A relational extension of the notion of motifs: application to the common 3D protein substructures searching problem

The geometrical configurations of atoms in protein structures can be viewed as approximate relations among them. Then, finding similar common substructures within a set of protein structures belongs to a new class of problems that generalizes that of finding repeated motifs. The novelty lies in the addition of constraints on the motifs in terms of relations that must hold between pairs of positions of the motifs. We will hence denote them as relational motifs. For this class of problems, we present an algorithm that is a suitable extension of the KMR paradigm and, in particular, of the KMRC as it uses a degenerate alphabet. Our algorithm contains several improvements that become especially useful when-as it is required for relational motifs-the inference is made by partially overlapping shorter motifs, rather than concatenating them. The efficiency, correctness and completeness of the algorithm is ensured by several non-trivial properties that are proven in this paper. The algorithm has been applied in the important field of protein common 3D substructure searching. The methods implemented have been tested on several examples of protein families such as serine proteases, globins and cytochromes P450 additionally. The detected motifs have been compared to those found by multiple structural alignments methods.