Modeling Uniaxial Nonuniform Cell Proliferation

  • Alexander Lai De Oliveira
  • Benjamin J. BinderEmail author


Growth in biological systems occurs as a consequence of cell proliferation fueled by a nutrient supply. In general, the nutrient gradient of the system will be nonconstant, resulting in biased cell proliferation. We develop a uniaxial discrete cellular automaton with biased cell proliferation using a probability distribution which reflects the nutrient gradient of the system. An explicit probability mass function for the displacement of any tracked cell under the cellular automaton model is derived and verified against averaged simulation results; this displacement distribution has applications in predicting cell trajectories and evolution of expected site occupancies.


Nonuniform growth Discrete model Cellular automata 



BJB contribution was supported by the Australian Research Council’s Discovery Project DP160102644.


  1. Alber M, Chen N, Glimm T, Lushnikov PM (2006) Multiscale dynamics of biological cells with chemotactic interactions: from a discrete stochastic model to a continuous description. Phys Rev E 73:051901MathSciNetCrossRefGoogle Scholar
  2. Baker RE, Yates CA, Erban R (2010) From microscopic to macroscopic descriptions of cell migration on growing domains. Bull Math Biol 72:719–762MathSciNetCrossRefzbMATHGoogle Scholar
  3. Binder BJ, Landman KA (2009) Exclusion processes on a growing domain. J Theor Biol 259:541–551MathSciNetCrossRefzbMATHGoogle Scholar
  4. 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 78:031912CrossRefGoogle Scholar
  5. Codling EA, Plank MJ, Benhamou S (2008) Random walk models in biology. J R Soc Interface 5(25):813–834CrossRefGoogle Scholar
  6. Crampin EJ, Gaffney EA, Maini PK (1999) Reaction and diffusion on growing domains: scenarios for robust pattern formation. Bull Math Biol 61:1093–1120CrossRefzbMATHGoogle Scholar
  7. Crampin EJ, Hackborn WW, Maini PK (2002) Pattern formation in reaction–diffusion models with nonuniform domain growth. Bull Math Biol 64:747–769CrossRefzbMATHGoogle Scholar
  8. Crampin EJ, Maini PK (2001) Modelling biological pattern formation: the role of domain growth. Comments Theor Biol 6:229–249Google Scholar
  9. Deroulers C, Aubert M, Badoual M, Grammaticos B (2009) Modeling tumor cell migration: from microscopic to macroscopic models. Phys Rev E 79:031917MathSciNetCrossRefGoogle Scholar
  10. Du H, Ayouz M, Lv P, Perré P (2018) A lattice-based system for modeling fungal mycelial growth in complex environments. Physica A 511:191–206CrossRefGoogle Scholar
  11. Goerke U, Chamberlain AHL, Crilly EA, McDonald PJ (2000) Model for water transport into powdered xanthan combining gel swelling and vapor diffusion. Phys Rev E 62:5353CrossRefGoogle Scholar
  12. Hywood JD, Hackett-Jones EJ, Landman KA (2013) Modeling biological tissue growth: discrete to continuum representations. Phys Rev E 88:032704CrossRefGoogle Scholar
  13. Johnson NL, Kemp AW, Kotz S (2005) Univariate discrete distributions, 3rd edn. Wiley, New JerseyCrossRefzbMATHGoogle Scholar
  14. Kansal AR, Torquato S, Harsh Iv GR, Chiocca EA, Deisboeck TS (2000) Cellular automaton of idealized brain tumor growth dynamics. BioSystems 55:119–127CrossRefGoogle Scholar
  15. Kondo S, Asai R (1995) A reaction–diffusion wave on the skin of the marine angelfish Pomacanthus. Nature 376:765–768CrossRefGoogle Scholar
  16. Kulesa PM, Cruywagen GC, Lubkin SR, Maini PK, Sneyd J, Ferguson MWJ, Murray JD (1996) On a model mechanism for the spatial patterning of teeth primordia in the alligator. J Theor Biol 180:287–296CrossRefGoogle Scholar
