  We study stochastic dynamics of a point and extended inclusion within a one
dimensional confined active viscoelastic gel. We show that the dynamics of a
point inclusion can be described by a Langevin equation with a confining
potential and multiplicative noise. Using a systematic adiabatic elimination
over the fast variables, we arrive at an overdamped equation with a proper
definition of the multiplicative noise. To highlight various features and to
appeal to different biological contexts, we treat the inclusion in turn as a
rigid extended element, an elastic element and a viscoelastic (Kelvin-Voigt)
element. The dynamics for the shape and position of the extended inclusion can
be described by coupled Langevin equations. Deriving exact expressions for the
corresponding steady state probability distributions, we find that the active
noise induces an attraction to the edges of the confining domain. In the
presence of a competing centering force, we find that the shape of the
probability distribution exhibits a sharp transition upon varying the amplitude
of the active noise. Our results could help understanding the positioning and
deformability of biological inclusions, eg. organelles in cells, or nucleus and
cells within tissues.
