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Emergence of spatial patterns in a mathematical model for the co-culture dynamics of epithelial-like and mesenchymal-like cells

1. Department of Mathematical Sciences, Politecnico di Torino, Corso Duca degli Abruzzi 24,10129 Torino, Italy
2. School of Mathematics and Statistics, University of St Andrews, North Haugh, St Andrews, Fife, KY16 9SS, United Kingdom

Accumulating evidence indicates that the interaction between epithelial and mesenchymal cells plays a pivotal role in cancer development and metastasis formation. Here we propose an integro-differential model for the co-culture dynamics of epithelial-like and mesenchymal-like cells. Our model takes into account the effects of chemotaxis, adhesive interactions between epithelial-like cells, proliferation and competition for nutrients. We present a sample of numerical results which display the emergence of spots, stripes and honeycomb patterns, depending on parameters and initial data. These simulations also suggest that epithelial-like and mesenchymal-like cells can segregate when there is little competition for nutrients. Furthermore, our computational results provide a possible explanation for how the concerted action between epithelial-cell adhesion and mesenchymal-cell spreading can precipitate the formation of ring-like structures, which resemble the fibrous capsules frequently observed in hepatic tumours.

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Keywords Mathematical modelling; epithelial-mesenchymal interactions; pattern formation; segregation patterns; ring patterns

Citation: Marcello Delitala, Tommaso Lorenzi. Emergence of spatial patterns in a mathematical model for the co-culture dynamics of epithelial-like and mesenchymal-like cells. Mathematical Biosciences and Engineering, 2017, 14(1): 79-93. doi: 10.3934/mbe.2017006


