  The formation of protein patterns inside cells is generically described by
reaction-diffusion models. The study of such systems goes back to Turing, who
showed how patterns can emerge from a homogenous steady state when two reactive
components have different diffusivities (e.g. membrane-bound and cytosolic
states). However, in nature, systems typically develop in a heterogeneous
environment, where upstream protein patterns affect the formation of protein
patterns downstream. Examples for this are the polarization of Cdc42 adjacent
to the previous bud-site in budding yeast, and the formation of an
actin-recruiter ring that forms around a PIP3 domain in macropinocytosis. This
suggests that previously established protein patterns can serve as a template
for downstream proteins and that these downstream proteins can 'sense' the edge
of the template. A mechanism for how this edge sensing may work remains
elusive. Here we demonstrate and analyze a generic and robust edge-sensing
mechanism, based on a two-component mass-conserving reaction-diffusion (McRD)
model. Our analysis is rooted in a recently developed theoretical framework for
McRD systems, termed local equilibria theory. We extend this framework to
capture the spatially heterogeneous reaction kinetics due to the template. This
enables us to graphically construct the stationary patterns in the phase space
of the reaction kinetics. Furthermore, we show that the protein template can
trigger a regional mass-redistribution instability near the template edge,
leading to the accumulation of protein mass, which eventually results in a
stationary peak at the template edge. We show that simple geometric criteria on
the reactive nullcline's shape predict when this edge-sensing mechanism is
operational. Thus, our results provide guidance for future studies of
biological systems, and for the design of synthetic pattern forming systems.
