SpringerLink
Log in

Introduction

  • Chapter
  • First Online:
Non-Local Cell Adhesion Models

Part of the book series: CMS/CAIMS Books in Mathematics ((CMS/CAIMS BM,volume 1))

  • 452 Accesses

Abstract

Cellular adhesion is one of the most important interaction forces in tissues. Cells adhere to each other, to other cells, and to the extracellular matrix (ECM). Cell adhesion is responsible for the formation of tissues, membranes, vasculature, muscle tissue, as well as cell movement and cancer spread.

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
EUR 29.95
Price includes VAT (Germany)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
EUR 71.68
Price includes VAT (Germany)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
EUR 90.94
Price includes VAT (Germany)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free ship** worldwide - see info
Hardcover Book
EUR 128.39
Price includes VAT (Germany)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free ship** worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. M. Adioui, O. Arino, N. El Saadi, A nonlocal model of phytoplankton aggregation. Nonlinear Anal. Real World Appl. 6(4), 593–607 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. B. Alberts, Molecular Biology of the Cell: Reference Edition, 5th edn. (Garland Science, New York, 2008)

    Google Scholar 

  3. W. Alt, Models for mutual attraction and aggregation of motile individuals, in Mathematics in Biology and Medicine. Lecture Notes in Biomathematics (Springer, Berlin, 1985), pp. 33–38

    Google Scholar 

  4. W. Alt, Degenerate diffusion equations with drift functionals modelling aggregation. Nonlinear Anal. Theory Methods Appl. 9(8), 811–836 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  5. V. Andasari, M.A.J. Chaplain, Intracellular modelling of cell-matrix adhesion during cancer cell invasion. Math. Model. Nat. Phenom. 7(1), 29–48 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. A.R.A. Anderson, A hybrid mathematical model of solid tumour invasion: the importance of cell adhesion. Math. Med. Biol. 22(2), 163–86 (2005)

    Article  MATH  Google Scholar 

  7. N.J. Armstrong, K.J. Painter, J.A. Sherratt, A continuum approach to modelling cell-cell adhesion. J. Theor. Biol. 243(1), 98–113 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. N.J. Armstrong, K.J. Painter, J.A. Sherratt, Adding adhesion to a chemical signaling model for somite formation. Bull. Math. Biol. 71(1), 1–24 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. P.B. Armstrong, Light and electron microscope studies of cell sorting in combinations of chick embryo neural retina and retinal pigment epithelium. Wilhelm Roux Arch. Entwickl. Org. 168(2), 125–141 (1971)

    Article  Google Scholar 

  10. R. Bailo, J.A. Carrillo, J. Hu, Fully discrete positivity-preserving and energy-dissipating schemes for aggregation-diffusion equations with a gradient-flow structure. Commun. Math. Sci. 18(5), 1259–1303 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  11. R. Bailo, J.A. Carrillo, H. Murakawa, M. Schmidtchen, Convergence of a fully discrete and energy-dissipating finite-volume scheme for aggregation-diffusion equations (2020). Preprint, ar**v:2002.10821

    Google Scholar 

  12. J. Barré, J.A. Carrillo, P. Degond, D. Peurichard, E. Zatorska, Particle interactions mediated by dynamical networks: assessment of macroscopic descriptions. J. Nonlinear Sci. 28(1), 235–268 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  13. J. Barré, P. Degond, E. Zatorska, Kinetic theory of particle interactions mediated by dynamical networks. Multiscale Model. Simul. 15(3), 1294–1323 (2017)

    Article  MathSciNet  Google Scholar 

  14. A.J. Bernoff, C.M. Topaz, Nonlocal aggregation models: a primer of swarm equilibria. SIAM Rev. 55(4), 709–747 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. V. Bitsouni, M.A.J. Chaplain, R. Eftimie, Mathematical modelling of cancer invasion: The multiple roles of tgf/β pathway on tumour proliferation and cell adhesion. Math. Models Methods Appl. Sci. 27(10), 1929–1962 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  16. I. Borsi, A. Fasano, M. Primicerio, T. Hillen, A non-local model for cancer stem cells and the tumour growth paradox. Math. Med. Biol. 34(1), 59–75 (2017)

