  We use state of the art hydrodyamical simulations from the Sherwood, EAGLE
and Illustris projects to examine the signature of $M_{\rm z=0}\simeq
10^{14}M_{\odot}$ protoclusters observed in Ly-$\alpha$ absorption at $z\simeq
2.4$. We find there is a weak correlation between the mass overdensity,
$\delta_{\rm m}$, and the Ly-$\alpha$ effective optical depth relative to the
mean, $\delta_{\tau_\textrm{eff}}$, averaged over $15~h^{-1}\rm\,cMpc$ scales,
although scatter in the $\delta_{\rm m}$--$\delta_{\tau_\textrm{eff}}$ plane
means it is not possible to uniquely identify large scale overdensities with
strong Ly-$\alpha$ absorption. Although all protoclusters are associated with
large scale mass overdensities, most sight lines through protoclusters in a
$\sim 10^{6}$ $\rm cMpc^{3}$ volume probe the low column density Ly-$\alpha$
forest. A small subset of sight lines that pass through protoclusters exhibit
coherent, strong Ly-$\alpha$ absorption on $15h^{-1}\rm\,cMpc$ scales, although
these correspond to a wide range in mass overdensity. Assuming perfect removal
of contamination by Ly-$\alpha$ absorbers with damping wings, more than half of
the remaining sight lines with $\delta_{\tau_{\rm eff}}>3.5$ trace
protoclusters. It is furthermore possible to identify a model dependent
$\delta_{\tau_{\rm eff}}$ threshold that selects only protoclusters. However,
such regions are rare: excluding absorption caused by damped systems, less than
0.1 per cent of sight lines that pass through a protocluster have
$\delta_{\tau_{\rm eff}}>3.5$, meaning that any protocluster sample selected in
this manner will also be highly incomplete. On the other hand, coherent regions
of Ly-$\alpha$ absorption also provide a promising route for identifying and
studying filamentary environments at high redshift.
