We review part of our recent ab initio molecular dynamics study on the Ti-based Ziegler-Natta supported heterogeneous catalysis of alpha-olefins. The results for the insertion of ethylene in the metal-carbon bond are discussed as a fundamental textbook example of polymerization processes. Comparison with the few experimental data available has shown that simulation can reproduce activation barriers and the overall energetics of the reaction with sufficient accuracy. This puts these quantum dynamical simulations in a new perspective as a virtual laboratory where the microscopic picture of the catalysis, which represents an important issue that still escapes experimental probes, can be observed and understood. These results are then discussed in comparison with a V-based catalyst in order to figure out analogies and differences with respect to the industrially more successful Ti-based systems.