The phylogenetic Kantorovich-Rubinstein metric for environmental sequence samples

Area of Science:

• Microbial ecology
• Bioinformatics
• Computational biology

Background:

• Microbial community surveys commonly use bulk nucleic acid sequencing.
• Comparative methods are essential for assessing differences between microbial communities.
• UniFrac is a widely used, though somewhat ad hoc, phylogenetic distance metric for microbial ecology.

Purpose of the Study:

• To provide a rigorous mathematical foundation for phylogenetic distance metrics used in microbial community analysis.
• To establish the relationship between weighted UniFrac and optimal transport theory.
• To develop computable methods for hypothesis testing in microbial community comparisons.

Main Methods:

• Equating metagenomic samples with empirical distributions on a reference phylogenetic tree.
• Identifying the weighted UniFrac distance as the Kantorovich-Rubinstein (earth mover's) distance.
• Developing integral representations for the distance and its generalizations.
• Approximating p-values using Gaussian process functionals for permutation tests.

Main Results:

• The weighted UniFrac distance is mathematically equivalent to the earth mover's distance between sample distributions on a phylogenetic tree.
• Computable integral forms and L(p) generalizations of the metric were developed.
• A Gaussian process functional approximation for permutation test p-values was established.
• The L(2)-case relates to an analysis-of-variance decomposition.

Conclusions:

• This work provides a strong theoretical basis for weighted UniFrac, connecting it to optimal transport.
• The developed methods offer improved analytical tools for comparing microbial communities.
• The findings facilitate more robust statistical inference in microbial ecology studies.