HyperDeepTAD: a topologically associated domains detection method based on multiway chromatin interaction data and deep learning

Abstract

Accurate identification of topologically associated domains (TADs), the basic units of 3D genomes, is crucial for deciphering gene regulatory mechanisms. Existing methods rely mostly on Hi-C pairwise contact signals, and struggle to extract synergistic interactions among multiple loci, limiting boundary accuracy. In contrast, multiway chromatin interaction data can overcome the limitations of binary interactions by directly capturing high-order interactions that occur simultaneously across multiple regions, thereby providing critical data support for a more comprehensive analysis of three-dimensional chromatin organization.This study proposes HyperDeepTAD: which involves constructing hypergraphs from multiway chromatin interaction data to retain high-order information, inputting transition probability matrices into dynamic convolutional networks to capture local features, obtaining candidate boundaries via BiGRU and residual connections, screening with hypergraph clustering coefficients and using cosine similarity to obtain hierarchical TADs. Experiments show it that outperforms Hi-C-based methods in metrics such as boundary enrichment. The source code is available from https://github.com/liukaihua0213-source/HyperDeepTAD.

Supplementary Information

The online version contains supplementary material available at 10.1186/s12864-026-12738-3.

Background

The genome folds into a three-dimensional (3D) structure, which facilitates the smooth execution of gene regulatory processes within the cell nucleus. Currently, chromatin conformation capture technologies such as Hi-C [1] and ChIA-PET [2] can characterize genome-wide chromatin organization, including A/B compartments [3], topologically associated domains (TADs), and chromatin loops [4–6]. These technologies are constrained by the inherent limitations of paired-end sequencing strategies, and can identify only two-way contacts between chromatin loci; In contrast, multiway chromatin interaction data generated by long-read sequencing technologies can directly capture multiway contacts among three or more chromatin loci [7, 8].

In recent years, several emerging technologies including SPRITE [9], Pore-C [10] and HiPore-C [11] have advanced the study of high-order 3D chromatin interactions among multiple genomic loci in single-cell nuclei. Quinodoz et al., using SPRITE, identified two nuclear body-associated transchromosomal high-order interaction hubs within cell nuclei, revealing both the global regulatory role of nuclear bodies in 3D genome organization and the tight link between high-order interactions and chromatin functional states [9]. Deshpande et al. used Pore-C to identify cooperative effects—specifically, sets of genomic loci with interaction frequencies significantly above background—and showed that enhancers and promoters, via multiway interactions, are enriched in active chromatin regions and associated with highly transcribed genes [10]. Zhong et al. employed HiPore-C to elucidate links between allele-specific topological features and 3D genome architecture [11]. Collectively, these findings demonstrate that high-order interactions are detectable in 3D genomics research—a capability beyond that of conventional experiments, which only directly capture pairwise interactions [12–14].

However, in the field of TAD identification, the application of these emerging technologies remains constrained by notable bottlenecks: most studies still convert captured multiway chromatin interaction data into pairwise contact signals and then analyze these signals using established methods, rather than directly leveraging the intrinsic features of multiway chromatin interaction data for TAD calling. This processing approach inevitably results in the loss of high-order information; thus, there is an urgent need to develop a TAD identification method directly based on multiway chromatin interaction data.

This study proposes the HyperDeepTAD method, which is designed for topological associated domain (TAD) identification based on multiway chromatin interaction data (Pore-C read set as an example used in the following parts). HyperDeepTAD first constructs hypergraphs on the basis of aligned Pore-C reads to model synergistic interactions among multiple genomic loci, thereby preserving the multinode interaction information inherent in Pore-C reads. It then generates subhypergraphs to enable localized analysis of these multiway interactions derived from Pore-C data. For TAD boundary determination, the method calculates node transition probabilities to quantify dynamic interaction patterns, employs a dynamic CNN module [15] to capture local features of potential TAD boundaries, utilizes a BiGRU module [16] to mine long-range dependencies in genomic sequences, and incorporates residual connections [17] to reinforce feature associations. TAD boundary identification is framed as a binary classification problem to enable optimized solutions. Following the identification of candidate TAD boundaries via the deep learning network, HyperDeepTAD refines these boundaries through hypergraph clustering coefficient calculations. It then derives hierarchical TADs by using cosine similarity to quantify interaction pattern similarities. Compared with traditional methods that rely on pairwise contact signals, this approach avoids high-order information loss and has significant advantages in terms of key metrics including the biological characteristics of TAD boundaries.

Results

Overview of the HyperDeepTAD framework

HyperDeepTAD identifies TADs from aligned Pore-C read data through the following steps:

1. Constructing and Partitioning Hypergraphs: HyperDeepTAD first constructs a weighted hypergraph on the basis of aligned Pore-C read data. This hypergraph serves as a foundational structure for preserving multilocus synergistic interaction information in Pore-C reads. After hypergraph construction, the method obtains subhypergraphs for each node and calculates the transition probability matrix corresponding to each node (Fig. 1A).

