  We recall our recent description of quark parton densities of the proton at
$Q^2=4 GeV^2$ in terms of Fermi-Dirac distributions parametrized with very few
free parameters. We have also proposed some simple assumptions to relate
unpolarized and polarized quark parton densities which lead to a fair
description of the spin-dependent structure functions $xg^p_1(x,Q^2)$ and
$xg^n_1(x,Q^2)$ at low $Q^2$. We will show the predictions we obtain after a
straightforward DGLAP $Q^2$ evolution and comparison in a much broader $x$ and
$Q^2$ range, with several recent and accurate deep-inelastic scattering data.
In particular, we will see that we get an excellent agreement with the sharp
rise of $F^{ep}_2(x,Q^2)$ for small $x$, recently observed at HERA. Finally, we
give several predictions for lepton pair and gauge boson production in $pp$ and
$pn$ collisions at high energies which will be tested in the future at RHIC.
