## Abstract

Cell–cell adhesion is essential for biological development: cells migrate to their target sites, where cell–cell adhesion enables them to aggregate and form tissues. Here, we extend analysis of the model of cell migration proposed by Anguige and Schmeiser (J. Math. Biol. 58(3):395–427, 2009) that incorporates both cell–cell adhesion and volume filling. The stochastic space-jump model is compared to two deterministic counterparts (a system of stochastic mean equations and a non-linear partial differential equation), and it is shown that the results of the deterministic systems are, in general, qualitatively similar to the mean behaviour of multiple stochastic simulations. However, individual stochastic simulations can give rise to behaviour that varies significantly from that of the mean. In particular, individual simulations might admit cell clustering when the mean behaviour does not. We also investigate the potential of this model to display behaviour predicted by the differential adhesion hypothesis by incorporating a second cell species, and present a novel approach for implementing models of cell migration on a growing domain.

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## Acknowledgements

RNT would like to thank the Centre for Mathematical Biology, University of Oxford, for the opportunity to carry out this research, the Nuffield Foundation for the bursary that allowed this research to begin, and BBSRC for research funding via the Genes to Organisms doctoral training award. He would also like to thank Endre Suli, Michael Thompson and Nik Cunniffe for helpful discussions and support. CAY would like to thank Christ Church College, Oxford, for a Junior Research Fellowship.

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## Appendix: Transition Probabilities on a Non-uniform Domain

### Appendix: Transition Probabilities on a Non-uniform Domain

When the domain has unequal compartment sizes, the distance that cells jump between neighbouring compartments varies. If this distance is large, then the transition rate between the compartments concerned should be reduced.

We note that, if we consider a stochastic system with transition probabilities of the form

where diffusivity *D* is constant, we can derive the PDE

as discussed by Baker et al. (2010).

If we consider approximating a solution to this PDE on a non-uniform domain (Morton and Mayers 2005), we obtain

where the *h*
_{
i
}s are defined as in Sect. 2.1. Assuming that the number of cells evolves according to

we deduce that

with a similar expression obtainable for \(T_{i}^{-}\). For ease of notation, we define

The above consideration motivates the pre-factor in our transition rates on a non-uniform domain, for *i*=2,3,…,*k*−1 (where the cell density in compartment *i* is *n*
_{
i
}/*S*
_{
i
}), given by

with similar expressions derived for *i*=1 and *i*=*k*. Strictly speaking, we have assumed that the compartment edges are halfway between the lattice points.

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Thompson, R.N., Yates, C.A. & Baker, R.E. Modelling Cell Migration and Adhesion During Development.
*Bull Math Biol* **74**, 2793–2809 (2012). https://doi.org/10.1007/s11538-012-9779-0

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DOI: https://doi.org/10.1007/s11538-012-9779-0

### Keywords

- Mathematical modelling
- Cell–cell adhesion
- Differential adhesion
- Cell sorting
- Domain growth