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A Nonlocal Model for Contact Attraction and Repulsion in Heterogeneous Cell Populations

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Abstract

Instructing others to move is fundamental for many populations, whether animal or cellular. In many instances, these commands are transmitted by contact, such that an instruction is relayed directly (e.g. by touch) from signaller to receiver: for cells, this can occur via receptor–ligand mediated interactions at their membranes, potentially at a distance if a cell extends long filopodia. Given that commands ranging from attractive to repelling can be transmitted over variable distances and between cells of the same (homotypic) or different (heterotypic) type, these mechanisms can clearly have a significant impact on the organisation of a tissue. In this paper, we extend a system of nonlocal partial differential equations (integrodifferential equations) to provide a general modelling framework to explore these processes, performing linear stability and numerical analyses to reveal its capacity to trigger the self-organisation of tissues. We demonstrate the potential of the framework via two illustrative applications: the contact-mediated dispersal of neural crest populations and the self-organisation of pigmentation patterns in zebrafish.

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Notes

  1. In fact the size of this total signal also varies with the space dimension \(n\), since the integral of \(\tilde{\Omega }\) is along a single ray but the cell signals along all rays. For simplicity we assume the parameter \(\mu \) implicitly incorporates this dimensional dependency—here we generally restrict to one dimension.

  2. In the numerics, we initially apply a small random perturbation to the uniform steady state value at each spatial grid point: set \(u(x_i,0) = U + \varepsilon (x_i)\) for \(i = 1\ldots N_x\), where \(N_x\) defines the number of spatial grid points in the discretisation. \(\varepsilon (x_i)\) is initially sampled from a uniform distribution in a range \(\pm 1\%U\) and subsequently normalised to ensure \(\frac{1}{N_x} \sum _{i=1}^{N_x} u(x_i,0) = U\).

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Authors and Affiliations

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

    K. J. Painter, J. M. Bloomfield & J. A. Sherratt

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

    A. Gerisch

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  1. K. J. Painter
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  2. J. M. Bloomfield
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  3. J. A. Sherratt
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  4. A. Gerisch
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Painter, K.J., Bloomfield, J.M., Sherratt, J.A. et al. A Nonlocal Model for Contact Attraction and Repulsion in Heterogeneous Cell Populations. Bull Math Biol 77, 1132–1165 (2015). https://doi.org/10.1007/s11538-015-0080-x

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