Soft inclusion in a confined fluctuating active gel

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, e.g., organelles in cells, or nucleus and cells within tissues.