  Employing the four-band tight-binding model we study theoretically the doping
dependence of the spin response in the normal state of novel Fe-based pnictide
superconductors. We show that the commensurate spin density wave (SDW)
transition that arises due to interband scattering between the hole
$\alpha$-pockets and the electron $\beta$-pockets disappears already at the
doping concentration $x \approx 0.04$ reflecting the evolution of the Fermi
surfaces. Correspondingly, with further increase of the doping the
antiferromagnetic fluctuations are suppressed for $x > 0.1$ and the
Im$\chi({\bf Q_{AFM}},\omega)$ becomes nearly temperature independent. At the
same time, we observe that the uniform susceptibility deviates from the
Pauli-like behavior and is increasing with increasing temperature reflecting
the activation processes for the $\alpha$-Fermi surfaces up to temperatures of
about T=800K. With increase of the doping the absolute value of the uniform
susceptibility lowers and its temperature dependence changes. In particular, it
is a constant at low temperatures and then decreases with increasing
temperature. We discuss our results in a context of recent experimental data.
