Discussion and Outlook

  • Andreas Deutsch
  • Sabine Dormann
Part of the Modeling and Simulation in Science, Engineering and Technology book series (MSSET)


In contrast to “continuous systems” and their canonical description with partial differential equations, there is no standard model for describing interactions of discrete objects, particularly of interacting discrete biological cells. In this book, cellular automata are proposed as models for spatially extended systems of interacting cells. Cellular automata are neither a replacement for (or discretization of) traditional (continuous) mathematical models nor preliminary mathematical models but constitute a proper class of discrete mathematical models – discrete in space, time, and state space, for which analytical tools already exist or can be developed in the future.


  1. Anderson, A. R. A., M. A. J. Chaplain, and K. A. Rejniak, eds. 2008. Single-Cell-Based Models in Biology and Medicine. Boston: Birkhauser.Google Scholar
  2. Arlotti, L., A. Deutsch, and M. Lachowicz. 2005. On a discrete Boltzmann-type model of swarming. Mathematical and Computer Modelling 41(10): 1193–1201.MathSciNetCrossRefzbMATHGoogle Scholar
  3. Arratia, R. 1983. The motion of a tagged particle in the simple symmetric exclusion system on Z. Annals of Probability 11: 362–373.MathSciNetCrossRefzbMATHGoogle Scholar
  4. Bleckmann, H. 1993. Role of the lateral line in fish behaviour. In Behaviour of Teleost Fishes, ed. T. J. Pitcher, 201–246. New York, London: Chapman and Hall.CrossRefGoogle Scholar
  5. Boerlijst, M. 1994. Selfstructuring: A Substrate for Evolution. Ph.D. thesis, University, Utrecht.Google Scholar
  6. Boghaert, E., D. C. Radisky, and C. M. Nelson. 2014. Lattice-based model of ductal carcinoma in situ suggests rules for breast cancer progression to an invasive state. PLoS Computational Biology 10:e1003997.CrossRefGoogle Scholar
  7. Börner, U., A. Deutsch, H. Reichenbach, and M. Bär. 2002. Rippling patterns in aggregates of myxobacteria arise from cell-cell collisions. Physical Reviews Letters 89: 078101.CrossRefGoogle Scholar
  8. Böttger, K., H. Hatzikirou, A. Voß-Böhme, E. A. Cavalcanti-Adam, M. A. Herrero, and A. Deutsch. 2015. An emerging Allee effect is critical for tumor initiation and persistence. PLoS Computational Biology 11(9):e1004366.CrossRefGoogle Scholar
  9. Bouré, O., N. Fatès, and V. Chevrier. 2012. First steps on asynchronous lattice-gas models with an application to a swarming rule. In ACRI 2012, LNCS 7495, 633–642. Berlin/Heidelberg: Springer.Google Scholar
  10. Buder, T., A. Deutsch, B. Klink, and A. Voß-Böhme. 2015. Model-based evaluation of spontaneous tumor regression in pilocytic astrocytoma. PLoS Computational Biology 11(12):e1004662.CrossRefGoogle Scholar
  11. Bussemaker, H., A. Deutsch, and E. Geigant. 1997. Mean-field analysis of a dynamical phase transition in a cellular automaton model for collective motion. Physical Review Letters 78: 5018–5021.CrossRefGoogle Scholar
  12. Bussemaker, H. J. 1996. Analysis of a pattern forming lattice-gas automaton: mean-field theory and beyond. Physical Review E 53(2): 1644–1661.CrossRefGoogle Scholar
  13. Byrne, H. M., and M. A. J. Chaplain. 1996. Modelling the role of cell–cell adhesion in the growth and development of carcinomas. Mathematical and Computer Modelling 24: 1–17.MathSciNetCrossRefzbMATHGoogle Scholar
  14. Capcarrere, M. S. 2002. Cellular Automata and other Cellular Systems: Design and Evolution. Ph.D. thesis, Swiss Federal Institute of Technology, Lausanne.Google Scholar
  15. Capcarrere, M. S., A. Tettamanzi, and M. Sipper. 1998. Statistical study of a class of cellular evolutionary algorithms. Evolutionary Computation 7(3): 255–274.CrossRefGoogle Scholar
  16. Chopard, B., and M. Droz. 1998. Cellular Automata Modeling of Physical Systems. New York: Cambridge University Press.CrossRefzbMATHGoogle Scholar
