$\boxed{\frac{720^{\circ}}{7}}$

Let $U$, $V$, $W$ be points on the sides $DE$, $FG$, $GA$, respectively, dividing respective side in $2 : 1$ ratio. Then, it is easy to see that $PQRUSVW$ becomes also a regular heptagon. Hence these $7$ points lie on the circumference of a same circle, and divides the circumference into $7$ arcs of equal length. From this it follows that $\angle PRS$ is $\frac{1}{2}$ of the angle subtended by the arc $PWVS$ of the circum-circle of this heptagon at its center, and thus $\angle PRS = \frac{1}{2} \cdot \frac{3}{7} \cdot 360^{\circ}$. Similarly, we have $\angle QSR = \frac{1}{2} \cdot \frac{1}{7} \cdot 360^{\circ}$. Consequently, we get $\angle PTS = \angle PRS + \angle QSR = \frac{720^{\circ}}{7}$.