  Several cellular activities, such as directed cell migration, are coordinated
by an intricate network of biochemical reactions which lead to a polarised
state of the cell, in which cellular symmetry is broken, causing the cell to
have a well defined front and back. Recent work on balancing biological
complexity with mathematical tractability resulted in the proposal and
formulation of a famous minimal model for cell polarisation, known as the wave
pinning model. In this study, we present a three-dimensional generalisation of
this mathematical framework through the maturing theory of coupled bulk-surface
semilinear partial differential equations in which protein compartmentalisation
becomes natural. We show how a local perturbation over the surface can trigger
propagating reactions, eventually stopped in a stable profile by the interplay
with the bulk component. We describe the behavior of the model through
asymptotic and local perturbation analysis, in which the role of the geometry
is investigated. The bulk-surface finite element method is used to generate
numerical simulations over simple and complex geometries, which confirm our
analysis, showing pattern formation due to propagation and pinning dynamics.
The generality of our mathematical and computational framework allows to study
more complex biochemical reactions and biomechanical properties associated with
cell polarisation in multi-dimensions.
