Isolating Patterns in Open Reaction–Diffusion Systems

Abstract

Realistic examples of reaction–diffusion phenomena governing spatial and spatiotemporal pattern formation are rarely isolated systems, either chemically or thermodynamically. However, even formulations of ‘open’ reaction–diffusion systems often neglect the role of domain boundaries. Most idealizations of closed reaction–diffusion systems employ no-flux boundary conditions, and often patterns will form up to, or along, these boundaries. Motivated by boundaries of patterning fields related to the emergence of spatial form in embryonic development, we propose a set of mixed boundary conditions for a two-species reaction–diffusion system which forms inhomogeneous solutions away from the boundary of the domain for a variety of different reaction kinetics, with a prescribed uniform state near the boundary. We show that these boundary conditions can be derived from a larger heterogeneous field, indicating that these conditions can arise naturally if cell signalling or other properties of the medium vary in space. We explain the basic mechanisms behind this pattern localization and demonstrate that it can capture a large range of localized patterning in one, two, and three dimensions and that this framework can be applied to systems involving more than two species. Furthermore, the boundary conditions proposed lead to more symmetrical patterns on the interior of the domain and plausibly capture more realistic boundaries in developmental systems. Finally, we show that these isolated patterns are more robust to fluctuations in initial conditions and that they allow intriguing possibilities of pattern selection via geometry, distinct from known selection mechanisms.

Biological action at a distance: Correlated pattern formation in adjacent tessellation domains without communication

“…When the control parameters were perturbed by up to 50% in each domain, the correlations appeared to diverge a little, but not enough to reject the null hypothesis that they were drawn from the same distribution ( p = 0.31). Moreover, the effects shown in Fig 5 are not sensitive to the particular choice of pattern-forming system, as confirmed via a sweep through the relevant parameters of an alternative system that does not include a non-linear diffusion coupling term of the type that is parameterised by χ in Eq 1 (see S1 Fig ; [ 22 , 23 ]).…”

Modern perspectives on near-equilibrium analysis of Turing systems

“…Throughout, we will mostly review work which studies models of the form (2.1) where the diffusion coefficients and/or the reaction kinetics explicitly depend on space x and/or time t. Heterogeneity can also appear due to inhomogeneous or more complex (e.g. Dirichlet or Robin) boundary conditions [78][79][80][81].…”

“…Beyond Turing's work on phyllotaxis, spatial heterogeneity was also used by Gierer & Meinhardt in their classical work on pattern formation [83]. Heterogeneity in developmental settings has been suggested as key for organizing different regions along cell boundaries based on sharp variations in gene expression [79,[84][85][86]. More recent work has used spatial heterogeneity in reaction-diffusion models to relate Turing-type pattern formation to other patterning theories, such as positional information [87][88][89].…”

Modern Perspectives on Near-Equilibrium Analysis of Turing Systems

“…From the point of view of near-equilibrium analysis, both of these scenarios require a novel definition of a base state in order to define what one means by an emergent pattern. Throughout, we will mostly review work which studies models of the form (2.1) where the diffusion coefficients and/or the reaction kinetics explicitly depend on space x and/or time t. Heterogeneity can also appear due to inhomogeneous or more complex (e.g., Dirichlet or Robin) boundary conditions [44,113,129,186].…”

“…Beyond Turing's work on phyllotaxis, spatial heterogeneity was also used by Gierer and Meinhardt in their classical work on pattern formation [68]. Heterogeneity in developmental settings has been suggested as key for organising different regions along cell boundaries based on sharp variations in gene expression [91,113,136,137]. More recent work has used spatial heterogeneity in reaction-diffusion models to relate Turing-type pattern formation to other patterning theories, such as positional information [74,115,204].…”