A reaction–diffusion wave on the skin of the marine angelfish Pomacanthus

Kondo, Shigeru; Asai, Rihito

doi:10.1038/376765a0

Cited by 757 publications

(591 citation statements)

References 12 publications

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“…1 (called frequency-doubling when transformed to the unit interval). Note that the insertion of peaks here is consistent with the description of Kondo and Asai (1995) for the growing angelfish. In the case of general nonuniform domain growth, (X, t) is a non-separable function of X and t: the convection term is not removed under a uniform transformation and may therefore influence pattern formation in the system.…”

Section: Uniform Domain Growthsupporting

confidence: 86%

“…The numerical computations of Kondo and Asai (1995), subsequently confirmed and extended by Painter et al (1999), show that the sequence of patterns observed on the angelfish is reproduced by reaction-diffusion schemes on growing domains, where the pattern wavelength is conserved when the domain length is doubled. This sequence of periodic patterns has been called mode-doubling.…”

Section: Introductionmentioning

confidence: 64%

“…A reaction-diffusion mechanism has been proposed by Kondo and Asai (1995) to account for the changes in pigmentation patterns on the marine angelfish Pomacanthus. In the skin of these fish the pattern changes dynamically during growth of the animal, unlike mammalian coat markings which simply enlarge in proportion to the body size.…”

Section: Introductionmentioning

confidence: 99%

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Pattern Formation in Reaction–Diffusion Models with Nonuniform Domain Growth

Crampin

Hackborn²,

Maini

2002

Bulletin of Mathematical Biology

165

192

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Recent examples of biological pattern formation where a pattern changes qualitatively as the underlying domain grows have given rise to renewed interest in the reaction-diffusion (Turing) model for pattern formation. Several authors have now reported studies showing that with the addition of domain growth the Turing model can generate sequences of patterns consistent with experimental observations. These studies demonstrate the tendency for the symmetrical splitting or insertion of concentration peaks in response to domain growth. This process has also been suggested as a mechanism for reliable pattern selection. However, thus far authors have only considered the restricted case where growth is uniform throughout the domain. In this paper we generalize our recent results for reaction-diffusion pattern formation on growing domains to consider the effects of spatially nonuniform growth. The purpose is twofold: firstly to demonstrate that the addition of weak spatial heterogeneity does not significantly alter pattern selection from the uniform case, but secondly that sufficiently strong nonuniformity, for example where only a restricted part of the domain is growing, can give rise to sequences of patterns not seen for the uniform case, giving a further mechanism for controlling pattern selection. A framework for modelling is presented in which domain expansion and boundary * Author to whom correspondence should be addressed. E-mail: crampin@maths.ox.ac. (apical) growth are unified in a consistent manner. The results have implications for all reaction-diffusion type models subject to underlying domain growth.

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Section: Uniform Domain Growthsupporting

confidence: 86%

Section: Introductionmentioning

confidence: 64%

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Pattern Formation in Reaction–Diffusion Models with Nonuniform Domain Growth

Crampin

Hackborn²,

Maini

2002

Bulletin of Mathematical Biology

165

192

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“…In the model domain growth plays a central role in establishing the order in which tooth precursors appear. Recently, stemming from work by Kondo and Asai (1995), various authors have considered the effect of growth in numerical simulations of reaction-diffusion models for the skin patterns on species of fish during their development from juvenile to adult forms (Varea et al, 1997;Painter et al, 1999). Here the pattern change is commensurate with growth and shape change, and many different pattern behaviours are observed in response to the growing domain before the final domain geometry is achieved.…”

Section: 2mentioning

confidence: 99%

Reaction and Diffusion on Growing Domains: Scenarios for Robust Pattern Formation

Crampin¹,

Gaffney²,

Maini³

1999

Bulletin of Mathematical Biology

333

388

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We investigate the sequence of patterns generated by a reaction-diffusion system on a growing domain. We derive a general evolution equation to incorporate domain growth in reaction-diffusion models and consider the case of slow and isotropic domain growth in one spatial dimension. We use a self-similarity argument to predict a frequency-doubling sequence of patterns for exponential domain growth and we find numerically that frequency-doubling is realized for a finite range of exponential growth rate. We consider pattern formation under different forms for the growth and show that in one dimension domain growth may be a mechanism for increased robustness of pattern formation.

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“…In contrast, however, the stripe patterns on fish skin, studied by Kondo and Asai (1995), change shape as fish grow. The number of stripes tends to increase with body size, but the width of each stripe and the distance between them remains almost unchanged.…”

Section: Introductionmentioning

confidence: 98%

Origin of directionality in the fish stripe pattern

Shoji

Mochizuki

Iwasa

et al. 2003

Developmental Dynamics

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The formation of stripe patterns in animal skin has been explained by the reaction-diffusion (RD) system, a hypothetical chemical reaction proposed by A. Turing. Although animal stripes usually have directionality, the RD model alone cannot explain how the direction is specified. To investigate the mechanism regulating the direction of stripes, we studied stripe pattern formation in two species of Genicanthus during sexual conversion. These species share almost identical morphologic properties, except for their stripe direction. In both species, spots transiently arise at random positions and then combine and rearrange to form directional stripes. Computational analysis has shown that diffusion anisotropy is very effective at specifying the direction of stripes formed by the RD system. Model simulations reproduce the transient dynamics of directional pattern formation observed in fish as well as the resulting stripes. In cases where the magnitude and direction of diffusion anisotropy of the substances are identical, the resulting stripes are not directional. However, if they differ, stripes become directional. As only a small difference in anisotropy is required for this effect, any kind of structure with directional conformation might cause a marked change in stripe direction. Scales are the most likely candidate structure for generating anisotropic interactions in skin. Developmental Dynamics 226:627-633, 2003.

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A reaction–diffusion wave on the skin of the marine angelfish Pomacanthus

Cited by 757 publications

References 12 publications

Pattern Formation in Reaction–Diffusion Models with Nonuniform Domain Growth

Pattern Formation in Reaction–Diffusion Models with Nonuniform Domain Growth

Reaction and Diffusion on Growing Domains: Scenarios for Robust Pattern Formation

Origin of directionality in the fish stripe pattern

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