Abstract
Mechanical cell competition is important during tissue development, cancer invasion, and tissue ageing. Heterogeneity plays a key role in practical applications since cancer cells can have different cell stiffness and different proliferation rates than normal cells. To study this phenomenon, we propose a one-dimensional mechanical model of heterogeneous epithelial tissue dynamics that includes cell-length-dependent proliferation and death mechanisms. Proliferation and death are incorporated into the discrete model stochastically and arise as source/sink terms in the corresponding continuum model that we derive. Using the new discrete model and continuum description, we explore several applications including the evolution of homogeneous tissues experiencing proliferation and death, and competition in a heterogeneous setting with a cancerous tissue competing for space with an adjacent normal tissue. This framework allows us to postulate new mechanisms that explain the ability of cancer cells to outcompete healthy cells through mechanical differences rather than an intrinsic proliferative advantage. We advise when the continuum model is beneficial and demonstrate why naively adding source/sink terms to a continuum model without considering the underlying discrete model may lead to incorrect results.
Similar content being viewed by others
References
Antman SS (2005) Nonlinear problems of elasticity, vol 107. Applied mathematical sciences. Springer, New York
Armstrong NJ, Painter KJ, Sherratt JA (2006) A continuum approach to modelling cell-cell adhesion. J Theor Biol 243(1):98–113. https://doi.org/10.1016/j.jtbi.2006.05.030
Baker RE, Parker A, Simpson MJ (2019) A free boundary model of epithelial dynamics. J Theor Biol 481:61–74. https://doi.org/10.1016/j.jtbi.2018.12.025
Basan M, Risler T, Joanny J, Sastre-Garau X, Prost J (2009) Homeostatic competition drives tumor growth and metastasis nucleation. HFSP J 3:265–272. https://doi.org/10.2976/1.3086732
Bodnar M, Velazquez J (2005) Derivation of macroscopic equations for individual cell-based models: a formal approach. Math Method Appl Sci 28:1757–1779. https://doi.org/10.1002/mma.638
Bras-Pereira C, Moreno E (2018) Mechanical cell competition. Curr Opin Cell Biol 51:15–21. https://doi.org/10.1016/j.ceb.2017.10.003
Cadard C, Venkova L, Recho P, Lagomarsino MC, Piel M (2019) The physics of cell-size regulation across timescales. Nat Phys 15:993–1004. https://doi.org/10.1038/s41567-019-0629-y
Delarue M, Montel F, Vignjevic D, Prost J, Joanny J, Cappello G (2014) Compressive stress inhibits proliferation in tumor spheroids through a volume limitation. Biophys J 107:1821–1828. https://doi.org/10.1016/j.bpj.2014.08.031
El-Hachem M, McCue SW, Simpson MJ (2020) A sharp-front moving boundary for malignant invasion. Physica D 412:132639. https://doi.org/10.1016/j.physd.2020.132639
Evans DJ, Morriss G (2008) Statistical mechanics of nonequilibrium liquids. Cambridge University Press, Cambridge, p 69
Evans ND, Oreffo ROC, Healy E, Thurner PJ, Man YH (2013) Epithelial mechanobiology, skin wound healing, and the stem cell niche. J Mech Behav Biomed 28:397–409. https://doi.org/10.1016/j.jmbbm.2013.04.023
Fletcher AG, Osterfield M, Baker RE, Shvartsman SY (2014) Vertex models of epithelial morphogenesis. Biophys J 106:2291–2304. https://doi.org/10.1016/j.bpj.2013.11.4498