  17. Mahmoud H (2008) Pólya urn models. CRC Press, Boca RatonCrossRefzbMATHGoogle Scholar
  18. Osborne JM, Walter A, Kershaw SK, Mirams GR, Fletcher AG, Pathmanathan P, Gavaghan D, Jensen OE, Maini PK, Byrne HM (2010) A hybrid approach to multi-scale modelling of cancer. Philos Trans R Soc A 368:5013–5028MathSciNetCrossRefzbMATHGoogle Scholar
  19. Othmer HG, Stevens A (1997) Aggregation, blowup, and collapse: the abc’s of taxis in reinforced random walks. SIAM J Appl Math 57(1081):1044–1081MathSciNetzbMATHGoogle Scholar
  20. Painter KJ (1997) Chemotaxis as a mechanism for morphogenesis. Ph.D. thesis, University of OxfordGoogle Scholar
  21. Painter KJ (2009) Continuous models for cell migration in tissues and applications to cell sorting via differential chemotaxis. Bull Math Biol 71:1117–1147MathSciNetCrossRefzbMATHGoogle Scholar
  22. Penington CJ, Hughes BD, Landman KA (2011) Building macroscale models from microscale probabilistic models: a general probabilistic approach for nonlinear diffusion and multispecies phenomena. Phys Rev E 84:041120CrossRefGoogle Scholar
  23. Pillay S, Byrne HM, Maini PK (2017) Modeling angiogenesis: a discrete to continuum description. Phys Rev E 95:012410MathSciNetCrossRefGoogle Scholar
  24. Ross JV, Binder BJ (2014) Approximating spatially exclusive invasion processes. Phys Rev E 89:052709CrossRefGoogle Scholar
  25. Ross RJH, Baker RE, Yates CA (2016) How domain growth is implemented determines the long-term behavior of a cell population through its effect on spatial correlations. Phys Rev E 94:012408CrossRefGoogle Scholar
  26. Ross RJH, Yates CA, Baker RE (2015) Inference of cell–cell interactions from population density characteristics and cell trajectories on static and growing domains. Math Biosci 264:108–118MathSciNetCrossRefzbMATHGoogle Scholar
  27. Simon BR, Liable JP, Pflaster D, Yuan Y, Krag MH (1996) A poroelastic finite element formulation including transport and swelling in soft tissue structures. J Biomech Eng 118:1–9CrossRefGoogle Scholar
  28. Tam A, Green JEF, Balasuriya S, Tek EL, Gardner JM, Sundstrom JF, Jiranek V, Binder BJ (2018) Nutrient-limited growth with non-linear cell diffusion as a mechanism for floral pattern formation in yeast biofilms. J Theor Biol 448:122–141MathSciNetCrossRefzbMATHGoogle Scholar
  29. Tronnolone H, Tam A, Szenczi Z, Green JEF, Balasuriya S, Tek EL, Gardner JM, Sundstrom JF, Jiranek V, Oliver SG, Binder BJ (2018) Diffusion-limited growth of microbial colonies. Sci Rep 8:5992Google Scholar
  30. Turner S, Sherratt JA, Painter KJ, Savill NJ (2004) From a discrete to a continuous model of biological cell movement. Phys Rev E 69:021910MathSciNetCrossRefGoogle Scholar
  31. Vulin C, Di Meglio JM, Lindner AB, Daerr A, Murray A, Hersen P (2014) Growing yeast into cylindrical colonies. Biophys J 106:2214–2221CrossRefGoogle Scholar
  32. Yates CA (2014) Discrete and continuous models for tissue growth and shrinkage. J Theor Biol 350:37–48MathSciNetCrossRefGoogle Scholar
  33. Yates CA, Baker RE, Erban R, Maini PK (2012) Going from microscopic to macroscopic on nonuniform growing domains. Phys Rev E 86:021921CrossRefGoogle Scholar
  34. Yates CA, Parker A, Baker RE (2015) Incorporating pushing in exclusion-process models of cell migration. Phys Rev E 91:052711CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2019

Authors and Affiliations

  • Alexander Lai De Oliveira
    • 1
  • Benjamin J. Binder
    • 1
    Email author
  1. 1.School of Mathematical SciencesUniversity of AdelaideAdelaideAustralia

Personalised recommendations