2. Determining TAD boundaries: For each transition probability matrix (corresponding to a node), HyperDeepTAD uses dynamic convolution and BiGRU to extract features, and introduces a residual connection mechanism to fuse the local features captured by the dynamic convolution module with the sequential features extracted by the BiGRU. Then, it determines whether the transition probability matrix corresponds to a TAD boundary (Fig. 1B).

3. Obtaining TAD: For each candidate boundary node, HyperDeepTAD performs screening and filtering by calculating the magnitudes of the bin’s upstream clustering coefficient, downstream clustering coefficient, and cross-boundary clustering coefficient in the hypergraph. This method subsequently assembles these filtered genomic regions via cosine similarity to obtain nested TADs (Fig. 1C).

Fig. 1.

Overall workflow of HyperDeepTAD. A Constructing and Partitioning Hypergraphs. B Determining TAD boundaries. C Obtaining TAD. D Dynamic convolution network. E Hierarchical attention

In the experimental phase, we first assessed the efficacy of HyperDeepTAD in predicting TAD boundaries, followed by a comparison with five other TAD detection methods—TopDom [18], deDoc [19], MSTD [20], CATAD [21], and deepTAD [22]. Using a panel of indicators from three cell lines (GM12878, K562, and mES) including the TAD number and size, average peak value, fold change, boundary marker ratio, MoC, and TADadjR², we conducted a comprehensive evaluation encompassing boundary detection accuracy, structural integrity, biological relevance, consistency with reference TADs, and the goodness of fit of the predicted structures. Specifically, TAD number and size characterize basic structural features and completeness of identified domains; average peak value, fold change, and boundary marker ratio capture regulatory element enrichment / depletion patterns near boundaries to reflect biological relevance; MoC quantifies concordance with reference annotations; and TADadjR² assesses the extent to which predicted architectures align with known structural patterns.

Number and size of TADs

As shown in Fig. 3A and B; Table 1, the TAD characteristics of HyperDeepTAD exhibit resolution dependence: as the resolution increases from 25 kb to 100 kb, the number of TADs continues to decrease, whereas the average size increases. This pattern is consistent with the “scale resolution” logic of the genomic structure [23], reflecting the algorithmic adaptability of HyperDeepTAD to different resolutions. Similarly, we analyzed the number and size of topologically associating domains (TADs) on chromosome 22 of the K562 cell line and chromosome 19 of the mouse embryonic stem cell (mES) line to verify the robustness of our method across different cell lines. Detailed results are provided in Supplementary Tables S14 and S16 and Supplementary Figures S15 and S17.

Fig. 3.

TAD identification performance. A Number of TAD boundaries identified by several methods on chromosome 22 in the GM12878 cell line. B Size of TAD boundaries identified by several methods have been used to detect chromosome 22 in the GM12878 cell line. C Consistency metric MoC values between each pair of methods on chromosomes 20–22 at 25 kb resolution in the GM12878 cell line. D Measurements by TADadjR2 between pairs of loci within genomic distances (0–1 M) on chr22 in the GM12878 cell line

Table 1.

Number and average size of TADs detected on chr22 at different resolutions in the GM12878 cell line

Resolution		HyperDeepTAD	TopDom	deDoc	MSTD	CATAD	deepTAD
25KB	#TAD	227	139	128	560	46	190
	Av.size(MB)	0.712	0.248	0.536	0.112	0.786	0.330
50KB	#TAD	113	77	136	254	40	111
	Av.size(MB)	1.353	0.453	0.524	0.242	0.881	0.523
100KB	#TAD	65	37	107	108	28	39
	Av.size(MB)	2.366	0.959	0.686	0.530	1.282	1.254

TAD boundary quantitative metrics

To determine biological relevance, we systematically computed three key metrics (boundary marker ratio, average peak value, and fold change) for a panel of epigenetic and genetic elements—including RNA polymerase II, H3K4me3, H3K36me3, H3K9me3, CTCF, RAD21, SMC3, and SINE [24, 25]. The Boundary Marker Ratio quantifies the abundance of these elements at boundaries relative to nonboundary regions; the average peak value reflects enrichment intensity at boundaries; and the fold change denotes fold differences in enrichment compared with the genomic background. Together, these metrics elucidate the distribution levels and degrees of regulatory elements at TAD boundaries, which is critical for understanding their functional importance.

The average peak value is an indicator for quantifying density, and specifically refers to a measure of the concentration of occurrence frequencies of such elements near boundaries. It is calculated by statistically analyzing the distribution characteristics of regulatory elements within the boundary regions of topologically associated domains in the genome [24] (Eq. 1).

1

where n denotes the number of TAD boundaries detected in the chromosome, and denotes the average frequency of regulatory elements per 10 kb within a 20 kb range centered on the TAD boundary.