    MathSciNet  MATH  Google Scholar 

  17. B. Brandolini, P. Freitas, C. Nitsch, C. Trombetti, Sharp estimates and saturation phenomena for a nonlocal eigenvalue problem. Adv. Math. 228(4), 2352–2365 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. N.F. Britton, Reaction-Diffusion Equations and Their Applications to Biology (Academic Press, London, 1986)

    MATH  Google Scholar 

  19. G.W. Brodland, The differential interfacial tension hypothesis (DITH): a comprehensive theory for the self-rearrangement of embryonic cells and tissues. J. Biomech. Eng. 124(2), 188 (2002)

    Google Scholar 

  20. G.W. Brodland, H.H. Chen, The mechanics of cell sorting and envelopment. J. Biomech. 33(7), 845–51 (2000)

    Article  Google Scholar 

  21. P.L. Buono, R. Eftimie, Symmetries and pattern formation in hyperbolic versus parabolic models of self-organised aggregation. J. Math. Biol. 71(4), 847–881 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. M. Burger, M. Di Francesco, S. Fagioli, A. Stevens, Sorting phenomena in a mathematical model for two mutually attracting/repelling species. SIAM J. Math. Anal. 50(3), 3210–3250 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  23. A. Buttenschön, T. Hillen, A. Gerisch, K.J. Painter, A space-jump derivation for non-local models of cell-cell adhesion and non-local chemotaxis. J. Math. Biol. 76(1), 429–456 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  24. H.M. Byrne, M.A.J. Chaplain, Modelling the role of cell-cell adhesion in the growth and development of carcinomas. Math. Comput. Model. 24(12), 1–17 (1996)

    Article  MATH  Google Scholar 

  25. H.M. Byrne, D. Drasdo, Individual-based and continuum models of growing cell populations: a comparison. J. Math. Biol. 58, 657–687 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. J.A. Carrillo, X. Chen, Q. Wang, Z.A. Wang, L. Zhang, Phase transitions and bump solutions of the Keller–Segel model with volume exclusion. SIAM J. Appl. Math. 80(1), 232–261 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  27. J.A. Carrillo, A. Chertock, Y. Huang, A finite-volume method for nonlinear nonlocal equations with a gradient flow structure (2014). Preprint, ar**v:1402.4252

    Google Scholar 

  28. J.A. Carrillo, R.S. Gvalani, G.A. Pavliotis, A. Schlichting, Long-time behaviour and phase transitions for the Mckean–Vlasov equation on the torus. Arch. Ration. Mech. Anal. 235(1), 635–690 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  29. M.A.J. Chaplain, M. Lachowicz, Z. Szymanska, D. Wrzosek, Mathematical modelling of cancer invasion: the importance of cell-cell adhesion and cell-matrix adhesion. Math. Models Methods Appl. Sci. 21(04), 719–743 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  30. H.H. Chen, G.W. Brodland, Cell-level finite element studies of viscous cells in planar aggregates. J. Biomech. Eng. 122(4), 394–401 (2000)

    Article  Google Scholar 

  31. R.H. Chisholm, T. Lorenzi, A. Lorz, A. Larsen, L. 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, 930–939 (2015)

    Article  Google Scholar 

  32. M.G. Crandall, P.H. Rabinowitz, Nonlinear Sturm-Liouville eigenvalue problems and topological degree. J. Math. Mech. 19(12), 1083–1102 (1970)

