  We carried out a comprehensive analysis of the chemical properties of the
interstellar medium (ISM) and the stellar population (SP) of current normal
galaxies and their progenitors in a hierarchical clustering scenario. We
compared the results with observations of Damped Lyman-$\alpha$ systems (DLAs)
under the hypothesis that, at least, part of the observed DLAs could originate
in the building blocks of today normal galaxies. We used a hydrodynamical
cosmological code which includes star formation and chemical enrichment.
Galaxy-like objects are identified at $z=0$ and then followed back in time.
Random line-of-sights (LOS) are drawn through these structures in order to
mimic Damped Lyman $\alpha$ systems. We then analysed the chemical properties
of the ISM and SP along the LOS. We found that the progenitors of current
galaxies in the field with mean $L <0.5 L^* $ and virial circular velocity of
$100-250 {\rm km/sec}$ could be the associated DLA galaxies. For these systems
we detected a trend for $<L/L^*>$ to increase with redshift.(Abridged)
