  Biological membranes are host to proteins and molecules which may form
domain-like structures resulting in spatially-varying material properties.
Vesicles with such heterogeneous membranes can exhibit intricate shapes at
equilibrium and rich dynamics when placed into a flow. Under the assumption of
small deformations we develop a reduced order model to describe the
fluid-structure interaction between a viscous background shear flow and an
inextensible membrane in two dimensions with spatially varying bending
stiffness and spontaneous curvature. Material property variations of a critical
magnitude, relative to the flow rate and internal/external viscosity contrast,
can set off a qualitative change in the vesicle dynamics. A membrane of nearly
constant bending stiffness or spontaneous curvature undergoes a small amplitude
swinging motion (which includes tangential tank-treading), while for large
enough material variations the dynamics pass through a regime featuring
tumbling and periodic phase-lagging of the membrane material, and ultimately
for very large material variation to a rigid body tumbling behavior. Distinct
differences are found for even and odd spatial modes of domain distribution.
Full numerical simulations are used to probe the theoretical predictions, which
appear valid even when studying substantially deformed membranes.
