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Journal of Mathematical Biology

, Volume 76, Issue 1–2, pp 429–456 | Cite as

A space-jump derivation for non-local models of cell–cell adhesion and non-local chemotaxis

  • Andreas Buttenschön
  • Thomas Hillen
  • Alf Gerisch
  • Kevin J. Painter
Article

Abstract

Cellular adhesion provides one of the fundamental forms of biological interaction between cells and their surroundings, yet the continuum modelling of cellular adhesion has remained mathematically challenging. In 2006, Armstrong et al. proposed a mathematical model in the form of an integro-partial differential equation. Although successful in applications, a derivation from an underlying stochastic random walk has remained elusive. In this work we develop a framework by which non-local models can be derived from a space-jump process. We show how the notions of motility and a cell polarization vector can be naturally included. With this derivation we are able to include microscopic biological properties into the model. We show that particular choices yield the original Armstrong model, while others lead to more general models, including a doubly non-local adhesion model and non-local chemotaxis models. Finally, we use random walk simulations to confirm that the corresponding continuum model represents the mean field behaviour of the stochastic random walk.

Keywords

Cell movement Cell–cell adhesion Non-local models 

Mathematics Subject Classification

92C17 35Q92 35R09 

Notes

Acknowledgements

AB was supported by NSERC, Alberta Innovates and PIMS. TH was supported by NSERC. AG thanks the Isaac Newton Institute for Mathematical Sciences for its hospitality during the programme Coupling Geometric PDEs with Physics for Cell Morphology, Motility and Pattern Formation; EPSRC EP/K032208/1. KJP thanks the Politecnico di Torino for a Visiting Professor position.

