Regular Article

A universal route to pattern formation in multicellular systems

¹ MACSI, Department of Mathematics and Statistics, University of Limerick, Limerick V94 T9PX, Ireland
² Department of Mathematics and naXys, Namur Institute for Complex Systems, University of Namur, rempart de la Vierge 8, 5000 Namur, Belgium
³ Dipartimento di Fisica e Astronomia, Università di Firenze, INFN and CSDC, Via Sansone 1, 50019 Sesto Fiorentino, Firenze, Italy
⁴ Mathematical Institute, University of Oxford, Woodstock Rd, OX2 6GG Oxford, UK

^a e-mail: malbor.asllani@unamur.be

Received: 21 April 2020
Received in final form: 31 May 2020
Published online: 13 July 2020

Abstract

A general framework for the generation of long wavelength patterns in multi-cellular (discrete) systems is proposed, which extends beyond conventional reaction-diffusion (continuum) paradigms. The standard partial differential equations of reaction-diffusion framework can be considered as a mean-field like ansatz which corresponds, in the biological setting, to sending to zero the size (or volume) of each individual cell. By relaxing this approximation and, provided a directionality in the flux is allowed for, we demonstrate here that instability leading to spatial pattern formation can always develop if the (discrete) system is large enough, namely, composed of sufficiently many cells, the units of spatial patchiness. The macroscopic patterns that follow the onset of the instability are robust and show oscillatory or steady state behavior.