4. In rectangle $A B C D$, $A B=2, A D=1$, point $P$ on side $D C$ (including points $D, C$) and point $Q$ on the extension of $C B$ (including point $B$) satisfy $|\overrightarrow{D P}|=|\overrightarrow{B Q}|$, then the dot product $\overrightarrow{P A} \cdot \overrightarrow{P Q}$ of vector $\overrightarrow{P A}$ and vector $\overrightarrow{P Q}$ has a minimum value of $\qquad$.