As shown in Table 2, at the 25 kb resolution, HyperDeepTAD outperformed the comparative methods in terms of enrichment scores for six types of core functional markers, including CTCF, H3K4me3, and PolII. At 50 kb and 100 kb resolutions, most indicators maintained leading or suboptimal performance (see Supplementary Material S18 and S19 for detailed results). These cross-scale advantages indicate that the predicted TAD boundaries accurately capture functional units anchored by CTCF, cyclized by cohesin, and enriched with transcriptional machinery, which aligns with the core biological logic of three-dimensional chromatin structure construction and gene expression regulation. In the K562 cell line and mES cell line, this method also showed advantages for certain proteins (Supplementary Material S20, S21 and S22).

Table 2.

Average peak values of 8 related types of biological evidence near TAD boundaries obtained via different methods, on chromosomes 20–22 with a resolution of 25 kb in the GM12878 cell line

	CTCF	H3K4me3	H3K36me3	PolII	RAD21	SINE	SMC3	H3K9me3
HyperDeepTAD	0.287	0.183	0.189	0.218	0.113	0.887	0.26	0.034
TopDom	0.248	0.155	0.185	0.184	0.1	0.894	0.207	0.033
deDoc	0.248	0.138	0.168	0.168	0.088	0.896	0.202	0.037
MSTD	0.119	0.059	0.092	0.071	0.047	0.441	0.088	0.021
CATAD	0.24	0.15	0.187	0.156	0.097	0.898	0.201	0.02
deepTAD	0.237	0.137	0.176	0.165	0.092	0.888	0.194	0.032

The values in bold indicate the best performance in each column

Fold change is an indicator for quantifying differences in the distribution of regulatory elements, and specifically measures the relative magnitude of changes in the density or quantity of regulatory elements between “regions close to TAD boundaries” and “regions far from TAD boundaries” in the genome, thereby reflecting the strength of the boundary’s influence on the distribution of regulatory elements [25] (Eq. 2):

2

where n denotes the number of TAD boundaries detected within the chromosome; denotes the frequency of biological evidence per 10 kb within a 20 kb range centered on the TAD boundary; and denotes the frequency of biological evidence per 10 kb in the bilateral regions 500 kb away from the boundary.

As shown in Table 3, the TAD boundaries identified by HyperDeepTAD from the 25 kb data were significantly enriched in core structural regulatory factors (CTCF, RAD21, SMC3) and transcriptionally active markers (H3K4me3, Pol II). Extending to 50 kb and 100 kb resolutions, the enrichment levels of the aforementioned core markers remain the highest or sub-optimal among the comparative methods (see Supplementary Materials S23 and S24 for detailed results). In the K562 cell line and mES cell line, this method also has advantages for certain proteins (Supplementary Material S25、S26 and S27).

Table 3.

Fold change in 8 related biological features near TAD boundaries obtained via different methods, on chromosomes 20–22 with a resolution of 25 kb in the GM12878 cell line

	CTCF	H3K4me3	H3K36me3	PolII	RAD21	SINE	SMC3	H3K9me3
HyperDeepTAD	0.802	0.551	0.233	0.542	1.292	-0.014	0.994	-0.31
TopDom	0.642	0.342	0.293	0.387	1.137	-0.004	0.74	-0.267
deDoc	0.654	0.224	0.179	0.155	1.044	0.018	0.705	-0.098
MSTD	0.648	0.115	0.422	0.121	0.969	-0.014	0.608	-0.154
CATAD	0.507	0.109	0.123	-0.118	1.009	0.011	0.597	-1.439
deepTAD	0.651	0.221	0.226	0.245	1.037	0.006	0.69	-0.154

The values in bold indicate thebest performance in each column

The boundary marker ratio is used to measure the degree of association between TAD boundaries and regulatory elements. It is calculated by counting the proportion of all TAD boundaries that contain peaks of biological evidence for regulatory elements within a certain range [24, 25]. A higher ratio implies that more TAD boundaries participate in functions such as gene expression regulation by enriching regulatory elements.(Eq. 3):

3

where B is the set of TAD boundaries, with each boundary marked by a peak of biological evidence within 20 kb upstream and 20 kb downstream of the boundary; and n denotes the total number of detected TAD boundaries.

As shown in Table 4; Fig. 2, at the 25 kb resolution, HyperDeepTAD has six indicators that are higher than those of the other methods; whereas at 50 kb and 100 kb resolutions, as many as seven indicators are higher than those of the other methods (see Supplementary Material S28 and S29 for detailed results). The remaining indicators that are not higher also remain in a sub-optimal state, which fully demonstrates the accuracy of HyperDeepTAD in boundary identification. In the K562 cell line and mES cell line, this method also has advantages for certain proteins (Supplementary Material S30、S31 and S32).

Table 4.

