On the heterogeneity of reaction-diffusion generated pattern

Berding, C.

doi:10.1007/bf02459700

Cited by 5 publications

(2 citation statements)

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“…To compare these dispersion relations against numerical simulations of the full nonlinear system, we compute a heterogeneity functional determining how far a solution is from a homogeneous state (Berding 1987 ). For simplicity, and because the surface layer is of primary interest in synthetic pattern formation within bacterial colonies, we only consider the heterogeneity of the activator in the surface.…”

Section: Numerical Exploration Of Example Systemsmentioning

confidence: 99%

Turing Patterning in Stratified Domains

et al. 2020

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Reaction–diffusion processes across layered media arise in several scientific domains such as pattern-forming E. coli on agar substrates, epidermal–mesenchymal coupling in development, and symmetry-breaking in cell polarization. We develop a modeling framework for bilayer reaction–diffusion systems and relate it to a range of existing models. We derive conditions for diffusion-driven instability of a spatially homogeneous equilibrium analogous to the classical conditions for a Turing instability in the simplest nontrivial setting where one domain has a standard reaction–diffusion system, and the other permits only diffusion. Due to the transverse coupling between these two regions, standard techniques for computing eigenfunctions of the Laplacian cannot be applied, and so we propose an alternative method to compute the dispersion relation directly. We compare instability conditions with full numerical simulations to demonstrate impacts of the geometry and coupling parameters on patterning, and explore various experimentally relevant asymptotic regimes. In the regime where the first domain is suitably thin, we recover a simple modulation of the standard Turing conditions, and find that often the broad impact of the diffusion-only domain is to reduce the ability of the system to form patterns. We also demonstrate complex impacts of this coupling on pattern formation. For instance, we exhibit non-monotonicity of pattern-forming instabilities with respect to geometric and coupling parameters, and highlight an instability from a nontrivial interaction between kinetics in one domain and diffusion in the other. These results are valuable for informing design choices in applications such as synthetic engineering of Turing patterns, but also for understanding the role of stratified media in modulating pattern-forming processes in developmental biology and beyond.

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Section: Numerical Exploration Of Example Systemsmentioning

confidence: 99%

Turing Patterning in Stratified Domains

et al. 2020

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“…More precise solutions, showing that a pattern will actually emerge at the steady state, are presented in the application part. However, a further clarification might be possible here with the help of the heterogeneity function (Berding, 1987;Murray, 2003), which is defined as…”

Section: State-costate Pathsmentioning

confidence: 99%