  • [1] W. Alt, Biased random walk models for chemotaxis and related diffusion approximations, J. Math. Biol., 9 (1980): 147-177.
  • [2] A. R. A. Anderson,M. Chaplain,E. Newman,R. Steele,E. Thompson, Mathematical modelling of tumour invasion and metastasis, J. Theor. Med., 2 (2000): 129-154.
  • [3] N. Bellomo, A. Bellouquid and M. Delitala, Methods and tools of the mathematical kinetic theory toward modeling complex biological systems, in Transport Phenomena and Kinetic Theory, Eds. C. Cercignani and E. Gabetta, Birkhäuser (Boston), (2007), 175-193
  • [4] N. Bellomo,M. Delitala, From the mathematical kinetic, and stochastic game theory for active particles to modelling mutations, onset, progression and immune competition of cancer cells, Phys. Life Rev., 5 (2008): 183-206.
  • [5] N. Bellomo,A. Bellouquid,J. Nieto,J. Soler, Modelling chemotaxis from L2-closure moments in kinetic theory of active particles, Discrete Contin. Dyn. Systems B, 18 (2013): 847-863.
  • [6] R. Callard,A. J. George,J. Stark, Cytokines, chaos, and complexity, Immunity, 11 (1999): 507-513.
  • [7] F. Cerreti,B. Perthame,C. Schmeiser,M. Tang,N. Vauchelet, Waves for an hyperbolic Keller-Segel model and branching instabilities, Math. Models and Meth. in Appl. Sci., 21 (2011): 825-842.
  • [8] F. A. C. C. Chalub,P. A. Markowich,B. Perthame,C. Schmeiser, Kinetic models for chemotaxis and their drift-diffusion limits, Monatsh. Math., 142 (2004): 123-141.
  • [9] A. Chauviere,T. Hillen,L. Preziosi, Modeling cell movement in anisotropic and heterogeneous network tissues, Netw. Heterog. Media, 2 (2007): 333-357.
  • [10] R. H. Chisholm,B. D. Hughes,K. A. Landman,M. Zaman, Analytic study of three-dimensional single cell migration with and without proteolytic enzymes, Cell. Mol. Bioeng, 6 (2013): 239-249.
  • [11] R. H. Chisholm,B. D. Hughes,K. A. Landman, Building a morphogen gradient without diffusion in a growing tissue, PLoS ONE, 5 (2010): e12857.
  • [12] R. H. Chisholm,T. Lorenzi,A. Lorz,A. K. Larsen,L. Neves de Almeida,A. Escargueil,J. Clairambault, Emergence of drug tolerance in cancer cell populations: An evolutionary outcome of selection, non-genetic instability and stress-induced adaptation, Cancer Res., 75 (2015): 930-939.
  • [13] S. Cui,A. Friedman, Analysis of a mathematical model of the growth of necrotic tumors, J. Math. Anal. Appl., 255 (2001): 636-677.
  • [14] J. C. Dallon,J. A. Sherratt,P. K. Maini, Mathematical modelling of extracellular matrix dynamics using discrete cells: fiber orientation and tissue regeneration, J. Theoret. Biol., 199 (1999): 449-471.
  • [15] M. Delitala,T. Lorenzi, A mathematical model for the dynamics of cancer hepatocytes under therapeutic actions, J. Theoret. Biol., 297 (2012): 88-102.
  • [16] M. Delitala,T. Lorenzi, A mathematical model for progression and heterogeneity in colorectal cancer dynamics, Theor. Popul. Biol., 79 (2011): 130-138.
  • [17] R. Dickinson, A generalized transport model for biased cell migration in an anisotropic environment, J. Math. Biol., 40 (2000): 97-135.
  • [18] R. Erban,H. G. Othmer, Taxis equations for amoeboid cells, J. Math. Biol., 54 (2007): 847-885.
  • [19] P. Friedl,K. Wolf, Tumour-cell invasion and migration: Diversity and escape mechanims, Nat. Rev. Cancer, 3 (2003): 362-374.
  • [20] N. Gavert,A. Ben-Ze'ev, Epithelial-mesenchymal transition and the invasive potential of tumors, Trends Mol. Med., 14 (2008): 199-209.
  • [21] T. Hillen,K. Painter, A user's guide to PDE Models for chemotaxis, J. Math. Biol., 58 (2009): 183-217.
  • [22] T. Hillen, M5 mesoscopic and macroscopic models for mesenchymal motion, J. Math. Biol., 53 (2006): 585-616.
  • [23] D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, Part Ⅱ, Jahresber. Dtsch. Math.-Ver., 106 (2004): 51-69.
  • [24] D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, Part Ⅰ, Jahresber. Dtsch. Math.-Ver., 105 (2003): 103-165.
  • [25] M. Ishizaki,K. Ashida,T. Higashi,H. Nakatsukasa,T. Kaneyoshi,K. Fujiwara,K. Nouso,Y. Kobayashi,M. Uemura,S. Nakamura,T. Tsuji, The formation of capsule and septum in human hepatocellular carcinoma, Virchows Arch., 438 (2001): 574-580.
  • [26] A. J. Kabla, Collective cell migration: Leadership, invasion and segregation, Journal of the Royal Society Interface, 77 (2012): 3268-3278.
  • [27] R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations Philadelphia, SIAM, 2007.
  • [28] A. Lorz,T. Lorenzi,J. Clairambault,A. Escargueil,B. Perthame, Effects of space structure and combination therapies on phenotypic heterogeneity and drug resistance in solid tumors, Bulletin of Mathematical Biology, 77 (2015): 1-22.
  • [29] A. Lorz,T. Lorenzi,M. E. Hochberg,J. Clairambault,B. Perthame, Populational adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapies, Math. Model. Numer. Anal., 47 (2013): 377-399.
  • [30] E. Mahes,E. Mones,V. Nameth,T. Vicsek, Collective motion of cells mediates segregation and pattern formation in co-cultures, PLoS ONE, 7 (2012): e31711.
  • [31] J. Murray, null, Mathematical Biology Ⅱ: Spatial Models and Biochemical Applications, 3rd edn, Springer, New York, 2003.
  • [32] G. Naldi, L. Pareschi and G. Toscani (Eds. ), Mathematical Modeling of Collective Behavior in Socio-economic and Life Sciences Birkhäuser, Basel, 2010.
  • [33] K. J. Painter, Continuous models for cell migration in tissues and applications to cell sorting via differential chemotaxis, Bull. Math. Biol., 71 (2009): 1117-1147.
  • [34] K. J. Painter, Modelling cell migration strategies in the extracellular matrix, J. Math. Biol., 58 (2009): 511-543.
  • [35] K. J. Painter,T. Hillen, Volume-filling and quorum sensing in models for chemosensitive movement, Canad. Appl. Math. Quart., 10 (2003): 501-543.
  • [36] B. Perthame, Transport Equations in Biology Birkhäuser, Basel, 2007.
  • [37] Z. Szymanska,C. M. Rodrigo,M. Lachowicz,M. A. J. Chaplain, Mathematical modelling of cancer invasion of tissue: The role and effect of nonlocal interactions, Math. Models and Meth. in Appl. Sci., 19 (2009): 257-281.
  • [38] C. Xue,H. J. Hwang,K. J. Painter,R. Erban, Travelling waves in hyperbolic chemotaxis equations, Bull. Math. Biol., 73 (2011): 1695-1733.
  • [39] M. Yilmaz,G. Christofori, EMT, the cytoskeleton, and cancer cell invasion, Cancer Metastasis Rev., 28 (2009): 15-33.
  • [40] F. van Zijl,S. Mall,G. Machat,C. Pirker,R. Zeillinger,A. Weinhäusel,M. Bilban,W. Berger,W. Mikulits, A human model of epithelial to mesenchymal transition to monitor drug efficacy in hepatocellular carcinoma progression, Mol. Cancer Ther., 10 (2011): 850-860.


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