On a Model Mechanism for the Spatial Patterning of Teeth Primordia in the Alligator

Kulesa, Paul M.; Cruywagen, G. C.; Lubkin, Sharon R.; Main, Patrick; Sneyd, James; Ferguson, M. W. J.; Murray, J. D.

doi:10.1006/jtbi.1996.0103

Cited by 53 publications

(44 citation statements)

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“…Here we develop a similar model from first principles, and find that the frequency-doubling sequence is a robust and pervasive solution behaviour. Kulesa et al (1996) have incorporated exponential domain growth into a model for the spatio-temporal sequence of initiation of tooth primordia in the Alligator mississippiensis. In the model domain growth plays a central role in establishing the order in which tooth precursors appear.…”

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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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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“…Nonetheless, domain growth is considered to be an important factor in numerous developmental scenarios, as exemplified by vertebrate limb development Shiota, 2000a, 2000b;Miura et al, 2006;Hentschel et al, 2004). It is also important in processes associated with biological growth, such as alligator tooth morphogenesis (Kulesa et al, 1996) and the reorganisation of fish-markings (Kondo and Asai, 1995). Domain growth is important theoretically due to another difficulty with the basic Turing model, namely that it exhibits sensitivity to fluctuations in the initial state of the system (Bard and Lauder, 1974;Bunow et al, 1980).…”

Section: Introductionmentioning

confidence: 99%

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

Lee

Gaffney

2010

Bull. Math. Biol.

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Turing's pattern formation mechanism exhibits sensitivity to the details of the initial conditions suggesting that, in isolation, it cannot robustly generate pattern within noisy biological environments. Nonetheless, secondary aspects of developmental selforganisation, such as a growing domain, have been shown to ameliorate this aberrant model behaviour. Furthermore, while in-situ hybridisation reveals the presence of gene expression in developmental processes, the influence of such dynamics on Turing's model has received limited attention. Here, we novelly focus on the Gierer-Meinhardt reaction diffusion system considering delays due the time taken for gene expression, while incorporating a number of different domain growth profiles to further explore the influence and interplay of domain growth and gene expression on Turing's mechanism. We find extensive pathological model behaviour, exhibiting one or more of the following: temporal oscillations with no spatial structure, a failure of the Turing instability and an extreme sensitivity to the initial conditions, the growth profile and the duration of gene expression. This deviant behaviour is even more severe than observed in previous studies of Schnakenberg kinetics on exponentially growing domains in the presence of gene expression (Gaffney and Monk in Bull. Math. Biol. 68:99-130, 2006). Our results emphasise that gene expression dynamics induce unrealistic behaviour in Turing's model for multiple choices of kinetics and thus such aberrant modelling predictions are likely to be generic. They also highlight that domain growth can no longer ameliorate the excessive sensitivity of Turing's mechanism in the presence of gene expression time delays. The above, extensive, pathologies suggest that, in the presence of gene expression, Turing's mechanism would generally require a novel and extensive secondary mechanism to control reaction diffusion patterning.

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“…Models such as these have been proposed to explain a wide variety of biological patterning processes, from the development of skin markings on the coats of mammals and fish [1,21] to the bifurcations of limbs [7] and the initiation of teeth in reptiles [17]. In these biological contexts, it is postulated that a reaction-diffusion system sets up a chemical "prepattern".…”

Section: Introductionmentioning

confidence: 99%

Pattern formation in spatially heterogeneous Turing reaction–diffusion models

Page

Maini

Monk

2003

Physica D: Nonlinear Phenomena

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The Turing reaction-diffusion model [Phil. Trans. R. Soc. 237 (1952) 37-72] for self-organised spatial pattern formation has been the subject of a great deal of study for the case of spatially homogeneous parameters. The case of parameters which vary spatially has received less attention. Here, we show that a simple step function heterogeneity in a kinetic parameter can lead to spatial pattern formation outside the classical Turing space parameter regime for patterning. This reduces the constraints on the model parameters, extending possible applications. Furthermore, it highlights the potential importance of boundaries during pattern formation.

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On a Model Mechanism for the Spatial Patterning of Teeth Primordia in the Alligator

Cited by 53 publications

References 0 publications

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

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

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

Pattern formation in spatially heterogeneous Turing reaction–diffusion models

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