  Nonequilibrium dynamics of biomembranes with active inclusions is considered.
The inclusions represent protein molecules which perform cyclic internal
conformational motions driven by the energy brought with ATP ligands. As
protein conformations cyclically change, this induces hydrodynamical flows and
also directly affects the local curvature of a membrane. On the other hand,
variations in the local curvature of the membrane modify the transitions rates
between conformational states in a protein, leading to a feedback in the
considered system. Moreover, active inclusions can move diffusively through the
membrane so that surface concentration varies. The kinetic description of this
system is constructed and the stability of the uniform stationary state is
analytically investigated. We show that, as the rate of supply of chemical
energy is increased above a certain threshold, this uniform state becomes
unstable and stationary or traveling waves spontaneously develop in the system.
Such waves are accompanied by periodic spatial variation of membrane curvature
and inclusion density. For typical parameter values, their characteristic
wavelengths are of the order of hundreds of nanometers. For traveling waves,
the characteristic frequency is of the order of a thousand Hz or less.