Fletcher AG, Cooper F, Baker RE (2017) Mechanocellular models of epithelial morphogenesis. Phil Trans R Soc B 327:20150519. https://doi.org/10.1098/rstb.2015.0519
Fozard JA, Byrne HM, Jensen OE, King JR (2010) Continuum approximations of individual-based models for epithelial monolayers. Math Med Biol 27:39–74. https://doi.org/10.1093/imammb/dqp015
Galle J, Loeffler M, Drasdo D (2005) Modeling the effect of deregulated proliferation and apoptosis on the growth dynamics of epithelial cell populations in vitro. Biophys J 88:62–75. https://doi.org/10.1529/biophysj.104.041459
Geritz SAH, Kisdi A (2012) Mathematical ecology: why mechanistic models? J Math Biol 65:1411–1415. https://doi.org/10.1007/s00285-011-0496-3
Goriely A (2017) The mathematics and mechanics of growth. Springer, New York
Gudipaty SA, Lindblom J, Loftus PD, Redd MJ, Edes K, Davey CF, Krishnegowda F, Rosenblatt J (2017) Mechanical stretch triggers rapid epithelial cell division through piezo1. Nature 543:118–121. https://doi.org/10.1038/nature21407
Haass NK (2015) Dynamic tumor heterogeneity in melanoma therapy: how do we address this in a novel model system? Melanoma Manag 2(2):93–95. https://doi.org/10.2217/mmt.15.1
Han YL, Pegoraro AF, Li H, Li K, Yuan Y, Xu G, Gu Z, Sun J, Hao Y, Gupta SK, Li Y, Tang W, Kang H, Teng L, Fredberg JJ, Guo M (2019) Cell swelling, softening and invasion in a three-dimensional breast cancer model. Nat Phys 16:101–108. https://doi.org/10.1038/s41567-019-0680-8
Hanahan D, Weinberg RA (2011) Hallmarks of cancer: the next generation. Cell 144:646–674. https://doi.org/10.1016/j.cell.2011.02.013
Holmes WR, Edelstein-Keshet L (2016) Analysis of a minimal Rho-GTPase circuit regulating cell shape. Phys Biol 13:046001. https://doi.org/10.1088/1478-3975/13/4/046001
Johnston ST, Simpson MJ, Plank MJ (2013) Lattice-free descriptions of collective motion with crowding and adhesion. Phys Rev E 88:062720. https://doi.org/10.1103/PhysRevE.88.062720
Landman KA, Simpson MJ, Slater JL, Newgreen DF (2005) Diffusive and chemotactic cellular migration: smooth and discontinuous traveling wave solutions. SIAM J Appl Math 65:1420–1442. https://doi.org/10.1137/040604066
Lee S, Morishita Y (2017) Possible roles of mechanical cell elimination intrinsic to growing tissues from the perspective of tissue growth efficiency and homeostasis. PLOS Comput Biol 13:e1005651. https://doi.org/10.1371/journal.pcbi.1005651
Lekka M (2016) Discrimination between normal and cancerous cells using AFM. BioNanoSci 6:65–80. https://doi.org/10.1007/s12668-016-0191-3
Levayer R (2019) Solid stress, competition for space and cancer: The opposing roles of mechanical cell competition in tumour initiation and growth. Semin Cancer Biol. https://doi.org/10.1016/j.semcancer.2019.05.004
Lighthill MJ (1958) An introduction to fourier analysis and generalised functions. Cambridge University Press, Cambridge
Lorenzi T, Murray P, Ptashnyk M (2019) From individual-based mechanical models of multicellular systems to free-boundary problems. Preprint on arXiv:1903.06590
Tse JM, Cheng G, Tyrrell JA, Wilcox-Adelman SA, Boucher Y, Jain RK, Munn LL (2011) Mechanical compression drives cancer cells toward invasive phenotype. Proc Natl Acad Sci USA 109:911–916. https://doi.org/10.1073/pnas.1118910109
Matamoro-Vidal A, Levayer R (2019) Multiple influences of mechanical forces on cell competition. Curr Biol 29:R762–R774. https://doi.org/10.1016/j.cub.2019.06.030