    MathSciNet  MATH  Google Scholar 

  33. F.A. Davidson, N. Dodds, Spectral properties of non-local differential operators. Appl. Anal. 85(6–7), 717–734 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  34. F.A. Davidson, N. Dodds, Spectral properties of non-local uniformly-elliptic operators. Electron. J. Differ. Equ. 126, 1–15 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  35. F.A. Davidson, N. Dodds, Existence of positive solutions due to non-local interactions in a class of nonlinear boundary value problems. Methods Appl. Anal. 14(1), 15–28 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  36. M. Delgado, I.B.M. Duarte, A. Suarez, Nonlocal problem arising from the birth-jump processes. Proc. R. Soc. Edinb. Sect. A Math. 149(2), 1–23 (2018)

    MathSciNet  Google Scholar 

  37. P. Domschke, D. Trucu, A. Gerisch, M.A.J. Chaplain, Mathematical modelling of cancer invasion: implications of cell adhesion variability for tumour infiltrative growth patterns. J. Theor. Biol. 361C, 41–60 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  38. D. Drasdo, S. Höhme, A single-cell-based model of tumor growth in vitro: monolayers and spheroids. Phys. Biol. 2(3), 133–147 (2005)

    Article  Google Scholar 

  39. A. Ducrot, P. Magal, Asymptotic behavior of a non-local diffusive logistic equation. SIAM J. Math. Anal. 46, 1731–1753 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  40. L. Edelstein-Keshet, Mathematical Models in Biology (SIAM, Philadelphia, 2005)

    Book  MATH  Google Scholar 

  41. R. Eftimie, G. de Vries, M.A. Lewis, Complex spatial group patterns result from different animal communication mechanisms. Proc. Natl. Acad. Sci. USA 104(17), 6974–6979 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  42. R. Eftimie, G. De Vries, M.A. Lewis, F. Lutscher, Modeling group formation and activity patterns in self-organizing collectives of individuals. Bull. Math. Biol. 69(5), 1537–1565 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  43. P.C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, vol. 28 (Springer, Berlin, 1979)

    MATH  Google Scholar 

  44. R.A. Foty, M.S. Steinberg, Cadherin-mediated cell-cell adhesion and tissue segregation in relation to malignancy. Int. J. Dev. Biol. 48(5–6), 397–409 (2004)

    Article  Google Scholar 

  45. P. Freitas, Bifurcation and stability of stationary solutions of nonlocal scalar reaction-diffusion equations. J. Dyn. Differ. Equ. 6(4), 613–629 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  46. P. Freitas, A nonlocal Sturm–Liouville eigenvalue problem. Proc. R. Soc. Edinb. Math. 124(01), 169–188 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  47. P. Freitas, M. Vishnevskii, Stability of stationary solutions of nonlocal reaction-diffusion equations in m-dimensional space. Differ. Integral Equ. 13(1–3), 265–288 (2000)

    MathSciNet  MATH  Google Scholar 

  48. P. Friedl, R. Mayor, Tuning collective cell migration by cell-cell junction regulation. Cold Spring Harb. Perspect. Biol. 9, a029199 (2017)

    Article  Google Scholar 

  49. A. Gerisch, Numerical methods for the simulation of taxis diffusion reaction systems. Ph.D. Thesis, Martin-Luther-Universitat Halle-Wittenberg, 2001

    Google Scholar 

  50. A. Gerisch, On the approximation and efficient evaluation of integral terms in PDE models of cell adhesion. IMA J. Numer. Anal. 30(1), 173–194 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  51. A. Gerisch, M.A.J. Chaplain, Mathematical modelling of cancer cell invasion of tissue: local and non-local models and the effect of adhesion. J. Theor. Biol. 250(4), 684–704 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  52. A. Gerisch, K.J. Painter, Mathematical modeling of cell adhesion and its applications to developmental biology and cancer invasion, in Cell Mechanics: From Single Scale-Based Models to Multiscale Modelling, ed. by A. Chauviére, L. Preziosi, C. Verdier (CRC Press, Boca Raton, 2010), pp. 319–350

