A Comprehensive Network Atlas Reveals That Turing Patterns Are Common but Not Robust

Scholes, Natalie S.; Schnoerr, David; Isalan, Mark; Stumpf, Michael

doi:10.1016/j.cels.2019.09.010

Cited by 15 publications

(54 citation statements)

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“…For the analysis of multi-model inference from mechanistic models we can distill two points: (i) ensembles of mechanistic models that are reasonably defined (Aijo & Bonneau, 2016) (i.e. their construction incorporates any available mechanistic insights; duplicate models are avoided; the model is predictive and can be used to generate data that can be com-pared with real observations/data) can be combined with the aid of model selection criteria or Bayesian posterior model probabilities with relative ease and safety; (ii) the inclusion of "nuisance models" can hamper ensemble approaches if they come to predominate the model universe M. Such situations could become more likely as model spaces are explored exhaustively (Scholes et al, 2019) or automatically (Sunnåker et al, 2014). Because of the formalism connecting different model selection criteria, Eqn.…”

Section: Discussionmentioning

confidence: 99%

Multi-Model and Network Inference Based on Ensemble Estimates: Avoiding the Madness of Crowds

Stumpf

2019

Preprint

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The development of mathematical models of biological systems has largely relied on a mix of biological intuition, mathematical expediency, and comparisons with data. Models are hard to develop and hard to validate. Recent progress in theoretical systems biology, applied mathematics and computational statistics has, however, led to a proliferation of models and methods that allow us to compare quantitatively the performance of different candidate models at describing a particular biological system or process. Model selection has been applied with great success to problems where a small number -typically less than 10 -of models are compared, but recently studies have started to consider thousands and even millions of candidate models. As formal computational and automated systems increasingly aim to replace or augment biological intuition this trend is likely to continue. Often no single model is chosen as the best, but a large number of models is found to be compatible with data, and these are then often studied as an ensemble. In other situations several prediction algorithms are used to make ensemble predictions, for example, about the presence or absence of molecular interactions. Using a set of test-cases I outline when such ensemble estimators can be relied upon, and when they are likely to fail. If model or predictor ensembles are not carefully constructed and curated, predictions and estimates based on ensembles can perform worse than good individual predictors.

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Section: Discussionmentioning

confidence: 99%

Multi-Model and Network Inference Based on Ensemble Estimates: Avoiding the Madness of Crowds

Stumpf

2019

Preprint

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“…an activator subsystem), generalizing the notion of activator-inhibitor interactions. A variety of recent approaches have analysed thesem-component systems from the perspective of reaction networks, in order to determine motifs that permit pattern formation [184,185], and to explore questions of robustness in larger networks [186]. Given that the general theoretical problem is difficult to analyse, it is important to work with experimentalists on specific systems [124,187,188].…”

Section: More General Systemsmentioning

confidence: 99%

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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“…The first required condition is that the fixed point of the non-spatial system where the dynamics with diffusion are set to zero, dAdt=ffalse(A,Bfalse) and dBdt=gfalse(A,Bfalse), must be stable in the sense that the Jacobian matrix corresponding to equations (2.5) and (2.6), J=(normal∂fnormal∂Anormal∂fnormal∂Bnormal∂gnormal∂Anormal∂gnormal∂B), must have all eigenvalues with negative real part at the fixed point [10]. If the real part of the leading eigenvalue becomes positive when diffusion is turned on (DA,DB>0), that is, if the fixed points become unstable in the diffusive system, then we speak of a Turing instability—more precisely a Turing type 1 instability, as there are other types of instabilities which are discussed in the literature [11]—and we expect to see the formation of a spatially extended TP.…”

Section: Turing-pattern Mechanismmentioning

confidence: 99%

“…In the notation of the Gierer–Meinhardt model (2.1) and (2.2), a TP is observed for a set of reaction parameters θ0=false(ρA,ρB,kA,kB,μA,ρfalse) and a set of diffusion parameters θD=false(DA,DBfalse) when θ=false(θ0,θDfalse)∈Ωθ0, which is the subset of the whole permissible parameter space Ωθ for which TPs are observed for the model. Finding parameter regions Ωθ0 for which TPs are observed can be computationally demanding [11].…”

Section: Turing-pattern Mechanismmentioning

confidence: 99%

“…To estimate parametric robustness we can simply explore the size of the parameter space of the non-diffusion (intracellular) parameters for which we can obtain a single stable stationary state (in [11], we found that bi-stable and multi-stable systems require further analysis), which can become unstable for suitable choices of the diffusion parameters; the fraction of the parameter space for which this is the case is denoted by ρ0. The fraction of the diffusion parameter space that renders a stable state unstable once diffusion has been turned on, ρD is then a measure of the intercellular robustness.…”

Section: Robustness: a Pragmatic Perspectivementioning

confidence: 99%

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Turing pattern design principles and their robustness

Vittadello

Leyshon

Schnoerr

et al. 2021

Phil. Trans. R. Soc. A.

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Turing patterns have morphed from mathematical curiosities into highly desirable targets for synthetic biology. For a long time, their biological significance was sometimes disputed but there is now ample evidence for their involvement in processes ranging from skin pigmentation to digit and limb formation. While their role in developmental biology is now firmly established, their synthetic design has so far proved challenging. Here, we review recent large-scale mathematical analyses that have attempted to narrow down potential design principles. We consider different aspects of robustness of these models and outline why this perspective will be helpful in the search for synthetic Turing-patterning systems. We conclude by considering robustness in the context of developmental modelling more generally. This article is part of the theme issue ‘Recent progress and open frontiers in Turing’s theory of morphogenesis’.

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A Comprehensive Network Atlas Reveals That Turing Patterns Are Common but Not Robust

Cited by 15 publications

References 0 publications

Multi-Model and Network Inference Based on Ensemble Estimates: Avoiding the Madness of Crowds

Multi-Model and Network Inference Based on Ensemble Estimates: Avoiding the Madness of Crowds

Modern perspectives on near-equilibrium analysis of Turing systems

Turing pattern design principles and their robustness

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