  The distribution of inclusion-rich domains in membranes with active two-state
inclusions is studied by simulations. Our study shows that typical size of
inclusion-rich domains ($L$) can be controlled by inclusion activities in
several ways. When there is effective attraction between state-1 inclusions, we
find: (i) Small domains with only several inclusions are observed for
inclusions with time scales ($\sim 10^{-3} {\rm s}$) and interaction energy
[$\sim \mathcal{O}({\rm k_BT})$] comparable to motor proteins. (ii) $L$ scales
as 1/3 power of the lifetime of state-1 for a wide range of parameters. (iii)
$L$ shows a switch-like dependence on state-2 lifetime $k_{12}^{-1}$. That is,
$L$ depends weakly on $k_{12}$ when $k_{12} < k_{12}^*$ but increases rapidly
with $k_{12}$ when $k_{12} > k_{12}^*$, the crossover $k_{12}^*$ occurs when
the diffusion length of a typical state-2 inclusion within its lifetime is
comparable to $L$. (iv) Inclusion-curvature coupling provides another length
scale that competes with the effects of transition rates.
