Topological Sequence Analysis of Genomes: Category theory Approaches

TL;DR

CTSA introduces a category-theoretic, multiscale approach to genomic and proteomic sequence analysis by modeling sequences as resolution categories and computing persistent homology of derived substructure complexes. It merges alignment-free scalability with structure-aware information via anchored subsequences and a hierarchy of k-mer spaces, offering robust topological embeddings. In protein–nucleic acid binding prediction, CTSA achieves Pearson correlation $r=0.709$ and RMSE $1.29$ kcal/mol, and in SARS-CoV-2 variant clustering it attains 100% accuracy, outperforming multiple baselines. The results suggest that topological features from category-theoretic representations capture informative sequence structure across tasks and could generalize to other domains and language-model integrations.

Abstract

Sequence data, such as DNA, RNA, and protein sequences, exhibit intricate, multi-scale structures that pose significant challenges for conventional analysis methods, particularly those relying on alignment or purely statistical representations. In this work, we introduce category-based topological sequence analysis (CTSA ) of genomes. CTSA models a sequence as a resolution category, capturing its hierarchical structure through a categorical construction. Substructure complexes are then derived from this categorical representation, and their persistent homology is computed to extract multi-scale topological features. Our models depart from traditional alignment-free approaches by incorporating structured mathematical formalisms rooted in sequence topology. The resulting topological signatures provide informative representations across a variety of tasks, including the phylogenetic analysis of SARS-CoV-2 variants and the prediction of protein-nucleic acid binding affinities. Comparative studies were carried out against six state-of-the-art methods. Experimental results demonstrate that CTSA achieves excellent and consistent performance in these tasks, suggesting its general applicability and robustness. Beyond sequence analysis, the proposed framework opens new directions for the integration of categorical and homological theories for biological sequence analysis.

Figures (3)

• Figure 1: Overview of the CTSA framework. A nucleic acid sequence is first encoded into a resolution category that captures its multi-scale structure algebraically. Based on this, a resolution complex is constructed as a topological realization of the sequence. Persistent homology is then applied to extract multiscale topological features, forming the CTSA embeddings. Protein sequences are embedded using ESM2. For interaction prediction tasks, CTSA and ESM2 features are concatenated and used in supervised models. For clustering tasks, only CTSA features are used. Performance is evaluated through cross-validation and comparative variant analysis.
• Figure 2: Overview and evaluation of the CTSA protein-nucleic acid binding affinity prediction framework. a-j Scatter plots of predicted versus experimental binding affinities for each fold in one representative round of 10-fold cross-validation, illustrating the predictive accuracy of the model. k Fold-wise Pearson correlation results across 20 rounds of cross-validation for CTSA. m Fold-wise RMSE across 20 rounds of cross-validation for CTSA. l and n Comparison of average Pearson correlation and RMSE of CTSA and baseline methods, respectively, over all 20 rounds. These plots together demonstrate the robustness and superior performance of CTSA in modeling sequence-based interactions.
• Figure 3: Comparison of alignment-free methods for phylogenetic analysis and feature space clustering of 44 complete SARS-CoV-2 genomes. The sequences are categorized by known variants: Alpha, Beta, Gamma, Delta, Lambda, Mu, GH/490R, and Omicron. a Phylogenetic tree generated by CTSA, showing hierarchical relationships among sequences based on topological features. b PCA visualization of CTSA feature embeddings used in a, illustrating their separability. c-g Phylogenetic trees produced by five baseline methods-NVM, FFP-JS, FFP-KL, Markov K-String (Markov), and Fourier Power Spectrum (FPS).

Theorems & Definitions (4)

• Definition 2.1
• Example 3.1
• Definition 3.1
• Example 3.2