Bulletin of Mathematical Biology

, Volume 80, Issue 4, pp 738–757 | Cite as

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

  • Oleksii M. Matsiaka
  • Catherine J Penington
  • Ruth E. Baker
  • Matthew J. Simpson
Original Article


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.


Cell migration Scratch assay Stochastic model Langevin equation Continuum model Mean-field approximation 



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.

Supplementary material

11538_2018_398_MOESM1_ESM.pdf (1005 kb)
Supplementary material 1 (pdf 1005 KB)


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Copyright information

© Society for Mathematical Biology 2018

Authors and Affiliations

  1. 1.Mathematical SciencesQueensland University of TechnologyBrisbaneAustralia
  2. 2.Department of MathematicsMacquarie UniversitySydneyAustralia
  3. 3.Mathematical Institute, Radcliffe Observatory QuarterUniversity of OxfordOxfordUnited Kingdom

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