  17. Czirók, A., A. Deutsch, and M. Wurzel. 2003. Individual-based models of cohort migration in cell cultures. In Models of Polymer and Cell Dynamics, eds. W. Alt, M. Chaplain, M. Griebel, and J. Lenz. Basel. Birkhäuser.Google Scholar
  18. de Roos, A. M., E. McCauley, and W. G. Wilson. 1998. Pattern formation and the spatial scale of interaction between predators and their prey. Theoretical Population Biology 53: 108–130.CrossRefzbMATHGoogle Scholar
  19. Deutsch, A., and A. T. Lawniczak. 1999. Probabilistic lattice models of collective motion and aggregation; from individual to collective dynamics. Mathematical Biosciences 156: 255–269.MathSciNetCrossRefzbMATHGoogle Scholar
  20. Dieterich, P., R. Klages, R. Preuss, and A. Schwab. 2008. Anomalous dynamics of cell migration. Proceedings of the National Academy of Sciences of the United States of America 105: 459–463.CrossRefGoogle Scholar
  21. Drasdo, D., and M. Löffler. 2001. Individual-based models to growth and folding in one-layered tissues: intestinal crypts and early development. Nonlinear Analysis: Theory 47: 245–256.MathSciNetCrossRefzbMATHGoogle Scholar
  22. Durrett, R., and S. Levin. 1994a. The importance of being discrete (and spatial). Theoretical Population Biology 46: 363–394.CrossRefzbMATHGoogle Scholar
  23. Edelstein-Keshet, L., and B. Ermentrout. 1990. Models for contact-mediated pattern formation: cells that form parallel arrays. Journal of Mathematical Biology 29: 33–58.MathSciNetCrossRefzbMATHGoogle Scholar
  24. Fischer, K. H., and J. A. Hertz. 1993. Spin Glasses. Cambridge: Cambridge University Press.Google Scholar
  25. Fletcher, A. G., M. Osterfield, R. E. Baker, and S. Y. Shvartsman. 2014. Vertex models of epithelial morphogenesis. Biophysical Journal 106: 2291–2304.CrossRefGoogle Scholar
  26. Frisch, U., B. Hasslacher, and Y. Pomeau. 1986. Lattice-gas automata for the Navier-Stokes equation. Physical Review Letters 56(14): 1505–1509.CrossRefGoogle Scholar
  27. Gao, X., J. T. McDonald, L. Hlatky, and H. Enderling. 2013. Acute and fractionated irradiation differentially modulate glioma stem cell division kinetics. Cancer Research 73(5): 1481–1490.CrossRefGoogle Scholar
  28. Geigant, E. 1999. Nichtlineare Integro-Differential-Gleichungen zur Modellierung interaktiver Musterbildungsprozesse auf S 1. Ph.D. thesis, University of Bonn, Bonn.Google Scholar
  29. Glazier, J. A., and F. Graner. 1993. Simulation of the differential adhesion driven rearrangement of biological cells. Physical Review E 47(3): 2128–2154.CrossRefGoogle Scholar
  30. Graner, F., and J. A. Glazier. 1992. Simulation of biological cell sorting using a two-dimensional extended Potts model. Physical Review Letters 69(13): 2013–2016CrossRefGoogle Scholar
  31. Grygierzec, W., A. Deutsch, W. Schubert, M. Friedenberger, and L. Philipsen. 2004. Modelling tumour cell population dynamics based on molecular adhesion assumptions. Journal of Biological Systems 12: 273–288.CrossRefzbMATHGoogle Scholar
  32. Hardy, J., Y. Pomeau, and O. de Pazzis. 1973. Time evolution of a two-dimensional model system. i. invariant states and time correlation functions. Journal of Mathematical Physics 14: 1746.Google Scholar
  33. Harris, T. E. 1965. Diffusion with collisions between particles. Journal of Applied Probability 2: 323–338.MathSciNetCrossRefzbMATHGoogle Scholar
  34. Kadanoff, L. P. 1986. On two levels. Physics Today Sept.: 7–9.Google Scholar
  35. Köhn-Luque, A., W. de Back, J. Starruß, A. Mattiotti, A. Deutsch, J. M. Perez-Pomares, and M. A. Herrero. 2011. Early embryonic vascular patterning by matrix-mediated paracrine signalling: a mathematical model study. PLOS One 6(9):e24175.CrossRefGoogle Scholar
  36. Levin, S. A. 1974. Dispersion and population interactions. The American Naturalist 108: 207–228.CrossRefGoogle Scholar
  37. Levin, S. A. 1992. The problem of pattern and scale. Ecology 73(6): 1943–1967.CrossRefGoogle Scholar
