Modern perspectives on near-equilibrium analysis of Turing systems

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’.

Geometry-induced patterns through mechanochemical coupling

“…The system considered here involves a dynamic interplay between the local total density and the metric, which leads to (self-organized) non-uniform growth rates and thereby rich pattern-forming dynamics. Classical Turing models have also been studied for non-uniformly growing lines [71], where, for example, one segment of the line is assumed to grow at a different rate from that of the remaining portion, which can effectively be described as a piecewise uniformly growing line. It was found that this leads to asymmetric pattern formation and peak-splitting of patterns, which can be interpreted as regional patterns in analogy to our work.…”

“…Notably, these classical Turing models have been mainly studied in the quasi-stationary limit [70,71], where one assumes that the pattern-forming dynamics unfolds on a much smaller time scale than domain growth. While such an assumption is reasonable at larger scales, such as in the context of morphogenesis, the time scales of growth and pattern formation are generally not far apart in an intracellular context.…”

Patterning, From Conifers to Consciousness: Turing’s Theory and Order From Fluctuations

“…A final, perennial objection to Turing’s reaction-diffusion mechanism is a supposed lack of robustness, that pattern formation depends on the parameters being adjusted within a narrow range. While this is true to a degree of 2-component models, more recent work has shown that having more components, especially if some are non-diffusing ( Marcon et al, 2016 ; Diego et al, 2018 ; Landge et al, 2020 ; Krause et al, 2021 ), and discrete rather than continuous systems ( Leyshon et al, 2021 ), yields models far more robust than previously supposed possible, and there is now a better understanding of how pattern stability is maintained in the non-linear regime ( Subramanian and Murray, 2021 ). The burden of past misconceptions concerning kinetic theories has thus now, in large part, been removed.…”

Pattern Propagation Driven by Surface Curvature

“…tern, which is static on a flat plane, was assumed to remain static irrespective of the surface geometry. However, this assumption has not been validated so far [31].…”