References

  1. Alberts B (2008) Molecular biology of the cell. Garland Science, New YorkGoogle Scholar
  2. Andasari V, Gerisch A, Lolas G, South AP, Chaplain MAJ (2011) Mathematical modeling of cancer cell invasion of tissue: biological insight from mathematical analysis and computational simulation. J Math Biol 63(1):141–171MathSciNetCrossRefzbMATHGoogle Scholar
  3. Armstrong NJ, Painter KJ, Sherratt JA (2006) A continuum approach to modelling cell–cell adhesion. J Theor Biol 243(1):98–113MathSciNetCrossRefGoogle Scholar
  4. Armstrong NJ, Painter KJ, Sherratt JA (2009) Adding adhesion to a chemical signaling model for somite formation. Bull Math Biol 71(1):1–24MathSciNetCrossRefzbMATHGoogle Scholar
  5. Bell G (1978) Models for the specific adhesion of cells to cells. Science 200(4342):618–627CrossRefGoogle Scholar
  6. Beysens DA, Forgacs G, Glazier JA (2000) Cell sorting is analogous to phase ordering in fluids. Proc Natl Acad Sci 97(17):9467–9471CrossRefGoogle Scholar
  7. Brodland GW, Chen HH (2000) The mechanics of cell sorting and envelopment. J Biomech 33(7):845–851CrossRefGoogle Scholar
  8. Calvo J, Campos J, Caselles V, Sánchez O, Soler J (2015) Flux-saturated porous media equations and applications. EMS Surv Math Sci 2(1):131–218MathSciNetCrossRefzbMATHGoogle Scholar
  9. Chaplain MAJ, Lachowicz M, Szymanska Z, Wrzosek D (2011) Mathematical modelling of cancer invasion: the importance of cell–cell adhesion and cell-matrix adhesion. Math Model Methods Appl Sci 21(04):719–743MathSciNetCrossRefzbMATHGoogle Scholar
  10. Charras G, Sahai E (2014) Physical influences of the extracellular environment on cell migration. Nat Rev Mol Cell Biol 15(12):813–824CrossRefGoogle Scholar
  11. Danuser G, Allard J, Mogilner A (2013) Mathematical modeling of eukaryotic cell migration: insights beyond experiments. Annu Rev Cell Dev Biol 29:501–528CrossRefGoogle Scholar
  12. Davies JA (2013) Mechanisms of morphogenesis. Academic, CambridgeGoogle Scholar
  13. Desgrosellier JS, Cheresh DA (2010) Integrins in cancer: biological implications and therapeutic opportunities. Nat Rev Cancer 10(1):9–22CrossRefGoogle Scholar
  14. Dolak Y (2004) Advection dominated models for chemotaxis. PhD thesis, University of ViennaGoogle Scholar
  15. Domschke P, Trucu D, Gerisch A, Chaplain MAJ (2014) Mathematical modelling of cancer invasion: implications of cell adhesion variability for tumour infiltrative growth patterns. J Theor Biol 361:41–60MathSciNetCrossRefzbMATHGoogle Scholar
  16. Dormann D, Weijer CJ (2001) Propagating chemoattractant waves coordinate periodic cell movement in dictyostelium slugs. Development 128(22):4535–4543Google Scholar
  17. Dyson J, Gourley SA, Villella-Bressan R, Webb GF (2010) Existence and asymptotic properties of solutions of a nonlocal evolution equation modeling cell–cell adhesion. SIAM J Math Anal 42(4):1784–1804MathSciNetCrossRefzbMATHGoogle Scholar
  18. Erban R, Chapman JS, Maini PK (2007) A practical guide to stochastic simulations of reaction-diffusion processes. arXiv:0704.1908, pp 24–29
  19. Estrada R, Kanwal RP (1993) Asymptotic analysis: a distributional approach. Birkhäuser, BostonzbMATHGoogle Scholar
  20. Friedl P, Alexander S (2011) Cancer invasion and the microenvironment: plasticity and reciprocity. Cell 147(5):992–1009CrossRefGoogle Scholar
  21. Geiger B, Spatz JP, Bershadsky AD (2009) Environmental sensing through focal adhesions. Nat Rev Mol Cell Biol 10(1):21–33CrossRefGoogle Scholar
  22. Gerisch A (2010) On the approximation and efficient evaluation of integral terms in PDE models of cell adhesion. IMA J Numer Anal 30(1):173–194MathSciNetCrossRefzbMATHGoogle Scholar
  23. Gerisch A, Chaplain MAJ (2008) Mathematical modelling of cancer cell invasion of tissue: local and non-local models and the effect of adhesion. J Theor Biol 250(4):684–704MathSciNetCrossRefGoogle Scholar
