Abstract
In the nearly seven decades since the publication of Alan Turing’s work on morphogenesis, enormous progress has been made in understanding both the mathematical and biological aspects of his proposed reaction–diffusion theory. Some of these developments were nascent in Turing’s paper, and others have been due to new insights from modern mathematical techniques, advances in numerical simulations and extensive biological experiments. Despite such progress, there are still important gaps between theory and experiment, with many examples of biological patterning where the underlying mechanisms are still unclear. Here, we review modern developments in the mathematical theory pioneered by Turing, showing how his approach has been generalized to a range of settings beyond the classical two-species reaction–diffusion framework, including evolving and complex manifolds, systems heterogeneous in space and time, and more general reaction-transport equations. While substantial progress has been made in understanding these more complicated models, there are many remaining challenges that we highlight throughout. We focus on the mathematical theory, and in particular linear stability analysis of ‘trivial’ base states. We emphasize important open questions in developing this theory further, and discuss obstacles in using these techniques to understand biological reality.This article is part of the theme issue ‘Recent progress and open frontiers in Turing’s theory of morphogenesis’.
Highlights
Alan Turing’s Chemical Basis of Morphogenesis [1] has had profound impacts within mathematics, physics, chemistry and biology, especially in developmental and ecological settings [2], as evidenced by the other articles in this theme issue
This review will focus on these mathematical developments, concentrating on ideas which naturally generalize those present in Turing’s work, as well as on open problems using these ideas to elucidate principles of self-organization in developmental biology and beyond
Turing instabilities are often understood as a route towards symmetry-breaking of homogeneous states, and pattern formation is often viewed as the emergence of spatial structure from a uniform background or initial state
Summary
Alan Turing’s Chemical Basis of Morphogenesis [1] has had profound impacts within mathematics, physics, chemistry and biology, especially in developmental and ecological settings [2], as evidenced by the other articles in this theme issue. While Turing’s mathematical ideas about morphogenesis have been heavily extended, including a much wider class of models and exploring them via numerical simulation, one of the simplest approaches to analysing reaction-diffusion-like systems is the one employed by Turing: linear instability analysis of simple base states (which we refer to as near-equilibrium analysis) [3,4]. Spatial dynamics and related approaches study solutions of time-independent reaction–diffusion systems on R by thinking of the spatial variable as ‘time’, and considering the two-component system as a flow in R4, where one can make use of all of the machinery for such systems, such as numerical continuation [6,7,8]. We discuss both modelling and mathematical difficulties in going beyond two-species systems. Throughout, we hope to give a sense of how far we have come from Turing’s initial insight, and how much there is left to do in understanding these kinds of models, and their relationships to biological reality
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