution parameter for the vertex model.

Load-bearing premise

A consistent microscopic-to-mesoscopic stress tensor can be defined for the vertex model without additional physical constraints fixing how tensions must be shared between cells.

What would settle it

Running a vertex-model simulation with a known, fixed tension distribution and directly comparing the computed stress field against an independent measurement of local stresses would reveal whether different distribution choices produce measurably distinct results.

Figures

Figures reproduced from arXiv: 2606.30401 by Leonardo G. Brunnet, Paulo C. Godolphim, Rodrigo Soto.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Force decomposition at edge [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Definition of the edge region [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) Outer and inner integration contours associated [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
read the original abstract

Mechanical stresses are fundamental regulators in biological tissues, where the vertex model (VM) is pivotal for theoretical and force-inference studies. Yet, no uniform expression for the stress tensor exists for the VM. Here we provide a microscopic derivation of it, linking mesoscopic stresses to the VM forces. The stress field presents a freedom on how tensions are distributed across cells, which allows previous expressions to emerge as particular realizations of the field and suggests a link between mesoscopic stresses and cytoskeletal force-transmission architectures in real cells.

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

0 major / 2 minor

Summary. The paper provides a microscopic derivation of the stress tensor field in the vertex model (VM) for tissues. It links the mesoscopic stresses directly to the forces in the VM and identifies a freedom in the distribution of tensions across cells. This freedom is shown to allow previous expressions for the stress tensor to arise as particular realizations of the general field, with a suggested connection to cytoskeletal force transmission in real cells.

Significance. If the derivation is correct, the result supplies a unifying framework for stress calculations in the VM, a model central to theoretical studies of epithelial mechanics and image-based force inference. Explicitly recovering prior expressions as special cases and tying the freedom to possible cellular architectures would strengthen consistency across the literature and offer a route to more flexible modeling of force transmission.

minor comments (2)
  1. The abstract states that previous expressions 'emerge as particular realizations of the field,' but the main text should include an explicit side-by-side comparison (e.g., in a dedicated subsection or table) showing the parameter choices that recover at least two earlier formulas.
  2. Notation for the tension-distribution freedom (introduced after the microscopic derivation) should be defined once with a clear symbol and then used consistently; occasional rephrasing as 'arbitrary distribution' risks ambiguity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation to accept the manuscript. No major comments were provided.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from VM forces

full rationale

The paper frames its central result as a microscopic derivation of the stress tensor field directly from the vertex model forces, with the reported freedom in tension distribution explicitly positioned to recover prior expressions as special cases rather than as a fitted input or self-definition. No load-bearing self-citations, uniqueness theorems imported from the authors, or ansatzes smuggled via citation are indicated. The derivation chain begins from the VM forces and produces a general field, satisfying the criteria for an independent result against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review limits visibility into parameters or assumptions; the tension-distribution freedom may function as an implicit choice that must be fixed by additional rules not stated here.

axioms (1)
  • domain assumption The vertex model provides a faithful microscopic representation of cell-edge forces in tissues
    Required to interpret the derived stress field as biologically relevant.

pith-pipeline@v0.9.1-grok · 5612 in / 991 out tokens · 49489 ms · 2026-06-30T03:35:50.716750+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

63 extracted references

  1. [1]

    Heisenberg and Y

    C.-P. Heisenberg and Y. Bella¨ ıche, Forces in tissue mor- phogenesis and patterning, Cell153, 948 (2013)

    2013

  2. [2]

    Goodwin and C

    K. Goodwin and C. M. Nelson, Mechanics of develop- ment, Developmental Cell56, 240 (2021)

    2021

  3. [3]

    X. Di, X. Gao, L. Peng, J. Ai, X. Jin, S. Qi, H. Li, K. Wang, and D. Luo, Cellular mechanotransduction in health and diseases: from molecular mechanism to thera- peutic targets, Sig Transduct Target Ther8, 282 (2023)

    2023

  4. [4]

