The Dynamics of Turing Patterns for Morphogen-Regulated Growing Domains with Cellular Response Delays

Lee, S. Seirin; Gaffney, Eamonn A.; Baker, Ruth E.

doi:10.1007/s11538-011-9634-8

Cited by 36 publications

(25 citation statements)

References 44 publications

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“…However, in the case in which developmental time is fast we are able to suggest that, although delays are able to produce aberrant effects in deterministic reaction-diffusion systems, there are certain specific conditions, e.g., glycolysis kinetics with reversible-ligand binding, where pathologies in delayed Turing systems are ameliorated. This contrasts with the conclusions previously drawn [26,28,54,59] emphasizing the structural sensitivity of delayed Turning patterning. In addition, there are also conditions under which stochasticity is able to act against the dominant dynamics, thereby impairing the pattern formation mechanism (see Sec.…”

Section: Discussioncontrasting

confidence: 99%

“…Due to these vastly different effects of noise it would be interesting to further investigate delayed stochastic systems by simulating them on growing domains. Although stochastic systems on growing domains may not be able to generate robust transition sequences, we might speculate that the noise could disrupt the homogeneous steady state and initiate pattern formation, thus removing the pattern breakdown which has been observed in the deterministic case [54,59].…”

Section: Discussionmentioning

confidence: 99%

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Effects of intrinsic stochasticity on delayed reaction-diffusion patterning systems

et al. 2012

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Cellular gene expression is a complex process involving many steps, including the transcription of DNA and translation of mRNA; hence the synthesis of proteins requires a considerable amount of time, from ten minutes to several hours. Since diffusion-driven instability has been observed to be sensitive to perturbations in kinetic delays, the application of Turing patterning mechanisms to the problem of producing spatially heterogeneous differential gene expression has been questioned. In deterministic systems a small delay in the reactions can cause a large increase in the time it takes a system to pattern. Recently, it has been observed that in undelayed systems intrinsic stochasticity can cause pattern initiation to occur earlier than in the analogous deterministic simulations. Here we are interested in adding both stochasticity and delays to Turing systems in order to assess whether stochasticity can reduce the patterning time scale in delayed Turing systems. As analytical insights to this problem are difficult to attain and often limited in their use, we focus on stochastically simulating delayed systems. We consider four different Turing systems and two different forms of delay. Our results are mixed and lead to the conclusion that, although the sensitivity to delays in the Turing mechanism is not completely removed by the addition of intrinsic noise, the effects of the delays are clearly ameliorated in certain specific cases.

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Section: Discussioncontrasting

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Effects of intrinsic stochasticity on delayed reaction-diffusion patterning systems

et al. 2012

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“…[7]). In an effort to understand the effect of such gene expression time-delays on pattern formation, various new two-component activator-inhibitor RD models with a fixed time delay in the reaction kinetics were developed based on various hypothethical sub-cellular gene expression processes in [7], [15], [16], [17] (see also the survey [18]). In these studies, pattern formation aspects of the time-delayed RD models were examined through a Turing-type linear stability around a homogeneous steady-state and from large-scale numerical computations of the PDE system on both fixed and slowly growing spatial domains.…”

Section: (Communicated By Shin-ichiro Ei)mentioning

confidence: 99%

A time-delay in the activator kinetics enhances the stability of a spike solution to the gierer-meinhardt model

Fadai¹,

Ward²,

Wei³

2018

Discrete &Amp; Continuous Dynamical Systems - B

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We study the spectrum of a new class of nonlocal eigenvalue problems (NLEPs) that characterize the linear stability properties of localized spike solutions to the singularly perturbed two-component Gierer-Meinhardt (GM) reaction-diffusion (RD) system with a fixed time-delay T in only the nonlinear autocatalytic activator kinetics. Our analysis of this model is motivated by the computational study of Seirin Lee et al. [Bull. Math. Bio., 72(8), (2010)] on the effect of gene expression time delays on spatial patterning for both the GM and some related RD models. For various limiting forms of the GM model, we show from a numerical study of the associated NLEP, together with an analytical scaling law analysis valid for large delay T , that a time-delay in only the activator kinetics is stabilizing in the sense that there is a wider region of parameter space where the spike solution is linearly stable than when there is no time delay. This enhanced stability behavior with a delayed activator kinetics is in marked contrast to the de-stabilizing effect on spike solutions of having a time-delay in both the activator and inhibitor kinetics. Numerical results computed from the RD system with delayed activator kinetics are used to validate the theory for the 1-D case. Introduction. For activator-inhibitor two-component reaction-diffusion (RD)systems, it is a well-known result, originating from Turing [21], that a small perturbation of a spatially uniform steady-state solution can become unstable when the diffusivity ratio is large enough. This initial instability then leads to the generation of large-amplitude stable spatial patterns. Although this mechanism for the development of spatially inhomogeneous patterns is well-understood, and has been applied to a broad range of specific RD systems (cf. [7]) and modeling scenarios on various spatial scales, what is less well-understood is the effect on pattern development of any time-delays in the reaction kinetics. Although there are now general results for the linear stability of spatially uniform steady-states under the effect of a

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“…However, the response delay between cell proliferation signalling and its subsequent effect upon growth are governed by the time scales of the cell cycle, i.e. of the order of a day or more, and such an extensive retardation also has the potential to profoundly affect the Turing instability in the same manner as gene expression [35]. In fact, when tissue growth is near-uniform and the influence of morphogen concentrations on tissue growth is small, domain growth response delays do not noticeably affect pattern formation.…”

Section: Reactant-controlled Domain Growth and Feedback Delay Dynamicsmentioning

confidence: 99%

Turing's model for biological pattern formation and the robustness problem

et al. 2012

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One of the fundamental questions in developmental biology is how the vast range of pattern and structure we observe in nature emerges from an almost uniformly homogeneous fertilized egg. In particular, the mechanisms by which biological systems maintain robustness, despite being subject to numerous sources of noise, are shrouded in mystery. Postulating plausible theoretical models of biological heterogeneity is not only difficult, but it is also further complicated by the problem of generating robustness, i.e. once we can generate a pattern, how do we ensure that this pattern is consistently reproducible in the face of perturbations to the domain, reaction time scale, boundary conditions and so forth. In this paper, not only do we review the basic properties of Turing's theory, we highlight the successes and pitfalls of using it as a model for biological systems, and discuss emerging developments in the area.

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The Dynamics of Turing Patterns for Morphogen-Regulated Growing Domains with Cellular Response Delays

Cited by 36 publications

References 44 publications

Effects of intrinsic stochasticity on delayed reaction-diffusion patterning systems

Effects of intrinsic stochasticity on delayed reaction-diffusion patterning systems

A time-delay in the activator kinetics enhances the stability of a spike solution to the gierer-meinhardt model

Turing's model for biological pattern formation and the robustness problem

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