  In the last decade a Quasi-Particle Model (QPM) has been developed to study
charm quark dynamics in ultra-relativistic heavy-ion collisions supplying a
satisfactory description of the main observables for $D$ meson and providing an
estimate of the space-diffusion coefficient $D_s(T)$ from the phenomenology. In
this paper, we extend the approach to bottom quarks describing their
propagation in the quark-gluon plasma within an event-by-event full Boltzmann
transport approach followed by a coalescence plus fragmentation hadronization.
We find that QPM approach is able to correctly predict the first available data
on $R_{AA}(p_T)$ and $v_{2}(p_T)$ of single-electron from B decays without any
parameter modification w.r.t. the charm. We show also predictions for
centralities where data are not yet available for both $v_{2}(p_T)$ and
$v_{3}(p_T)$. Moreover, we discuss the significant breaking of the expected
scaling of the thermalization time $\tau_{th}$ with $M_Q/T$, discussing the
evolution with mass of $D_s(T)$ to better assess the comparison to lQCD
calculations. We find that at $T=T_c$ charm quark $D_s(T)$ is about a factor of
2 larger than the asymptotic value for $M \rightarrow \infty$, while bottom
$D_s(T)$ is only a $20-25\%$ higher. This implies a $D_{s}(T)$ which is
consistent within the current uncertainty to the most recent lattice QCD
calculations with dynamical quarks for $M \rightarrow \infty$.
