  We present a nonequlibrium approach for the study of a flexible bilayer whose
two components induce distinct curvatures. In turn, the two components are
interconverted by an externally promoted reaction. Phase separation of the two
species in the surface results in the growth of domains characterized by
different local composition and curvature modulations. This domain growth is
limited by the effective mixing due to the interconversion reaction, leading to
a finite characteristic domain size. In addition to these effects, first
introduced in our earlier work [Phys. Rev. E {\bf 71}, 051906 (2005)], the
important new feature is the assumption that the reactive process actively
affects the local curvature of the bilayer. Specifically, we suggest that a
force energetically activated by external sources causes a modification of the
shape of the membrane at the reaction site. Our results show the appearance of
a rich and robust dynamical phenomenology that includes the generation of
traveling and/or oscillatory patterns. Linear stability analysis, amplitude
equations and numerical simulations of the model kinetic equations confirm the
occurrence of these spatiotemporal behaviors in nonequilibrium reactive
bilayers.