  38. Li, J. F., and J. Lowengrub. 2014. The effects of cell compressibility, motility and contact inhibition on the growth of tumor cell clusters using the cellular Potts model. Journal of Theoretical Biology 343: 79–91.CrossRefGoogle Scholar
  39. Liggett, T. M. 1985. Interacting Particle Systems. New York: Springer.CrossRefzbMATHGoogle Scholar
  40. Liggett, T. M. 1999. Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. New York: Springer.CrossRefzbMATHGoogle Scholar
  41. Marée, A. F. M., and P. Hogeweg. 2001. How amoeboids self-organize into a fruiting body: multicellular coordination in Dictyostelium discoideum. Proceedings of the National Academy of Sciences of the United States of America 98: 3879–3883.Google Scholar
  42. Mente, C., I. Prade, L. Brusch, G. Breier, and A. Deutsch. 2011. Parameter estimation with a novel gradient-based optimization method for biological lattice-gas cellular automaton models. Journal of Mathematical Biology 63: 173–200.MathSciNetCrossRefzbMATHGoogle Scholar
  43. Mente, C., A. Voß-Böhme, and A. Deutsch. 2015. Analysis of individual cell trajectories in lattice-gas cellular automaton models for migrating cell populations. Bulletin of Mathematical Biology 77(4): 1–38.MathSciNetCrossRefzbMATHGoogle Scholar
  44. Merks, R. M. H., and J. A. Glazier. 2006. Dynamic mechanisms of blood vessel growth. Nonlinearity 19(1):C1–C10.MathSciNetCrossRefzbMATHGoogle Scholar
  45. Merks, R. M. H., E. D. Perryn, A. Shirinifard, and J. A. Glazier. 2008. Contact-inhibited chemotaxis in de novo and sprouting blood-vessel growth. PLoS Computational Biology 49:e1000163.MathSciNetCrossRefGoogle Scholar
  46. Meyer-Hermann, M. 2002. A mathematical model for the germinal center morphology and affinity maturation. Journal of Theoretical Biology 216: 273–300.CrossRefGoogle Scholar
  47. Meyer-Hermann, M., A. Deutsch, and M. Or-Guil. 2001. Recycling probability and dynamical properties of germinal center reactions. Journal of Theoretical Biology 210: 265–285.CrossRefGoogle Scholar
  48. Nava-Sedeño, J. M., H. Hatzikirou, F. Peruani, and A. Deutsch. 2017. Extracting cellular automation rules from physical Langevin equation models for single and collective cell migration. Journal of Mathematical Biology. doi:10.1007/s00285-017-1106-9.zbMATHGoogle Scholar
  49. Painter, K. J., J. M. Bloomfield, J. A. Sherratt, and A. Gerisch. 2015. A nonlocal model for contact attraction and repulsion in heterogeneous cell populations. Bulletin of Mathematical Biology 77: 1132–1165.MathSciNetCrossRefzbMATHGoogle Scholar
  50. Peruani, F., A. Deutsch, and M. Bär. 2006. Non-equilibrium clustering of self-propelled rods. Physical Review E 74(3): 030904.CrossRefGoogle Scholar
  51. Peruani, F., T. Klauss, A. Deutsch, and A. Voß-Böhme. 2011. Traffic jams, gliders, and bands in the quest for collective motion of self-propelled particles. Physical Review Letters 106(12): 128101.CrossRefGoogle Scholar
  52. Perumpanani, A. J., J. A. Sherratt, J. Norbury, and H. M. Byrne. 1996. Biological inferences from a mathematical model of malignant invasion. Invasion and Metastasis 16: 209–221.Google Scholar
  53. Qian, Y., D. d’Humières, and P. Lallemand. 1992. Diffusion simulation with a deterministic one-dimensional lattice-gas model. Journal of Statistical Physics 68(3/4): 563–573.MathSciNetCrossRefzbMATHGoogle Scholar
  54. Reher, D., B. Klink, A. Deutsch, and A. Voß-Böhme. 2017. Cell adhesion heterogeneity reinforces tumour cell dissemination: novel insights from a mathematical model. Biology Direct 12: 18CrossRefGoogle Scholar
  55. Rejniak, K. A. 2005. A single-cell approach in modeling the dynamics of tumormicroregions. Mathematical Biosciences and Engineering 2: 643–655.MathSciNetCrossRefzbMATHGoogle Scholar
  56. Rubenstein, B., and L. Kaufman. 2008. The role of extracellular matrix in glioma invasion: a cellular Potts model approach. Biophysical Journal 95: 5661–5680.CrossRefGoogle Scholar
