Hysteresis-driven pattern formation in reaction-diffusion-ODE systems

Köthe, Alexandra; Marciniak–Czochra, Anna; Izumi, Teruo

doi:10.3934/dcds.2020170

Cited by 24 publications

(26 citation statements)

References 26 publications

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“…Therefore, we phenomenologically summarize HyWnt3 inhibition by a diffusive HyWnt3 inhibitor possibly including Sp5 and HmTSP. Similar dynamics may result from multistability in the intracellular signaling [73][74][75], or a negative feedback loop stemming from mechano-chemical interactions [66,76]. The reduced mathematical representation of the underlying mechanisms is sufficient for the purpose of this paper, since we focus on the role of HAS-7 in the pattern formation process.…”

Section: Mathematical Modelmentioning

confidence: 99%

The Wnt-specific astacin proteinase HAS-7 restricts head organizer formation in Hydra

et al. 2021

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Background The Hydra head organizer acts as a signaling center that initiates and maintains the primary body axis in steady state polyps and during budding or regeneration. Wnt/beta-Catenin signaling functions as a primary cue controlling this process, but how Wnt ligand activity is locally restricted at the protein level is poorly understood. Here we report a proteomic analysis of Hydra head tissue leading to the identification of an astacin family proteinase as a Wnt processing factor. Results Hydra astacin-7 (HAS-7) is expressed from gland cells as an apical-distal gradient in the body column, peaking close beneath the tentacle zone. HAS-7 siRNA knockdown abrogates HyWnt3 proteolysis in the head tissue and induces a robust double axis phenotype, which is rescued by simultaneous HyWnt3 knockdown. Accordingly, double axes are also observed in conditions of increased Wnt activity as in transgenic actin::HyWnt3 and HyDkk1/2/4 siRNA treated animals. HyWnt3-induced double axes in Xenopus embryos could be rescued by coinjection of HAS-7 mRNA. Mathematical modelling combined with experimental promotor analysis indicate an indirect regulation of HAS-7 by beta-Catenin, expanding the classical Turing-type activator-inhibitor model. Conclusions We show the astacin family protease HAS-7 maintains a single head organizer through proteolysis of HyWnt3. Our data suggest a negative regulatory function of Wnt processing astacin proteinases in the global patterning of the oral-aboral axis in Hydra.

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Section: Mathematical Modelmentioning

confidence: 99%

The Wnt-specific astacin proteinase HAS-7 restricts head organizer formation in Hydra

et al. 2021

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“…If the system exhibits Turing instability, the same mechanism which destabilizes the spatially homogeneous steady state also induces instability of all close-to-equilibrium patterns in the limit system [23,24,49,50]. The underlying loss of compactness results in emergence of a larger class of stationary patterns that may exhibit spatial irregularity not observed in the model with a small but strictly positive diffusion [24,25]. Consequently, the pattern selection process changes significantly.…”

Section: Far-from-equilibrium Patternsmentioning

confidence: 99%

“…This is to highlight the methodology, rather than the specific modelling context. Indeed, a pattern forming process as described here, can take place in a wide variety of models including systems of reaction–diffusion equations [20], non-local models of integro-differential equations [21,22], degenerated reaction–diffusion-ODE systems [23–25] or fourth-order partial differential equation models accounting for evolution of thin elastic structures [17,26]. Hence, it seems useful to adapt a rather abstract approach and describe the system as the evolution of a state variable ufalse(x,tfalse)∈double-struckRn determined by the evolution equation Δtu=Ffalse[ufalse], where the functional F encodes the evolution rules.…”

Section: Introductionmentioning

confidence: 99%

Beyond Turing: far-from-equilibrium patterns and mechano-chemical feedback

Veerman

Mercker

Marciniak–Czochra

2021

Phil. Trans. R. Soc. A.

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Turing patterns are commonly understood as specific instabilities of a spatially homogeneous steady state, resulting from activator–inhibitor interaction destabilized by diffusion. We argue that this view is restrictive and its agreement with biological observations is problematic. We present two alternatives to the classical Turing analysis of patterns. First, we employ the abstract framework of evolution equations to enable the study of far-from-equilibrium patterns. Second, we introduce a mechano-chemical model, with the surface on which the pattern forms being dynamic and playing an active role in the pattern formation, effectively replacing the inhibitor. We highlight the advantages of these two alternatives vis-à-vis the classical Turing analysis, and give an overview of recent results and future challenges for both approaches. This article is part of the theme issue ‘Recent progress and open frontiers in Turing’s theory of morphogenesis’.

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“…for x ∈ [0, 1], supplemented with homogeneous Neumann boundary condition for u was introduced in [18] in order to understand the role of bistability and hysteresis in pattern formation (see also [12] for more comments). We refer the reader to [13] for a construction of stable discontinuous stationary solutions of the one dimensional problem (1.7). Here, in Section 7, we apply the general theory from this work to extend results from [13] on the N-dimensional version of model (1.7).…”

Section: Introductionmentioning

confidence: 99%

Stable discontinuous stationary solutions to reaction-diffusion-ODE systems

Cygan¹,

Karch²,

Marciniak–Czochra³

et al. 2021

Preprint

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A general system of n ordinary differential equations coupled with one reaction-diffusion equation, considered in a bounded N -dimensional domain, with no-flux boundary condition is studied in a context of pattern formation. Such initial boundary value problems may have different types of stationary solutions. In our parallel work [Instability of all regular stationary solutions to reaction-diffusion-ODE systems (2021)], regular (i.e. sufficiently smooth) stationary solutions are shown to exist, however, all of them are unstable. The goal of this work is to construct discontinuous stationary solutions to general reaction-diffusion-ODE systems and to find sufficient conditions for their stability.

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Hysteresis-driven pattern formation in reaction-diffusion-ODE systems

Cited by 24 publications

References 26 publications

The Wnt-specific astacin proteinase HAS-7 restricts head organizer formation in Hydra

The Wnt-specific astacin proteinase HAS-7 restricts head organizer formation in Hydra

Beyond Turing: far-from-equilibrium patterns and mechano-chemical feedback

Stable discontinuous stationary solutions to reaction-diffusion-ODE systems

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