  In this work we study the properties of protoclusters of galaxies by
employing the MUSIC set of hydrodynamical simulations, featuring a mass-limited
sample of 282 resimulated clusters with available merger trees up to high
redshift, and we trace the cluster formation back to $z$ = 1.5, 2.3 and 4. We
study the features and redshift evolution of the mass and the spatial
distribution for all the cluster progenitors and for the protoclusters, which
we define as the most massive progenitors of the clusters identified at $z$ =
0. A natural extension to redshifts larger than 1 is applied to the estimate of
the baryon content also in terms of gas and stars budgets: no remarkable
variations with redshift are discovered. Furthermore, motivated by the proven
potential of Sunyaev-Zel'dovich surveys to blindly search for faint distant
objects, we focus on the scaling relation between total object mass and
integrated Compton $y$-parameter, and we check for the possibility to extend
the mass-observable paradigm to the protocluster regime, far beyond the
redshift of 1, to account for the properties of the simulated objects. We find
that the slope of this scaling law is steeper than what expected for a
self-similarity assumption among these objects, and it increases with redshift
mainly for the synthetic clusters where radiative processes, such as radiative
cooling, heating processes of the gas due to UV background, star formation and
supernovae feedback, are included. We use three different criteria to account
for the dynamical state of the protoclusters, and find no significant
dependence of the scaling parameters from the level of relaxation. Based on
this, we exclude that the dynamical state is the cause of the observed
deviations from self-similarity.
