Skip to main content
SpringerLink
Log in

Dynamics of an Expanding Cell Monolayer

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

This work considers an expansion of a dense monolayer of cells: a collective multicellular phenomenon, where cells divide, grow, and maintain contacts with their neighbors. During migration, cells display complex behavior, adjusting both their division rate and their growth after division to the local mechanical environment. Experimental observations show that cells near the edge of the expanding monolayer are larger and move faster than cells deep inside the colony. To explain these observations and describe cell migration patterns, we formulate a spatio-temporal theoretical model for multicellular dynamics in terms of the cell area distribution; the model includes cell growth after division and effective pressure. Numerical simulations of the model predict both the speed of invasion and the width of the outer proliferative rim; these predictions are in a good agreement with experimental observations. Theoretical analysis yields the equation for density of cells and reveals a novel type of propagating front with compact support. The velocity of front propagation (monolayer expansion) is derived analytically and its dependence on all the relevant parameters is determined.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Murray, J.D.: Mathematical Biology. Springer, New York (2002)

    Book  MATH  Google Scholar 

  2. Hegedus, B., Zach, J., Czirok, A., Lovey, J., Vicsek, T.: Homeostatic competition drives tumor growth and metastasis nucleation. J. Neuro-Oncol. 67, 147 (2004)

    Article  Google Scholar 

  3. Khain, E., Katakowski, M., Hopkins, S., Szalad, A., Zheng, X., Jiang, F., Chopp, M.: Collective behavior of brain tumor cells: the role of hypoxia. Phys. Rev. E 83, 031920 (2011)

    Article  ADS  Google Scholar 

  4. Liang, C.-C., Park, A.Y., Guan, J.-L.: In vitro scratch assay: a convenient and inexpensive method for analysis of cell migration in vitro. Nat. Protoc. 2, 329 (2007)

    Article  Google Scholar 

  5. Khain, E., Katakowski, M., Charteris, N., Jiang, F., Chopp, M.: Migration of adhesive glioma cells: front propagation and fingering. Phys. Rev. E 86, 011904 (2012)

    Article  ADS  Google Scholar 

  6. Fisher, R.A.: The wave of advance of advantageous genes. Ann. Eugen. 7, 355 (1937)

    Article  MATH  Google Scholar 

  7. Kolmogorov, A., Petrovsky, I., Piscounov, N.: A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem. Bull. Moscow Univ. Math. Mech. 1, 1 (1937)

    Google Scholar 

  8. Khain, E., Sander, L.M., Schneider-Mizell, C.M.: The role of cell-cell adhesion in wound healing. J. Stat. Phys. 128, 209 (2007)

    Article  ADS  MATH  Google Scholar 

  9. Khain, E., Sander, L.M.: Generalized Cahn-Hilliard equation for biological applications. Phys. Rev. E 77, 051129 (2008)

    Article  ADS  Google Scholar 

  10. Simpson, M.J., Towne, C., Sean McElwain, D.L., Upton, Z.: Migration of breast cancer cells: understanding the roles of volume exclusion and cell-to-cell adhesion. Phys. Rev. E 82, 041901 (2010)

    Article  ADS  Google Scholar 

  11. Johnston, S.T., Baker, R.E., Simpson, M.J.: A new and accurate continuum description of moving fronts. New J. Phys. 19, 033010 (2017)

    Article  ADS  Google Scholar 

  12. Vittadello, S.T., McCue, S.W., Gunasingh, G., Haass, N.K., Simpson, M.J.: Mathematical models for cell migration with real-time cell cycle dynamics. Biophys. J. 114, 1241 (2018)

    Article  ADS  Google Scholar 

  13. Charteris, N., Khain, E.: Modeling chemotaxis of adhesive cells: stochastic lattice approach and continuum description. New J. Phys. 16, 025002 (2014)

    Article  ADS  Google Scholar 

  14. Bobadilla, A.V.P., Carraro, T., Byrne, H.M., Maini, P.K., Alarcon, T.: Age structure can account for delayed logistic proliferation of scratch assays. Bull. Math. Biol. 81, 2706 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  15. d’Alessandro, J., Solon, A.P., Hayakawa, Y., Anjard, C., Detcheverry, F., Rieu, J.P., Riviere, C.: Contact enhancement of locomotion in spreading cell colonies. Nat. Phys. 13, 999 (2017)

