DM particle is $8.47 \times 10^{8} \ h^{-1} M_{\odot}$ and the initial mass of the gas particle is $1.53 \times 10^{8} \ h^{-1} M_{\odot}$.

The sample includes all the objects in the high-resolution Lagrangian regions with $M_{500} > 7 \times 10^{13}  E(z)^{-1} h^{-1} M_{\odot}$ with $E(z)=H(z)/H_0=(\Omega_M\times(1+z)^3+\Omega_{\Lambda})^{1/2}$. The number of [[[agn]{}]{}]{} clusters selected at any redshift of interest and the corresponding mass range are presented in Fig. \[fig:a2\]. Notice that at the highest redshifts, $z=1, z=1.5$ and $z=2$, our sample includes 74, 60, and 36 objects. The smallest systems at $z=2$ contains more than $3.5 \times 10^4$ particles providing good estimates of global quantities. We do not extend the sample to smaller systems, even if we would have obtained numerically robust global measures, because none of the planned missions will reach such small masses (see @borm.etal.2014 for e-ROSITA, @pointecouteau.etal.2013 for Athena and @benson.etal.2014 for SPT-3G).

Method of Analysis {#Method of Analysis}
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Computing ICM Quantities
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In the following, we briefly describe how we compute the relevant quantities from our simulated datasets. [**Masses.**]{} The total mass, $M$, is calculated by summing the contribution of all the species of particles (dark-matter, gas, and stars) within $R_{500}$. For the gas mass, ${M_{\rm g}}$, 