  Recent advances in high-resolution experimental methods have highlighted the
significance of cell signal pathway crosstalk and localised signalling activity
in the development and disease of numerous biological systems. The
investigation of multiple signal pathways often introduces different methods of
cell-cell communication, i.e. contact-based or diffusive signalling, which
generates both a spatial and temporal dependence on cell behaviours. Motivated
by cellular mechanisms that control cell-fate decisions in developing bilayer
tissues, we use dynamical systems coupled with multilayer graphs to analyse the
role of signalling polarity and pathway crosstalk in fine-grain pattern
formation of protein activity. Specifically, we study how multilayer graph edge
structures and weights influence the layer-wise (laminar) patterning of cells
in bilayer structures, which are commonly found in glandular tissues. We
present sufficient conditions for existence, uniqueness and instability of
homogeneous cell states in the large-scale spatially discrete dynamical system.
Using methods of pattern templating by graph partitioning to generate quotient
systems in combination with concepts from monotone dynamical systems, we
exploit the extensive dimensionality reduction to provide existence conditions
for the polarity required to induce fine-grain laminar patterns with multiple
spatially dependent intracellular components. We then explore the spectral
links between the quotient and large-scale dynamical systems to extend the
laminar patterning criteria from existence to convergence for sufficiently
large amounts of cellular polarity in the large-scale dynamical system,
independent of spatial dimension and number of cells in the tissue.