  24. Gerisch A, Painter KJ (2010) Mathematical modeling of cell adhesion and its applications to developmental biology and cancer invasion. In: Chauvière A, Preziosi L, Verdier C (eds) Cell mechanics: from single scale-based models to multiscale model. CRC Press, Boca Raton, pp 319–350CrossRefGoogle Scholar
  25. Gillespie DT (2007) Stochastic simulation of chemical kinetics. Annu Rev Phys Chem 58:35–55CrossRefGoogle Scholar
  26. Hillen T (2002) Hyperbolic models for chemosensitive movement. Math Model Methods Appl Sci 12(07):1007–1034MathSciNetCrossRefzbMATHGoogle Scholar
  27. Hillen T (2005) On the \(L^2\)-moment closure of transport equations: the general case. Discret Contin Dyn Syst 5(2):299–318MathSciNetCrossRefzbMATHGoogle Scholar
  28. Hillen T, Painter KJ (2009) A user’s guide to PDE models for chemotaxis. J Math Biol 58(1–2):183–217MathSciNetCrossRefzbMATHGoogle Scholar
  29. Hillen T, Painter KJ (2013) Transport and anisotropic diffusion models for movement in oriented habitats. In: Lewis MA, P Maini SP (eds) Dispersal, individual movement and spatial ecology, vol 2071. Springer, Heidelberg, pp 177–222CrossRefGoogle Scholar
  30. Hillen T, Painter KJ, Schmeiser C (2007) Global existence for chemotaxis with finite sampling radius. Discret Contin Dyn Syst Ser B 7(1):125–144MathSciNetCrossRefzbMATHGoogle Scholar
  31. Horstmann D (2003) From 1970 until present : the Keller–Segel model in chemotaxis and its consequences. Jahresber Dtsch Math Ver 105(3):103–165MathSciNetzbMATHGoogle Scholar
  32. Hughes BD (1995) Random walks and random environments: random walks, 1st edn. Oxford Science Publications, OxfordzbMATHGoogle Scholar
  33. Jilkine A, Edelstein-Keshet L (2011) A comparison of mathematical models for polarization of single eukaryotic cells in response to guided cues. PLoS Comput Biol 7(4):e1001,121MathSciNetCrossRefGoogle Scholar
  34. Johnston ST, Simpson MJ, Baker RE (2012) Mean-field descriptions of collective migration with strong adhesion. Phys Rev E 85(5):051,922CrossRefGoogle Scholar
  35. Lauffenburger D (1989) A simple model for the effects of receptor–mediated cell-substratum adhesion on cell migration. Chem Eng Sci 44(9):1903–1914CrossRefGoogle Scholar
  36. Lauffenburger DA, Horwitz AF (1996) Cell migration: a physically integrated molecular process. Cell 84(3):359–369CrossRefGoogle Scholar
  37. Leckband D (2010) Design rules for biomolecular adhesion: lessons from force measurements. Annu Rev Chem Biomol Eng 1:365–389CrossRefGoogle Scholar
  38. Li L, Nørrelkke SF, Cox EC (2008) Persistent cell motion in the absence of external signals: a search strategy for eukaryotic cells. PLoS One 3(5):e2093CrossRefGoogle Scholar
  39. Middleton AM, Fleck C, Grima R (2014) A continuum approximation to an off-lattice individual-cell based model of cell migration and adhesion. J Theor Biol 359:220–232MathSciNetCrossRefGoogle Scholar
  40. Mogilner A, Edelstein-Keshet L (1999) A non-local model for a swarm. J Math Biol 38(6):534–570MathSciNetCrossRefzbMATHGoogle Scholar
  41. Mombach JC, Glazier JA, Raphael RC, Zajac M (1995) Quantitative comparison between differential adhesion models and cell sorting in the presence and absence of fluctuations. Phys Rev Lett 75(11):2244–2247CrossRefGoogle Scholar
  42. Nishiya N, Kiosses WB, Han J, Ginsberg MH (2005) An alpha4 integrin-paxillin-arf-gap complex restricts rac activation to the leading edge of migrating cells. Nat Cell Biol 7(4):343–352CrossRefGoogle Scholar
  43. Othmer HG, Hillen T (2002) The diffusion limit of transport equations ii: chemotaxis equations. SIAM J Appl Math 62:1222–1250MathSciNetCrossRefzbMATHGoogle Scholar
  44. Othmer HG, Dunbar S, Alt W (1988) Models of dispersal in biological systems. J Math Biol 26:263–298MathSciNetCrossRefzbMATHGoogle Scholar
  45. Ou C, Zhang Y (2013) Traveling wavefronts of nonlocal reaction-diffusion models for adhesion in cell aggregation and cancer invasion. Can Appl Math Q 21(1):21–62MathSciNetzbMATHGoogle Scholar
