Vertex stability and topological transitions in vertex models of foams and epithelia

  • Meryl A. Spencer
  • Zahera Jabeen
  • David K. Lubensky
Regular Article


In computer simulations of dry foams and of epithelial tissues, vertex models are often used to describe the shape and motion of individual cells. Although these models have been widely adopted, relatively little is known about their basic theoretical properties. For example, while fourfold vertices in real foams are always unstable, it remains unclear whether a simplified vertex model description has the same behavior. Here, we study vertex stability and the dynamics of T1 topological transitions in vertex models. We show that, when all edges have the same tension, stationary fourfold vertices in these models do indeed always break up. In contrast, when tensions are allowed to depend on edge orientation, fourfold vertices can become stable, as is observed in some biological systems. More generally, our formulation of vertex stability leads to an improved treatment of T1 transitions in simulations and paves the way for studies of more biologically realistic models that couple topological transitions to the dynamics of regulatory proteins.

Graphical abstract


Living systems: Multicellular Systems 


  1. 1.
    J.D. Axelrod, Sem. Cell Dev. Biol. 20, 964 (2009)CrossRefGoogle Scholar
  2. 2.
    T. Lecuit, P.F. Lenne, E. Munro, Annu. Rev. Cell Dev. Biol. 27, 157 (2011)CrossRefGoogle Scholar
  3. 3.
    F. Bosveld, I. Bonnet, B. Guirao, S. Tlili, Z. Wang, A. Petitalot, R. Marchand, P.L. Bardet, P. Marcq, F. Graner et al., Science 336, 724 (2012)ADSCrossRefGoogle Scholar
  4. 4.
    A. Classen, K. Anderson, E. Marois, S. Eaton, Dev. Cell 9, 805 (2005)CrossRefGoogle Scholar
  5. 5.
    Y. Mao, A.L. Tournier, P.A. Bates, J.E. Gale, N. Tapon, B.J. Thompson, Genes Dev. 25, 131 (2011)CrossRefGoogle Scholar
  6. 6.
    E. Assémat, E. Bazelliéres, E. Pallesi-Pocachard, A.L. Bivic, D. Massey-Harroche, Biomembranes 1778, 614 (2008)CrossRefGoogle Scholar
  7. 7.
    A.G. Fletcher, M. Osterfield, R.E. Baker, S.Y. Shvartsman, Biophys. J. 106, 2291 (2014)ADSCrossRefGoogle Scholar
  8. 8.
    A.G. Fletcher, J.M. Osborne, P.K. Maini, D.J. Gavaghan, Progr. Biophys. Mol. Biol. 113, 299 (2013)CrossRefGoogle Scholar
  9. 9.
    S. Schilling, M. Willecke, T. Aegerter-Wilmsen, O.A. Cirpka, K. Basler, C. von Mering, PLoS Comput. Biol. 7, e1002025 (2011)ADSCrossRefGoogle Scholar
  10. 10.
    R. Farhadifar, J.C. Röper, B. Algouy, S. Eaton, F. Jülicher, Curr. Biol. 17, 2095 (2007)CrossRefGoogle Scholar
  11. 11.
    P.A. Raymond, S.M. Colvin, Z. Jabeen, M. Nagashima, L.K. Barthel, J. Hadidjojo, L. Popova, V.R. Pejaver, D.K. Lubensky, PLoS ONE 9, e85325 (2014)ADSCrossRefGoogle Scholar
  12. 12.
    M. Rauzi, P.F. Lenne, T. Lecuit, Nature 468, 1110 (2010)ADSCrossRefGoogle Scholar
  13. 13.
    A.M. Greiner, H. Chen, J.P. Spatz, R. Kemkemer, PLoS ONE 8, e77328 (2013)ADSCrossRefGoogle Scholar
  14. 14.
    D. Weaire, S. Hutzler, The Physics of Foams (Clarendon Press, 1999)Google Scholar
  15. 15.
    H. Honda, H. Yamanaka, M. Dan-Sohkawa, J. Theor. Biol. 106, 423 (1984)CrossRefGoogle Scholar
  16. 16.
    H. Honda, H. Yamanaka, G. Eguchi, J. Embryol. Exp. Morphol. 98, 1 (1986)Google Scholar
  17. 17.
    T. Okuzono, K. Kawasaki, Phys. Rev. E 51, 1246 (1995)ADSCrossRefGoogle Scholar
  18. 18.
    H. Frost, C. Thompson, C. Howe, J. Whang, Scr. Metall. 22, 65 (1988)CrossRefGoogle Scholar
  19. 19.
    Y. Ishimoto, Y. Morishita, Phys. Rev. E 90, 052711 (2014)ADSCrossRefGoogle Scholar
  20. 20.
