  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
polarisation. We develop a modelling framework for bi-layer 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.
