  An analytical approach is proposed to study the evolution of the star-forming
galaxy (SFG) main sequence (MS) and the fraction of dust-obscured SF up to
$z\sim4$. Far-ultraviolet (FUV) and infrared (IR) star formation rates, SFRs,
are described as conditional probability functions of $M_{\ast}$. We convolve
them with the galaxy stellar mass function (GSMF) of SFGs to derive the FUV and
IR LFs. The 2 SF modes formalism is used to describe starburst galaxies. By
fitting observed FUV and IR LFs, the parametrization of SFR$_{\rm
FUV}-M_{\ast}$ and SFR$_{\rm IR}-M_{\ast}$ are constrained. Our derived
SFR$_{\rm FUV+IR}-M_{\ast}$ reproduces the evolution of the MS as compared to
other observational inferences. At any redshift, we find that the sSFR$_{\rm
FUV+IR}-M_{\ast}$ relation for MS SFGs approaches to a power law at the
high-mass end. At lower masses, it bends and eventually the slope sign changes
from negative to positive at very low masses. At $z\sim0$, this change of sign
is at $M_{\ast}\sim5\times10^{8}{\rm M}_{\odot}$ close to dust-obscured SF
regime, $M_{\ast}\sim6\times10^{8}{\rm M}_{\odot}$. The slope sign change is
related to the knee of the FUV LF. Our derived dust-obscured fractions agree
with previous determinations at $0\leq z\leq2.5$. Dust-obscured fractions
depend strongly on mass with almost no dependence with redshift at
$z\gtrsim1.2$. At $z\lesssim0.75$ high-mass galaxies become more "transparent"
compared to their high redshift counterparts. On the opposite, low- and
intermediate-mass galaxies have become more obscured by dust. The joint
evolution of the GSMF and the FUV and IR LFs is a promising approach to study
mass growth and dust formation/destruction mechanisms.

Astrophysics of Galaxies