Journal of Mathematical Biology

, Volume 75, Issue 5, pp 1075–1100 | Cite as

Extracting cellular automaton rules from physical Langevin equation models for single and collective cell migration

  • J. M. Nava-SedeñoEmail author
  • H. Hatzikirou
  • F. Peruani
  • A. Deutsch


Cellular automata (CA) are discrete time, space, and state models which are extensively used for modeling biological phenomena. CA are “on-lattice” models with low computational demands. In particular, lattice-gas cellular automata (LGCA) have been introduced as models of single and collective cell migration. The interaction rule dictates the behavior of a cellular automaton model and is critical to the model’s biological relevance. The LGCA model’s interaction rule has been typically chosen phenomenologically. In this paper, we introduce a method to obtain lattice-gas cellular automaton interaction rules from physically-motivated “off-lattice” Langevin equation models for migrating cells. In particular, we consider Langevin equations related to single cell movement (movement of cells independent of each other) and collective cell migration (movement influenced by cell-cell interactions). As examples of collective cell migration, two different alignment mechanisms are studied: polar and nematic alignment. Both kinds of alignment have been observed in biological systems such as swarms of amoebae and myxobacteria. Polar alignment causes cells to align their velocities parallel to each other, whereas nematic alignment drives cells to align either parallel or antiparallel to each other. Under appropriate assumptions, we have derived the LGCA transition probability rule from the steady-state distribution of the off-lattice Fokker-Planck equation. Comparing alignment order parameters between the original Langevin model and the derived LGCA for both mechanisms, we found different areas of agreement in the parameter space. Finally, we discuss potential reasons for model disagreement and propose extensions to the CA rule derivation methodology.


Lattice-gas cellular automata Langevin equations Polar alignment Nematic alignment Ferromagnetic interaction Liquid-crystal interaction Single cell migration Collective cell migration Fokker-Planck equation 

Mathematics Subject Classification

60J20 60K35 82B44 82C21 92B25 



The authors thank the Centre for Information Services and High Performance Computing (ZIH) at TU Dresden for providing an excellent infrastructure. The authors would like to thank Rainer Klages, Anja Voß-Böhme, Jörn Starruß, and Fabian Rost for their helpful comments and fruitful discussions. Andreas Deutsch acknowledges support from Stiftung Deutsche Krebshilfe through grant No. 70112014. This work was funded by the German Research Foundation (Deutsche Forschungsgemeinschaft) within the projects SFB-TR 79 “Materials for tissue regeneration within systemically altered bones” and Research Cluster of Excellence “Center for Advancing Electronics Dresden” (cfaed). This work was possible thanks to the joint scholarship program DAAD-CONACYT-Regierungsstipendien (50017046) by the German Academic Exchange Service and the National Council on Science and Technology of Mexico. H. Hatzikirou was supported by the German Federal Ministry of Education and research (BMBF) for the eMED project SYSIMIT (01ZX1308D) and the ERACoSysMed project SYSMIFTA (031L0085B).

Supplementary material

285_2017_1106_MOESM1_ESM.pdf (316 kb)
Supplementary material 1 (pdf 316 KB)


