A universal route to pattern formation in multicellular systems
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THE EUROPEAN PHYSICAL JOURNAL B
Regular Article
A universal route to pattern formation in multicellular systems Malbor Asllani 1,2,a , Timoteo Carletti 2 , Duccio Fanelli 3 , and Philip K. Maini 4 1 2
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MACSI, Department of Mathematics and Statistics, University of Limerick, Limerick V94 T9PX, Ireland Department of Mathematics and naXys, Namur Institute for Complex Systems, University of Namur, rempart de la Vierge 8, 5000 Namur, Belgium Dipartimento di Fisica e Astronomia, Universit` a di Firenze, INFN and CSDC, Via Sansone 1, 50019 Sesto Fiorentino, Firenze, Italy Mathematical Institute, University of Oxford, Woodstock Rd, OX2 6GG Oxford, UK Received 21 April 2020 / Received in final form 31 May 2020 Published online 13 July 2020 c EDP Sciences / Societ`
a Italiana di Fisica / Springer-Verlag GmbH Germany, part of Springer Nature, 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.
1 Introduction Self-organization, the ability of a system of microscopically interacting entities to shape macroscopically ordered structures, is ubiquitous in Nature. Spatio-temporal patterns are observed in a plethora of applications, encompassing different fields and scales. Examples of emerging patterns are the spots and stripes on the coat or skin of animals [1,2], the spatial distribution of vegetation in arid areas [3], the organization of colonies of insects in host-parasitoid systems [4] and the architecture of large complex ecosystems [5]. In the early 1950s, Alan Turing laid down, in a seminal paper [6], the mathematical basis of pattern formation, the discipline that aims at explaining the richness and diversity of forms displayed in Nature. Turing’s idea paved the way for a whole field of investigation and fertilized a cross-disciplinary perspective to yield a universally accepted paradigm of self-organization [7]. The onset of pattern formation on a bounded spatial domain originates from the loss of stability of an unpatterned equilibrium. To start with, Turing proposed a minimal model composed of at least two chemicals, hereby termed species. The species were assumed to diffuse across an ensemble of cells, adjacent to each other and organized in a closed ring, as depicted in Figure 1a. One of the species should
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