  Cell membranes phase separate into ordered ${\rm L_o}$ and disordered ${\rm
L_d}$ domains depending on their compositions. This membrane
compartmentalization is heterogeneous and regulates the localization of
specific proteins related to cell signaling and trafficking. However, it is
unclear how the heterogeneity of the membranes affects the diffusion and
localization of proteins in ${\rm L_o}$ and ${\rm L_d}$ domains. Here, using
Langevin dynamics simulations coupled with the phase-field (LDPF) method, we
investigate several tens of milliseconds-scale diffusion and localization of
proteins in heterogeneous biological membrane models showing phase separation
into ${\rm L_o}$ and ${\rm L_d}$ domains. The diffusivity of proteins exhibits
temporal fluctuations depending on the field composition. Increases in
molecular concentrations and domain preference of the molecule induce
subdiffusive behavior due to molecular collisions by crowding and confinement
effects, respectively. Moreover, we quantitatively demonstrate that the protein
partitioning into the ${\rm L_o}$ domain is determined by the difference in
molecular diffusivity between domains, molecular preference of domain, and
molecular concentration. These results pave the way for understanding how
biological reactions caused by molecular partitioning may be controlled in
heterogeneous media. Moreover, the methodology proposed here is applicable not
only to biological membrane systems but also to the study of diffusion and
localization phenomena of molecules in various heterogeneous systems.