  1. Alber MS, Jiang Y, Kiskowski MA (2004) Lattice gas cellular automation model for rippling and aggregation in myxobacteria. Phys D 191(3):343–358CrossRefzbMATHGoogle Scholar
  2. Angelini TE, Hannezo E, Trepat X, Marquez M, Fredberg JJ, Weitz DA (2011) Glass-like dynamics of collective cell migration. Proc Natl Acad Sci USA 108(12):4714–4719CrossRefGoogle Scholar
  3. Belmonte JM, Thomas GL, Brunnet LG, De Almeida RM, Chaté H (2008) Self-propelled particle model for cell-sorting phenomena. Phys Rev Lett 100(24):248702CrossRefGoogle Scholar
  4. Berg HC, Brown DA (1972) Chemotaxis in Escherichia coli analysed by three-dimensional tracking. Nature 239(5374):500–504CrossRefGoogle 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. B Math Biol 74(2):474–490MathSciNetCrossRefzbMATHGoogle Scholar
  6. Binder BJ, Landman KA, Simpson MJ, Mariani M, Newgreen DF (2008) Modeling proliferative tissue growth: a general approach and an avian case study. Phys Rev E 78(3):031912CrossRefGoogle Scholar
  7. Bonner JT (1998) A way of following individual cells in the migrating slugs of Dictyostelium discoideum. Proc Natl Acad Sci USA 95(16):9355–9359CrossRefGoogle Scholar
  8. Böttger K, Hatzikirou H, Voss-Böhme A, Cavalcanti-Adam EA, Herrero MA, Deutsch A (2015) An emerging Allee effect is critical for tumor initiation and persistence. PLOS Comput Biol 11(9):e1004366CrossRefGoogle Scholar
  9. Bovin SA, Chibotaru LF, Ceulemans A (2001) The quantum structure of carbon tori. J Mol Catal A-Chem 166(1):47–52CrossRefGoogle 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(26):5018–5021CrossRefGoogle Scholar
  11. Chauvière A, Hatzikirou H, Kevrekidis IG, Lowengrub JS, Cristini V (2012) Dynamic density functional theory of solid tumor growth: preliminary models. AIP Adv 2(1):011210CrossRefGoogle Scholar
  12. Christofides A, Tanyi B, Christofides S, Whobrey D, Christofides N (1999) The optimal discretization of probability density functions. Comput Stat Data Anal 31(4):475–486CrossRefzbMATHGoogle Scholar
  13. Deutsch A, Dormann S (2005) Cellular automaton modeling of biological pattern formation: characterization, applications, and analysis, 2nd edn. Birkhauser, BostonzbMATHGoogle Scholar
  14. d’Humières D, Lallemand P, Frisch U (1986) Lattice gas models for 3D hydrodynamics. Europhys Lett 2(4):291CrossRefGoogle Scholar
  15. Dickinson RB, Tranquillo RT (1993) A stochastic model for adhesion-mediated cell random motility and haptotaxis. J Math Biol 31(6):563–600CrossRefzbMATHGoogle Scholar
  16. Dieterich P, Klages R, Preuss R, Schwab A (2008) Anomalous dynamics of cell migration. Proc Natl Acad Sci USA 105(2):459–463CrossRefGoogle Scholar
  17. Doi M, Edwards SF (1986) The theory of polymer dynamics. Claredon, OxfordGoogle Scholar
  18. Dunn GA, Brown AF (1987) A unified approach to analysing cell motility. J Cell Sci 1987:81–102CrossRefGoogle Scholar
  19. Fischman DA (1967) An electron microscope study of myofibril formation in embryonic chick skeletal muscle. J Cell Biol 32(3):557–575CrossRefGoogle Scholar
  20. Fraser LM, Foulkes WMC, Rajagopal G, Needs RJ, Kenny SD, Williamson AJ (1996) Finite size effects and Coulomb interactions in quantum Monte Carlo calculations for homogeneous systems with periodic boundary conditions. Phys Rev B 53(4):1814CrossRefGoogle Scholar
  21. Freiser MJ (1970) Ordered states of a nematic liquid. Phys Rev Lett 24(19):1041CrossRefGoogle Scholar
  22. Frisch U, Hasslacher B, Pomeau Y (1986) Lattice-gas automata for the Navier-Stokes equation. Phys Rev Lett 56(14):1505CrossRefGoogle Scholar
  23. Gardiner CW (1985) Handbook of stochastic methods. Springer, BerlinGoogle Scholar
  24. Griffiths RB (1967) Correlations in Ising ferromagnets. I. J Math Phys 8(3):478–483CrossRefGoogle Scholar
  25. Grønbech-Jensen N, Beardmore KM, Pincus P (1998) Interactions between charged spheres in divalent counterion solution. Phys A 261(1–2):74–81CrossRefGoogle Scholar
  26. Hardy J, Pomeau Y, De Pazzis O (1973) Time evolution of a two-dimensional model system. I. Invariant states and time correlation functions. J Math Phys 14(12):1746–1759MathSciNetCrossRefGoogle Scholar
  27. Hatzikirou H, Basanta D, Simon M, Schaller K, Deutsch A (2012) ‘Go or Grow’: the key to the emergence of invasion in tumour progression? Math Med Biol 29(1):49–65MathSciNetCrossRefzbMATHGoogle Scholar
  28. Kansal AR, Torquato S, Harsh GR, Chiocca EA, Deisboeck TS (2000) Simulated brain tumor growth dynamics using a three-dimensional cellular automaton. J Theor Biol 203(4):367–382CrossRefGoogle Scholar
