Background: DNA methylation is a key epigenetic mechanism whose dynamics are closely linked to ageing. Modern epigenetic clocks predict biological age based on the average methylation level. The concept of “epigenetic rejuvenation” posits that at early stages of development, the biological age of the embryo may decrease, reaching a minimum (“ground zero”) at the gastrulation stage. However, standard averaging methods may not account for important rearrangements in the internal structure of methylation. Objective: To apply the apparatus of information theory and topological data science to the analysis of scNMT-seq data and to test whether DNA methylation entropy decreases from stage E4.5 to E6.5, which would correspond to an approach towards the biological zero state. Methods: Publicly available scNMT-seq data (GSE121690) were analyzed. Five entropy measures were calculated for each cell (Shannon, Renyi, Tsallis, LZ-complexity, local gradient entropy (entropy of variations in the smoothed histogram of the methylation distribution), and persistent entropy (PE)—a topological complexity measure). For the five-dimensional entropy feature space, a Rips complex was constructed, and persistence diagrams H_0 and H_1 were computed. Results: All five entropy measures decreased significantly, with LZ complexity showing the largest relative reduction (−28.4%) and the strongest independent signal (partial r = −0.181). Among all the complexity measures studied, LZ complexity exhibited the largest relative reduction, underscoring its heightened sensitivity to the progressive ordering of the epigenetic landscape. Notably, the ternary encoding of LZ complexity showed strong correlation with Shannon entropy (r = 0.71), indicating that algorithmic complexity, when accounting for partial methylation states, aligns closely with statistical entropy while retaining sensitivity to spatial order. The consistency of results across binary and ternary encodings confirms the robustness of LZ complexity as a structural biomarker. Persistent entropy confirmed the general dynamics (decrease from 15.91 to 14.89, p = 0.01). Topological analysis of the multidimensional space revealed a qualitative reorganization: at stage E6.5, stable cyclic structures (H₁) emerge, while at E4.5 the space is dominated by a single large-scale cycle. Null model validation confirmed that the observed H₁-cycles are genuine topological features rather than random fluctuations. Comprehensive topological characterization showed that normalized persistent entropy increases from 0.846 to 0.882 (p < 0.001), while maximum persistence decreases from 0.446 to 0.218 (p < 0.001), reflecting a transition from a homogeneous state to structured diversification—multiple, evenly distributed cycles corresponding to distinct cell lineages. Consistent with this, regional disorder (RE/RD) at the single-cell level decreases from E4.5 to E6.5 (RE: −25.5%, RD: −27.4%, p < 10⁻¹³), while global entropy also decreases, together painting a picture of epigenetic rejuvenation as ordered consolidation at the whole-genome scale. An SVM model trained on 15 entropy and structural features achieved stage classification with an accuracy of 93.4% and AUC of 0.981, confirming the diagnostic potential of the approach. Conclusions: The decrease in DNA methylation entropy from E4.5 to E6.5 corresponds to an approach to “ground zero”—the point of minimum biological age in embryogenesis—and supports the hypothesis of a link between decreasing entropy and epigenetic rejuvenation. The addition of topological analysis reveals the hidden organization of epigenetic diversity, showing that ordering does not lead to homogenization but is accompanied by the formation of distinguishable cell lineages.

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.

Background/Objectives: DNA methylation dynamics are closely linked to ageing. While epigenetic clocks predict biological age from average methylation levels, they may miss structural rearrangements. The "epigenetic rejuvenation" hypothesis posits that embryonic biological age decreases toward a minimum ("ground zero") at gastrulation. This study tested whether DNA methylation entropy decreases from stage E4.5 to E6.5. Methods: Publicly available single-cell nucleosome, methylation and transcription sequencing (scNMT-seq) data (GSE121690; 364 mouse cells) were analysed. Five entropy measures were calculated: Shannon, Renyi, Tsallis, Lempel-Ziv (LZ) complexity, and local gradient entropy. Persistent entropy (PE) and topological analysis (Rips complex, persistence diagrams H₀ and H₁) were computed. A support vector machine (SVM) classifier was trained to distinguish developmental stages. Results: Mean Shannon entropy decreased from 0.841 (E4.5) to 0.805 (E6.5; p < 0.01). All five entropy measures and PE decreased significantly. LZ complexity showed the largest reduction (−28.4% binary, −11.4% ternary; p < 0.001). Ternary LZ correlated with Shannon entropy (r = 0.71). PE decreased from 15.91 to 14.89 (p = 0.01). Topological analysis revealed reorganisation: E6.5 showed stable H₁-cycles (9 intervals), absent at E4.5 (2 intervals), reflecting lineage-specific diversification. Regional entropy and disorder decreased (RE: −25.5%, RD: −27.4%, p < 10⁻¹³). The SVM achieved 93.4% accuracy (AUC = 0.981). Conclusions: The decrease in DNA methylation entropy from E4.5 to E6.5 supports epigenetic rejuvenation as an approach to "ground zero." Topological analysis revealed a qualitative reorganisation: E6.5 showed stable H₁-cycles (9 intervals) that were largely absent at E4.5 (2 intervals). Comprehensive characterisation with balanced subsampling revealed a significant decrease in max persistence (0.446 → 0.218, p = 0.03) and total persistence (2.170 → 1.070, p < 0.001), indicating a transition from a single dominant cycle to multiple smaller, evenly distributed cycles.

