Scale-Dependent Concordance of Poincaré and Tree-Based TAD Hierarchies and Its Coupling to A/B Compartmentalization in Isogenic Breast Cancer Cells

Background. Hyperbolic embedding of Hi-C matrices into a Poincaré disc recovers the full TAD hierarchy from a single embedding. Two questions remain open: whether the hierarchy is method-independent, and whether the radial coordinate acquires new biological meaning upon neoplastic transformation.Results. Using the isogenic MCF10A/MCF7 pair at 40 kb (chr17, chr1, chr19), we show that (i) the Poincaré hierarchy is stable across lines and chromosomes (loss 1.4–2.9; six levels; ~16.7% of loci per level); (ii) concordance with an independent tree-based method (OnTAD) is level- and line-dependent: negligible for small TADs in MCF10A (ARI = 0.001, within noise) but detectable in MCF7 (ARI = 0.022, 16 SD above chance), and consistently detectable for large TADs (ARI = 0.047 for Lv.5–6, 36–48 SD above chance); (iii) mean TAD size increases monotonically from Lv.1 to Lv.6 (Spearman ρ = +0.39 to +0.70), quantitatively confirming scale encoding; (iv) boundaries are partially conserved between MCF10A and MCF7 (Jaccard 0.20–0.59), with reorganization concentrated at the largest domains; and (v) in MCF7 the radial coordinate correlates with the A/B compartment eigenvector (ρ = −0.43 chr17, −0.41 chr1), a coupling that is weak or unstable in MCF10A; (vi) a dimensionless hierarchy-shift metric—the median radius difference Δ r̃—is positive on all three chromosomes (chr17 +0.020, chr1 +0.013, chr19 +0.047, all p < 0.001 by permutation test) and is reproduced at the gene level (ΔTHgene > 0, all p < 0.05), indicating a redistribution of the hierarchy toward the periphery rather than a simple expansion of large domains.Conclusions. The Poincaré hierarchy is scale-encoding and shows scale- and line-dependent concordance with an independent method, and it acquires a compartmental signature in the transformed state. These results extend the Poincaré framework toward comparative, method-agnostic analysis of 3D chromatin organization.