Modeling pattern formation in hydra: a route to understanding essential steps in development

Meinhardt, Hans

doi:10.1387/ijdb.113483hm

Cited by 76 publications

(113 citation statements)

References 66 publications

(107 reference statements)

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“…Numerical computations have confirmed this behaviour [19]. This is in agreement with the pattern of multiple spikes covering the whole interval computed in our paper (see Figures 4-6).…”

Section: Resultssupporting

confidence: 91%

Stable spike clusters for the one-dimensional Gierer–Meinhardt system

Wei

Winter

2016

Eur. J. Appl. Math

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We consider the Gierer-Meinhardt system with precursor inhomogeneity and two small diffusivities in an intervalτ > 0 and τ is independent of ε.A spike cluster is the combination of several spikes which all approach the same point in the singular limit. We rigorously prove the existence of a steady-state spike cluster consisting of N spikes near a non-degenerate local minimum point t 0 of the smooth positive inhomogeneity μ(x), i.e. we assume that μ (t 0 ) = 0, μ (t 0 ) > 0 and we have μ(t 0 ) > 0. Here, N is an arbitrary positive integer. Further, we show that this solution is linearly stable. We explicitly compute all eigenvalues, both large (of order O (1)) and small (of order o (1)). The main features of studying the Gierer-Meinhardt system in this setting are as follows: (i) it is biologically relevant since it models a hierarchical process (pattern formation of small-scale structures induced by a pre-existing large-scale inhomogeneity); (ii) it contains three different spatial scales two of which are small: the O(1) scale of the precursor inhomogeneity μ(x), the O( √ D) scale of the inhibitor diffusivity and the O(ε) scale of the activator diffusivity; (iii) the expressions can be made explicit and often have a particularly simple form.

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“…Numerical computations have confirmed this behaviour [19]. This is in agreement with the pattern of multiple spikes covering the whole interval computed in our paper (see Figures 4-6).…”

Section: Resultssupporting

confidence: 91%

Stable spike clusters for the one-dimensional Gierer–Meinhardt system

Wei

Winter

2016

Eur. J. Appl. Math

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“…In this form, the Wnt molecules are much more mobile. Although it has not yet been shown that the differently processed Wnt molecules have different functions, from our model we would expect that the slowly diffusing variety is involved in the selfenhancement while the long-ranging variety undermines this self-enhancement [27]. The following set of equations describes a highly simplified version of this reaction scheme.…”

Section: The Wnt Pathway: the Conversion Of An Activator Into An Inhimentioning

confidence: 99%

Turing's theory of morphogenesis of 1952 and the subsequent discovery of the crucial role of local self-enhancement and long-range inhibition

2012

Self Cite

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In his pioneering work, Alan Turing showed that de novo pattern formation is possible if two substances interact that differ in their diffusion range. Since then, we have shown that pattern formation is possible if, and only if, a self-enhancing reaction is coupled with an antagonistic process of longer range. Knowing this crucial condition has enabled us to include nonlinear interactions, which are required to design molecularly realistic interactions. Different reaction schemes and their relation to Turing's proposal are discussed and compared with more recent observations on the molecular-genetic level. The antagonistic reaction may be accomplished by an inhibitor that is produced in the activated region or by a depletion of a component that is used up during the self-enhancing reaction. The autocatalysis may be realized by an inhibition of an inhibition. Activating molecules can be processed into molecules that have an inhibiting function; patterning of the Wnt pathway is proposed to depend on such a mechanism. Three-component systems, as discussed in Turing's paper, are shown to play a major role in the generation of highly dynamic patterns that never reach a stable state.

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“…Many of the bioinformatic approaches applied to precision medicine are currently concerned with diagnosis and treatment response prediction by molecular stratification of patients using genetic and more generally omic profiling. However, computational mathematical models are being generated to predict mode-of-action and responses-to-treatments (perturbations) not just at the molecular level but across all levels of biological organisation, including molecular, gene regulatory networks, signal transduction pathways and metabolic networks, cell populations, tissue level and whole organism models [10,[35][36][37][38]. In addition, models are being generated to account for pharmacokinetics and pharmacodynamics to analyse drug action, and even human-population level pharmacogenomics models of disease risk [10,39,40].…”

Section: Precision Medicine Approachesmentioning

confidence: 99%

Problems, challenges and promises: perspectives on precision medicine

Duffy¹

2015

Brief Bioinform

110

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Conflicts of interest:The author discloses no potential conflicts of interest. Summary key points•Introduction of precision medicine/systems medicine concepts and approaches• Review of some of the challenges to the clinical implementation of precision medicine• Outline of methodologies being employed to overcome these challenges• Examination of the claims that precision medicine has overpromised Author biographyDavid J. Duffy is primarily an experimentalist, based at Systems Biology Ireland in University College Dublin. His research interests include cancer biology and evolutionary developmental biology and the application of omic technologies and systems medicine approaches to address questions in these fields. AbstractThe precision medicine (systems medicine) concept promises to achieve a shift to future

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Modeling pattern formation in hydra: a route to understanding essential steps in development

Cited by 76 publications

References 66 publications

Stable spike clusters for the one-dimensional Gierer–Meinhardt system

Stable spike clusters for the one-dimensional Gierer–Meinhardt system

Turing's theory of morphogenesis of 1952 and the subsequent discovery of the crucial role of local self-enhancement and long-range inhibition

Problems, challenges and promises: perspectives on precision medicine

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