  Membrane phase behavior induced by the binding of curvature-inducing proteins
is studied by a combination of analytical and numerical approaches. In thermal
equilibrium under the detailed balance between binding and unbinding, the
membrane exhibits three phases: an unbound uniform flat phase (U), a bound
uniform flat phase (B), and a separated/corrugated phase (SC). In the SC phase,
the bound proteins form hexagonally-ordered bowl-shaped domains. The
transitions between the U and SC phases and between the B and SC phases are
second order and first order, respectively. At a small spontaneous curvature of
the protein or high surface tension, the transition between B and SC phases
becomes continuous. Moreover, a first-order transition between the U and B
phases is found at zero spontaneous curvature driven by the Casimir-like
interactions between rigid proteins. Furthermore, nonequilibrium dynamics is
investigated by the addition of active binding and unbinding at a constant
rate. The active binding and unbinding processes alter the stability of the SC
phase.

Soft Condensed Matter