Compositional representation of protein sequences and the number of Eulerian loops

An amino acid sequence of a protein may be decomposed into consecutive overlapping strings of length K. How unique is the converse, i.e., reconstruction of amino acid sequences using the set of K-strings obtained in the decomposition? This problem may be transformed into the problem of counting the number of Eulerian loops in an Euler graph, though the well-known formula must be modified. By exhaustive enumeration and by using the modified formula, we show that the reconstruction is unique at K ≥ 5 for an overwhelming majority of the proteins in pdb.seq database. The corresponding Euler graphs provide a means to study the structure of repeated segments in protein sequences.