Matsiaka OM, Penington CJ, Baker RE, Simpson MJ (2018) Discrete and continuum approximations for collective cell migration in a scratch assay with cell size dynamics. Bull Math Biol 80:738–757. https://doi.org/10.1007/s11538-018-0398-2
Moulton DE, Lessinnes T, Goriely A (2013) Morphoelastic rods: part I a single growing elastic rod. J Mech Phys Solids 61(2):398–427. https://doi.org/10.1016/j.jmps.2012.09.017
Murphy RJ, Buenzli PR, Baker RE, Simpson MJ (2019) A one-dimensional individual-based mechanical model of cell movement in heterogeneous tissues and its coarse-grained approximation. Proc R Soc A 475:20180838. https://doi.org/10.1098/rspa.2018.0838
Murphy RJ, Buenzli PR, Baker RE, Simpson MJ (2021) Travelling waves in a free boundary mechanobiological model of an epithelial tissue. Appl Math Lett 111:106636. https://doi.org/10.1016/j.aml.2020.106636
Murray PJ, Edwards CM, Tindall MJ, Maini PK (2009) From a discrete to a continuum model of cell dynamics in one dimension. Phys Rev E 80:031912. https://doi.org/10.1103/PhysRevE.80.031912
Murray PJ, Edwards CM, Tindall MJ, Maini PK (2012) Classifying general non-linear force laws in cell-based models via the continuum limit. Phys Rev E 85:021921. https://doi.org/10.1103/PhysRevE.85.021921
Murray PJ, Kang J, Mirams GR, Shin S, Byrne HM, Maini PK, Cho KH (2010) Modelling spatially regulated \(\beta \)-catenin dynamics and invasion in intestinal crypts. Biophys J 99:716–725. https://doi.org/10.1016/j.bpj.2010.05.016
Murray PJ, Walter A, Fletcher AG, Edwards CM, Tindall MJ, Maini PK (2011) Comparing a discrete and continuum model of the intestinal crypt. Phys Biol 8:026011. https://doi.org/10.1088/1478-3975/8/2/026011
O’Dea RD, King JR (2012) Continuum limits of pattern formation in hexagonal-cell monolayers. J Math Biol 64:579–610. https://doi.org/10.1007/s00285-011-0427-3
Osborne JM, Fletcher AG, Pitt-Francis JM, Maini PK, Gavaghan DJ (2017) Comparing individual-based approaches to modelling the self-organization of multicellular tissues. PLOS Comput Biol 13:e1005387. https://doi.org/10.1371/journal.pcbi.1005387
Pathmanathan P, Cooper J, Fletcher AG, Mirams G, Murray PJ, Osborne JM, Pitt-Francis J, Walter A, Chapman SJ (2009) A computational study of discrete mechanical tissue models. Phys Biol 6:036001. https://doi.org/10.1088/1478-3975/6/3/036001
Penta R, Ambrosi D, Shipley RJ (2014) Effective governing equations for poroelastic growing media. Q J Mech Appl Math 67:69–91. https://doi.org/10.1093/qjmam/hbt024
Plodinec M, Loparic M, Monnier C, Obermann E, Zanetti-Dallenbach R, Oertle P, Hyotyla J, Aebi U, Bentires-Alj M, Lim R, Schoenberger C (2012) The nanomechanical signature of breast cancer. Nat Nanotechnol 7:757–764. https://doi.org/10.1038/nnano.2012.167
Powell K (2019) These secret battles between your body’s cells might just save your life. Nature 574:310–312. https://doi.org/10.1038/d41586-019-03060-y
Puliafito A, Hufnagel L, Neveu P, Streichan S, Sigal A, Kuchnir Fygenson D, Shraiman BI (2012) Collective and single cell behavior in epithelial contact inhibition. Proc Natl Acad Sci USA 109:739–744. https://doi.org/10.1073/pnas.1007809109
Recho P, Ranft J, Marcq P (2016) One-dimensional collective migration of a proliferating cell monolayer. Soft Matter 8:2381–2391. https://doi.org/10.1039/C5SM02857D
Ross SM (1996) Stochastic processes, 2nd edn. Wiley, New York
