scholarly journals Bespoke Turing Systems

2021 ◽  
Vol 83 (5) ◽  
Author(s):  
Thomas E. Woolley ◽  
Andrew L. Krause ◽  
Eamonn A. Gaffney
Keyword(s):  
Reaction Kinetics ◽  
Reaction Diffusion ◽  
Turing Instability ◽  
Spatial Dimension ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Spatial And Temporal Heterogeneity ◽  
Spatial Dimensions ◽  
Functional Forms ◽  
Turing Systems

AbstractReaction–diffusion systems are an intensively studied form of partial differential equation, frequently used to produce spatially heterogeneous patterned states from homogeneous symmetry breaking via the Turing instability. Although there are many prototypical “Turing systems” available, determining their parameters, functional forms, and general appropriateness for a given application is often difficult. Here, we consider the reverse problem. Namely, suppose we know the parameter region associated with the reaction kinetics in which patterning is required—we present a constructive framework for identifying systems that will exhibit the Turing instability within this region, whilst in addition often allowing selection of desired patterning features, such as spots, or stripes. In particular, we show how to build a system of two populations governed by polynomial morphogen kinetics such that the: patterning parameter domain (in any spatial dimension), morphogen phases (in any spatial dimension), and even type of resulting pattern (in up to two spatial dimensions) can all be determined. Finally, by employing spatial and temporal heterogeneity, we demonstrate that mixed mode patterns (spots, stripes, and complex prepatterns) are also possible, allowing one to build arbitrarily complicated patterning landscapes. Such a framework can be employed pedagogically, or in a variety of contemporary applications in designing synthetic chemical and biological patterning systems. We also discuss the implications that this freedom of design has on using reaction–diffusion systems in biological modelling and suggest that stronger constraints are needed when linking theory and experiment, as many simple patterns can be easily generated given freedom to choose reaction kinetics.

scholarly journals Pattern formation from spatially heterogeneous reaction–diffusion systems

2021 ◽  
Vol 379 (2213) ◽  
Author(s):  
Robert A. Van Gorder
Keyword(s):  
Pattern Formation ◽  
Spatial Heterogeneity ◽  
Heterogeneous Systems ◽  
Reaction Diffusion ◽  
Turing Instability ◽  
Instability Criterion ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Spatially Homogeneous ◽  
Turing Systems

First proposed by Turing in 1952, the eponymous Turing instability and Turing pattern remain key tools for the modern study of diffusion-driven pattern formation. In spatially homogeneous Turing systems, one or a few linear Turing modes dominate, resulting in organized patterns (peaks in one dimension; spots, stripes, labyrinths in two dimensions) which repeats in space. For a variety of reasons, there has been increasing interest in understanding irregular patterns, with spatial heterogeneity in the underlying reaction–diffusion system identified as one route to obtaining irregular patterns. We study pattern formation from reaction–diffusion systems which involve spatial heterogeneity, by way of both analytical and numerical techniques. We first extend the classical Turing instability analysis to track the evolution of linear Turing modes and the nascent pattern, resulting in a more general instability criterion which can be applied to spatially heterogeneous systems. We also calculate nonlinear mode coefficients, employing these to understand how each spatial mode influences the long-time evolution of a pattern. Unlike for the standard spatially homogeneous Turing systems, spatially heterogeneous systems may involve many Turing modes of different wavelengths interacting simultaneously, with resulting patterns exhibiting a high degree of variation over space. We provide a number of examples of spatial heterogeneity in reaction–diffusion systems, both mathematical (space-varying diffusion parameters and reaction kinetics, mixed boundary conditions, space-varying base states) and physical (curved anisotropic domains, apical growth of space domains, chemicalsimmersed within a flow or a thermal gradient), providing a qualitative understanding of how spatial heterogeneity can be used to modify classical Turing patterns. This article is part of the theme issue ‘Recent progress and open frontiers in Turing’s theory of morphogenesis’.

scholarly journals Reaction-Diffusion Systems: Self-Balancing Diffusion and the Use of the Extent of Reaction as a Descriptor of Reaction Kinetics

2019 ◽  
Author(s):  
Miloslav Pekař
Keyword(s):  
Reaction Kinetics ◽  
Reaction Diffusion ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
And Diffusion ◽  
Reaction And Diffusion

Self-balancing diffusion is a concept which restricts the introduction of extents of reactions. This concept is analyzed in detail for mass- and molar-based balances of reaction-diffusion mixtures, in relation to non-self-balancing cases, and with respect to its practical consequences. A note on a recent generalization of the concept of reaction and diffusion extents is also included.<br>

scholarly journals Turing Patterning in Stratified Domains

2020 ◽  
Vol 82 (10) ◽  
Author(s):  
Andrew L. Krause ◽  
Václav Klika ◽  
Jacob Halatek ◽  
Paul K. Grant ◽  
Thomas E. Woolley ◽  
...  
Keyword(s):  
Diffusion Processes ◽  
Reaction Diffusion ◽  
Turing Instability ◽  
Cell Polarization ◽  
The Other ◽  
Modeling Framework ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Coupling Parameters ◽  
Pattern Forming

Abstract 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.

Multistability and Stochastic Phenomena in the Distributed Brusselator Model

2019 ◽  
Vol 15 (1) ◽  
Author(s):  
Alexander Kolinichenko ◽  
Lev Ryashko
Keyword(s):  
Reaction Diffusion ◽  
Turing Instability ◽  
Spatial Structures ◽  
Reaction Diffusion Systems ◽  
Corporate Influence ◽  
Basic Model ◽  
Brusselator Model ◽  
Diffusion Systems ◽  
Random Disturbances ◽  
Stochastic Phenomena

Abstract An influence of random disturbances on the pattern formation in reaction–diffusion systems is studied. As a basic model, we consider the distributed Brusselator with one spatial variable. A coexistence of the stationary nonhomogeneous spatial structures in the zone of Turing instability is demonstrated. A numerical parametric analysis of shapes, sizes of deterministic pattern–attractors, and their bifurcations is presented. Investigating the corporate influence of the multistability and stochasticity, we study phenomena of noise-induced transformation and generation of patterns.

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