    Article  Google Scholar 

  16. Johnston, S.T., Shah, E.T., Chopin, L.K., McElwain, D.L.S., Simpson, M.J.: Estimating cell diffusivity and cell proliferation rate by interpreting IncuCyte ZOOM assay data using the Fisher-Kolmogorov model. BMC Syst. Biol. 9, 38 (2015)

    Article  Google Scholar 

  17. Alert, R., Trepat, X.: Physical models of collective cell migration. Annu. Rev. Condens. Matter Phys. 11, 77 (2020)

    Article  Google Scholar 

  18. Alert, R., Trepat, X.: Living cells on the move. Phys. Today 74(6), 30 (2021)

    Article  Google Scholar 

  19. Trepat, X., Wasserman, M.R., Angelini, T.E., Millet, E., Weitz, D.A., Butler, J.P., Fredberg, J.J.: Physical forces during collective cell migration. Nat. Phys. 5, 426 (2009)

    Article  Google Scholar 

  20. Shraiman, B.I.: Mechanical feedback as a possible regulator of tissue growth. Proc. Natl. Acad. Sci. USA 102, 3318 (2005)

    Article  ADS  Google Scholar 

  21. Basan, M., Risler, T., Joanny, J.-F., Sastre-Garau, X., Prost, J.: Homeostatic competition drives tumor growth and metastasis nucleation. HFSP J. 3, 265 (2009)

    Article  Google Scholar 

  22. Recho, P., Ranft, J., Marcq, P.: One-dimensional collective migration of a proliferating cell monolayer. Soft Matter 12, 2381 (2016)

    Article  ADS  Google Scholar 

  23. Puliafito, A., Hufnagel, L., Neveu, P., Streichan, S., Sigal, A., Fygenson, D.K., Shraiman, B.I.: Collective and single cell behavior in epithelial contact inhibition. Proc. Natl. Acad. Sci. USA 109, 739 (2012)

    Article  ADS  Google Scholar 

  24. Streichan, S.J., Hoerner, C.R., Schneidt, T., Holzer, D., Hufnagel, L.: Spatial constraints control cell proliferation in tissues. Proc. Natl. Acad. Sci. USA 111, 5586 (2014)

    Article  ADS  Google Scholar 

  25. Reffay, M., Petitjean, L., Coscoy, S., Grasland-Mongrain, E., Amblard, F., Buguin, A., Silberzan, P.: Orientation and polarity in collectively migrating cell structures: statics and dynamics. Biophys. J. 100, 2566 (2011)

    Article  ADS  Google Scholar 

  26. Silberzan, P.: Imposing and releasing confinement to an epithelium. Talk at the KITP Conference on Active Processes in Living and Nonliving Matter, 2014. http://online.kitp.ucsb.edu/online/bioacter-c14/silberzan/; private communication (2020)

  27. Gauquelin, E., Tlili, S., Gay, C., Peyret, G., Mege, R.M., Fardin, M.A., Ladoux, B.: Influence of proliferation on the motions of epithelial monolayers invading adherent strips. Soft Matter 15, 2798 (2019)

    Article  Google Scholar 

  28. Tlili, S., Gauquelin, E., Li, B., Cardoso, O., Ladoux, B., Delanoe-Ayari, H., Graner, F.: Collective cell migration without proliferation: density determines cell velocity and wave velocity. R. Soc. Open Sci. 5, 172421 (2018)

    Article  ADS  Google Scholar 

  29. Heinrich, M.A., Alert, R., LaChance, J.M., Zajdel, T.J., Košmrlj, A., Cohen, D.J.: Size-dependent patterns of cell proliferation and migration in freely-expanding epithelia. eLife 9, e58945 (2020)

    Article  Google Scholar 

  30. Khain, E., Tsimring, L.S.: Effective pressure and cell area distribution in a confined monolayer. Fluid Dyn. Res. 50, 051413 (2018)