    G. Salbreux, L.K. Barthel, P.A. Raymond, D.K. Lubensky, PLoS Comput. Biol. 8, e1002618 (2012)ADSCrossRefGoogle Scholar
  21. 21.
    K. Sherrard, F. Robin, P. Lemaire, E. Munro, Curr. Biol. 20, 1499 (2010)CrossRefGoogle Scholar
  22. 22.
    H. Chen, G. Brodland, J. Biomech. Engin. Trans. ASME 122, 394 (2000)CrossRefGoogle Scholar
  23. 23.
    E. Hannezo, J. Prost, J.F. Joanny, Proc. Natl. Acad. Sci. U.S.A. 111, 27 (2014)ADSCrossRefGoogle Scholar
  24. 24.
    B. Aigouy, R. Farhadifar, D.B. Staple, A. Sagner, J.C. Röper, F. Jülicher, S. Eaton, Cell 142, 773 (2010)CrossRefGoogle Scholar
  25. 25.
    J.T. Blankenship, S.T. Backovic, J.S.P. Sanny, O. Weitz, J.A. Zallen, Dev. Cell 11, 459 (2006)CrossRefGoogle Scholar
  26. 26.
    M. Rauzi, P. Verant, T. Lecuit, P.F. Lenne, Nat. Cell Biol. 10, 1401 (2008)CrossRefGoogle Scholar
  27. 27.
    R.P. Simone, S. DiNardo, Development 137, 1385 (2010)CrossRefGoogle Scholar
  28. 28.
    P.L. Bardet, B. Guirao, C. Paoletti, F. Serman, V. Léopold, F. Bosveld, Y. Goya, V. Mirouse, F. Graner, Y. Bellaiche, Dev. Cell 25, 534 (2013)CrossRefGoogle Scholar
  29. 29.
    G. Trichas, A.M. Smith, N. White, V. Wilkins, T. Watanabe, A. Moore, B. Joyce, J. Sugnaseelan, T.A. Rodriguez, D. Kay et al., PLoS Biol. 10, e1001256 (2012)CrossRefGoogle Scholar
  30. 30.
    M. Tamada, J.A. Zallen, Dev. Cell 35, 151 (2015)CrossRefGoogle Scholar
  31. 31.
    M.J. Harding, H.F. McGraw, A. Nechiporuk, Development 141, 2549 (2014)CrossRefGoogle Scholar
  32. 32.
    H. Honda, Int. Rev. Cytol. 81, 191 (1983)CrossRefGoogle Scholar
  33. 33.
    T. Nagai, H. Honda, Philos. Mag. B 81, 699 (2001)ADSCrossRefGoogle Scholar
  34. 34.
    S. Ishihara, K. Sugimura, S.J. Cox, I. Bonnet, Y. Bellaiche, F. Graner, Eur. Phys. J. E 36, 9859 (2013)CrossRefGoogle Scholar
  35. 35.
    P. Spahn, R. Reuter, PLoS ONE 8, e75051 (2013)ADSCrossRefGoogle Scholar
  36. 36.
    X. Du, M. Osterfield, S.Y. Shvartsman, Phys. Biol. 11, 066007 (2014)ADSCrossRefGoogle Scholar
  37. 37.
    D.B. Staple, R. Farhadifar, J. Röper, B. Aigouy, S. Eaton, F. Jülicher, Eur. Phys. J. E 33, 117 (2010)CrossRefGoogle Scholar
  38. 38.
    D. Bi, J.H. Lopez, J.M. Schwarz, M.L. Manning, Nat. Phys. 11, 1074 (2015)CrossRefGoogle Scholar
  39. 39.
    K.T. Sakurai, T. Kojima, T. Aigaki, S. Hayashi, Dev. Biol. 309, 126 (2007)CrossRefGoogle Scholar
  40. 40.
    K. Kawasaki, T. Nagai, K. Nakashima, Philos. Mag. B 60, 399 (1989)ADSCrossRefGoogle Scholar
  41. 41.
    G. Odell, G. Oster, P. Alberch, B. Burnside, Dev. Biol. 85, 446 (1981)CrossRefGoogle Scholar
  42. 42.
    W. Karush, PhD Thesis, Univ. of Chicago (1939)Google Scholar
  43. 43.
    H. Kuhn, A. Tucker, Proc. Second Berkeley Symp. Math. Stat. Probab, pp. 481--492 (1951)Google Scholar
  44. 44.
    P. Campinho, M. Behrndt, J. Ranft, T. Risler, N. Minc, C.P. Heisenberg, Nat. Cell Biol. 15, 65 (2013)CrossRefGoogle Scholar
  45. 45.
    M. Zajac, G. Jones, J. Glazier, Phys. Rev. Lett. 85, 2022 (2000)ADSCrossRefGoogle Scholar
  46. 46.
    M. Rauzi, P. Verant, T. Lecuit, P.F. Lenne, Nat. Cell Biol. 10, 1401 (2008)CrossRefGoogle Scholar
  47. 47.
    Y. Mao, A.L. Tournier, A. Hoppe, L. Kester, B.J. Thompson, N. Tapon, EMBO J. 32, 2790 (2013)CrossRefGoogle Scholar
  48. 48.
    M. Durand, H.A. Stone, Phys. Rev. Lett. 97, 226101 (2006)ADSCrossRefGoogle Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Meryl A. Spencer
    • 1
  • Zahera Jabeen
    • 1
  • David K. Lubensky
    • 1
  1. 1.Department of PhysicsUniversity of MichiganAnn ArborUSA

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