Boundary marker ratio of 8 related biological lines near TAD boundaries obtained via different methods, on chromosomes 20–22 with a resolution of 25 kb in the GM12878 cell line

	CTCF	H3K4me3	H3K36me3	PolII	RAD21	SINE	SMC3	H3K9me3
HyperDeepTAD	0.694	0.431	0.333	0.424	0.372	0.993	0.654	0.141
TopDom	0.683	0.413	0.338	0.419	0.371	0.985	0.636	0.151
deDoc	0.661	0.371	0.299	0.38	0.339	0.972	0.608	0.145
MSTD	0.171	0.101	0.073	0.096	0.081	0.271	0.148	0.036
CATAD	0.64	0.422	0.334	0.377	0.341	0.984	0.597	0.084
deepTAD	0.637	0.379	0.3	0.381	0.322	0.982	0.579	0.127

The values in bold indicate thebest performance in each column

Fig. 2.

TAD enrichment analysis. A Enrichment plots of 6 proteins on chromosome 1 in the K562 cell line. B Enrichment plots of 6 proteins on chromosome 22 in the GM12878 cell line. C Comparison of 3 proteins on chromosome 22 in the GM12878 cell line with other methods

TAD concordance measure and segmentation quality assessment

To evaluate the similarity between TADs identified by different methods, we use the consistency metric (MoC) to compare cluster partitions [24]. The MoC is defined as follows(Eq. 4):

4

where P and Q are the TAD comparison partitions composed of andrespectively; and denote the sizes of two independent TADs; and |||| is the size of the overlapping part of the two TADs. The range of the MoC is from 0 to 1.

The MoC quantifies the similarity of prediction results among different TAD detection methods by calculating the weighted sum of ratios between the overlapping area of TAD intervals identified by different methods and the size of their respective intervals, normalizing with the number of partitions, and mapping the results to the range of 0 to 1. This approach provides a quantitative basis for analyzing the consistency of TAD identification and structural stability among algorithms. As shown in Fig. 3C, the diagonal line represents the consistency measure of the method itself, with a theoretical value of 1. Focusing on HyperDeepTAD, its MoC with deDoc and deepTAD exceeds 90% at 25 kb resolution, and maintains high consistency at 50 kb and 100 kb resolutions (Supplementary Material S33). Even when compared with methods such as TopDom and CATAD, it has relatively high MoC values. This high degree ot consensus across algorithms and resolutions intuitively confirms the conservative characteristics of TADs.

is a specialized tool designed to quantify the correlation between dynamic changes in Hi-C signals and transcriptional activity domain structures. Its core function involves analyzing contact frequency differences within TAD regions through statistical models. Specifically, in the Hi-C data, the contact frequencies between different TADs exhibited significant variations due to chromatin spatial partitioning. Even when two genomic sites are geographically close, those belonging to different TADs may have significantly lower contact frequencies than sites within the same TAD. measures the proportion of “contact frequency changes caused by TAD partitioning” relative to overall signal variations, thereby quantifying the structural integrity of TADs. The Theis defined as follows(Eq. 5):

5

epresents the observed contact frequency of the i-th bin in the chromatin interaction matrix at a specific genomic distance, serving as the raw experimental data for calculation; denotes the predicted contact frequency derived from TAD classification; is the global mean contact frequency at the current genomic distance, which acts as the baseline for calculating total variation; and n represents the total number of bins involved in the calculation at the current genomic distance [26].

From the analysis of the TADadjR² curve in Fig. 3D and the data in Table 5, HyperDeepTAD has an advantage in explanatory power across genome-wide distance ranges: its curve consistently lies above those of methods such as TopDom and deDoc within the genomic distance of 0–1 Mb, and even significantly leads in some intervals. TADadjR² measures the “proportion of Hi-C signal variation explained by TAD segmentation.” The high value of HyperDeepTAD indicates that, whether in short-range or long-range, the TADs segmented by it can more accurately distinguish the signal patterns of high intradomain contact and low interdomain contact. Additionally, it has certain advantages in identifying nested TADs (Supplementary Material S34、S35 and S36).

Table 5.

The average value measured by TADadjR2 between pairs of loci within genomic distances (0–1 M) at 25 kb resolution on chromosomes 20–22 in GM12878 cells

	HyperDeepTAD	TopDom	deDoc	MSTD	CATAD	deepTAD
Chr20	0.812	0.751	0.802	0.764	0.762	0.819
Chr21	0.702	0.604	0.718	0.666	0.696	0.736
Chr22	0.772	0.58	0.753	0.654	0.649	0.698

The comprehensive analysis above, reveals that the boundaries identified by HyperDeepTAD exhibit strong performance in terms of the average peak value, fold change, and boundary marker ratio. These characteristics collectively indicate that the boundaries predicted by HyperDeepTAD are more closely associated with specific histone modifications; this close association highlights HyperDeepTAD’s ability to capture biologically meaningful TAD boundaries. Leveraging this advantage, HyperDeepTAD provides support for a deeper understanding of gene regulatory mechanisms and chromatin organization.

