Star formation in galaxies is inefficient, and understanding how star formation is regulated in galaxies is one of the most fundamental challenges of contemporary astrophysics. Radiative cooling, feedback from supernovae and active galactic nuclei, large-scale dynamics and dissipation of turbulent energy act over various time and spatial scales, and all regulate star formation in a complex gas cycle... This paper presents the physics implemented in a new semi-analytical model of galaxy formation and evolution: G.A.S.. The fundamental underpinning of our new model is the development of a multi-phase interstellar medium in which energy produced by supernovae and active galactic nuclei maintains an equilibrium between the diffuse, hot, stable gas and a cooler, clumpy, low-volume filling factor gas. The hot gas is susceptible to thermal and dynamical instabilities. We include a description of how turbulence leads to the formation of giant molecular clouds through an inertial turbulent energy cascade. We explicitly model the evolution of the velocity dispersion at different scales of the cascade and account for thermal instabilities in the hot halo gas. We show that rapid and multiple exchanges between diffuse and unstable gas phases strongly regulates star-formation rates in galaxies because only a small fraction of the unstable gas is forming stars. For high mass halos, cooling is naturally regulated by the growth of thermal instabilities, so we do not need to implement strong AGN feedback. The characteristic timescales describing the gas cycle are in good agreement with observations. Results are in good agreement with the observed stellar mass function from z=6.0 to z=0.5. Thermal instabilities and the cascade of turbulent energy in the dense gas phase introduce a delay between gas accretion and star formation, which keeps galaxy growth inefficient in the early Universe. (read more)