Advertisement

Bulletin of Mathematical Biology

, Volume 77, Issue 4, pp 660–697 | Cite as

Analysis of Individual Cell Trajectories in Lattice-Gas Cellular Automaton Models for Migrating Cell Populations

  • Carsten Mente
  • Anja Voss-Böhme
  • Andreas Deutsch
Original Article

Abstract

Collective dynamics of migrating cell populations drive key processes in tissue formation and maintenance under normal and diseased conditions. Collective cell behavior at the tissue level is typically characterized by considering cell density patterns such as clusters and moving cell fronts. However, there are also important observables of collective dynamics related to individual cell behavior. In particular, individual cell trajectories are footprints of emergent behavior in populations of migrating cells. Lattice-gas cellular automata (LGCA) have proven successful to model and analyze collective behavior arising from interactions of migrating cells. There are well-established methods to analyze cell density patterns in LGCA models. Although LGCA dynamics are defined by cell-based rules, individual cells are not distinguished. Therefore, individual cell trajectories cannot be analyzed in LGCA so far. Here, we extend the classical LGCA framework to allow labeling and tracking of individual cells. We consider cell number conserving LGCA models of migrating cell populations where cell interactions are regulated by local cell density and derive stochastic differential equations approximating individual cell trajectories in LGCA. This result allows the prediction of complex individual cell trajectories emerging in LGCA models and is a basis for model–experiment comparisons at the individual cell level.

Keywords

Individual cell trajectories Migrating cell populations Lattice-gas cellular automata Stochastic differential equations 

Notes

Acknowledgments

The authors thank Dr. Elisabetta Ada Cavalcanti-Adam, Max Planck Institute for Intelligent Systems, and Katrin Böttger, TU Dresden, for useful discussions and for critically reading the manuscript. This work was financially supported by the Virtual Liver initiative (http://www.virtual-liver.de), funded by the German Ministry of Education and Research (BMBF). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