The solvent-accessible surface area (SASA) of amino acid residues is a crucial parameter for protein structure analysis; however, precise computational methods such as FreeSASA are computationally expensive. As an alternative, empirical approximations based on residue interaction network (RIN) graphs can offer high speed while maintaining acceptable accuracy. In this study, we propose and validate three empirical functions for estimating relative SASA—approx_sasa, surface_score, and exp_sasa—using node degree as the sole argument. We present a comparative analysis of two graph construction approaches: the classical Cα-graph (8 Å threshold) and the heavy-atom graph (HAG, 5.0 Å threshold). Parameters were calibrated on a dataset of 509 protein structures (128,794 residues) using the true relative SASA calculated by the FreeSASA library. An extended set of 11 topological features was also developed and validated. Ensemble models (Random Forest, XGBoost) achieved a best performance of MAE = 0.057 ± 0.033 and Pearson r = 0.915 ± 0.080 on HAG, outperforming graph neural networks (GCN, GAT, GraphSAGE) in this setting. The empirical formulas demonstrate extreme computational efficiency (0.008 ms per structure), ~26,000× faster than FreeSASA, making them suitable for large-scale pipelines requiring both speed and interpretability. Random Forest on HAG is recommended for applications requiring maximum accuracy, while GraphSAGE on HAG is a viable deep learning alternative.

We present EPI-OrGenHm (Epigenetic Ordered Gene Homeostasis) — a closed-loop computational framework integrating DNA methylation dynamics, gene expression regulation, virtual perturbation action, and entropy-based anomaly detection within a unified GPU-accelerated environment. The model is calibrated on public scNMT-seq data (GSE121690, 364 mouse cells, 43 genes, 9362 CpG sites) using a previously developed structural-entropy approach. The regulatory network combines STRING DB (functional interactions) and TRRUST (directed regulatory links). The key methodological innovation is the implementation of negative feedback in the "expression → methylation" loop, which stabilizes the system and prevents methylation collapse under virtual demethylating perturbation. A sigmoid age coordinate with a quadratic penalty for extreme hypomethylation is introduced for quantifying relative epigenetic state changes. Results (model coordinates): Under the current parameterization, the baseline model age coordinate is 95.7 a.u. The maximum reduction of this coordinate is 2.75 a.u. (to 92.96 a.u.) at perturbation strength S = 0.179. The model-predicted window with zero entropy-flagged dysregulation is S ∈ [0, 0.357]. Above S = 0.393, 1.3–1.9% of cells are flagged by the entropy-based classifier. A pronounced nonlinearity is observed: 20% of the maximum perturbation (S = 0.10) produces 88.4% of the maximum effect. Entropy analysis (Shannon, LZ-complexity, persistent entropy, KL-divergence, spatial entropy) confirms these metrics as sensitive markers of structural epigenetic changes. The OrGenHm metric is introduced as a theoretical lower bound (M̄ < 0.185) for catastrophic loss of ordered gene homeostasis; in the present simulations this bound is never reached, and early dysregulation is detected by entropy thresholds instead. EPI-OrGenHm enables statistically robust in silico screening of virtual perturbation regimes (n = 1000 cells per dose, CV of optimal strength < 5% across LOOCV folds) and generates hypotheses for target genes and parameter ranges. Michaelis–Menten fitting (R² = 0.696) with leave-one-dose-out cross-validation (RMSE = 0.575 a.u.) confirms ranking stability. All dose effects are statistically significant (p < 0.001, Bonferroni-corrected Welch's t-test, n = 1000).Scope statement: The model is calibrated on embryonic mouse cells and predicts relative changes in a model-specific epigenetic coordinate, not absolute human age or clinical outcomes. Translation to aging somatic tissue requires recalibration on aged-tissue data and experimental validation.

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.