  Studies of galaxy protoclusters yield insights into galaxy cluster formation
complementary to those obtained via `archaeological' studies of present-day
galaxy clusters. Submillimetre-selected galaxies (SMGs) are one class of
sources used to find high-redshift protoclusters. However, due to the rarity of
protoclusters (and thus the large simulation volume required) and the
complexity of modeling dust emission from galaxies, the relationship between
SMGs and protoclusters has not been adequately addressed in the theoretical
literature. In this work, we apply the L-GALAXIES semi-analytic model (SAM) to
the Millennium N-body simulation. We assign submillimetre (submm) flux
densities to the model galaxies using a scaling relation from previous work, in
which dust radiative transfer was performed on high-resolution galaxy zoom
simulations. We find that the fraction of model galaxies that are submm-bright
is higher in protocluster cores than in both protocluster `outskirts' and the
field; the fractions for the latter two are similar. This excess is not driven
by an enhanced starburst frequency. Instead, the primary reason is that
overdense environments have a relative overdensity of high-mass halos and thus
`oversample' the high-mass end of the star formation main sequence relative to
less-dense environments. The fraction of SMGs that are optically bright is
dependent on stellar mass and redshift but independent of environment. The
fraction of galaxies for which the majority of star formation is dust-obscured
is higher in protocluster cores, primarily due to the dust-obscured fraction
being correlated with stellar mass. Our results can be used to guide and
interpret multi-wavelength studies of galaxy populations in protoclusters.
