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The idea that structural disorder might be a novel mechanism of protein interaction is widespread in the Literature, although the number of statistically significant structural studies supporting this is surprisingly low. At variance with previous works, our conclusions rely exclusively on a large-scale analysis of all the 134337 X-ray crystallographic structures of the Protein Data Bank averaged over clusters of almost identical protein sequences. In this work, we explore the complexity of the organization of all the interaction interfaces observed when a protein lies in alternative complexes, showing that interfaces progressively add up in a hierarchical way. We further investigate the connection of this complexity with different measures of structural disorder: the standard missing residues and a new definition, called "soft disorder", that covers all the flexible and structurally amorphous residues of a protein. We show evidences that both the interaction interfaces and the soft disordered regions tend to involve roughly the same amino-acids of the protein, and preliminary results suggesting that soft disorder spots those surface regions where new interfaces are progressively accommodated by complex formation. Our results suggest that disordered regions not only carry crucial information about the location of alternative interfaces within complexes, but also of the order of the assembly. We verify these hypotheses in several examples. We finally compare our measures of disorder with several disorder predictors, showing that these latter are optimized to predict the residues that are missing in all the alternative structures of a protein, and they are not able to catch the progressive evolution of the disordered regions upon complex formation. Yet, the predicted residues, if not missing, tend to be characterized as soft disordered.