Samuel MS, Lopez J, McGhee E, Croft D, Strachan D, Timpson P, Munro J, Schroder E, Zhou J, Brunton V, Baker N, Clevers H, Sansom O, Anderson K, Weaver V, Olson M (2011) Actomyosin-mediated cellular tension drives increased tissue stiffness and \(\beta \)-catenin activation to induce epidermal hyperplasia and tumor growth. Cancer Cell 19:776–791. https://doi.org/10.1016/j.ccr.2011.05.008
Serra-Picamal X, Conte V, Vincent R, Anon E, Tambe DT, Bazellieres E, Butler JP, Fredberg JJ, Trepat X (2012) Mechanical waves during tissue expansion. Nat Phys 8:628–634. https://doi.org/10.1038/nphys2355
Shraiman BI (2005) Mechanical feedback as a possible regulator of tissue growth. Proc Natl Acad Sci USA 102:3318–3323. https://doi.org/10.1073/pnas.0404782102
Trepat X, Wasserman MR, Angelini TE, Millet E, Weitz DA, Butler JP, Fredberg JJ (2009) Physical forces during collective cell migration. Nat Phys 5:426–430. https://doi.org/10.1038/nphys1269
Trepat X, Sahai E (2018) Mesoscale physical principles of collective cell organisation. Nat Phys 14:671–682. https://doi.org/10.1038/s41567-018-0194-9
Tsuboi A, Ohsawa S, Umetsu D, Sando Y, Kuranaga E, Igaki T, Fujimoto K (2018) Competition for space is controlled by apoptosis-induced change of local epithelial topology. Curr Biol 28:2115–2128. https://doi.org/10.1016/j.cub.2018.05.029
Van Meurs P, Marondotti M (2019) Discrete-to-continuum limits of particles with an annihilation rule. SIAM J Appl Math 79:1940–1966. https://doi.org/10.1137/18M1236058
Vittadello ST, McCue SW, Gunasingh G, Haass NK, Simpson MJ (2020) A novel mathematical model of heterogeneous cell proliferation. To appear J Math Biol. arXiv:2003.03024
Wagstaff L, Goschorksa M, Kozyrska K, Duclos G, Kucinski I, Chessel A, Hampton-O‘Neil L, Bradshaw CR, Allen GE, Rawlins EL, Silberzan P, Carazo-Salas RE, Piddini E (2016) Mechanical cell competition kills cells via induction of lethal p53 levels. Nat Commun 7:11373. https://doi.org/10.1038/ncomms11373
Wyatt T, Baum B, Charras G (2016) A question of time: tissue adaptation to mechanical forces. Curr Opin Cell Biol 38:68–73. https://doi.org/10.1016/j.ceb.2016.02.012
Yereniuk MA, Olson SD (2019) Global density analysis for an off-lattice agent-based model. SIAM J Appl Math 79:1700–1721. https://doi.org/10.1137/18M1186939
Zmurchok C, Bhaskar D, Edelstein-Keshet L (2018) Coupling mechanical tension and GTPase signaling to generate cell and tissue dynamics. Phys Biol 15:046004. https://doi.org/10.1088/1478-3975/aab1c0
Zmurchok C, Holmes WR (2020) Simple Rho GTPase dynamics generate a complex regulatory landscape associated with cell shape. Biophys J 118:1438–1454. https://doi.org/10.1016/j.bpj.2020.01.035
Acknowledgements
This work was funded by the Australian Research Council (DP170100474,DP180101797). R.E.B is a Royal Society Wolfson Research Merit Award holder, would like to thank the Leverhulme Trust for a Research Fellowship, and also acknowledges the BBSRC for funding via grant no. BB/R000816/1. We thank the two referees for their helpful comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Electronic supplementary material
Below is the link to the electronic supplementary material.
Rights and permissions
About this article
Cite this article
Murphy, R.J., Buenzli, P.R., Baker, R.E. et al. Mechanical Cell Competition in Heterogeneous Epithelial Tissues. Bull Math Biol 82, 130 (2020). https://doi.org/10.1007/s11538-020-00807-x
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11538-020-00807-x