To further demonstrate the advantages of our method, we present a heatmap of a region on chr22 of the GM12878 cell line, as shown in Fig. 4 below, where TADs identified by each method are marked with blue boxes. This figure clearly illustrates the differences in the number, size, and boundaries of TADs detected by different methods, reflecting each method’s distinct perspective on chromosomal architecture and facilitating a multi-level understanding of the 3D genome structure.Detailed values of the average peak values, fold change rates, and boundary marking rates in this region are provided in Supplementary Files S37, S38 and S39.

Fig. 4.

Heatmap and TADs on chr22 (1200bin–1300bin) at 25 kb from GM12878

Discussion

In the field of 3D genome TAD identification, traditional methods have long relied on pairwise chromatin contact signals provided by Hi-C sequencing. This reliance prevents them from capturing synergistic interaction information among multiple genomic loci—a type of information crucial for accurately deciphering the functional relevance of TAD boundaries and the complex regulatory logic of chromatin spatial organization, directly limiting the accuracy of boundary identification and biological interpretability.

The HyperDeepTAD method proposed in this study successfully addresses this core limitation by innovatively combining aligned multiway chromatin interaction data with a deep learning framework. On the one hand, constructing a hypergraph based on aligned multiway chromatin interaction data fully retains high-order interaction information among multiple genomic regions, laying a data foundation that more closely aligns with real chromatin interaction scenarios for subsequent analyses. On the other hand, the integrated feature extraction architecture of dynamic convolutional networks and BiGRU further enables the accurate capture of local structural features and long-range dependency relationships.

In the context of existing research, the experimental results of this method offer notable value: In three classic model cell lines GM12878, K562, and mES, the TAD boundaries identified by HyperDeepTAD show relatively high average peak values, fold change, and boundary marker ratios for core functional markers involved in chromatin structure maintenance, such as CTCF, RAD21, and SMC3. This finding indicates that the boundaries identified by this method are consistent with the distribution of real chromatin functional loci, with good biological relevance. Moreover, across different analysis scales including 25 kb, 50 kb, and 100 kb, the method stably identifies TAD structures conforming to the scaling law of chromatin spatial organization. It also achieves a good measure of concordance with mainstream methods such as TopDom and deepTAD, and shows favorable performance in the TADadjR² index, providing reliable technical support for subsequent analyses of chromatin spatial organization and research on gene regulatory mechanisms.

However, this method has a notable limitation, namely a strong dependence on multiway chromatin interaction data, and all current experimental validations and practical applications of HyperDeepTAD rely entirely on such data. Compared with traditional sequencing technologies based on pairwise contact signals such as Hi-C, multiway chromatin interaction sequencing technologies face significant barriers in experimental operation and cost control.

To tackle this core issue, we plan to investigate multi-faceted solutions in future research to overcome the single data type reliance and broaden the method’s application in 3D genome research: (1) optimize the model architecture and feature extraction logic to develop adaptation modules for mainstream Hi-C and Micro-C data, enabling compatible analysis of multiple sequencing data types; (2) build a cross-data type transfer learning model, training the base model on existing multiway chromatin interaction data and fine-tuning it with limited Hi-C data to reduce single data type dependence; (3) adopt data augmentation and multi-omics integration strategies, simulating long-read sequencing features and integrating ChIP-seq, ATAC-seq and other genomic data to boost the model’s TAD identification performance on conventional sequencing data.

Conclusion

In summary, by effectively utilizing high-order interaction information, HyperDeepTAD significantly improves the accuracy and biological significance of TAD identification, providing more reliable methodological support for in-depth analysis of three-dimensional genomic structure and gene regulatory mechanisms, and opening a new path for the mining and application of high-order chromatin interaction data. In the future, it can be further extended to more cell lines and species to explore its application value in studies on disease-related chromatin structural variations.

Methods

Dataset generation

The Pore-C data and Hi-C data used in this study correspond; for specific conversion methods, please refer to the HiPore-C article [11]. The data accession number for this study is: GSE202539. The accession number for mES cells is: GSE114242 [9]. Specific data such as ChIP-seq data can be found in Supplementary Materials S1 and S2.

Constructing and partitioning hypergraphs

A hypergraph is an extension of graph theory. This data structure enables us to model multiway relationships, where edges can connect multiple nodes rather than just two nodes [27, 28]. The representation of a hypergraph is: H=(V, E), where V={v₁,v₂,.,v_N} |V|=N is the set of nodes and E={e₁,e₂,.,e_M} |E|=M is the set of hyperedges. In addition, a hypergraph is generally represented by its incidence matrix .

