  The structure of polyelectrolytes is highly sensitive to small changes in the
interactions between its monomers. In particular, interactions mediated by
counterions play a significant role, and are affected by both specific
molecular effects and generic concentration effects. The ability of
coarse-grained models to reproduce the structural properties of an atomic model
is thus a challenging task. Our present study compares the ability of different
kinds of coarse-grained models: (i)~to reproduce the structure of an atomistic
model of a polyelectrolyte (the sodium polyacrylate), (ii)~to reproduce the
variations of this structure with the number of monomers and with the
concentration of the different species. We show that the adequate scalings of
the gyration radius of the polymer~$R_{\rm g}$ with the number of monomers~$N$
and with the box size~$L_{\rm box}$ are only obtained, first, if the monomer
charges and the counterions are explicitely described, and second, if an
attractive Lennard-Jones contribution is added to the interaction between
distant monomers. Also, we show that implicit ion models are relevant only to
the high electrostatic screening regime.
