Exploring geometry of genome space via Grassmann manifolds

It is important to understand the geometry of genome space in biology. After transforming genome sequences into frequency matrices of the chaos game representation (FCGR), we regard a genome sequence as a point in a suitable Grassmann manifold by analyzing the column space of the corresponding FCGR. To assess the sequence similarity, we employ the generalized Grassmannian distance, an intrinsic geometric distance that differs from the traditional Euclidean distance used in the classical k-mer frequency-based methods. With this method, we constructed phylogenetic trees for various genome datasets, including influenza A virus hemagglutinin gene, Orthocoronavirinae genome, and SARS-CoV-2 complete genome sequences. Our comparative analysis with multiple sequence alignment and alignment-free methods for large-scale sequences revealed that our method, which employs the subspace distance between the column spaces of different FCGRs (FCGR-SD), outperformed its competitors in terms of both speed and accuracy. In addition, we used low-dimensional visualization of the SARS-CoV-2 genome sequences and spike protein nucleotide sequences with our methods, resulting in some intriguing findings. We not only propose a novel and efficient algorithm for comparing genome sequences but also demonstrate that genome data have some intrinsic manifold structures, providing a new geometric perspective for molecular biology studies.