Skip to content

Research at St Andrews

A hybrid discrete-continuum approach to model Turing pattern formation

Research output: Contribution to journalArticlepeer-review

DOI

Open Access permissions

Open

Abstract

Since its introduction in 1952, with a further refinement in 1972 by Gierer and Meinhardt, Turing’s (pre-)pattern theory (the chemical basis of morphogenesis) has been widely applied to a number of areas in developmental biology, where evolving cell and tissue structures are naturally observed. The related pattern formation models normally comprise a system of reaction-diffusion equations for interacting chemical species (morphogens), whose heterogeneous distribution in some spatial domain acts as a template for cells to form some kind of pattern or structure through, for example, differentiation or proliferation induced by the chemical pre-pattern. Here we develop a hybrid discrete-continuum modelling framework for the formation of cellular patterns via the Turing mechanism. In this framework, a stochastic individual-based model of cell movement and proliferation is combined with a reaction-diffusion system for the concentrations of some morphogens. As an illustrative example, we focus on a model in which the dynamics of the morphogens are governed by an activator-inhibitor system that gives rise to Turing pre-patterns. The cells then interact with the morphogens in their local area through either of two forms of chemically-dependent cell action: Chemotaxis and chemically-controlled proliferation. We begin by considering such a hybrid model posed on static spatial domains, and then turn to the case of growing domains. In both cases, we formally derive the corresponding deterministic continuum limit and show that that there is an excellent quantitative match between the spatial patterns produced by the stochastic individual-based model and its deterministic continuum counterpart, when sufficiently large numbers of cells are considered. This paper is intended to present a proof of concept for the ideas underlying the modelling framework, with the aim to then apply the related methods to the study of specific patterning and morphogenetic processes in the future.
Close

Details

Original languageEnglish
Pages (from-to)7442-7479
JournalMathematical Biosciences and Engineering
Volume17
Issue number6
DOIs
Publication statusPublished - 29 Oct 2020

    Research areas

  • Cell pattern formation, Turing patterns, Hybrid models, Individual-based models, Reaction-diffusion systems

Discover related content
Find related publications, people, projects and more using interactive charts.

View graph of relations

Related by author

  1. Bridging the gap between individual-based and continuum models of growing cell populations

    Chaplain, M. A. J., Lorenzi, T. & Macfarlane, F. R., Jan 2020, In: Journal of Mathematical Biology. 80, 1-2, p. 343-371

    Research output: Contribution to journalArticlepeer-review

  2. Quantitative predictive modelling approaches to understanding rheumatoid arthritis: a brief review

    Macfarlane, F. R., Chaplain, M. A. J. & Eftimie, R., 27 Dec 2019, In: Cells. 9, 1, 26 p., 74.

    Research output: Contribution to journalReview articlepeer-review

  3. Development of a coupled simulation toolkit for computational radiation biology based on Geant4 and CompuCell3D

    Liu, R., Higley, K., Swat, M., Chaplain, M. A. J., Powathil, G. & Glazier, J., 18 Dec 2020, In: Physics in Medicine and Biology.

    Research output: Contribution to journalArticlepeer-review

Related by journal

  1. Parameter sensitivity analysis for biochemical reaction networks

    Minas, G. & Rand, D. A., 6 May 2019, In: Mathematical Biosciences and Engineering. 16, 5, p. 3965-3987 23 p.

    Research output: Contribution to journalArticlepeer-review

  2. Emergence of spatial patterns in a mathematical model for the co-culture dynamics of epithelial-like and mesenchymal-like cells

    Delitala, M. & Lorenzi, T., Feb 2017, In: Mathematical Biosciences and Engineering. 14, 1, p. 79-93

    Research output: Contribution to journalArticlepeer-review

  3. Spatio-temporal models of synthetic genetic oscillators

    Macnamara, C. K. & Chaplain, M. A. J., Feb 2017, In: Mathematical Biosciences and Engineering. 14, 1, p. 249-262 14 p.

    Research output: Contribution to journalArticlepeer-review

  4. Visualisation of the numerical solution of partial differential equation systems in three space dimensions and its importance for mathematical models in biology

    Enderling, H., Anderson, A. R. A., Chaplain, M. A. J. & Rowe, G. W. A., Oct 2006, In: Mathematical Biosciences and Engineering. 3, 4, p. 571-582 12 p.

    Research output: Contribution to journalArticlepeer-review

ID: 270703973

Top