    Chapter  MATH  Google Scholar 

  53. S.F. Gilbert, M.J.F. Barresi, Developmental Biology, 12th edn. (Oxford University Press, Oxford, 2020)

    Google Scholar 

  54. J.A. Glazier, F. Graner, Simulation of the differential adhesion driven rearrangement of biological cells. Phys. Rev. E 47(3), 2128–2154 (1993)

    Article  Google Scholar 

  55. F. Graner, Can surface adhesion drive cell-rearrangement? Part I: Biological cell-sorting. J. Theor. Biol. 164(4), 455–476 (1993)

    Google Scholar 

  56. F. Graner, J.A. Glazier, Simulation of biological cell sorting using a two-dimensional extended Potts model. Phys. Rev. Lett. 69(13), 2013–2016 (1992)

    Article  Google Scholar 

  57. F. Graner, Y. Sawada, Can surface adhesion drive cell rearrangement? Part II: a geometrical model. J. Theor. Biol. 164(4), 477–506 (1993)

    Google Scholar 

  58. M.L. Graves, J.A. Cipollone, P. Austin, E.M. Bell, J.S. Nielsen, C.B. Gilks, K.M. McNagny, C.D. Roskelley, The cell surface mucin podocalyxin regulates collective breast tumor budding. Breast Cancer Res. 18(1), 11 (2016)

    Google Scholar 

  59. J.M. Halbleib, W.J. Nelson, Cadherins in development: Cell adhesion, sorting, and tissue morphogenesis. Genes Dev. 20(23), 3199–3214 (2006)

    Article  Google Scholar 

  60. D. Hanahan, R.A. Weinberg, The hallmarks of cancer. Cell 100(1), 57–70 (2000)

    Article  Google Scholar 

  61. D. Hanahan, R.A. Weinberg, Hallmarks of cancer: the next generation. Cell 144(5), 646–674 (2011)

    Article  Google Scholar 

  62. A.K. Harris, Is cell sorting caused by differences in the work of intercellular adhesion? A critique of the Steinberg hypothesis. J. Theor. Biol. 61(2), 267–285 (1976)

    Google Scholar 

  63. T.J. Healey, H.J. Kielhöfer, Symmetry and nodal properties in the global bifurcation analysis of quasi-linear elliptic equations. Arch. Ration. Mech. Anal. 113(4), 299–311 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  64. T. Hillen, H. Enderling, P. Hahnfeldt, The tumor growth paradox and immune system-mediated selection for cancer stem cells. Bull. Math. Biol. 75(1), 161–184 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  65. T. Hillen, B. Greese, J. Martin, G. de Vries, Birth-jump processes and application to forest fire spotting. J. Biol. Dyn. 9(Suppl. 1), 104–127 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  66. T. Hillen, C. Painter, K.J. Schmeiser, Global existence for chemotaxis with finite sampling radius. Discrete Contin. Dyn. Syst. Ser. B 7(1), 125–144 (2006)

    MathSciNet  MATH  Google Scholar 

  67. T. Hillen, M. Painter, K.J. Winkler, Global solvability and explicit bounds for a non-local adhesion model. Eur. J. Appl. Math. 29, 645–684 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  68. S. Hoehme, D. Drasdo, Biomechanical and nutrient controls in the growth of mammalian cell populations. Math. Popul. Stud. 17(3), 166–187 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  69. D. Horstmann, From 1970 until present : the Keller-Segel model in chemotaxis and its consequences. Jahresber. Dtsch. Math. Vereinigung 105(3), 103–165 (2003)

