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

The hierarchical organization of topological associating domains (TADs) is a funda-mental feature of three-dimensional genome architecture, yet its systematic character-ization remains challenging due to the scale-dependent nature of conventional detection methods. Here, we apply hyperbolic embedding into a Poincaré disc to analyze TAD hierarchies from Hi-C data. In this framework, genomic loci are 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 pa-rameter tuning across resolutions. 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. 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 hierar-chical chromatin rearrangements associated with cancer.