  We study a bulk-surface coupled system that describes the processes of
lipid-phase separation and lipid-cholesterol interaction on cell membranes, in
which cholesterol exchange between cytosol and cell membrane is also
incorporated. The PDE system consists of a surface Cahn-Hilliard equation for
the relative concentration of saturated/unsaturated lipids and a surface
diffusion-reaction equation for the cholesterol concentration on the membrane,
together with a diffusion equation for the cytosolic cholesterol concentration
in the bulk. The detailed coupling between bulk and surface evolutions is
characterized by a mass exchange term $q$. For the system with a physically
relevant singular potential, we first prove the existence, uniqueness and
regularity of global weak solutions to the full bulk-surface coupled system
under suitable assumptions on the initial data and the mass exchange term $q$.
Next, we investigate the large cytosolic diffusion limit that gives a reduction
of the full bulk-surface coupled system to a system of surface equations with
non-local contributions. Afterwards, we study the long-time behavior of global
solutions in two categories, i.e., the equilibrium and non-equilibrium models
according to different choices of the mass exchange term $q$. For the full
bulk-surface coupled system with a decreasing total free energy, we prove that
every global weak solution converges to a single equilibrium as $t\to +\infty$.
For the reduced surface system with a mass exchange term of reaction type, we
establish the existence of a global attractor.
