Mode Transitions in a Model Reaction–Diffusion System Driven by Domain Growth and Noise

Barrass, Iain; Crampin, Edmund J.; Maini, Philip K.

doi:10.1007/s11538-006-9106-8

Cited by 43 publications

(51 citation statements)

References 17 publications

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“…We note that asymmetric spot splitting sequences have been observed on a growing domain in two-dimensional space [17]. Barrass et al [18] have clarified how the breakdown of the mode-doubling sequence occurs by computing a global bifurcation diagram for stripe solutions. However, in the work of [18], as well as in the other literature cited above, no analysis of the alternate splitting that is typically observed on slowly growing domains has been carried out, and the general mechanism of symmetry-breaking splitting dynamics, including the alternate splitting process, remains an open problem.…”

Section: Introductionmentioning

confidence: 78%

“…Barrass et al [18] have clarified how the breakdown of the mode-doubling sequence occurs by computing a global bifurcation diagram for stripe solutions. However, in the work of [18], as well as in the other literature cited above, no analysis of the alternate splitting that is typically observed on slowly growing domains has been carried out, and the general mechanism of symmetry-breaking splitting dynamics, including the alternate splitting process, remains an open problem. It should be emphasized that it is quite difficult to predict the behavior of solutions based only on information concerning each fixed domain size, as illustrated by Fig.1.…”

Section: Introductionmentioning

confidence: 99%

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A mathematical mechanism for instabilities in stripe formation on growing domains

Ueda

Nishiura

2012

Physica D: Nonlinear Phenomena

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A cascade process involving stripe splitting in reaction-diffusion systems with isotropically growing one-dimensional domains is studied. Such cascades, propagating from a smaller domain to a larger domain, have been proposed as an answer to the criticism that the Turing mechanism lacks robustness because many stable patterns can coexist on a large domain and, therefore, the final state is very sensitive to the initial conditions. In contrast, if the system starts with a small domain, very few stable patterns are possible, which limits the sensitivity to the initial conditions. In order to show the validity and limitations of this scenario, we clarify the underlying mathematical mechanism that drives various types of stripe-splitting via a reduction from partial differential equations to ordinary differential equations, as well as investigating global arrangements of the set of n-mode stripe branches with D n -symmetry of the stripe locations. The mathematical simplification above allows us to reveal how each n-mode stripe branch is destabilized as the domain grows and to characterize the associated eigenprofiles that actually determine the manner of splitting at the infinitesimal level. We find that all the D n -symmetry-breaking instabilities typically occur simultaneously up to leading order before the saddle-node point of the n-mode stripe branch is reached. The instability with the largest real part is of the alternate type: every other peak splits at the infinitesimal level. A symmetry-preserving instability appears at the saddle-node point, which drives the simultaneous type of splitting, i.e., mode-doubling. Due to competition between these two types of instabilities, the problem depends subtly on the growth speed. Alternate splitting typically arises for slow growth and simultaneous splitting for fast growth. For intermediate growth rates, the manner of splitting becomes mixed and sensitive to fluctuations.

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Section: Introductionmentioning

confidence: 78%

Section: Introductionmentioning

confidence: 99%

A mathematical mechanism for instabilities in stripe formation on growing domains

Ueda

Nishiura

2012

Physica D: Nonlinear Phenomena

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“…1 D and H). Another specific property of the stationary pattern formed by the RD mechanism is the dynamic regulation of the pattern (34). Therefore, if the original stripe width was artificially changed, the adjacent stripes are expected to orient themselves to recover the original width.…”

Section: Experiments 1: Regeneration Of Stripes Without An Inherent Prmentioning

confidence: 99%

“…A characteristic property of the stationary patterns generated by RD mechanism is the ability to self-regulate the pattern and their robustness against perturbations (34). For instance, in numerical simulations, if enlargement of the field is introduced by increasing the number of cells in which the reaction occurs, the stripes or spots do not enlarge accordingly; in fact, the intrinsic size is retained by the splitting or insertion of new stripes or spots (1).…”

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confidence: 99%

Pattern regulation in the stripe of zebrafish suggests an underlying dynamic and autonomous mechanism

Yamaguchi¹,

Yoshimoto²,

Kondo³

2007

Proc. Natl. Acad. Sci. U.S.A.

