Revealing the Shape of Genome Space via \({k}\)-mer Topology

Abstract. Despite decades of research, understanding the structure and shape of genome space remains a major challenge due to the high similarity, variability, and evolutionary plasticity among species, genes, and other biological entities. We introduce [Formula: see text]-mer topology, a novel computational framework for capturing the shape of genome space through topological data analysis. This approach employs persistent Laplacians, a recent spectral theory extension of persistent homology, to track the evolution of topological features across [Formula: see text]-mer frequency distributions in genomes. We define a new topological genetic distance based on both topological invariants and nonharmonic spectral information, enabling the construction of phylogenetic trees that reflect underlying genome geometry. Our method achieves superior classification and clustering performance across a diverse set of benchmark datasets, including mammalian mitochondrial genomes, SARS-CoV-2 variants, Ebola virus, influenza genes, and bacterial genomes. [Formula: see text]-mer topology reveals fundamental geometric patterns in genomic data and offers fresh perspectives on evolutionary relationships and genomic organization. Relevance to Life Sciences. This work addresses a central problem in evolutionary biology: how to quantitatively represent and compare the genomic content of diverse organisms in a way that reflects their biological relatedness. By analyzing the shape of [Formula: see text]-mer frequency spaces in viral and bacterial genomes, we uncover structural patterns that correspond to meaningful biological groupings. Our topological genetic distance matrix enables phylogenetic reconstructions that align with known evolutionary histories and also highlight novel groupings that may warrant further biological investigation. Notably, our analysis of SARS-CoV-2 variants sheds light on vaccine escape patterns, revealing structural separations between pre- and post-vaccine strains. These findings suggest the potential of topological methods in guiding public health responses, improving genomic surveillance, and identifying targets for future experimental study. Mathematical Content. The core of our approach involves a novel application of persistent Laplacians, a spectral theory extension of algebraic topology that incorporates both harmonic and nonharmonic spectral features. Genomic sequences are represented as points in a high-dimensional [Formula: see text]-mer frequency space, from which a family of simplicial complexes is constructed across filtration scales. Persistent Laplacians are computed for each filtration, and their spectra are used to define topological invariants and spectral signatures of genomic structure. A new distance metric is introduced that captures both local and global structural information. Our analysis combines tools from algebraic topology, spectral theory, and geometric analysis to provide a multiscale, interpretable, and biologically meaningful representation of genome space.