    MathSciNet  MATH  Google Scholar 

  70. S.B. Hsu, J. López-Gómez, L. Mei, M. Molina-Meyer, A nonlocal problem from conservation biology. SIAM J. Math. Anal. 46(6), 4035–4059 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  71. T. Ikeda, Standing pulse-like solutions of a spatially aggregating population model. Jpn. J. Appl. Math. 2(1), 111–149 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  72. T. Ikeda, T. Nagai, Stability of localized stationary solutions. Jpn. J. Appl. Math. 4(1), 73–97 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  73. D. Iron, M.J. Ward, A metastable spike solution for a nonlocal reaction-diffusion model. SIAM J. Appl. Math. 60(3), 778–802 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  74. K.L. Johnson, K. Kendall, A.D. Roberts, Surface energy and the contact of elastic solids. Proc. R. Soc. A Math. Phys. Eng. Sci. 324(1558), 301–313 (1971)

    Google Scholar 

  75. R. Klages, Anomalous Transport: Foundations and Applications (Wiley, Hoboken, 2008)

    Book  Google Scholar 

  76. H. Knútsdóttir, E. Pálsson, L. Edelstein-Keshet, Mathematical model of macrophage-facilitated breast cancer cells invasion. J. Theor. Biol. 357, 184–199 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  77. M. Kot, M.A. Lewis, M.G. Neubert, Integrodifference equations, in Encyclopedia of Theoretical Ecology, ed. by A. Hastings, L. Gross (University of California Press, Berkeley, 2012), pp. 382–384

    Google Scholar 

  78. J.S. Lowengrub, H.B. Frieboes, F. **, Y.-L. Chuang, X. Li, P. Macklin, S.M. Wise, V. Cristini, Nonlinear modelling of cancer: bridging the gap between cells and tumours. Nonlinearity 23(1), R1–R91 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  79. J. López-Gómez, On the structure and stability of the set of solutions of a nonlocal problem modeling ohmic heating. J. Dyn. Differ. Equ. 10(4), 537–566 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  80. P. Macklin, S. McDougall, A.R.A. Anderson, M.A.J. Chaplain, V. Cristini, J.S. Lowengrub, Multiscale modelling and nonlinear simulation of vascular tumour growth. J. Math. Biol. 58(4–5), 765–798 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  81. J. Martin, T. Hillen, The spotting distribution of wildfires. Appl. Sci. 6(6), 177 (2016)

    Google Scholar 

  82. P. McMillen, S.A. Holley, Integration of cell-cell and cell-ECM adhesion in vertebrate morphogenesis. Curr. Opin. Cell Biol. 36, 48–53 (2015)

    Article  Google Scholar 

  83. A. Mogilner, Modelling spatio-angular patterns in cell biology. Ph.D. Thesis, University of British Columbia, 1995

    Google Scholar 

  84. A. Mogilner, L. Edelstein-Keshet, A non-local model for a swarm. J. Math. Biol. 38(6), 534–570 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  85. H. Murakawa, H. Togashi, Continuous models for cell-cell adhesion. J. Theor. Biol. 374, 1–12 (2015)

    Article  MATH  Google Scholar 

  86. J.D. Murray, Mathematical Biology (Springer, Berlin, 1993)

    Book  MATH  Google Scholar 

  87. T. Nagai, Some nonlinear degenerate diffusion equations with a nonlocally convective term in ecology. Hiroshima Math. J. 13(1), 165–202 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  88. T. Nagai, M. Mimura, Asymptotic behavior for a nonlinear degenerate diffusion equation in population dynamics. SIAM J. Appl. Math. 43(3), 449–464 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  89. T. Nagai, M. Mimura, Some nonlinear degenerate diffusion equations related to population dynamics. J. Math. Soc. Jpn. 35(3), 539–562 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  90. M.G. Neubert, H. Caswell, Demography and dispersal: calculation and sensitivity analysis of invasion speed for structured populations. Ecology 81(6), 1613–1628 (2000)

    Article  Google Scholar 

  91. M.A. Nieto, The ins and outs of the epithelial to mesenchymal transition in health and disease. Annu. Rev. Cell Dev. Biol. 27, 347–376 (2011)

    Article  Google Scholar 

  92. A. Okubo, S.A. Levin, Diffusion and Ecological Problems: Modern Perspectives, vol. 14 (Springer Science & Business Media, New York, 2001)