181

191

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The mechanism by which animal markings are formed is an intriguing problem that has remained unsolved for a long time. One of the most important questions is whether the positional information for the pattern formation is derived from a covert prepattern or an autonomous mechanism. In this study, using the zebrafish as the model system, we attempted to answer this classic question. We ablated the pigment cells in limited areas of zebrafish skin by using laser irradiation, and we observed the regeneration of the pigmentation pattern. Depending on the area ablated, different patterns regenerated in a specific time course. The regenerated patterns and the transition of the stripes during the regeneration process suggest that pattern formation is independent of the prepattern; furthermore, pattern formation occurs by an autonomous mechanism that satisfies the condition of ''local self-enhancement and long-range inhibition.'' Because the zebrafish is the only striped animal for which detailed molecular genetic studies have been conducted, our finding will facilitate the identification of the molecular and cellular mechanisms that underlie skin pattern formation.local self-enhancement and long-range inhibition ͉ pigment patterns ͉ reaction-diffusion mechanism A nimals exhibit a wide spectrum of pigment patterns. Although it has been attracting considerable interest for long time (1, 2), the mechanism underlying the emergence of these patterns remains largely unknown.In his pioneering paper in 1952 (3), Alan Turing showed that various spatial patterns arise in a system in which two substances react and diffuse at different rates. This system is now commonly termed as the reaction-diffusion (RD) ¶ system, and it has become the standard for theoretically studying of biological pattern formation (4, 5). To form the stationary patterns, the system needs to satisfy a necessary condition, ''local selfenhancement and long-range inhibition'' (6-9). Recent molecular genetic experiments corroborated the existence of such condition during the events in biological pattern formation (9-11).Using computer simulations, theoretical studies have shown that a variety of characteristic animal skin patterns can be reproduced by RD mechanism (12-15). Nevertheless, the similarity of the patterns made by the computer simulation is not enough to prove that the RD mechanism underlies the pigment pattern (16,17). Therefore, it is desired to identify the molecular network that controls the pigment pattern formation.Recent studies have focused on the stripe pattern of zebrafish as the model system to investigate the mechanism of animal pigmentation (18)(19)(20). Many genes related to the pigmentation of zebrafish have been identified (21-27), and investigations on the functions of these genes are underway (28-33). However, a strong theoretical premise is necessary to develop a complex molecular network that can explain the pattern formation. If it is shown that the RD mechanism underlies the stripes of zebrafish, the mechanism can be used as the ...

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“…In the simpler context of a one-dimensional domain, the stability problem for equilibrium spike patterns for (1.1) with b = 0 has been studied analytically in [23] and [47]. Moreover, in certain parameter regimes it has been shown numerically in [5], [10], and [19], that spike patterns for (1.1) can undergo self-replication in a slowly growing one-dimensional domain.…”

Section: Introductionmentioning

confidence: 99%

Spot Self-Replication and Dynamics for the Schnakenburg Model in a Two-Dimensional Domain

2008

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The dynamical behavior of multi-spot solutions in a two-dimensional domain Ω is analyzed for the two-component Schnakenburg reaction-diffusion model in the singularly perturbed limit of small diffusivity ε for one of the two components. In the limit ε → 0, a quasi-equilibrium spot pattern in the region away from the spots is constructed by representing each localized spot as a logarithmic singularity of unknown strength Sj for j = 1, . . . , K at unknown spot locations xj ∈ Ω for j = 1, . . . , K. A formal asymptotic analysis, which has the effect of summing infinite logarithmic series in powers of −1/ log ε, is then used to derive an ODE differential algebraic system (DAE) for the collective coordinates Sj and xj for j = 1, . . . , K, which characterizes the slow dynamics of a spot pattern. This DAE system involves the Neumann Green's function for the Laplacian. By numerically examining the stability thresholds for a single spot solution, a specific criterion in terms of the source strengths Sj , for j = 1, . . . , K, is then formulated to theoretically predict the initiation of a spot-splitting event. The analytical theory is illustrated for spot patterns in the unit disk and the unit square, and is compared with full numerical results computed directly from the Schnakenburg model.

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Mode Transitions in a Model Reaction–Diffusion System Driven by Domain Growth and Noise

Cited by 43 publications

References 17 publications

A mathematical mechanism for instabilities in stripe formation on growing domains

A mathematical mechanism for instabilities in stripe formation on growing domains

Pattern regulation in the stripe of zebrafish suggests an underlying dynamic and autonomous mechanism

Spot Self-Replication and Dynamics for the Schnakenburg Model in a Two-Dimensional Domain

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