our previous paper that the model is able to produce Tully-Fisher relation, cold gas and disk size distributions, and B-band galaxy luminosity functions (from $z=0$ to $z\approx 4$) in good agreement with observations. As for the growth of SMBHs with the associated the AGN emission, and for the corresponding AGN feedback, we adopt the models by Cavaliere & Vittorini (2000) and by Lapi, Cavaliere & Menci (2005), respectively. Here we recall their basic points.

The Growth of SMBHs
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The growth of SMBH is implemented as described in Menci et al. (2003); the accretion of cold gas is triggered by galaxy encounters (both fly-by and merging), which destabilize part of the avaiable cold gas. The rate of interactions is (Menci et al. 2003) $$\tau_r^{-1}=n_T\,\Sigma (r_t,v,V_{rel})\,V_{rel}.$$ Here $n_T$ is the number density of galaxies in the same halo and $V_{rel}$ is their relative velociyt. The encounters effective for angular momentum transfer require 1) the interaction time to be comparable with the internal oscillation time in the involved galaxies (resonance), ii) the orbital specific energy of the partners not to exceed the sum of their specific internal gravitational energies. The cross section $\Sigma$ for such encounters is given by Saslaw (1995) in 