  We study the force generation by a set of parallel actin filaments growing
against an elastic membrane. The elastic membrane tries to stay flat and any
deformation from this flat state, either caused by thermal fluctuations or due
to protrusive polymerization force exerted by the filaments, costs energy. We
study two lattice models to describe the membrane dynamics. In one case, the
energy cost is assumed to be proportional to the absolute magnitude of the
height gradient (gradient model) and in the other case it is proportional to
the square of the height gradient (Gaussian model). For the gradient model we
find that the membrane velocity is a non-monotonic function of the elastic
constant $\mu$, and reaches a peak at $\mu=\mu^\ast$. For $\mu < \mu^\ast$ the
system fails to reach a steady state and the membrane energy keeps increasing
with time. For the Gaussian model, the system always reaches a steady state and
the membrane velocity decreases monotonically with the elastic constant $\nu$
for all nonzero values of $\nu$. Multiple filaments give rise to protrusions at
different regions of the membrane and the elasticity of the membrane induces an
effective attraction between the two protrusions in the Gaussian model which
causes the protrusions to merge and a single wide protrusion is present in the
system. In both the models, the relative time-scale between the membrane and
filament dynamics plays an important role in deciding whether the shape of
elasticity-velocity curve is concave or convex. Our numerical simulations agree
reasonably well with our analytical calculations.
