Complex morphogenesis of surfaces: theory and experiment on coupling of reaction–diffusion patterning to growth

Harrison, Lionel G.; Wehner, Stephan; Holloway, David M.

doi:10.1039/b103246c

Cited by 42 publications

(49 citation statements)

References 15 publications

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“…For all three models, the majority of the final patterns were for dichotomous branching, but a substantial minority (e.g. about 20% for the Brusselator model) ended with annular pattern, and one model gave another variant also [tabulation, (Harrison et al, 2001)]. These results led to the bifurcation analysis reported here for two of the models.…”

Section: Introductionmentioning

confidence: 59%

“…The analysis described here leads to basins of attraction indicating that, starting from random noise about the patternless steady state, patterning events should give dichotomous branching with 84% probability and annular pattern with 16% probability. Numerical solution of the dynamics (Harrison et al, 2001) further from the Turing instability boundary gave, in a set of 103 computations, 82% branched and 18% annular.…”

Section: Pattern Formation In the Brusselator Modelmentioning

confidence: 99%

“…in infinite two-dimensional planar surfaces favouring stripes over spots where the Brusselator favours spots. [Computations with radii optimal for l > 3, reported in (Harrison et al, 2001), showed that, for the hyperchirality model, complex patterns appear that seem to be showing the striping tendency. For l = 2, but on the complete sphere, see (Hunding and Billing, 1981;Hunding and Brons, 1990).…”

Section: Introductionmentioning

confidence: 99%

“…As a start to three-dimensional modelling of patterning and growth, a study was done (Harrison et al, 2001) comprising a large number of computations (to apparent steady state) for each of three different reaction-diffusion models having substantially different nonlinearities, on a hemispherical shell domain of fixed size, optimal for l = 3. For all three models, the majority of the final patterns were for dichotomous branching, but a substantial minority (e.g.…”

Section: Introductionmentioning

confidence: 99%

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Reaction-Diffusion Models of Growing Plant Tips: Bifurcations on Hemispheres

Nagata

Harrison

Wehner

2003

Bulletin of Mathematical Biology

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We study two chemical models for pattern formation in growing plant tips. For hemisphere radius and parameter values together optimal for spherical surface harmonic patterns of index l = 3, the Brusselator model gives an 84% probability of dichotomous branching pattern and 16% of annular pattern, while the hyperchirality model gives 88% probability of dichotomous branching and 12% of annular pattern. The models are two-morphogen reaction-diffusion systems on the surface of a hemispherical shell, with Dirichlet boundary conditions. Bifurcation analysis shows that both models give possible mechanisms for dichotomous branching of the growing tips. Symmetries of the models are used in the analysis.

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Section: Introductionmentioning

confidence: 59%

Section: Pattern Formation In the Brusselator Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Reaction-Diffusion Models of Growing Plant Tips: Bifurcations on Hemispheres

Nagata

Harrison

Wehner

2003

Bulletin of Mathematical Biology

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“…Modelling the morphogenesis of green algae, Harrison and Kolář (1988) and Holloway and Harrison (1999) couple a Turing mechanism to domain growth. Their study is extended to two-and threedimensional domains (Harrison et al, 2002). However, they also include a feedback mechanism to the kinetics such that the diffusion-driven instability operates only within history-dependent boundaries separated by 'inert' regions for which there is no pattern formation.…”

Section: Reactant-controlled Growthmentioning

confidence: 99%

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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