St Andrews Research Repository

St Andrews University Home
View Item 
  •   St Andrews Research Repository
  • University of St Andrews Research
  • University of St Andrews Research
  • University of St Andrews Research
  • View Item
  •   St Andrews Research Repository
  • University of St Andrews Research
  • University of St Andrews Research
  • University of St Andrews Research
  • View Item
  •   St Andrews Research Repository
  • University of St Andrews Research
  • University of St Andrews Research
  • University of St Andrews Research
  • View Item
  • Login
JavaScript is disabled for your browser. Some features of this site may not work without it.

A hybrid discrete-continuum approach to model Turing pattern formation

Thumbnail
View/Open
Macfarlane_2020_A_hybrid_discrete_MBE_7442_CCBY.pdf (14.65Mb)
Date
29/10/2020
Author
Macfarlane, Fiona Ruth
Chaplain, Mark Andrew Joseph
Lorenzi, Tommaso
Keywords
Cell pattern formation
Turing patterns
Hybrid models
Individual-based models
Reaction-diffusion systems
QA Mathematics
QH301 Biology
T-NDAS
Metadata
Show full item record
Altmetrics Handle Statistics
Altmetrics DOI Statistics
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.
Citation
Macfarlane , F R , Chaplain , M A J & Lorenzi , T 2020 , ' A hybrid discrete-continuum approach to model Turing pattern formation ' , Mathematical Biosciences and Engineering , vol. 17 , no. 6 , pp. 7442-7479 . https://doi.org/10.3934/mbe.2020381
Publication
Mathematical Biosciences and Engineering
Status
Peer reviewed
DOI
https://doi.org/10.3934/mbe.2020381
ISSN
1547-1063
Type
Journal article
Rights
Copyright © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0).
Description
Funding: MAJC gratefully acknowledges support of EPSRC Grant No. EP/N014642/1 (EPSRC Centre for Multiscale Soft Tissue Mechanics–With Application to Heart & Cancer)
Collections
  • University of St Andrews Research
URI
http://hdl.handle.net/10023/20961

Items in the St Andrews Research Repository are protected by copyright, with all rights reserved, unless otherwise indicated.

Advanced Search

Browse

All of RepositoryCommunities & CollectionsBy Issue DateNamesTitlesSubjectsClassificationTypeFunderThis CollectionBy Issue DateNamesTitlesSubjectsClassificationTypeFunder

My Account

Login

Open Access

To find out how you can benefit from open access to research, see our library web pages and Open Access blog. For open access help contact: openaccess@st-andrews.ac.uk.

Accessibility

Read our Accessibility statement.

How to submit research papers

The full text of research papers can be submitted to the repository via Pure, the University's research information system. For help see our guide: How to deposit in Pure.

Electronic thesis deposit

Help with deposit.

Repository help

For repository help contact: Digital-Repository@st-andrews.ac.uk.

Give Feedback

Cookie policy

This site may use cookies. Please see Terms and Conditions.

Usage statistics

COUNTER-compliant statistics on downloads from the repository are available from the IRUS-UK Service. Contact us for information.

© University of St Andrews Library

University of St Andrews is a charity registered in Scotland, No SC013532.

  • Facebook
  • Twitter