  29. Kawasaki K (1973) Simple derivations of generalized linear and nonlinear Langevin equations. J Phys A Math Gen 6(9):1289MathSciNetCrossRefGoogle Scholar
  30. Kiskowski MA, Alber MS, Thomas GL, Glazier JA, Bronstein NB, Pu J, Newman SA (2004) Interplay between activator-inhibitor coupling and cell-matrix adhesion in a cellular automaton model for chondrogenic patterning. Dev Biol 271(2):372–387CrossRefGoogle Scholar
  31. Langevin P (1908) Sur la théorie du mouvement Brownien. CR Acad Sci (Paris) 146(530–533):530zbMATHGoogle Scholar
  32. Lawniczak AT (1997) Lattice gas automata for diffusive-convective transport dynamics. Center for Nonlinear Studies, Newsletter No. 136, LALP-97-010Google Scholar
  33. Lebwohl PA, Lasher G (1972) Nematic-liquid-crystal order - a Monte Carlo calculation. Phys Rev A 6(1):426CrossRefGoogle Scholar
  34. Mallet DG, De Pillis LG (2006) A cellular automata model of tumor-immune system interactions. J Theor Biol 239(3):334–350MathSciNetCrossRefGoogle Scholar
  35. Mermin ND, Wagner H (1966) Abscence of ferromagnetism or antiferromagnetism in one-or two-dimensional Heisenberg models. Phys Rev Lett 17(22):1133CrossRefGoogle Scholar
  36. Metzler R (2000) Generalized Chapman-Kolmogorov equation: a unifying approach to the description of anomalous transport in external fields. Phys Rev E 62(5):6233MathSciNetCrossRefGoogle Scholar
  37. Meyer DA (1996) From quantum cellular automata to quantum lattice gases. J Stat Phys 85(5–6):551–574MathSciNetCrossRefzbMATHGoogle Scholar
  38. Newman JP, Sayama H (2008) Effect of sensory blind zones on milling behavior in a dynamic self-propelled particle model. Phys Rev E 78(1):011913CrossRefGoogle Scholar
  39. Ornstein LS, Zernike F (1914) Accidental deviations of density and opalescence at the critical point of a single substance. Proc Akad Sci (Amsterdam) 17:793Google Scholar
  40. Peruani F, Deutsch A, Bär M (2006) Nonequilibrium clustering of self-propelled rods. Phys Rev E 74(3):030904CrossRefGoogle Scholar
  41. Peruani F, Deutsch A, Bär M (2008) A mean-field theory for self-propelled particles interacting by velocity alignment mechanisms. Eur Phys J-Spec Top 157(1):111–122CrossRefGoogle Scholar
  42. Polin M, Tuval I, Drescher K, Gollub JP, Goldstein RE (2009) Chlamydomonas swims with two “gears” in a eukaryotic version of run-and-tumble locomotion. Science 325(5939):487–490CrossRefGoogle Scholar
  43. Simpson MJ, Merriefield A, Landman KA, Hughes BD (2007) Simulating invasion with cellular automata: connecting cell-scale and population-scale properties. Phys Rev E 76(2):021918CrossRefGoogle Scholar
  44. Smith JT, Elkin JT, Reichert WM (2006) Directed cell migration on fibronectin gradients: effect of gradient slope. Exp Cell Res 312(13):2424–2432CrossRefGoogle Scholar
  45. Spudich JL, Koshland DE Jr (1976) Non-genetic individuality: chance in the single cell. Nature 262(5568):467–471CrossRefGoogle Scholar
  46. Van Kampen NG (1981) Itō versus stratonovich. J Stat Phys 24(1):175–187CrossRefzbMATHGoogle Scholar
  47. Vicsek T, Czirók A, Ben-Jacob E, Cohen I, Shochet O (1995) Novel type of phase transition in a system of self-driven particles. Phys Rev Lett 75(6):1226MathSciNetCrossRefGoogle Scholar
  48. Wang KG, Dong LK, Wu XF, Zhu FW, Ko T (1994) Correlation effects, generalized Brownian motion and anomalous diffusion. Phys A 203(1):53–60CrossRefGoogle Scholar
  49. Weimar JR, Boon JP (1994) Class of cellular automata for reaction-diffusion systems. Phys Rev E 49(2):1749CrossRefGoogle Scholar
  50. Weitz S, Deutsch A, Peruani F (2015) Self-propelled rods exhibit a phase-separated state characterized by the presence of active stresses and the ejection of polar clusters. Phys Rev E 92(1):012322CrossRefGoogle Scholar
  51. Welch R, Kaiser D (2001) Cell behavior in traveling wave patterns of myxobacteria. Proc Natl Acad Sci USA 98(2):14907–14912CrossRefGoogle Scholar
  52. Wittkowski R, Löwen H (2012) Self-propelled Brownian spinning top: dynamics of a biaxial swimmer at low Reynolds numbers. Phys Rev E 85(2):021406CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Center for Information Services and High Performance ComputingTechnische Universität DresdenDresdenGermany
  2. 2.Department of Systems Immunology and Braunschweig Integrated Centre of Systems BiologyHelmholtz Center for Infection ResearchBraunschweigGermany
  3. 3.Laboratoire J. A. Dieudonné, UMR 7351 CNRS, Parc ValroseUniversité Côte d’AzurNice Cedex 02France

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