  We investigate a model of interacting dark matter and dark energy given by a
modified holographic Ricci cutoff in a spatially flat
Friedmann-Robertson-Walker (FRW) space-time. We consider a nonlinear
interaction consisting of a significant rational function of the total energy
density and its first derivative homogeneous of degree one and show that the
effective one-fluid obeys the equation of state of a relaxed Chaplygin gas. So
that, the universe is dominated by pressureless dark matter at early times and
undergoes an accelerated expansion in the far future driven by a strong
negative pressure. We apply the $\chi^{2}$-statistical method to the
observational Hubble data and the Union2 compilation of SNe Ia for constraining
the cosmological parameters and analyze the feasibility of the modified
holographic Ricci cutoff. By using the new $Om$ diagnostic method, we find that
the effective model differs substantially from the $\Lambda$--CDM one, because
it gets the accelerated expansion faster than the $\Lambda$--CDM model.
Finally, a new model with a third component decoupled from the interacting dark
sector is presented for studying bounds on the dark energy at early times.
