  The space-time dynamics of rigid inhomogeneities (inclusions) free to move in
a randomly fluctuating fluid bio-membrane is derived and numerically simulated
as a function of the membrane shape changes. Both vertically placed (embedded)
inclusions and horizontally placed (surface) inclusions are considered. The
energetics of the membrane, as a two-dimensional (2D) meso-scale continuum
sheet, is described by the Canham-Helfrich Hamiltonian, with the membrane
height function treated as a stochastic process. The diffusion parameter of
this process acts as the link coupling the membrane shape fluctuations to the
kinematics of the inclusions. The latter is described via Ito stochastic
differential equation. In addition to stochastic forces, the inclusions also
experience membrane-induced deterministic forces. Our aim is to simulate the
diffusion-driven aggregation of inclusions and show how the external inclusions
arrive at the sites of the embedded inclusions. The model has potential use in
such emerging fields as designing a targeted drug delivery system.
