Beyond Turing: far-from-equilibrium patterns and mechano-chemical feedback

Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences, vol. 379, pp. 20200278

Abstract

Turing patterns are commonly understood as specific instabilities of a spatially homogeneous steady state, resulting from activator–inhibitor interaction destabilized by diffusion. We argue that this view is restrictive and its agreement with biological observations is problematic. We present two alternatives to the classical Turing analysis of patterns. First, we employ the abstract framework of evolution equations to enable the study of far-from-equilibrium patterns. Second, we introduce a mechano-chemical model, with the surface on which the pattern forms being dynamic and playing an active role in the pattern formation, effectively replacing the inhibitor. We highlight the advantages of these two alternatives vis-à-vis the classical Turing analysis, and give an overview of recent results and future challenges for both approaches. This article is part of the theme issue ‘Recent progress and open frontiers in Turing’s theory of morphogenesis’.

Authors 3

1. Frits Veerman Aachen Mathematical Institute

   Leiden University

   Affiliation as printed

   University of Leiden, Mathematical Institute, Niels Bohrweg 1, Leiden 2333 CA, The Netherlands

2. Moritz Mercker

   Heidelberg University

   Affiliation as printed

   Institute for Applied Mathematics and Interdisciplinary Center for Scientific Computing (IWR), Heidelberg University, Im Neuenheimer Feld 205, Heidelberg 69120, Germany

3. Anna Marciniak‐Czochra corresponding

   Heidelberg University

   Affiliation as printed

   Institute for Applied Mathematics and Interdisciplinary Center for Scientific Computing (IWR), Heidelberg University, Im Neuenheimer Feld 205, Heidelberg 69120, Germany