  57. Sager, B., and D. Kaiser. 1993. Spatial restriction of cellular differentiation. Genes and Development 7: 1645–1653.CrossRefGoogle Scholar
  58. Savill, N. J., and P. Hogeweg. 1997. Modeling morphogenesis: from single cells to crawling slugs. Journal of Theoretical Biology 184: 229–235.CrossRefGoogle Scholar
  59. Schaller, G., and M. Meyer-Hermann. 2005. Multicellular tumor spheroid in an off-lattice Voronoi-Delaunay cell model. Physical Review E 71: 051910.MathSciNetCrossRefGoogle Scholar
  60. Schönfisch, B. 1997. Anisotropy in cellular automata. Biosystems 41: 29–41.CrossRefGoogle Scholar
  61. Schönfisch, B., and A. de Roos. 1999. Synchronous and asynchronous updating in cellular automata. Biosystems 51: 123–143.CrossRefGoogle Scholar
  62. Schubert, W. 1998. Molecular semiotic structures in the cellular immune system: key to dynamics and spatial patterning. In A Perspective Look at Nonlinear Physics; from Physics to Biology and Social Sciences, eds. J. Parisi, S. C. Müller, and W. Zimmermann, 197–206. Heidelberg: Springer.Google Scholar
  63. Schubert, W., C. L. Masters, and K. Beyreuther. 1993. APP + T-lymphocytes selectively sorted to endomysial tubes in polymyositis displace NCAM-expressing muscle fibers. European Journal of Cell Biology 62: 333–342.Google Scholar
  64. Shirinifard, A., J. S. Gens, B. L. Zaitlen, N. J. Popawski, M. Swat, and J. A. Glazier. 2009. 3D multi-cell simulation of tumor growth and angiogenesis. PLOS One 4(10):e7190.CrossRefGoogle Scholar
  65. Starruß, J., T. Bley, L. Søgaard-Andersen, and A. Deutsch. 2007. A new mechanism for collective migration in M. xanthus. Journal of Statistical Physics 128(1–2): 269–286.MathSciNetCrossRefzbMATHGoogle Scholar
  66. Starruß, J., W. de Back, L. Brusch, and A. Deutsch. 2014. Morpheus: a user-friendly modeling environment for multiscale and multicellular systems biology. Bioinformatics 30: 1331–1332.CrossRefGoogle Scholar
  67. Stevens, A., and F. Schweitzer. 1997. Aggregation induced by diffusing and nondiffusing media. In Dynamics of Cell and Tissue Motion, Chapter III.2, eds. W. Alt, A. Deutsch, and G. Dunn, 183–192. Basel: Birkhäuser.CrossRefGoogle Scholar
  68. Szabo, A., and R. M. H. Merks. 2013. Cellular Potts modeling of tumor growth, tumor invasion and tumor evolution. Frontiers in Oncology 3: 87.CrossRefGoogle Scholar
  69. Talkenberger, K., E. A. Cavalvanti-Adam, A. Deutsch, and A. Voß-Böhme. 2017. Amoeboid-mesenchymal migration plasticity promotes invasion only in complex heterogeneous microenvironments. Scientific Reports 7: article no. 9237.Google Scholar
  70. Turner, S., and J. A. Sherratt. 2002. Intercellular adhesion and cancer invasion: a discrete simulation using the extended Potts model. Journal of Theoretical Biology 216: 85–100.MathSciNetCrossRefGoogle Scholar
  71. Van Liedekerke, P., M. M. Palm, N. Jagiella, and D. Drasdo. 2015. Simulating tissue mechanics with agent-based models: concepts, perspectives and some novel results. Computational Particle Mechanics 2: 401–444.CrossRefGoogle Scholar
  72. Voß-Böhme, A. 2012. Multi-scale modeling in morphogenesis: a critical analysis of the cellular Potts model. PLOS One 7(9):e42852.CrossRefGoogle Scholar
  73. Voß-Böhme, A., and A. Deutsch. 2010. The cellular basis of cell sorting kinetics. Journal of Theoretical Biology 263(4): 419–436.MathSciNetCrossRefGoogle Scholar
  74. Wolfram, S. 1986a. Cellular automaton fluids 1: basic theory. Journal of Statistical Physics 45(3/4): 471–526.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Andreas Deutsch
    • 1
  • Sabine Dormann
    • 2
  1. 1.Centre for Information Services and High Performance ComputingTechnische Universität DresdenDresdenGermany
  2. 2.FB Mathematik/InformatikUniversität OsnabrückOsnabrückGermany

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