Insights from chemical systems into Turing-type morphogenesis

Konow, Christopher; Dolnik, Miloš; Epstein, Irving R.

doi:10.1098/rsta.2020.0269

Cited by 10 publications

(13 citation statements)

References 120 publications

(233 reference statements)

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“…A difficulty alluded to above is both the identification of morphogens, and the specification of morphogen kinetics. In chemical models of Turing systems, we often have a strong indication that the chemical kinetics can be captured by specific nonlinear functions [ 118 ]. However, such certainty is much harder to obtain in biological systems, especially due to the difficulty in operationalizing the abstract notion of morphogen .…”

Section: Discussionmentioning

confidence: 99%

“…There is also a large literature on reaction–diffusion systems with strongly localized heterogeneities [ 109 , 110 ], and numerous studies exploring spatially heterogeneous reaction–diffusion systems in chemical settings [ 61 , 111 – 117 ]. See [ 118 ] in this theme issue for a modern review of chemical approaches to studying Turing systems.…”

Section: Heterogeneitymentioning

confidence: 99%

“…More recently, it was shown that this frequency-doubling can depend somewhat sensitively on the kind of growth rates involved, even in a one-dimensional domain growing uniformly [ 134 ]. Chemical experiments have also been carried out on growing systems using photosensitive reactions [ 118 ].

Figure 6 A period-doubling cascade in an exponentially growing reaction–diffusion system.…”

Section: Heterogeneitymentioning

confidence: 99%

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Modern perspectives on near-equilibrium analysis of Turing systems

Krause

Gaffney

Maini

et al. 2021

Phil. Trans. R. Soc. A.

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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’.

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Section: Discussionmentioning

confidence: 99%

Section: Heterogeneitymentioning

confidence: 99%

Figure 6 A period-doubling cascade in an exponentially growing reaction–diffusion system.…”

Section: Heterogeneitymentioning

confidence: 99%

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Modern perspectives on near-equilibrium analysis of Turing systems

Krause

Gaffney

Maini

et al. 2021

Phil. Trans. R. Soc. A.

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“…It was mostly neglected, with only a few critical papers discussing it in the decades following the publication, in 1952, of The chemical basis of morphogenesis [ 2 ], though interest from both mathematicians and biologists increased in the 1970s following preliminary investigations of possible biological morphogens, as well as the increased use of computers to visualize Turing’s predicted patterns. In the late 1980s and early 1990s, chemical experiments demonstrated this pattern-forming mechanism in a laboratory setting [ 3 ]. The past two decades have seen an increase in mathematical, chemical and biological work extending, or relating to, Turing’s ideas on morphogenesis.…”

mentioning

confidence: 99%

“…Konow et al . [ 3 ] provide a comprehensive review of the theoretical and experimental literature on Turing patterns in chemical systems. Unlike the complexity and uncertainty of in vivo systems, chemical systems can exploit extremely well-characterized reaction kinetics and diffusion rates to design and engineer a range of pattern formation scenarios, from the interaction of coupled layers to growing domains using photosensitive reaction kinetics.…”

mentioning

confidence: 99%

Introduction to ‘Recent progress and open frontiers in Turing’s theory of morphogenesis’

Krause

Gaffney

Maini

et al. 2021

Phil. Trans. R. Soc. A.

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Elucidating pattern forming processes is an important problem in the physical, chemical and biological sciences. Turing's contribution, after being initially neglected, eventually catalysed a huge amount of work from mathematicians, physicists, chemists and biologists aimed towards understanding how steady spatial patterns can emerge from homogeneous chemical mixtures due to the reaction and diffusion of different chemical species. While this theory has been developed mathematically and investigated experimentally for over half a century, many questions still remain unresolved. This theme issue places Turing's theory of pattern formation in a modern context, discussing the current frontiers in foundational aspects of pattern formation in reaction-diffusion and related systems. It highlights ongoing work in chemical, synthetic and developmental settings which is helping to elucidate how important Turing's mechanism is for real morphogenesis, while highlighting gaps that remain in matching theory to reality. The theme issue also surveys a variety of recent mathematical research pushing the boundaries of Turing's original theory to more realistic and complicated settings, as well as discussing open theoretical challenges in the analysis of such models. It aims to consolidate current research frontiers and highlight some of the most promising future directions. This article is part of the theme issue ‘Recent progress and open frontiers in Turing’s theory of morphogenesis’.

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Concentration-Dependent Domain Evolution in Reaction–Diffusion Systems

2023

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Pattern formation has been extensively studied in the context of evolving (time-dependent) domains in recent years, with domain growth implicated in ameliorating problems of pattern robustness and selection, in addition to more realistic modelling in developmental biology. Most work to date has considered prescribed domains evolving as given functions of time, but not the scenario of concentration-dependent dynamics, which is also highly relevant in a developmental setting. Here, we study such concentration-dependent domain evolution for reaction–diffusion systems to elucidate fundamental aspects of these more complex models. We pose a general form of one-dimensional domain evolution and extend this to N-dimensional manifolds under mild constitutive assumptions in lieu of developing a full tissue-mechanical model. In the 1D case, we are able to extend linear stability analysis around homogeneous equilibria, though this is of limited utility in understanding complex pattern dynamics in fast growth regimes. We numerically demonstrate a variety of dynamical behaviours in 1D and 2D planar geometries, giving rise to several new phenomena, especially near regimes of critical bifurcation boundaries such as peak-splitting instabilities. For sufficiently fast growth and contraction, concentration-dependence can have an enormous impact on the nonlinear dynamics of the system both qualitatively and quantitatively. We highlight crucial differences between 1D evolution and higher-dimensional models, explaining obstructions for linear analysis and underscoring the importance of careful constitutive choices in defining domain evolution in higher dimensions. We raise important questions in the modelling and analysis of biological systems, in addition to numerous mathematical questions that appear tractable in the one-dimensional setting, but are vastly more difficult for higher-dimensional models.

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Insights from chemical systems into Turing-type morphogenesis

Cited by 10 publications

References 120 publications

Modern perspectives on near-equilibrium analysis of Turing systems

Modern perspectives on near-equilibrium analysis of Turing systems

Introduction to ‘Recent progress and open frontiers in Turing’s theory of morphogenesis’

Concentration-Dependent Domain Evolution in Reaction–Diffusion Systems

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