  We study the implications of coupling Langmuir kinetics (LK) process with the
dynamics of an extending lattice. The model that we consider couples
dynamically extending exclusion process (DEEP) with the process of random
attachment and detachment of particles in the bulk of the lattice. We explore a
dynamical regime where the boundary processes of lattice extension and particle
input compete with the bulk process of particle attachment and detachment. This
competition leads to scale invariant density profile of particles when
expressed in terms of the relative position in the growing lattice. It also
leads to phase coexistence and shocks in the bulk of the lattice. We use a
combination of Mean Field (MF) analysis and Monte Carlo simulations to
characterize the density profile in growing lattice and construct a MF phase
diagram. This study will have possible implications for transport and
patterning in growing fungal filament.
