Semi-analytical models (SAMs) are currently the best way to understand the formation of galaxies within the cosmic dark-matter structures. While they fairly well reproduce the local stellar mass functions, correlation functions and luminosity functions, they fail to match observations at high redshift (z > 3) in most cases, particularly in the low-mass range. The inconsistency between models and observations indicates that the history of gas accretion in galaxies, within their host dark-matter halo, and the transformation of gas into stars, are not well followed. Hereafter, we briefly present a new version of the GalICS semi-analytical model. We explore the impacts of classical mechanisms, such as supernova feedback or photoionization, on the evolution of the stellar mass assembly. Even with a strong efficiency, these two processes cannot explain the observed stellar mass function and star formation rate distribution and some other relations. We thus introduce an ad-hoc modification of the standard paradigm, based on the presence of a \textit{no-star-forming} gas component, and a concentration of the star-forming gas in galaxy discs. The main idea behind the existence of the no-star-forming gas reservoir is that only a fraction of the total gas mass in a galaxy is available to form stars. The reservoir generates a delay between the accretion of the gas and the star formation process. This new model is in much better agreement with the observations of the stellar mass function in the low-mass range than the previous models, and agrees quite well with a large set of observations, including the redshift evolution of the specific star formation rate. However, it predicts a large fraction of no-star-forming baryonic gas, potentially larger than observed, even if its nature has still to be examined in the context of the missing baryon problem.