The number of k-mer matches between two DNA sequences as a function of k and applications to estimate phylogenetic distances

We study the number N(k) of length-k word matches between pairs of evolutionarily related DNA sequences, as a function of k. We show that the Jukes-Cantor distance between two genome sequences—i.e. the number of substitutions per site that occurred since they evolved from their last common ancestor—can be estimated from the slope of a function F that depends on N(k) and that is affine-linear within a certain range of k. Integers k(min) and k(max) can be calculated depending on the length of the input sequences, such that the slope of F in the relevant range can be estimated from the values F(k(min)) and F(k(max)). This approach can be generalized to so-called Spaced-word Matches (SpaM), where mismatches are allowed at positions specified by a user-defined binary pattern. Based on these theoretical results, we implemented a prototype software program for alignment-free sequence comparison called Slope-SpaM. Test runs on real and simulated sequence data show that Slope-SpaM can accurately estimate phylogenetic distances for distances up to around 0.5 substitutions per position. The statistical stability of our results is improved if spaced words are used instead of contiguous words. Unlike previous alignment-free methods that are based on the number of (spaced) word matches, Slope-SpaM produces accurate results, even if sequences share only local homologies.