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Discrete and Continuum Approximations for Collective Cell Migration in a Scratch Assay with Cell Size Dynamics

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Abstract

Scratch assays are routinely used to study the collective spreading of cell populations. In general, the rate at which a population of cells spreads is driven by the combined effects of cell migration and proliferation. To examine the effects of cell migration separately from the effects of cell proliferation, scratch assays are often performed after treating the cells with a drug that inhibits proliferation. Mitomycin-C is a drug that is commonly used to suppress cell proliferation in this context. However, in addition to suppressing cell proliferation, mitomycin-C also causes cells to change size during the experiment, as each cell in the population approximately doubles in size as a result of treatment. Therefore, to describe a scratch assay that incorporates the effects of cell-to-cell crowding, cell-to-cell adhesion, and dynamic changes in cell size, we present a new stochastic model that incorporates these mechanisms. Our agent-based stochastic model takes the form of a system of Langevin equations that is the system of stochastic differential equations governing the evolution of the population of agents. We incorporate a time-dependent interaction force that is used to mimic the dynamic increase in size of the agents. To provide a mathematical description of the average behaviour of the stochastic model we present continuum limit descriptions using both a standard mean-field approximation and a more sophisticated moment dynamics approximation that accounts for the density of agents and density of pairs of agents in the stochastic model. Comparing the accuracy of the two continuum descriptions for a typical scratch assay geometry shows that the incorporation of agent growth in the system is associated with a decrease in accuracy of the standard mean-field description. In contrast, the moment dynamics description provides a more accurate prediction of the evolution of the scratch assay when the increase in size of individual agents is included in the model.

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References

  • Baker RE, Simpson MJ (2010) Correcting mean-field approximations for birth-death-movement processes. Phys Rev E 82:041905

    Article  MathSciNet  Google Scholar 

  • Binder BJ, Simpson MJ (2016) Cell density and cell size dynamics during in vitro tissue growth experiments: implications for mathematical models of collective cell behaviour. Appl Math Model 40:3438–3446

    Article  Google Scholar 

  • Binny RN, Plank MJ, James A (2015) Spatial moment dynamics for collective cell movement incorporating a neighbour-dependent directional bias. J R Soc Interface 12:20150228

    Article  Google Scholar 

  • Binny RN, Haridas P, James A, Law R, Simpson MJ, Plank MJ (2016) Spatial structure arising from neighbour dependent bias in collective cell movement. PeerJ 4:e1689

    Article  Google Scholar 

  • Binny RN, James A, Plank MJ (2016) Collective cell behaviour with neighbour-dependent proliferation, death and directional bias. Bull Math Biol 78(11):2277–2301

    Article  MathSciNet  MATH  Google Scholar 

  • Bolker B, Pacala SW (1997) Using moment equations to understand stochastically driven spatial pattern formation in ecological systems. Theor Popul Biol 52:179–197

    Article  MATH  Google Scholar 

  • Byrne H, Preziosi L (2003) Modelling solid tumour growth using the theory of mixtures. Math Med Biol 20:341–366

    Article  MATH  Google Scholar 

  • Callaghan T, Khain E, Sander LM, Ziff RM (2006) A stochastic model for wound healing. J Stat Phys 122(5):909–924

    Article  MathSciNet  MATH  Google Scholar 

  • Codling EA, Plank MJ, Benhamou S (2008) Random walk models in biology. J R Soc Interface 5:813–834

    Article  Google Scholar 

  • Dyson L, Maini PK, Baker RE (2012) Macroscopic limits of individual-based models for motile cell populations with volume exclusion. Phys Rev E 86:031903

    Article  Google Scholar 

  • Edmondson R, Broglie JJ, Adcock AF, Yang L (2014) Three-dimensional cell culture systems and their applications in drug discovery and cell-based biosensors. Assay Drug Dev Technol 12(4):207–218

    Article  Google Scholar 

  • Galle J, Loeffler M, Drasdo D (2005) Modeling the effect of deregulated proliferation and apoptosis on the growth dynamics of epithelial cell populations in vitro. Biophys J 88:62–75

    Article  Google Scholar 

  • Glenn HL, Messner J, Meldrum DR (2016) A simple non-perturbing cell migration assay insensitive to proliferation effects. Sci Rep 6:31694