    Book  MATH  Google Scholar 

  93. H.G. Othmer, T. Hillen, The diffusion limit of transport equations II: Chemotaxis equations. SIAM J. Appl. Math. 62, 1222–1250 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  94. C. Ou, Y. Zhang, Traveling wavefronts of nonlocal reaction-diffusion models for adhesion in cell aggregation and cancer invasion. Can. Appl. Math. Q. 21(1), 21–62 (2013)

    MathSciNet  MATH  Google Scholar 

  95. J.A. Owen, J. Punt, S.A. Stranford, Kuby Immunology (W. H. Freeman, New York, 2013)

    Google Scholar 

  96. J.M. Painter, K.J., Bloomfield, J.A. Sherratt, A. Gerisch, A nonlocal model for contact attraction and repulsion in heterogeneous cell populations. Bull. Math. Biol. 77(6), 1132–1165 (2015)

    Google Scholar 

  97. K.J. Painter, N.J. Armstrong, J.A. Sherratt, The impact of adhesion on cellular invasion processes in cancer and development. J. Theor. Biol. 264(3), 1057–1067 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  98. A. Paksa, J. Bandemer, B. Hoeckendorf, N. Razin, K. Tarbashevich, S. Minina, D. Meyen, A. Biundo, S.A. Leidel, N. Peyriéras, N.S. Gov, P.J. Keller, E. Raz, Repulsive cues combined with physical barriers and cell-cell adhesion determine progenitor cell positioning during organogenesis. Nat. Commun. 7, 1–14 (2016)

    Article  Google Scholar 

  99. E. Palsson, H.G. Othmer, A model for individual and collective cell movement in dictyostelium discoideum. Proc. Natl. Acad. Sci. USA 97(19), 10448–10453 (2000)

    Article  Google Scholar 

  100. A.J. Perumpanani, J.A. Sherratt, J. Norbury, H.M. Byrne, Biological inferences from a mathematical model for malignant invasion. Invasion Metastasis 16(4–5), 209–221 (1996)

    Google Scholar 

  101. A. Puliafito, L. Hufnagel, P. Neveu, S. Streichan, A. Sigal, D.K. Fygenson, B.I. Shraiman, Collective and single cell behavior in epithelial contact inhibition. Proc. Natl. Acad. Sci. USA 109(3), 739–744 (2012)

    Article  Google Scholar 

  102. P.H. Rabinowitz, Some global results for nonlinear eigenvalue problems. J. Funct. Anal. 513, 487–513 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  103. I. Ramis-Conde, M.A.J. Chaplain, A.R.A. Anderson, Mathematical modelling of cancer cell invasion of tissue. Math. Comput. Model. 47(5–6), 533–545 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  104. I. Ramis-Conde, D. Drasdo, A.R.A. Anderson, M.A.J. Chaplain, Modeling the influence of the e-cadherin-beta-catenin pathway in cancer cell invasion: a multiscale approach. Biophys. J. 95(1), 155–165 (2008)

    Article  Google Scholar 

  105. R. Schaaf, Stationary solutions of chemotaxis systems. Trans. Am. Math. Soc. 292(2), 531–531 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  106. D.K. Schlüter, I. Ramis-Conde, M.A.J. Chaplain, Computational modeling of single-cell migration: the leading role of extracellular matrix fibers. Biophys. J. 103(6), 1141–1151 (2012)

    Article  Google Scholar 

  107. D.K. Schlüter, I. Ramis-Conde, M.A.J. Chaplain, Multi-scale modelling of the dynamics of cell colonies: insights into cell-adhesion forces and cancer invasion from in silico simulations. J. R. Soc. Interface 12(103), 20141080 (2015)

    Google Scholar 

  108. M. Scianna, L. Preziosi, Cellular Potts Models: Multiscale Extensions and Biological Applications. Chapman & Hall/CRC Mathematical and Computational Biology (Taylor & Francis, Boca Raton, 2013)