    Campinho, M

    P. Campinho, M. Behrndt, J. Ranft, T. Risler, N. Minc, and C. P. Heisenberg, Tension-oriented cell divisions limit anisotropic tissue tension in epithelial spreading during zebrafish epiboly, T. Nat Cell Biol.15, 1405 (2013)

    2013

  5. [5]

    McBeath, D

    R. McBeath, D. M. Pirone, C. M. Nelson, K. Bhadriraju, and C. S. Chen, Cell shape, cytoskeletal tension, and rhoa regulate stem cell lineage commitment, Developmental Cell6, 483 (2004)

    2004

  6. [6]

    Keller, L

    R. Keller, L. A. Davidson, and D. R. Shook, How we are shaped: The biomechanics of gastrulation, Differentia- tion71, 171 (2003)

    2003

  7. [7]

    Saadaoui, D

    M. Saadaoui, D. Rocancourt, J. Roussel, F. Corson, and J. Gros, A tensile ring drives tissue flows to shape the gastrulating amniote embryo, Science367, 453 (2020)

    2020

  8. [8]

    Caldarelli, A

    P. Caldarelli, A. Chamolly, A. Villedieu, O. Alegria- Pr´ evot, C. Phan, J. Gros, and F. Corson, Self-organized tissue mechanics underlie embryonic regulation, Nature 633, 887 (2024)

    2024

  9. [9]

    A. G. Fletcher, M. Osterfield, R. E. Baker, and S. Y. Shvartsman, Vertex models of epithelial morphogenesis, Biophysical Journal106, 2291 (2014)

    2014

  10. [10]

    S. Alt, P. Ganguly, and G. Salbreux, Vertex models: from cell mechanics to tissue morphogenesis, Phil. Trans. R. Soc.372, 20150520 (2017)

    2017

  11. [11]

    Lange, F

    Z. Lange, F. Matth¨ aus, and M. Qiu, Vertex models cap- turing subcellular scales in epithelial tissues, PLOS Com- putational Biology21, e1012993 (2025)

    2025

  12. [12]

    Salbreux, L

    G. Salbreux, L. K. Barthel, P. A. Raymond, and D. K. Lubensky, Coupling mechanical deformations and planar cell polarity to create regular patterns in the zebrafish retina, PLoS Comput. Biol.8, e1002618 (2012)

    2012

  13. [13]

    Weliky and G

    M. Weliky and G. Oster, The mechanical basis of cell re- arrangement. I. epithelial morphogenesis during fundulus epiboly, Development109, 373 (1990)

    1990

  14. [14]

    P´ erez-Verdugo, G

    F. P´ erez-Verdugo, G. Reig, M. Cerda, M. L. Concha, and R. Soto, Geometrical characterization of active contrac- tion pulses in epithelial cells using the two-dimensional vertex model, J. R. Soc. Interface19, 20210851 (2022)

    2022

  15. [15]

    Weliky, S

    M. Weliky, S. Minsuk, R. Keller, and G. Oster, Noto- chord morphogenesis in Xenopus laevis: simulation of 6 cell behavior underlying tissue convergence and exten- sion, Development112, 1231 (1991)

    1991

  16. [16]

    Honda, N

    H. Honda, N. Motosugi, T. Nagai, M. Tanemura, and T. Hiiragi, Computer simulation of emerging asymmetry in the mouse blastocyst, Development135, 1407 (2008)

    2008

  17. [17]

    Y. Mao, A. L. Tournier, A. Hoppe, L. Kester, B. J. Thompson, and N. Tapon, Differential proliferation rates generate patterns of mechanical tension that orient tissue growth, The EMBO Journal32, 2790 (2013)

    2013

  18. [18]

    L. Sui, S. Alt, M. Weigert, N. Dye, S. Eaton, F. Jug, E. W. Myers, F. J¨ ulicher, G. Salbreux, and C. Dah- mann, Differential lateral and basal tension drive fold- ing of drosophila wing discs through two distinct mecha- nisms, Nat Commun9, 4620 (2018)

    2018

  19. [19]