    Article  Google Scholar 

  • Grada A, Otero-Vinas M, Prieto-Castrillo F, Obagi Z, Falanga V (2017) Research techniques made simple: analysis of collective cell migration using the wound healing assay. J Investig Dermatol 137:e11–e16

    Article  Google Scholar 

  • House T (2014) Algebraic moment closure for population dynamics on discrete structures. Bull Math Biol 77:646–659

    Article  MathSciNet  MATH  Google Scholar 

  • Jeon J, Quaranta V, Cummings PT (2010) An off-lattice hybrid discrete-continuum model of tumor growth and invasion. Biophys J 98:37–47

    Article  Google Scholar 

  • Jin W, Shah ET, Penington CJ, McCue SW, Chopin LK, Simpson MJ (2016) Reproducibility of scratch assays is affected by the initial degree of confluence: Experiments, modelling and model selection. J Theor Biol 390:136–145

    Article  MATH  Google Scholar 

  • Johnston ST, Shah ET, Chopin LK, McElwain DLS, Simpson MJ (2015) Estimating cell diffusivity and cell proliferation rate by interpreting IncuCyte ZOOM\(^{\rm TM}\) assay data using the Fisher-Kolmogorov model. BMC Syst Biol 9:38

    Article  Google Scholar 

  • Kaighn ME, Narayan KS, Ohnuki Y, Lechner JF, Jones LW (1979) Establishment and characterization of a human prostatic carcinoma cell line (PC-3). Invest Urol 17:16–23

    Google Scholar 

  • Keeling MJ, Rand DA, Morris AJ (1997) Correlation models for childhood epidemics. Proc R Soc Lond B Biol Sci 264:1149–1156

    Article  Google Scholar 

  • Khain E, Katakowski M, Hopkins S, Szalad A, Zheng X, Jiang F, Chopp M (2011) Collective behavior of brain tumor cells: the role of hypoxia. Phys Rev E 83:031920

    Article  Google Scholar 

  • Kirkwood JG (1935) Statistical mechanics of fluid mixtures. J Chem Phys 3(5):300–313

    Article  MATH  Google Scholar 

  • Levin SA, Whitfield M (1994) Patchiness in marine and terrestrial systems: from individuals to populations. Philos Trans Biol Sci 343:99–103

    Article  Google Scholar 

  • Liang CC, Park AY, Guan JL (2007) In vitro scratch assay: a convenient and inexpensive method for analysis of cell migration in vitro. Nat Protoc 2:329–333

    Article  Google Scholar 

  • Maini PK, McElwain DL, Leavesley DI (2004) Traveling wave model to interpret a wound-healing cell migration assay for human peritoneal mesothelial cells. Tissue Eng 10(3–4):475–82

    Article  Google Scholar 

  • Markham DC, Simpson MJ, Baker RE (2015) Choosing an appropriate modelling framework for analysing multispecies co-culture cell biology experiments. Bull Math Biol 77:713–734

    Article  MathSciNet  MATH  Google Scholar 

  • Matsiaka OM, Penington CJ, Baker RE, Simpson MJ (2017) Continuum approximations for lattice-free multi-species models of collective cell migration. J Theor Biol 422:1–11

    Article  MathSciNet  MATH  Google Scholar 

  • 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 359:220–232

    Article  MathSciNet  Google Scholar 

  • Murray PJ, Edwards CM, Tindall MJ, Maini PK (2012) Classifying general nonlinear force laws in cell-based models via the continuum limit. Phys Rev E 85:021921

    Article  Google Scholar 

  • Murrell DJ, Dieckmann U, Law R (2004) On moment closures for population dynamics in continuous space. J Theor Biol 229:421–432

    Article  Google Scholar 

  • Newman TJ, Grima R (2004) Many-body theory of chemotactic cell-cell interactions. Phys Rev E 70:051916

    Article  Google Scholar 

  • Nyegaard S, Christensen B, Rasmussen JT (2016) An optimized method for accurate quantification of cell migration using human small intestine cells. Metab Eng Commun 3:76–83

    Article  Google Scholar 

  • 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 84:041120

    Article  Google Scholar 

  • Plank MJ, Law R (2015) Spatial point processes and moment dynamics in the life sciences: a parsimonious derivation and some extensions. Bull Math Biol 77:586–613

