  Motivated by recent experiments showing the buckling of microtubules in
cells, we study theoretically the mechanical response of, and force propagation
along elastic filaments embedded in a non-linear elastic medium. We find that,
although embedded microtubules still buckle when their compressive load exceeds
the critical value $f_c$ found earlier, the resulting deformation is restricted
to a penetration depth that depends on both the non-linear material properties
of the surrounding cytoskeleton, as well as the direct coupling of the
microtubule to the cytoskeleton. The deformation amplitude depends on the
applied load $f$ as $(f- f_c)^{1/2}$. This work shows how the range of
compressive force transmission by microtubules can be as large as tens of
microns and is governed by the mechanical coupling to the surrounding
cytoskeleton.
