Similarity-preserving Metrics for Amino-acid Sequences

Sequence alignments and sequence similarity scores derived from them are the most common tools for comparing amino acid and DNA sequences. Different scoring schemes, from simple +6/-1 to PAM and BLOSUM scoring matrices have been devised for highlighting particular biological or evolutionary properties of the sequences to compare. However, most methods of classical, as well as of non-parametrical statistics, including a number of neural network approaches, rely on another measure of similarity between data: the distance measure. It would be therefore of an advantage, if we could derive a distance between two sequences from their similarity score. Intuitively, it is clear that these two measures are somehow related: the higher the similarity between sequences, the lower the distance between them should be. But, contrary to similarity score, which can be defined in a fairly arbitrary (although not always meaningful) manner, there are three elementary requirements for a distance measure. A Distance measure on a set D is a function d : D × D → R which is equal to zero iff both arguments are equal, symmetric and satisfies the Cauchy-Schwarz-Bunyakovskii inequality (the triangle inequality). As a consequence, such a function is always positive-semidefinite. Distances are most commonly defined on vector spaces, but are not limited to them. Any space on which a function with above properties is defined is called a metric space. A number of metrics for strings have been proposed, many of them not really being metrics, for failing on one or more of the requirements. A simple and computationally very effective " distance " measure for sequences is the feature distance [1]. A feature is a short substring, usually referred to as N-gram, N being its length. The feature distance is then computed as the number of features the two strings differ in. It must be noted that this measure is not a distance, for two different strings can have zero distance. For example, strings, AABA and ABAA contain the same bigrams, so with N = 2 the " distance " between them is zero. Another very common distance measure for strings is the Levenshtein distance [2]. It measures the minimum effort needed to transform one string into another, using basic edit operations: replacement , insertion and deletion of a symbol. Generally, each of these operations has a cost assigned to it, in which case the distance function is usually referred to as weighted Levenshtein distance. …