    Article  MathSciNet  MATH  Google Scholar 

  • Press WH, Flannery BP, Teukolsky SA, Vetterling WT (2007) Numerical recipes: the art of scientific computing, 3rd edn. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • QUT high performance computing. https://www.student.qut.edu.au/technology/research-computing/high-performance-computing. Date last accessed 1 Nov 2017

  • Riss T (2005) Selecting cell-based assays for drug discovery screening (2005). Cell Notes 13:16–21

    Google Scholar 

  • Sadeghi HM, Seitz B, Hayashi S, LaBree L, McDonnell PJ (1998) In vitro effects of Mitomycin-C on human keratocytes. J Refract Surg 14:534–540

    Google Scholar 

  • Shah ET, Upadhyaya A, Philp LK, Tang T, Skalamera D, Gunter J, Nelson CC, Williams ED, Hollier BG (2016) Repositioning “old” drugs for new causes: identifying new inhibitors of prostate cancer cell migration and invasion. Clin Exp Metastasis 33:385–399

    Article  Google Scholar 

  • Sharkey KJ, Fernandez C, Morgan KL, Peeler E, Thrush M, Turnbull JF, Bowers RG (2006) Pair-level approximations to the spatio-temporal dynamics of epidemics on asymmetric contact networks. J Math Biol 53:61–85

    Article  MathSciNet  MATH  Google Scholar 

  • Sharkey KJ, Kiss IZ, Wilkinson RR, Simon PL (2015) Exact equations for SIR epidemics on tree graphs. Bull Math Biol 77:614–645

    Article  MathSciNet  MATH  Google Scholar 

  • Sherratt JA, Murray JD (1990) Models of epidermal wound healing. Proc R Soc Lond B 241:29–36

    Article  Google Scholar 

  • Simpson MJ (2009) Depth-averaging errors in reactive transport modelling. Water Resour Res 45:W02505

    Article  Google Scholar 

  • Simpson MJ, Treloar KK, Binder BJ, Haridas P, Manton KJ, Leavesley DI, McElwain DLS, Baker RE (2013) Quantifying the roles of cell motility and cell proliferation in a circular barrier assay. J R Soc Interface 10:20130007

    Article  Google Scholar 

  • Singer A (2004) Maximum entropy formulation of the Kirkwood superposition approximation. J Chem Phys 121:3657–3666

    Article  Google Scholar 

  • Smith S, Cianci C, Grima R (2017) Macromolecular crowding directs the motion of small molecules inside cells. J R Soc Interface 14:20170047

    Article  Google Scholar 

  • Steinberg MS (1996) Adhesion in development: an historical review. Dev Biol 180:377–388

    Article  Google Scholar 

  • Tersoff J (1987) New empirical approach for the structure and energy of covalent systems. Phys Rev B 37(12):6991–7000

    Article  Google Scholar 

  • Treloar KK, Simpson MJ, Haridas P, Manton KJ, Leavesley DI, McElwain DLS, Baker RE (2013) Multiple types of data are required to identify the mechanisms influencing the spatial expansion of melanoma cell colonies. BMC Syst Biol 7:137

    Article  Google Scholar 

Download references

Acknowledgements

We appreciate support from the Australian Research Council (FT130100148, DP170100474). Ruth E. Baker is a Royal Society Wolfson Research Merit Award holder and a Leverhulme Research Fellow. Computational resources are provided by the High Performance Computing and Research Support Group at QUT. We also appreciate helpful comments provided by two referees.

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Authors and Affiliations

  1. Mathematical Sciences, Queensland University of Technology, Brisbane, Australia

    Oleksii M. Matsiaka & Matthew J. Simpson

  2. Department of Mathematics, Macquarie University, Sydney, Australia

    Catherine J Penington

  3. Mathematical Institute, Radcliffe Observatory Quarter, University of Oxford, Woodstock Road, Oxford, United Kingdom

    Ruth E. Baker

Authors
  1. Oleksii M. Matsiaka
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  2. Catherine J Penington
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  3. Ruth E. Baker
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  4. Matthew J. Simpson
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Correspondence to Matthew J. Simpson.

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Matsiaka, O.M., Penington, C.J., Baker, R.E. et al. Discrete and Continuum Approximations for Collective Cell Migration in a Scratch Assay with Cell Size Dynamics. Bull Math Biol 80, 738–757 (2018). https://doi.org/10.1007/s11538-018-0398-2

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  • DOI: https://doi.org/10.1007/s11538-018-0398-2

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