Turing Patterning in Stratified Domains

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.

Isolating Patterns in Open Reaction–Diffusion Systems

“…2.1 (Krause et al. 2020 ). Finally, we have restricted our attention to studying stationary solutions, but RDS can exhibit a wide variety of spatiotemporal behaviours, such as spatiotemporal chaos and spiral waves observed in excitable systems (Sánchez-Garduno et al.…”

Modern perspectives on near-equilibrium analysis of Turing systems

“…differential diffusion) compared to the classical case [54]. bind to a membrane [12], bacterial pattern formation on inert substrates [55,56], and epithelialmesenchymal signalling [57,58].…”

“…There have been recent efforts to study pattern formation in domains with multiple distinct regions. Examples of such systems include proteins which can both diffuse within a cell and bind to a membrane [ 12 ], bacterial pattern formation on inert substrates [ 55 , 56 ], and epithelial-mesenchymal signalling [ 57 , 58 ].…”

“…Such models can sometimes be reduced to bulk-surface models given appropriate distinguished limits and scaling assumptions (for example, in [ 75 ], the authors reduce two-dimensional spatially continuous regions to one-dimensional boundaries assuming thinness of these regions). Krause et al [ 56 ] analyse a model of a two-layered stratified system in two spatial dimensions, primarily focusing on the case of a standard two-component system in the surface coupled to an inert layer which only allows for diffusion. This was motivated by an experimental framework of bacteria placed on top of agar (for use in synthetic pattern formation), and the study investigated the impact of this inert bulk on the pattern-forming potential of the surface reaction–diffusion system.…”

“…In particular, if we consider the circular and spherical geometries analysed in [ 67 ], but extend the surface outwards (so that we have an annular region coupled to an interior spherical one), we can no longer use the power series ansatz involving classical eigenfunctions, as the radial and angular eigenfunctions do not decouple. In principle, the more elaborate approach used in [ 56 ] could be employed, though it would be cumbersome in terms of calculations. As far as we are aware, most multi-domain systems on generic geometries, both bulk-surface and stratified (bulk-bulk), will not admit analysis by classical series expansions of eigenfunctions, at least in a way that leads to polynomial dispersion relations.…”