6

In this work, a hypergraph is constructed on the basic of Pore-C multiway chromatin interaction data. Rows represent nodes, which are continuous segments or intervals of the genome of uniform length. The length of a node refers to the resolution; for each chromosome with length L and resolution r, the number of nodes is n = L/r. In this paper, the resolution defaults to 25 kb. The columns represent hyperedges, where a value of 1 is assigned if an aligned Pore-C read is within the node, and 0 otherwise (Eq. 6). W is a diagonal weight matrix, where its elements represent the number of occurrences of the same hyperedge. Specifically, for the obtained alignment files, first, via the MATCHA method [7], We filtered the aligned data using a Python script according to preset criteria, yielding a file containing read names and their mapped nodes, which was then sorted by node order. For frequency calculation, multi-node sets were decomposed into smaller node subsets of the minimum unit. A hypergraph weighted matrix (denoted as HW) was generated through statistical calculations, which is the product of the hypergraph incidence matrix and the weight diagonal matrix.

Additionally, to address the issues of data sparsity and noise interference, the weighted hypergraph is partitioned into 11*50 subhypergraphs. The specific operations are as follows: for the weighted hypergraph H^~, slide along the row direction with a step size S = 1 node to extract local windows W_localR^11*M. Then, columnwise sorting is performed, and each column is independently sorted in descending order (Eq. 7).

7

Finally, sparsity optimization is performed: the top K = 50 high-weight elements of each column are truncated to construct a compressed matrix.(Eq. 8)

8

In the complex network world of hypergraphs, interconnections between nodes often exceed the constraints of traditional binary edges, resulting in high-order structures of multibody cooperative interactions. To accurately characterize such “atypical” interactions, the hypergraph transition probability emerges—it focuses on the possibility of nodes jumping through hyperedges, incorporating not only the connection strength of nodes within hyperedges but also the use of node degrees and hyperedge degrees for normalization constraints [29]. In the random walk model of hypergraphs, the transition probability describes the probability of moving from the current node to other nodes through hyperedges [30]. Therefore, we compute the transition probability matrix for the subhypergraph W_compressed by each node (Eq. 9).

9

Here,denotes the connection strength of node v in hyperedge e, denotes the connection strength of node u in hyperedge e, d(v) is the node degree, and (e) is the degree of hyperedge. Through the above formula, the transition probability matrix P of each subhypergraph can be obtained, where the dimension of P is 11 × 11. For this process, refer to Fig. 1A.

Determining TAD boundaries

In this step, we identify TAD boundaries by integrating multiscale feature processing and bidirectional contextual modeling, specifically targeting an 11 × 11 input matrix. First, a two-layer dynamic convolution is used to extract local features [15], and the network splits into two core branches upon receiving the input: One branch is responsible for generating dynamic weights: Let the input be an 11 × 11 feature matrix XR^11*11. The input first undergoes average pooling for dimensionality reduction while retaining global information, resulting in a vector x_pool. It is then fed into a fully connected layer to transform the feature dimensions: (Eq. 10)

10

where W_fc and b_fc are learnable parameters. Finally, a Softmax layer generates four dynamic weights, forming a probability distribution:

11

The other branch consists of four parallel convolutional layers (Supplementary Material S7) (Eq. 11). Each convolutional layer takes the 11 × 11 matrix as input and processes it independently via its respective kernel K_i and bias b_i, generating feature maps that maintain the 11 × 11 spatial dimensions: (Eq. 12)

12

where Conv_i() denotes the i-th convolutional operation.

In the fusion stage, the dynamic weights are elementwise multiplied with the 11 × 11 feature maps from their corresponding convolutional branches, and the results are summed to obtain a fused feature map: (Eq. 13)

13

The fused feature map is subsequently fed into the final convolutional layer for further extraction and integration, which outputs an 11 × 11 feature matrix. This design adaptively fuses multiscale convolutional results through dynamic weights, enabling flexible adjustment of each branch’s contribution on the basis of the local features of the 11 × 11 input, thereby enhancing the ability to express features in different regions. For details of the process, refer to Fig. 1D.

Next, to capture the critical bidirectional dependencies of TAD boundaries, our method employs the bidirectional gated recurrent unit (BiGRU) [16]. A TAD boundary is inherently a genomic position with distinct structural features on both sides: its left side typically lies within a TAD with dense interactions, whereas the right side may belong to another TAD or an inter-structural region with sparse interactions. Therefore, boundary identification requires simultaneous reference to sequential features before (left internal patterns) and after (right inter-structural patterns) the target position.

For the 11 × 11 feature matrix output by dynamic convolution, we first flatten it into a sequential input S=[s₁,s₂,…,s₁₂₁]. The BiGRU captures long-range dependencies on the left side through a forward GRU (processing in sequence order). The forward GRU updates its hidden state at each step as follows: (Eq. 14)

14

where (H_d is the hidden dimension)encodes dependencies from the start of the sequence to position t.

It captures long-range dependencies on the right side through a backward GRU. The backward GRU processes the sequence in reverse order[s₁₂₁,…,s₁]and updates its hidden state as: (Eq. 15)

15

where encodes dependencies from the end of the sequence back to position t.

