  Active matter systems such as eukaryotic cells and bacteria continuously
transform chemical energy to motion. Hence living systems exert active stresses
on the complex environments in which they reside. One recurring aspect of this
complexity is the viscoelasticity of the medium surrounding living systems:
bacteria secrete their own viscoelastic extracellular matrix, and cells
constantly deform, proliferate, and self-propel within viscoelastic networks of
collagen. It is therefore imperative to understand how active matter modifies,
and gets modified by, viscoelastic fluids. Here, we present a two-phase model
of active nematic matter that dynamically interacts with a passive viscoelastic
polymeric phase and perform numerical simulations in two dimensions to
illustrate its applicability. Motivated by recent experiments we first study
the suppression of cell division by a viscoelastic medium surrounding the cell.
We further show that the self-propulsion of a model keratocyte cell is modified
by the polymer relaxation of the surrounding viscoelastic fluid in a
non-uniform manner and find that increasing polymer viscosity effectively
suppresses the cell motility. Lastly, we explore the hampering impact of the
viscoelastic medium on the generic hydrodynamic instabilities of active
nematics by simulating the dynamics of an active stripe within a polymeric
fluid. The model presented here can provide a framework for investigating more
complex dynamics such as the interaction of multicellular growing systems with
viscoelastic environments.