    R. J. Tetley, M. F. Staddon, D. Heller, A. Hoppe, S. Banerjee, and Y. Mao, Tissue fluidity promotes ep- ithelial wound healing, Nat. Phys.15, 1195 (2019)

    2019

  20. [20]

    Farhadifar, J

    R. Farhadifar, J. C. R¨ oper, B. Aigouy, S. Eaton, and F. J¨ ulicher, The influence of cell mechanics, cell-cell inter- actions, and proliferation on epithelial packing, Current Biology17, 2095 (2007)

    2095

  21. [21]

    D. B. Staple, R. Farhadifar, J. C. R¨ oper, B. Aigouy, S. Eaton, and F. J¨ ulicher, Mechanics and remodelling of cell packings in epithelia, The European Physical Journal E33, 117 (2010)

    2010

  22. [22]

    D. Bi, J. H. Lopez, J. M. Schwarz, and M. L. Manning, Energy barriers and cell migration in densely packed tis- sues, Soft Matter10, 1885 (2014)

    2014

  23. [23]

    D. Bi, J. H. Lopez, J. M. Schwarz, and M. L. Manning, A density-independent rigidity transition in biological tis- sues, Nature Physics11, 1074 (2015)

    2015

  24. [24]

    Arzash, I

    S. Arzash, I. Tah, A. J. Liu, and M. L. Manning, Rigidity of epithelial tissues as a double optimization problem, Phys. Rev. Res.7, 013157 (2025)

    2025

  25. [25]

    D. L. Barton, S. Henkes, C. J. Weijer, and R. Sknep- nek, Active vertex model for cell-resolution description of epithelial tissue mechanics, PLoS Comput Biol13, e1005569 (2017)

    2017

  26. [26]

    D. Bi, X. Yang, M. C. Marchetti, and M. L. Manning, Motility-driven glass and jamming transitions in biolog- ical tissues, Phys Rev X6, 021011 (2016)

    2016

  27. [27]

    X. Yang, D. Bi, M. Czajkowski, M. Merkel, M. L. Man- ning, and M. C. Marchetti, Correlating cell shape and cel- lular stress in motile confluent tissues, Proc. Natl. Acad. Sci. U.S.A.114, 12663 (2017)

    2017

  28. [28]

    N. Noll, M. Mani, I. Heemskerk, S. J. Streichan, and B. I. Shraiman, Active tension network model suggests an exotic mechanical state realized in epithelial tissues, Nature Physics13, 1221 (2017)

    2017

  29. [29]

    S. Kim, M. Pochitaloff, G. A. Stooke-Vaughan, and O. Camp` as, Embryonic tissues as active foams, Nature Physics17, 859 (2021)

    2021

  30. [30]

    L. P. Dow, T. Parmar, M. C. Marchetti, and B. L. Pruitt, Engineering tools for quantifying and manipu- lating forces in epithelia, Biophysics Reviews4, 021303 (2023)

    2023

  31. [31]

    Naik, Y.-E

    S. Naik, Y.-E. Keta, K. Pranjic-Ferscha, ´Edouard Han- nezo, S. Henkes, and C.-P. Heisenberg, Keratins coordi- nate tissue spreading by balancing spreading forces with tissue material properties, Nat Commun (2026)

    2026

  32. [32]

    Ishihara and K

    S. Ishihara and K. Sugimura, Bayesian inference of force dynamics during morphogenesis, Journal of Theoretical Biology313, 201 (2012)

    2012

  33. [33]

    K. K. Chiou, L. Hufnagel, and B. I. Shraiman, Mechani- cal stress inference for two dimensional cell arrays, PLoS Comput Biol.8, e1002512 (2012)

    2012

  34. [34]

    G. W. Brodland, J. H. Veldhuis, S. Kim, M. Perrone, D. Mashburn, and M. S. Hutson, Cellfit: A cellular force- inference toolkit using curvilinear cell boundaries, PLoS ONE9, e99116 (2014)

    2014

  35. [35]

    Roffay, C

    C. Roffay, C. J. Chan, B. Guirao, T. Hiiragi, and F. Graner, Inferring cell junction tension and pressure from cell geometry, Development148, dev192773 (2021)

