Bespoke Turing Systems

Woolley, Thomas E.; Krause, Andrew L.; Gaffney, Eamonn A.

doi:10.1007/s11538-021-00870-y

Cited by 33 publications

(35 citation statements)

References 93 publications

(144 reference statements)

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“…Our main aim with these examples is to present interesting bifurcation problems and how they can be studied by numerical continuation and bifurcation. Other recent applications of pde2path are given in, e.g., [6,9,26,41,44,55,62,69,75,78,82,83,86]. In Table 1 we collect some acronyms frequently used in PDEs and bifurcation theory, and some specific for this paper.…”

Section: Introductionmentioning

confidence: 99%

Continuation and Bifurcation in Nonlinear PDEs – Algorithms, Applications, and Experiments

Uecker

2021

Jahresber. Dtsch. Math. Ver.

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Numerical continuation and bifurcation methods can be used to explore the set of steady and time–periodic solutions of parameter dependent nonlinear ODEs or PDEs. For PDEs, a basic idea is to first convert the PDE into a system of algebraic equations or ODEs via a spatial discretization. However, the large class of possible PDE bifurcation problems makes developing a general and user–friendly software a challenge, and the often needed large number of degrees of freedom, and the typically large set of solutions, often require adapted methods. Here we review some of these methods, and illustrate the approach by application of the package to some advanced pattern formation problems, including the interaction of Hopf and Turing modes, patterns on disks, and an experimental setting of dead core pattern formation.

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Section: Introductionmentioning

confidence: 99%

Continuation and Bifurcation in Nonlinear PDEs – Algorithms, Applications, and Experiments

Uecker

2021

Jahresber. Dtsch. Math. Ver.

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“…The set of recent large-scale analyses into TP design principles [6,20,22,23,55] (and similar attempts at elucidating other design principles [52,56,57]) can help to distil design principles of…”

Section: Discussionmentioning

confidence: 99%

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 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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Bespoke Turing Systems

Cited by 33 publications

References 93 publications

Continuation and Bifurcation in Nonlinear PDEs – Algorithms, Applications, and Experiments

Continuation and Bifurcation in Nonlinear PDEs – Algorithms, Applications, and Experiments

Turing pattern design principles and their robustness

Modern perspectives on near-equilibrium analysis of Turing systems

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