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A space-jump derivation for non-local models of cell–cell adhesion and non-local chemotaxis

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Abstract

Cellular adhesion provides one of the fundamental forms of biological interaction between cells and their surroundings, yet the continuum modelling of cellular adhesion has remained mathematically challenging. In 2006, Armstrong et al. proposed a mathematical model in the form of an integro-partial differential equation. Although successful in applications, a derivation from an underlying stochastic random walk has remained elusive. In this work we develop a framework by which non-local models can be derived from a space-jump process. We show how the notions of motility and a cell polarization vector can be naturally included. With this derivation we are able to include microscopic biological properties into the model. We show that particular choices yield the original Armstrong model, while others lead to more general models, including a doubly non-local adhesion model and non-local chemotaxis models. Finally, we use random walk simulations to confirm that the corresponding continuum model represents the mean field behaviour of the stochastic random walk.

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Notes

  1. Implementation from libstc++ gcc 4.9.3 https://gcc.gnu.org/.

  2. www.numpy.org.

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Acknowledgements

AB was supported by NSERC, Alberta Innovates and PIMS. TH was supported by NSERC. AG thanks the Isaac Newton Institute for Mathematical Sciences for its hospitality during the programme Coupling Geometric PDEs with Physics for Cell Morphology, Motility and Pattern Formation; EPSRC EP/K032208/1. KJP thanks the Politecnico di Torino for a Visiting Professor position.

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  1. Kevin J. Painter

    Present address: Department of Mathematical Sciences, Politecnico di Torino, 10129, Turin, Italy

Authors and Affiliations

  1. Department of Mathematical and Statistical Sciences, Centre for Mathematical Biology, University of Alberta, Edmonton, AB, T6G 2G1, Canada

    Andreas Buttenschön & Thomas Hillen

  2. Fachbereich Mathematik, Technische Universität Darmstadt, Dolivostr. 15, 64293, Darmstadt, Germany

    Alf Gerisch

  3. Department of Mathematics and Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh, EH14 4AS, UK

    Kevin J. Painter

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  1. Andreas Buttenschön
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Correspondence to Andreas Buttenschön.

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Buttenschön, A., Hillen, T., Gerisch, A. et al. A space-jump derivation for non-local models of cell–cell adhesion and non-local chemotaxis. J. Math. Biol. 76, 429–456 (2018). https://doi.org/10.1007/s00285-017-1144-3

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