We study the space-time concentration asymptotics of radially decreasing solutions of the parabolic-elliptic Keller-Segel system of chemotaxis, in a ball or in the whole space. We show that, in dimensions $3\le n\le 9$, for any solution $u$ blowing at time $T$, there exists a nonflat backward self-similar solution U such that $u/U\to 1$ as $(x,t)\to(0,T)$. This macroscopic behavior is important from the physical point of view, since it gives a sharp description of the concentration phenomenon in the scale of the original space-time variables $(x,t)$, which was previously known only in the microscopic scale $|x| \le O(\sqrt{T-t})$. As a consequence, we obtain the sharp final profile: $\lim |x|^2 u(x,T) = L > 0$ as $x\to 0$, as well as the precise two-sided space-time global estimates (joint work with Loth Chabi).