Aberrant Behaviours of Reaction Diffusion Self-organisation Models on Growing Domains in the Presence of Gene Expression Time Delays

Lee, S. Seirin; Gaffney, Eamonn A.

doi:10.1007/s11538-010-9533-4

Cited by 25 publications

(19 citation statements)

References 51 publications

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“…Due to the delay, we must also define the populations for the time bins during the period −τ t < 0. Previously [26,54], these prior time points have simply been fixed at (small perturbations around) the steady state and, thus, products from the delayed equation are able to appear at the beginning of the simulation, t = 0. Physically, these initial conditions model an experiment that is kept at steady state during the initial time period, −τ t < 0.…”

Section: Methodsmentioning

confidence: 99%

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: Methodsmentioning

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%

“…The motivation for the study of this problem is the previous computational studies (cf. [7], [15], [16], [18]) of pattern formation in RD systems with a fixed time-delay in the reaction kinetics, which models time lags needed for the expression of genes. In contrast to the conclusion of our recent analysis of [6], where a time-delay was assumed in both the activator and inhibitor kinetics, we showed herein that a time-delay in only the activator kinetics has a stabilizing effect, in the sense that there is a larger region in parameter space where a spike solution is linearly stable than when there is no delayed reaction kinetics.…”

Section: 2mentioning

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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“…In addition, geneexpression time delays sensitize the timing of the onset of patterning to fluctuations in the initial conditions for ligand internalization models on growing domains [31]. Such delays in the onset and/or stabilization of pattern impose potentially severe constraints on the applicability of reaction -diffusion models that rely on regulated gene expression, regardless of domain growth.…”

Section: The Aberrant Patterning Lagmentioning

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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Aberrant Behaviours of Reaction Diffusion Self-organisation Models on Growing Domains in the Presence of Gene Expression Time Delays

Cited by 25 publications

References 51 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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