  46. Painter KJ (2009) Continuous models for cell migration in tissues and applications to cell sorting via differential chemotaxis. Bull Math Biol 71(5):1117–1147MathSciNetCrossRefzbMATHGoogle Scholar
  47. Painter KJ, Hillen T (2002) Volume-filling and quorum-sensing in models for chemosensitive movement. Can Appl Math Q 10(4):501–543MathSciNetzbMATHGoogle Scholar
  48. Painter KJ, Armstrong NJ, Sherratt JA (2010) The impact of adhesion on cellular invasion processes in cancer and development. J Theor Biol 264(3):1057–1067MathSciNetCrossRefGoogle Scholar
  49. Painter KJ, Bloomfield JM, Sherratt JA, Gerisch A (2015) A nonlocal model for contact attraction and repulsion in heterogeneous cell populations. Bull Math Biol 77(6):1132–1165MathSciNetCrossRefzbMATHGoogle Scholar
  50. Ridley AJ (2011) Life at the leading edge. Cell 145(7):1012–22CrossRefGoogle Scholar
  51. Ridley AJ, Schwartz MA, Burridge K, Firtel Ra, Ginsberg MH, Borisy G, Parsons JT, Horwitz AR (2003) Cell migration: integrating signals from front to back. Science 302(5651):1704–1709CrossRefGoogle Scholar
  52. Roussos ET, Condeelis JS, Patsialou A (2011) Chemotaxis in cancer. Nat Rev Cancer 11(8):573–587CrossRefGoogle Scholar
  53. Schienbein M, Franke K, Gruler H (1994) Random walk and directed movement: comparison between inert particles and self-organized molecular machines. Phys Rev E 49(6):5462–5471CrossRefGoogle Scholar
  54. Sherratt JA, Gourley SA, Armstrong NJ, Painter KJ (2008) Boundedness of solutions of a non-local reaction-diffusion model for adhesion in cell aggregation and cancer invasion. Eur J Appl Math 20(01):123MathSciNetCrossRefzbMATHGoogle Scholar
  55. Shi L, Yu Z, Mao Z, Xiao A (2014) A directed continuous time random walk model with jump length depending on waiting time. Sci World J 2014:1–4Google Scholar
  56. Stevens A, Othmer HG (1997) Aggregation, blowup, and collapse: the ABC’s of taxis in reinforced random walks. SIAM J Appl Math 57(4):1044–1081MathSciNetCrossRefzbMATHGoogle Scholar
  57. Théry M, Racine V, Piel M, Pépin A, Dimitrov A, Chen Y, Sibarita Jb, Bornens M (2006) Anisotropy of cell adhesive microenvironment governs cell internal organization and orientation of polarity. Proc Natl Acad Sci USA 103(52):19,771–19,776CrossRefGoogle Scholar
  58. Turner S, Sherratt JA, Painter KJ, Savill N (2004) From a discrete to a continuous model of biological cell movement. Phys Rev E 69(2):021,910MathSciNetCrossRefGoogle Scholar
  59. Van Kampen NG (2011) Stochastic processes in physics and chemistry. Elsevier Science, AmsterdamzbMATHGoogle Scholar
  60. Weiner OD (2002) Regulation of cell polarity during eukaryotic chemotaxis: the chemotactic compass. Curr Opin Cell Biol 14(2):196–202CrossRefGoogle Scholar
  61. Weiner OD, Servant G, Parent CA, Devreotes PN, Bourne HR (2000) Cell polarity in response to chemoattractants. In: Drubin DG (ed) Cell polarity, 1st edn. Oxford University Press, Oxford, pp 201–239Google Scholar
  62. White MD, Plachta N (2015) How adhesion forms the early mammalian embryo, 1st edn. Elsevier Inc, AmsterdamGoogle Scholar
  63. Winkler M, Hillen T, Painter KJ (2017) Global solvability and explicit bounds for a non-local adhesion model (submitted)Google Scholar
  64. Zaburdaev VY (2006) Random walk model with waiting times depending on the preceding jump length. J Stat Phys 123(4):871–881MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of Mathematical and Statistical Sciences, Centre for Mathematical BiologyUniversity of AlbertaEdmontonCanada
  2. 2.Fachbereich MathematikTechnische Universität DarmstadtDarmstadtGermany
  3. 3.Department of Mathematics and Maxwell Institute for Mathematical SciencesHeriot-Watt UniversityEdinburghUK
  4. 4.Department of Mathematical SciencesPolitecnico di TorinoTurinItaly

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