The fused outputs of the two integrate bidirectional contextual information: (Eq. 16)

16

where denotes concatenation, which avoids the limitation of unidirectional models that can capture only unidirectional information, thereby more comprehensively depicting structural differences around boundaries. The fused sequence is then reshaped back to an 11 × 11 spatial dimension, denoted as .

Additionally, to address the potential loss of local details in global dependency modeling, we introduce residual connections [17], aiming to synergistically integrate local details and global dependencies. Let F_convR^11*11 represent the local features extracted via dynamic convolution. Although BiGRU excels at capturing global sequential patterns, it may overlook key local features—such as differences in short-range interaction intensity and subtle local structural changes, which are critical for distinguishing real boundaries from random fluctuations. The residual connections retain these local details through elementwise addition: (Eq. 17)

17

This operation directly superimposes the outputs of the BiGRU with the local features extracted via convolution, preventing feature attenuation in deep networks caused by multiple transformations. This design enables the model to leverage both global sequential patterns and local structural evidence, significantly improving prediction accuracy.

Ultimately, the resulting feature vector is fed into two fully connected layers to perform classification and prediction. A dropout layer follows each fully connected layer, and the last fully connected layer employs a sigmoid function as its activation function. If the final output exceeds 0.5, the sample is deemed a TAD boundary; otherwise, it is not classified as one.

Obtaining TAD

Filtering false positive boundaries

In graph theory, the clustering coefficient measures how tightly vertices in a graph form clusters, specifically the tendency of a vertex’s neighbors to connect with one another. However, hypergraphs have edges that can include three or more nodes, rendering the traditional clustering coefficient of simple graphs inapplicable. To resolve this issue, R. Miyashita et al. proposed a new definition of the hypergraph clustering coefficient, which quantifies local connection density via pairwise relationships among nodes within hyperedges [31]. The most recent review of high-order network statistical metrics explicitly identified the “shared weight method” as the preferred approach for computing the clustering coefficient of weighted hypergraphs. The reason is: when weights carry physical meaning, normalizing the total weight of hyperedges shared by node pairs enables precise quantification of differences in local connection strength. This approach is well-suited for verifying the “internal cohesion and interregion looseness” characteristic of TAD boundaries. Thus, in this study, we compute the hypergraph clustering coefficient to filter false positive boundaries, as detailed below: (i) Neighbor node acquisition. (ii) Calculation of shared hyperedge weights. (iii) Clustering coefficient calculation.

Neighbor node acquisition

On the basis of the hypergraph structure, we identify upstream or downstream neighbors that share hyperedges with candidate boundary nodes. For details, refer to Fig. 1C. First, throughout the entire hypergraph, it iterates through all the hyperedges to find those containing the target node. Second, other nodes are extracted of from these hyperedges and filtered upstream or downstream; Third, neighbors with a distance to the target node ≤ max_distance are filtered out. Some studies have shown that the size of TADs is generally 180 kb–2mb [32] ; thus, in this study, max_distance is set to 80 for a resolution of 25 kb, 40 for 50 kb, and 20 for 100 kb.

Calculation of shared hyperedge weights

The sum of the hyperedge weights for all node pairs in a set of nodes is calculated, which is used to quantify the degree of connection tightness between nodes. For a node set, the sum of its hyperedge weights is (Eq. 18):

18

where E is the set of all the hyperedges, denotes the hyperedges that contain both and , and is the weight of node n in the hyperedge e (the weights of nodes i and j in the same hyperedge are the same).

Clustering coefficient calculation

Upstream clustering coefficient: This coefficient quantifies the tightness of connections among the upstream neighbors of the boundary node. Downstream clustering coefficient: Analogously, this coefficient quantifies the tightness of connections among the downstream neighbors of the boundary node.

Let N denote the neighbor set, where N = U represents the set of upstream neighbors and N = D the set of downstream neighbors. Let k=∣N∣ be the number of neighbors in N, and W(N) the sum of shared hyperedge weights among nodes in N. The unified formula for the clustering coefficient is given as follows (Eq. 19):

19

Cross-boundary clustering coefficient: This coefficient quantifies the tightness of cross-boundary connections between upstream and downstream neighbors. The smaller its value, the stronger the “isolation” of the boundary. Let the set of cross-boundary nodes be , then (Eq. 20):

20

where is the sum of the shared hyperedge weights among all node pairs in is the sum of the shared hyperedge weights among purely cross-boundary (upstream-downstream) node pairs (excluding connections within upstream neighbors and within downstream neighbors); = |U| the number of upstream neighbors in set U; and = |D| the number of downstream neighbors in set D.

Finally, false positive boundaries are filtered on the basis that the upstream/downstream clustering coefficient is greater than the cross-boundary clustering coefficient. When boundary nodes appear consecutively, the node with the lower cross-boundary clustering coefficient is selected.

