  We introduce a mathematical model for the mechanical behaviour of the
eukaryotic cell cytoskeleton. This discrete model involves a regular array of
pre-stressed protein filaments that exhibit resistance to enthalpic stretching,
joined at crosslinks to form a network. Assuming that the inter-crosslink
distance is much shorter than the lengthscale of the cell, we upscale the
discrete force balance to form a continuum system of governing equations and
deduce the corresponding macroscopic stress tensor. We use these discrete and
continuum models to analyse the imposed displacement of a bead placed in the
domain, characterising the cell rheology through the force-displacement curve.
We further derive an analytical approximation to the stress and strain fields
in the limit of small bead radius, predicting the net force required to
generate a given deformation and elucidating the dependency on the microscale
properties of the filaments. We apply these models to networks of the
intermediate filament vimentin and demonstrate good agreement between
predictions of the discrete, continuum and analytical approaches. In
particular, our model predicts that the network stiffness increases sublinearly
with the filament pre-stress and scales logarithmically with the bead size.
