Abstract
The modelling of collective migration has traditionally been undertaken in a continuous framework, with little reference to the individual-level mechanisms that give rise to such a concerted movement. One factor whose importance is now coming to light is that the individuals themselves occupy space in the domain, thus obstructing others from moving past them (volume exclusion). In this work, we systematically derive continuous descriptions of cellular migration with volume exclusion for a wide range of individual-based mechanisms and in one, two and three dimensions. We also consider subpopulations of migrating individuals, which may have different characteristics, such as differing sizes and speeds of migration. We demonstrate that volume exclusion is of particular importance when biased movement is included, and thus conclude that volume exclusion may have its greatest effect when considering directed migratory mechanisms such as chemotaxis.
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References
Baker RE, Simpson MJ (2010) Correcting mean-field approximations for birth–death–movement processes. Phys Rev E 82(4):041,905
Baker RE, Simpson MJ (2012) Models of collective cell motion for cell populations with different aspect ratio: diffusion, proliferation and travelling waves. Phys A 391(14):3729–3750
Binder B, Landman K (2012) Spatial analysis of multi-species exclusion processes: application to neural crest cell migration in the embryonic gut. Bull Math Biol 74:474–490
Bruna M, Chapman SJ (2012a) Diffusion of multiple species with excluded-volume effects. J Chem Phys 137(20):204,116
Bruna M, Chapman SJ (2012b) Excluded-volume effects in the diffusion of hard spheres. Phys Rev E 85(1):011,103
Callaghan T, Khain E, Sander L, Ziff R (2006) A stochastic model for wound healing. J Stat Phys 122:909–924
Dormann D, Weijer CJ (2006) Chemotactic cell movement during Dictyostelium development and gastrulation. Curr Opin Genet Dev 16(4):367–373
Dyson L, Maini P, Baker RE (2012) Macroscopic limits of individual-based models for motile cell populations with volume exclusion. Phys Rev E 86(3):031,903
Flache A, Hegselmann R (2001) Do irregular grids make a difference? Relaxing the spatial regularity assumption in cellular models of social dynamics. JASSS 4(4) . http://jasss.soc.surrey.ac.uk/4/4/6.html
Gillespie CS (2009) Moment-closure approximations for mass-action models. IET Syst Biol 3(1):52–58
Gillespie DT (1977) Exact stochastic simulation of coupled chemical reactions. J Phys Chem 81(25):2340–2361
Hillen T, Painter K (2009) A user’s guide to PDE models for chemotaxis. J Math Biol 58(1–2):183–217
Johnston ST, Simpson MJ, Baker RE (2012) Mean-field descriptions of collective migration with strong adhesion. Phys Rev E 85(5):051,922
Johnston ST, Simpson MJ, Plank MJ (2013) Lattice-free descriptions of collective motion with crowding and adhesion. Phys Rev E 88(6):062,720
Keller EF, Segel LA (1971) Model for chemotaxis. J Theor Biol 30(2):225–234
Khain E, Sander LM, Schneider-Mizell CM (2006) The role of cell–cell adhesion in wound healing. J Stat Phys 128(1–2):209–218
Landman KA, Pettet GJ, Newgreen DF (2003) Mathematical models of cell colonization of uniformly growing domains. Bull Math Biol 65:235–262
McLennan R, Dyson L, Prather KW, Morrison JA, Baker RE, Maini P, Kulesa PM (2012) Multiscale mechanisms of cell migration during development: theory and experiment. Development 139(16):2935–2944
Numerical Algorithms Groupd03pc—NAG Toolbox for MATLAB documentation (2013a). http://www.nag.co.uk/numeric/MB/manual64_23_1/pdf/D03/d03pc
Numerical Algorithms Group d03ra—NAG Toolbox for MATLAB documentation (2013b). http://www.nag.co.uk/numeric/MB/manual64_23_1/pdf/D03/d03ra
Painter KJ, Maini PK, Othmer HG (2000) A chemotactic model for the advance and retreat of the primitive streak in avian development. Bull Math Biol 62(3):501–525
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(4):041,120
Perthame B (2004) PDE models for chemotactic movements: parabolic, hyperbolic and kinetic. Appl Math 49(6):539–564
Plank MJ, Simpson MJ (2012) Models of collective cell behaviour with crowding effects: comparing lattice-based and lattice-free approaches. J R Soc Interface 9(76):2983–96
Plank MJ, Simpson MJ (2013) Lattice-free models of cell invasion: discrete simulations and travelling waves. Bull Math Biol 75(11):2150–2166
Simpson MJ, Zhang DC, Mariani M, Landman KA, Newgreen DF (2007) Cell proliferation drives neural crest cell invasion of the intestine. Dev Biol 302(2):553–568
Tremel A, Cai A, Tirtaatmadja N, Hughes B, Stevens G, Landman K, OConnor A (2009) Cell migration and proliferation during monolayer formation and wound healing. Chem Eng Sci 64(2):247–253. doi:10.1016/j.ces.2008.10.008
Trewenack AJ, Landman KA (2009) A traveling wave model for invasion by precursor and differentiated cells. Bull Math Biol 71:291–317
Zhang DC, Brinas IM, Binder BJ, Landman KA, Newgreen DF (2010) Neural crest regionalisation for enteric nervous system formation: implications for Hirschsprung’s disease and stem cell therapy. Dev Biol 339(2):280–294
Acknowledgments
The authors would like to thank Prof. Philip K. Maini for his assistance and many helpful discussions.
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Dyson, L., Baker, R.E. The importance of volume exclusion in modelling cellular migration. J. Math. Biol. 71, 691–711 (2015). https://doi.org/10.1007/s00285-014-0829-0
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DOI: https://doi.org/10.1007/s00285-014-0829-0