A convenient correspondence between k-mer-based metagenomic distances and phylogenetically-informed β-diversity measures

k-mer-based distances are often used to describe the differences between communities in metagenome sequencing studies because of their computational convenience and history of effectiveness. Although k-mer-based distances do not use information about taxon abundances, we show that one class of k-mer distances between metagenomes (the Euclidean distance between k-mer spectra, or EKS distances) are very closely related to a class of phylogenetically-informed β-diversity measures that do explicitly use both the taxon abundances and information about the phylogenetic relationships among the taxa. Furthermore, we show that both of these distances can be interpreted as using certain features of the taxon abundances that are related to the phylogenetic tree. Our results allow practitioners to perform phylogenetically-informed analyses when they only have k-mer data available and provide a theoretical basis for using k-mer spectra with relatively small values of k (on the order of 4-5). They are also useful for analysts who wish to know more of the properties of any method based on k-mer spectra and provide insight into one class of phylogenetically-informed β-diversity measures.