Applying the Poincaré Disc Method to Analyze the Hierarchical Structure of TADs in Hi-C Data

Background/Objectives: The hierarchical organization of topological associating domains (TADs) is a fundamental feature of three-dimensional genome architecture, yet its systematic characterization remains challenging due to the scale-dependent nature of conventional detection methods. Here, we applied hyperbolic embedding into a Poincaré disc to analyze TAD hierarchies from Hi-C data. Methods: Genomic loci were projected onto the disc such that the radial coordinate encodes hierarchical depth: subTADs localize near the center, while metaTADs shift toward the periphery. The method was validated on ten cell lines, including the isogenic MCF10A/MCF7 pair, and does not require manual parameter tuning across resolutions. Results: In the MCF10A/MCF7 isogenic system, neoplastic transformation was associated with a redistribution of the hierarchy: the number of small TADs (levels 1-3) decreased by 6.7%, whereas large TADs (levels 4-6) increased by 4.7%, with the largest domains (level 6) showing a 33.9% increase. Comparison with standard approaches-Insulation Score (IS) and Directionality Index (DI)-yielded an average Jaccard index of 0.568 and F1 score of 0.718 against DI. Conclusions: Unlike IS and DI, which operate at fixed scales, the proposed approach recovers the full TAD hierarchy from a single embedding, enabling cross-resolution and cross-cell-line comparisons without parameter reoptimization. These results demonstrate that the Poincaré disc method provides a robust, interpretable, and scale-invariant framework for detecting hierarchical chromatin rearrangements associated with cancer.