    2021

  36. [36]

    S. V. Plotnikov, B. Sabass, U. S. Schwarz, and C. M. Wa- terman, High-resolution traction force microscopy, Meth- ods in Cell Biology123, 367 (2014)

    2014

  37. [37]

    A. J. Ribeiro, A. K. Denisin, R. E. Wilson, and B. L. Pruitt, For whom the cells pull: Hydrogel and micropost devices for measuring traction forces, Methods94, 51 (2016)

    2016

  38. [38]

    Schierbaum, J

    N. Schierbaum, J. Rheinlaender, and T. E. Sch¨ affer, Combined atomic force microscopy (afm) and traction force microscopy (tfm) reveals a correlation between vis- coelastic material properties and contractile prestress of living cells, Soft Matter15, 1721 (2019)

    2019

  39. [39]

    Guevorkian and J

    K. Guevorkian and J. L. Maˆ ıtre, Micropipette aspiration: A unique tool for exploring cell and tissue mechanics in vivo, Methods in Cell Biology139, 187 (2017)

    2017

  40. [40]

    W. Kong, O. Loison, P. C. Shivakumar, E. H. Y. Chan, M. Saadaoui, C. Collinet, P. F. Lenne, and R. Cl´ ement, Experimental validation of force inference in epithelia from cell to tissue scale, Scientific Reports9, 50690 (2019)

    2019

  41. [41]

    Camp` as, T

    O. Camp` as, T. Mammoto, S. Hasso, R. A. Sperling, D. O’connell, A. G. Bischof, R. Maas, D. A. Weitz, L. Ma- hadevan, and D. E. Ingber, Quantifying cell-generated mechanical forces within living embryonic tissues, Nature Methods11, 183 (2014)

    2014

  42. [42]

    Serwane, A

    F. Serwane, A. Mongera, P. Rowghanian, D. A. Keal- hofer, A. A. Lucio, Z. M. Hockenbery, and O. Camp` as, In vivo quantification of spatially varying mechanical properties in developing tissues, Nature Methods14, 181 (2016)

    2016

  43. [43]

    A. A. Lucio, A. Mongera, E. Shelton, R. Chen, A. M. Doyle, and O. Camp` as, Spatiotemporal variation of en- dogenous cell-generated stresses within 3d multicellular spheroids, Scientific Reports7, 12022 (2017)

    2017

  44. [44]

    Trepat, M

    X. Trepat, M. Grabulosa, F. Puig, G. N. Maksym, D. Navajas, and R. Farr´ e, Viscoelasticity of human alve- olar epithelial cells subjected to stretch, Am. J. Physiol. Lung Cell. Mol. Physiol.287, L1025 (2004)

    2004

  45. [45]

    M. A. Garcia, E. Sadeghipour, L. Engel, W. J. Nelson, and B. L. Pruitt, Mems device for applying shear and tension to an epithelium combined with fluorescent live cell imaging, Journal of micromechanics and microengi- neering : structures, devices, and systems30, 125004 (2020)

    2020

  46. [46]

    Roshanzadeh, T

    A. Roshanzadeh, T. T. Nguyen, K. D. Nguyen, D. S. Kim, B. K. Lee, D. W. Lee, and E. S. Kim, Mechanoad- aptive organization of stress fiber subtypes in epithelial cells under cyclic stretches and stretch release, Scientific Reports10, 18684 (2020)

    2020

  47. [47]

    K. C. Hart, J. Y. Sim, M. A. Hopcroft, D. J. Co- hen, J. Tan, W. J. Nelson, and B. L. Pruitt, An easy- to-fabricate cell stretcher reveals density-dependent me- chanical regulation of collective cell movements in ep- ithelia, Cellular and Molecular Bioengineering14, 569 7 (2021)

    2021

  48. [48]

    Vanslambrouck, W

    M. Vanslambrouck, W. Thiels, J. Vangheel, C. V. Bavel, B. Smeets, and R. Jelier, Image-based force inference by biomechanical simulation, PLoS Comput Biol.20, e1012629 (2024)