    Google Scholar 

  109. J.A. Sherratt, S.A. Gourley, N.J. Armstrong, K.J. Painter, Boundedness of solutions of a non-local reaction-diffusion model for adhesion in cell aggregation and cancer invasion. Eur. J. Appl. Math. 20(01), 123–144 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  110. M.S. Steinberg, Reconstruction of tissues by dissociated cells. Science 141(3579), 401–408 (1963)

    Article  Google Scholar 

  111. M.S. Steinberg, Does differential adhesion govern self-assembly processes in histogenesis? Equilibrium configurations and the emergence of a hierarchy among populations of embryonic cells. J. Exp. Zool. 173(4), 395–433 (1970)

    Google Scholar 

  112. M.S. Steinberg, Differential adhesion in morphogenesis: a modern view. Curr. Opin. Genet. Dev. 17(4), 281–286 (2007)

    Article  Google Scholar 

  113. H.B. Taylor, A. Khuong, Z. Wu, Q. Xu, R. Morley, L. Gregory, A. Poliakov, W.R. Taylor, D.G. Wilkinson, Cell segregation and border sharpening by eph receptor–ephrin-mediated heterotypic repulsion. J. R. Soc. Interface 14(132), 20170338 (2017)

    Google Scholar 

  114. J.P. Taylor-King, R. Klages, R.A. Van Gorder, Fractional diffusion equation for an n-dimensional correlated Levy walk. Phys. Rev. E 94(1), 012104 (2016)

    Google Scholar 

  115. C.M. Topaz, A.L. Bertozzi, M.A. Lewis, A nonlocal continuum model for biological aggregation. Bull. Math. Biol. 68(7), 1601–1623 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  116. S. Turner, J.A. Sherratt, Intercellular adhesion and cancer invasion: a discrete simulation using the extended Potts model. J. Theor. Biol. 216(1), 85–100 (2002)

    Article  MathSciNet  Google Scholar 

  117. S. Turner, J.A. Sherratt, K.J. Painter, N. Savill, From a discrete to a continuous model of biological cell movement. Phys. Rev. E 69(2), 021910 (2004)

    Google Scholar 

  118. X. Wang, Q. Xu, Spiky and transition layer steady states of chemotaxis systems via global bifurcation and helly’s compactness theorem. J. Math. Biol. 66, 1241–1266 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  119. S. Watanabe, S. Matsumoto, T. Higurashi, N. Ono, Burgers equation with no-flux boundary conditions and its application for complete fluid separation. Physica D 331, 1–12 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  120. R.A. Weinberg, The Biology of Cancer (Garland Science, New York, 2013)

    Book  Google Scholar 

  121. H.V. Wilson, On some phenomena of coalescence and regeneration in sponges. J. Elisha Mitchell Sci. Soc. 23(4), 161–174 (1907)

    Google Scholar 

  122. T. **ang, A study on the positive nonconstant steady states of nonlocal chemotaxis systems. Discrete Contin. Dyn. Syst. Ser. B 18(9), 2457–2485 (2013)

    MathSciNet  MATH  Google Scholar 

  123. X. Zhang, L. Mei, On a nonlocal reaction-diffusion-advection system modeling phyto-plankton growth with light and nutrients. Discrete Contin. Dyn. Syst. Ser. B 17(1), 221–243 (2011)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of British Columbia, Vancouver, BC, Canada

    Andreas Buttenschön

  2. Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, Canada

    Thomas Hillen

Authors
  1. Andreas Buttenschön
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Thomas Hillen
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Buttenschön, A., Hillen, T. (2021). Introduction. In: Non-Local Cell Adhesion Models. CMS/CAIMS Books in Mathematics, vol 1. Springer, Cham. https://doi.org/10.1007/978-3-030-67111-2_1

Download citation

Publish with us

Policies and ethics

Search

Navigation