Pattern Matching and Pattern Discovery Algorithms for Protein Topologies

We describe algorithms for pattern matching and pattern learning in TOPS diagrams (formal descriptions of protein topologies). These problems can be reduced to checking for subgraph isomorphism and finding maximal common subgraphs in a restricted class of ordered graphs. We have developed a subgraph isomorphism algorithm for ordered graphs, which performs well on the given set of data. The maximal common subgraph problem then is solved by repeated subgraph extension and checking for isomorphisms. Despite the apparent inefficiency such approach gives an algorithm with time complexity proportional to the number of graphs in the input set and is still practical on the given set of data. As a result we obtain fast methods which can be used for building a database of protein topological motifs, and for the comparison of a given protein of known secondary structure against a motif database. 1 Biological motivation Once the structure of a protein has been determined, the next task for biologist is to find hypotheses about its function. One possible approach is pairwise comparison of the structure with the structures of proteins whose functions are already known. There are already several tools that allow such comparisons, for example DALI [7] (http://www.ebi.ac.uk/dali/) or CATH [11] (http://www.biochem.ucl.ac.uk/bsm/cath/). However there are two weaknesses with such approach. Firstly, as the number of proteins with given structure is growing the time needed to do such comparisons is also growing. Currently there are about 15000 protein structure descriptions deposited in the Protein ∗ Supported by a Wellcome Trust International Research Award Data Bank [1] (http://www.rcsb.org/pdb/), but in the future this number may grow significantly. Secondly, even if a similarity with one or more proteins has been found, it may not be apparent whether this may also imply functional similarity, especially if the similarity is not very strong. Another possibility is to try to use a similar approach at a structure level to that used for sequences in PROSITE database [6] (http://ca.expasy.org/prosite/). That is pre-compute a database of motifs for proteins with known structures – i.e. structural patterns which are associated with some particular protein function. This effectively requires comnputing the maximal common substructure for a set of structures. One such approach is that of CORA [10], based on multiple structural alignments of protein sequences for given CATH families. Both of these approaches have been successfully used for protein comparison on the sequence level. The main difficulty in adapting them to the structural level is the complexity of the necessary algorithms – whilst exact sequence comparison algorithms work in linear time, exact structure comparison algorithms may require exponential time and the situation only gets worse with algorithms for finding maximal common substructures. Another aspect of the problem is that it is far from clear which is the best way to define structure similarity. There are many possible approaches, which require different algorithmic methods and are likely to produce different results. Our work is aimed at the development of efficient comparison and maximal common substructure algorithms using TOPS diagrams for structural topology descriptions, defining structure similarity in a natural way that arises from such formalisation, and at the evaluation of usefulness of such approach. The drawback of such an approach is that TOPS diagrams are not very rich in information; however it has the advantage that it is still possible to define practical algorithms for this level of abstraction.