    2024

  49. [49]

    Jurado, J

    A. Jurado, J. Isensee, A. Hofemeier, L. J. Kr¨ uger, R. Wit- tkowski, R. Golestanian, P. Bittihn, and T. Betz, 3D dy- namic multiscale force and shape analysis of in-vivo elas- tic stress sensors, bioRxiv , 2025.01.22.633835 (2025)

    2025

  50. [50]

    Nestor-Bergmann, G

    A. Nestor-Bergmann, G. Goddard, S. Woolner, and O. E. Jensen, Relating cell shape and mechanical stress in a spatially disordered epithelium using a vertex-based model, Mathematical Medicine and Biology: A Journal of the IMA35, i1 (2018)

    2018

  51. [51]

    O. E. Jensen, E. Johns, and S. Woolner, Force networks, torque balance and airy stress in the planar vertex model of a confluent epithelium, Proc Math Phys Eng Sci476, 20190716 (2020)

    2020

  52. [52]

    P´ erez-Verdugo,Statistical Properties of a Non- Polarized Active Tissue, Ph.D

    F. P´ erez-Verdugo,Statistical Properties of a Non- Polarized Active Tissue, Ph.D. thesis, Universidad de Chile (2021)

    2021

  53. [53]

    S. Tong, N. Singh, R. Sknepnek, and A. Koˇ smrlj, Lin- ear viscoelastic properties of the vertex model for epithe- lial tissues, PLOS Computational Biology18, e1010135 (2022)

    2022

  54. [54]

    O. E. Jensen and C. K. Revell, Couple stresses and dis- crete potentials in the vertex model of cellular monolay- ers, Biomech Model Mechanobiol22, 1465–1486 (2023)

    2023

  55. [55]

    N. C. Admal and E. B. Tadmor, A unified interpreta- tion of stress in molecular systems, J Elast100, 63–143 (2010)

    2010

  56. [56]

    P. C. Godolphim, L. G. Brunnet, and R. Soto, Parameter degeneracy in the vertex model for tissues, Phys. Rev. E 112, 064406 (2025)

    2025

  57. [57]

    J. H. Irving and J. G. Kirkwood, The statistical mechan- ical theory of transport processes. iv. the equations of hydrodynamics, J. Chem. Phys.18, 817–829 (1950)

    1950

  58. [58]

    Soto,Kinetic Theory and Transport Phenomena(Ox- ford University Press, Oxford, 2016)

    R. Soto,Kinetic Theory and Transport Phenomena(Ox- ford University Press, Oxford, 2016)

    2016

  59. [59]

    Triguero-Platero, F

    G. Triguero-Platero, F. Ziebert, and L. L. Bonilla, Coarse-graining the vertex model and its response to shear, Phys. Rev. E108, 044118 (2023)

    2023

  60. [60]

    L. D. Landau and E. M. Lifshitz,Theory of Elasticity, 3rd ed., Course of Theoretical Physics, Vol. 7 (Butterworth- Heinemann, Oxford, 1986)

    1986

  61. [61]

    Guillot and T

    C. Guillot and T. Lecuit, Mechanics of epithelial tis- sue homeostasis and morphogenesis, Science340, 1185 (2013)

    2013

  62. [62]

    G. R. Kale, X. Yang, J. M. Philippe, M. Mani, P. F. Lenne, and T. Lecuit, Distinct contributions of tensile and shear stress on e-cadherin levels during morphogen- esis, Nature Communications9, 1 (2018)

    2018

  63. [63]

    (9) is only obtained if λc m =λ m

    Indeed, to obtain an expression free of degeneracy (that is, ifA 0c →A ′ 0c =A 0c +P 0/KAc, ∆Pc is unchanged) it is enough to have ∆Pc =P c −P m λc mPm with P m λc m = 1, but later, the correct force in Eq. (9) is only obtained if λc m =λ m. SUPPLEMENT AL MA TERIAL VER TEX FORCES The vertex variational force [Eq. (1) in the main text] can be expressed as ...