    Article  ADS  Google Scholar 

  31. Straetmans, J., Khain, E.: Modeling cell size dynamics in a confined nonuniform dense cell culture. J. Stat. Phys. 176, 299 (2019)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. Petitjean, L., Reffay, M., Grasland-Mongrain, E., Poujade, M., Ladoux, B., Buguin, A., Silberzan, P.: Velocity fields in a collectively migrating epithelium. Biophys. J. 98, 1790 (2010)

    Article  ADS  Google Scholar 

  33. Streichan, S.: Spatial constraints control cell proliferation in tissues, talk at the KITP Conference on Active Processes in Living and Nonliving Matter, 2014. http://online.kitp.ucsb.edu/online/bioacter-c14/streichan/

  34. Ginzberg, M.B., Kafri, R., Kirschner, M.: On being the right (cell) size. Science 348, 1245075 (2015)

    Article  Google Scholar 

  35. Heinrich, M. A., Alert, R., Wolf, A.E., Kosmrlj, A., Cohen, D.J.: Self-assembly of tessellated tissue sheets by growth and collision. Preprint at https://doi.org/10.1101/2021.05.06.442983 (2021)

  36. Khain, E., Sander, L.M.: Dynamics and pattern formation in invasive tumor growth. Phys. Rev. Lett. 96, 188103 (2006)

    Article  ADS  Google Scholar 

  37. Habbal, A., Barelli, H., Malandain, G.: Assessing the ability of the 2D Fisher-KPP equation to model cell-sheet wound closure. Math. Biosci. 252, 45 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  38. Serra-Picamal, X., Conte, V., Vincent, R., Anon, E., Tambe, D.T., Bazellieres, E., Butler, J.P., Fredberg, J.J., Trepat, X.: Mechanical waves during tissue expansion. Nat. Phys. 8, 628 (2012)

    Article  Google Scholar 

  39. Perez-Gonzalez, C., Alert, R., Blanch-Mercader, C., Gomez-Gonzalez, M., Kolodziej, T., Bazellieres, E., Casademunt, J., Trepat, X.: Active wetting of epithelial tissues. Nat. Phys. 15, 79 (2019)

    Article  Google Scholar 

  40. Mark, S., Shlomovitz, R., Gov, N.S., Poujade, M., Grasland-Mongrain, E., Silberzan, P.: Physical model of the dynamic instability in an expanding cell culture. Biophys. J. 98, 361 (2010)

    Article  Google Scholar 

  41. Basan, M., Joanny, J.F., Prost, J., Risler, T.: Undulation instability of epithelial tissues. Phys. Rev. Lett. 106, 158101 (2011)

    Article  ADS  Google Scholar 

  42. Risler, T., Basan, M.: Morphological instabilities of stratified epithelia: a mechanical instability in tumour formation. New J. Phys. 15, 065011 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  43. Tarle, V., Ravasio, A., Hakim, V., Gov, N.S.: Modeling the finger instability in an expanding cell monolayer. Integr. Biol. 7, 1218 (2015)

    Article  Google Scholar 

  44. Alert, R., Blanch-Mercader, C., Casademunt, J.: Active fingering instability in tissue spreading. Phys. Rev. Lett. 122, 088104 (2019)

    Article  ADS  Google Scholar 

  45. Yang, Y., Levine, H.: Leader-cell-driven epithelial sheet fingering. Phys. Biol. 17, 046003 (2020)

    Article  Google Scholar 

  46. Zehnder, S.M., Suaris, M., Bellaire, M.M., Angelini, T.E.: Cell volume fluctuations in MDCK monolayers. Biophys. J. 108, 247 (2015)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

This research, inspired by discussions in several KITP programs, was supported in part by the National Science Foundation under Grant No. NSF PHY-1748958.

Author information

Authors and Affiliations

  1. Department of Physics, Oakland University, Rochester, MI, 48309, USA

    Evgeniy Khain & John Straetmans

  2. College of Engineering, University of Michigan, Ann Arbor, MI, 48109, USA

    John Straetmans

Authors
  1. Evgeniy Khain
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. John Straetmans
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Evgeniy Khain.

Additional information

Communicated by Julien Tailleur.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Khain, E., Straetmans, J. Dynamics of an Expanding Cell Monolayer. J Stat Phys 184, 20 (2021). https://doi.org/10.1007/s10955-021-02812-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10955-021-02812-3

Keywords

Search

Navigation