References

  1. Alarcón T, Byrne HM, Maini PK (2003) A cellular automaton model for tumour growth in inhomogeneous environment. J Theor Biol 225(2):257–274CrossRefGoogle Scholar
  2. Arratia R (1983) The motion of a tagged particle in the simple symmetric exclusion system on z. Ann Prob 11:362–373MathSciNetCrossRefzbMATHGoogle Scholar
  3. Badoual M, Deroulers C, Aubert M, Grammaticos B (2010) Modelling intercellular communication and its effects on tumour invasion. Phys Biol 7(4):046013CrossRefGoogle Scholar
  4. Bergman AJ, Zygourakis K (1999) Migration of lymphocytes on fibronectin-coated surfaces: temporal evolution of migratory parameters. Biomaterials 20(23):2235–2244CrossRefGoogle Scholar
  5. Binder BJ, Landman KA, Newgreen DF, Simkin JE, Takahashi Y, Zhang D (2012) Spatial analysis of multi-species exclusion processes: application to neural crest cell migration in the embryonic gut. Bull Math Biol 74(2):474–490MathSciNetCrossRefzbMATHGoogle Scholar
  6. Bloomfield JM, Sherratt JA, Painter KJ, Landini G (2010) Cellular automata and integrodifferential equation models for cell renewal in mosaic tissues. J R Soc Interface 7(52):1525–1535CrossRefGoogle Scholar
  7. Böttger K, Hatzikirou H, Chauviere A, Deutsch A (2012) Investigation of the migration/proliferation dichotomy and its impact on avascular glioma invasion. Math Model Nat Phenom 7(1):105–135MathSciNetCrossRefzbMATHGoogle Scholar
  8. Böttger K, Hatzikirou H, Voss-Böhme A, Cavalcanti-Adam EA, Herrero MA, Deutsch A (2015) Emerging allee effect in tumor growth. Plos Comput Biol (in press)Google Scholar
  9. Bryant DM, Mostov KE (2008) From cells to organs: building polarized tissue. Nature Rev Mol Cell Biol 9:887–901CrossRefGoogle Scholar
  10. Bussemaker HJ, Deutsch A, Geigant E (1997) Mean-field analysis of a dynamical phase transition in a cellular automaton model for collective motion. Phys Rev Lett 78:5018–5021CrossRefGoogle Scholar
  11. Capasso V, Bakstein D (2012) An introduction to continuous-time stochastic processes. Birkhauser, SwitzerlandCrossRefzbMATHGoogle Scholar
  12. Carmeliet P (2003) Angiogenesis in health and disease. Nat Med 9(6):653–660CrossRefGoogle Scholar
  13. Chopard B, Ouared R, Deutsch A, Hatzikirou H, Wolf-Gladrow DA (2010) Lattice-gas cellular automaton models for biology: from fluids to cells. Acta Biotheor 58(4):329–340CrossRefGoogle Scholar
  14. Deutsch A, Dormann S (2005) Cellular automaton modeling of biological pattern formation. Birkhauser, SwitzerlandzbMATHGoogle Scholar
  15. Dormann S, Deutsch A (2002) Modeling of self-organized avascular tumor growth with a hybrid cellular automaton. Silico Biol 2(3):393–406Google Scholar
  16. Dormann S, Deutsch A, Lawniczak AT (2001) Fourier analysis of turing-like pattern formation in cellular automaton models. Future Gener Comp Sy 17(7):901–909CrossRefzbMATHGoogle Scholar
  17. Drasdo D, Höhme S (2005) A single-cell-based model of tumor growth in vitro: monolayers and spheroids. Phys Biol 2(3):133CrossRefGoogle Scholar
  18. Drasdo D, Kree R, McCaskill JS (1995) Monte carlo approach to tissue-cell populations. Phys Rev E 52(6):6635–6656CrossRefGoogle Scholar
  19. Ermentrout GB, Edelstein-Keshet L (1993) Cellular automata approaches to biological modeling. J Theor Biol 160(1):97–133CrossRefGoogle Scholar
  20. Friedl P, Gilmour D (2009) Collective cell migration in morphogenesis, regeneration and cancer. Nat Rev Mol Cell Biol 10:445–457CrossRefGoogle Scholar
  21. Galle J, Hoffmann M, Aust G (2009) From single cells to tissue architecturea bottom-up approach to modelling the spatio-temporal organisation of complex multi-cellular systems. J Math Biol 58:261–283MathSciNetCrossRefzbMATHGoogle Scholar
  22. Gardiner CW (1998) Handbook of stochastic methods. Springer, New YorkzbMATHGoogle Scholar
  23. Glazier JA, Graner F (1993) Simulation of the differential adhesion driven rearrangement of biological cells. Phys Rev E 47(3):2128–2154CrossRefGoogle Scholar
  24. Graner F, Glazier JA (1992) Simulation of biological cell sorting using a two-dimensional extended potts model. Phys Rev Lett 69:2013–2016CrossRefGoogle Scholar
  25. Hardy J, De Pazzis O, Pomeau Y (1976) Molecular dynamics of a classical lattice gas: transport properties and time correlation functions. Phys Rev A 13(5):1949–1962CrossRefGoogle Scholar
  26. Harris TE (1965) Diffusion with collisions between particles. J App Prob 2:323–338CrossRefGoogle Scholar
  27. Hatzikirou H, Basanta D, Simon M, Schaller K, Deutsch A (2006) Go or grow: the key to the emergence of invasion in tumour progression? Math Med Biol 29(1):49–65MathSciNetCrossRefGoogle Scholar
  28. Hatzikirou H, Böttger K, Deutsch A (2015) Model-based comparison of cell density-dependent cell migration strategies. Math Model Nat Phenom 10(1):94–107MathSciNetCrossRefGoogle Scholar
  29. Hatzikirou H, Brusch L, Deutsch A, Schaller C, Simon M (2006) Characterization of travelling front behaviour in a lattice gas cellular automaton model of glioma invasion. Math Mod Meth Appl Sci 15:1779–1794CrossRefGoogle Scholar
  30. Hatzikirou H, Brusch L, Schaller C, Simon M, Deutsch A (2010) Prediction of traveling front behavior in a lattice-gas cellular automaton model for tumor invasion. Comput Math Appl 59(7):2326–2339MathSciNetCrossRefzbMATHGoogle Scholar
  31. Knapp DM, Tower TT, Tranquillo RT, Barocas VH (1999) Estimation of cell traction and migration in an isometric cell traction assay. AIChE J 45(12):2628–2640CrossRefGoogle Scholar
  32. Liggett TM (1985) Interacting particle systems. Springer, New YorkCrossRefzbMATHGoogle Scholar
  33. Mente C, Prade I, Brusch L, Breier G, Deutsch A (2012) A lattice-gas cellular automaton model for in vitro sprouting angiogenesis. Acta Phys Pol B 5(1):99–115Google Scholar
  34. Merks RMH, Glazier JA (2005) A cell-centered approach to developmental biology. Phys A 352(1):113–130CrossRefGoogle Scholar
  35. Oelschläger K (1989) Many-particle systems and the continuum description of their dynamicsGoogle Scholar
  36. Pézeron G, Mourrain P, Courty S, Ghislain J, Becker TS, Rosa FM, David NB (2008) Live analysis of endodermal layer formation identifies random walk as a novel gastrulation movement. Curr Biol 18(4):276–281CrossRefGoogle Scholar
  37. Polyak K, Weinberg RA (2009) Transitions between epithelial and mesenchymal states: acquisition of malignant and stem cell traits. Nat Rev Cancer 9(4):265–273CrossRefGoogle Scholar
  38. Pomeau B, Hasslacher Y, Frisch U (1986) Lattice-gas automata for the Navier-stokes equation. Phys Rev Lett 56(14):1505–1509CrossRefGoogle Scholar
  39. Rejniak KA, Anderson ARA (2011) Hybrid models of tumor growth. Wiley Interdiscip Rev Syst Biol Med 3(1):115–125CrossRefGoogle Scholar
  40. Row RH, Maître J, Martin BL, Stockinger P, Heisenberg C, Kimelman D (2011) Completion of the epithelial to mesenchymal transition in zebrafish mesoderm requires spadetail. Dev Biol 354(1):102–110CrossRefGoogle Scholar
  41. Shreiber DI, Barocas VH, Tranquillo RT (2003) Temporal variations in cell migration and traction during fibroblast-mediated gel compaction. Biophys J 84(6):4102–4114CrossRefGoogle Scholar
  42. Simpson MJ, Landman KA, Hughes BD (2009) Pathlines in exclusion processes. Phys l Rev E 79(3):031920CrossRefGoogle Scholar
  43. Steeg PS (2006) Tumor metastasis: mechanistic insights and clinical challenges. Nat Med 12(8):895–904CrossRefGoogle Scholar
  44. Voss-Böhme A, Deutsch A (2010) The cellular basis of cell sorting kinetics. J Theor Biol 263(4):419–436CrossRefGoogle Scholar
  45. Wolf-Gladrow DA (2000) Lattice-gas cellular automata and lattice Boltzmann models: an introduction. Springer, New YorkCrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2015

Authors and Affiliations

  • Carsten Mente
    • 1
  • Anja Voss-Böhme
    • 1
  • Andreas Deutsch
    • 1
  1. 1.Technische Universität Dresden, Zentrum für Informationsdienste und HochleistungsrechnenDresdenGermany

Personalised recommendations