  We propose a model that leads to the formation of non-equilibrium finite-size
domains in a biological membrane. Our model considers the active conformational
change of the inclusions and the coupling between inclusion density and
membrane curvature. Two special cases with different interactions are studied
by Monte Carlo simulations. In case (i) exited state inclusions prefer to
aggregate. In case (ii) ground state inclusions prefer to aggregate. When the
inclusion density is not coupled to the local membrane curvature, in case (i)
the typical length scale ($\sqrt{M}$) of the inclusion clusters shows weak
dependence on the excitation rate ($K_{on}$) of the inclusions for a wide range
of $K_{on}$ but increases fast when $K_{on}$ becomes sufficiently large; in
case (ii) $\sqrt{M}\sim {K_{on}}^{-{1/3}}$ for a wide range of $K_{on}$. When
the inclusion density is coupled to the local membrane curvature, the curvature
coupling provides the upper limit of the inclusion clusters. In case (i) (case
(ii)), the formation of the inclusions is suppressed when $K_{off}$ ($K_{on}$)
is sufficiently large such that the ground state (excited state) inclusions do
not have sufficient time to aggregate. We also find that the mobility of an
inclusion in the membrane depends on inclusion-curvature coupling. Our study
suggests possible mechanisms that produce finite-size domains in biological
membranes.
