On Suboptimal Alignments of Biological Sequences

It is widely accepted that the optimal alignment between a pair of proteins or nucleic acid sequences that minimizes the edit distance may not necessarily re ect the correct biological alignment. Alignments of proteins based on their structures or of DNA sequences based on evolutionary changes are often di erent from alignments that minimize edit distance. However, in many cases (e.g. when the sequences are close), the edit distance alignment is a good approximation to the biological one. Since, for most sequences, the true alignment is unknown, a method that either assesses the signi cance of the optimal alignment, or that provides few \close" alternatives to the optimal one, is of great importance. A suboptimal alignment is an alignment whose score lies within the neighborhood of the optimal score. Enumeration of suboptimal alignments [Wa83, WaBy] is not very practical since there are many such alignments. Other approaches [Zuk, Vi, ViAr] that use only partial information about suboptimal alignments are more successful in practice. We present a method for representing all alignments whose score is within any given delta from the optimal score. It represents a large number of alignments by a compact graph which makes it easy to impose additional biological constraints and select one desirable alignment from this large set. We study the combinatorial nature of suboptimal alignments. We de ne a set of \canonical" suboptimal alignments, and argue that these are the essential ones since any other suboptimal alignment is a combination of few canonical ones. We then show how to e ciently enumerate suboptimal alignments in order of their score, and count their numbers. Examples are presented to motivate the problem. Since alignments are essentially (s; t)-paths in a directed acyclic graph with (possibly negative) weights on its edges, our solution gives an extremely simple method to enumerate all K shortest (or longest) paths from s to t in such graphs in increasing order, as well as all (s; t) paths that are within of the optimum, for any . We compare this solution with known algorithms that nd the K-best shortest paths in a graph. ? Supported by a Postdoctoral Fellowship from the Program in Mathematics and Molecular Biology of the University of California at Berkeley, under National Science Foundation Grant DMS-9720208