  Mixtures of lipids and cholesterol are commonly used as model systems for
studying the formation of liquid-ordered ($L_o$) domains in heterogeneous
biological membranes. The simplest model system exhibiting coexistence between
$L_o$ domains and a liquid-disordered ($L_d$) matrix is that of a binary
mixture of saturated lipids like DPPC and cholesterol (Chol). DPPC/Chol
mixtures have been investigated for decades both experimentally, theoretically,
and recently also by means of atomistic simulations. Here, we present a minimal
lattice model that captures the correct behavior of this mixture across
multiple scales. On the macroscopic scales, we present simulation results of
mixtures of thousands of lipids and Chol molecules which accurately reproduce
the phase diagram of the system. The simulations are conducted on timescales of
hundreds of microseconds and show the morphologies and dynamics of the domains.
On the molecular scales, the simulations reveal local structures similar to
those recently seen in atomistic simulations, including the formation of small
gel-like nano-domains within larger Chol-rich $L_o$ domains. The observed
multi-scale behavior is fully explained in terms of (i) a competition between
the packing enthalpy and configurational entropy of the lipid chains, and (ii)
the "dual effect" of the Chol on the membrane fluidity.