Obtain nested TADs

After obtaining the boundary set B: based on the aforementioned steps, since each TAD is bounded by two boundary nodes, we construct initial TAD sets X: [,.,] through paired combinations of boundaries in B, where represents a TAD bounded by two boundaries. In TAD analysis, cosine similarity has been proven to be an effective tool for capturing relationships between genomic regions. CATAD [21] uses cosine similarity to determine the similarity between prekernels (merging them if the similarity exceeds a threshold), and deepTAD [22] has also validated its applicability, which is highly aligned with our need to detect nested TAD structures. Therefore, this study adopts the cosine similarity used in CATAD to dentify nested TADs.

To analyze the associations between TADs via cosine similarity for detecting nested structures, it is first necessary to obtain the vector representation of each TAD. Therefore, this study introduces random walk techniques to generate feature vectors for TADs. LPAD [32] extracts node correlations from global chromosomal interactions via a random walk with a restart model; node2vec [33] explores flexible neighborhoods of nodes through biased random walks, learning mappings from nodes to a low-dimensional feature space to preserve network neighborhood properties; SpriteHyper2vec [30] more directly models high-order chromatin interactions via hypergraphs and generates node sequences via hypergraph random walks to learn low-dimensional features, successfully achieving SCSPRITE data imputation. These studies collectively validate the effectiveness of generating node feature vectors via random walks on hypergraphs, providing a basis for the method in this research.

Building on this, we obtain the feature vector of each TAD through hypergraph random walks, and then calculate the cosine similarity between the feature vectors of adjacent TADs. When the similarity exceeds the set threshold, a merging operation is performed on the two TADs, thereby achieving the detection of nested TAD structures. Through comparative experiments on multiple parameter combinations of walk counts (3–9) and walk lengths (5–30), with TADadjR²—the accuracy of nested TAD identification—as the evaluation metric, we identified the optimal configuration of “walk count = 7, walk length = 10”. This configuration enables the model to accurately capture the similarity features between TADs while maintaining computational efficiency, and the results are presented in Supplementary Figures S40 and S41.

Training

Positive and negative sample generation

Following the idea of constructing positive and negative samples via the deepTAD method [22], we built a standard dataset by identifying TAD boundaries with a co-occurrence count of at least 3 across six methods: CaTCH [34], deDoc [19], DI [35], TopDom [18], TADBD [36], and Arrowhead [37]. We then labeled the transition probability matrices of the aforementioned node pairs, converting TAD boundary identification into a binary classification problem.

To distinguish the contributions of the machine learning model architecture and training label quality to TAD calling performance, we first controlled for the consistency of training labels by adopting the same labels as deepTAD and only replacing the model architecture for comparative analysis. The results showed that with consistent training labels, our model significantly outperformed deepTAD in the enrichment intensity of key chromatin marks (e.g., CTCF, H3K4me3, and SMC3) at TAD boundaries, the ability to capture nested TAD structures, as well as the number, size distribution, and cross-resolution stability of identified TADs in the GM12878 and K562 cell lines. These findings directly indicate that the performance improvement stems from the model’s more efficient feature extraction and topological structure learning capabilities rather than differences in training labels. To further rule out the alternative explanation that the performance improvement solely originates from the high quality of the consensus labels themselves, we additionally designed a direct control experiment using consensus labels to verify the unique added value of our model. Specifically, we screened for model-specific TADs (defined as TAD structures predicted by the model with an overlap rate of less than 90% with consensus label TADs to ensure their independence from existing labels), and subsequent downstream analyses of chromatin mark association strength and gene co-expression levels revealed that these model-specific TADs still exhibited significantly high enrichment characteristics. These results demonstrate that the model’s performance improvement is not dependent on the high quality of consensus labels; instead, the model can mine biologically meaningful topological structures beyond the labels, which fully confirms the unique added value of the proposed framework.with detailed results available in Supplementary Files S48.

The final generated training samples have a three-dimensional tensor structure (Eq. 21).

21

where is an 11 × 11 feature matrix. is the genomic position of the central bin. y is the label, where 1 indicates a boundary and 0 indicates a nonboundary.

Since the feature difference between positive and negative samples at the boundary is small, and considering that the matrix has 11 rows, this study introduces a strategy with a safety distance of 5 to control the ratio of positive to negative samples at 1:1; although this strategy may lead to an increase in false positive samples, the subsequent filtering steps can effectively eliminate this phenomenon. We have performed runtime statistics for both data preprocessing and TAD generation, and the detailed results are provided in Supplementary File S11. It can be observed that the computational time of our method is mainly consumed in data preprocessing.

Training and validation samples

In this study, the training, validation, and test sets were all derived from GM12878 Pore-C multi-way chromatin interaction data [11] at a resolution of 25 kb. The data partitioning scheme adopted in the main text is as follows: chromosomes 1–12 were used as the training set, chromosomes 13–19 as the validation set, and chromosomes 20–22 as the test set. During the training process, 5-fold cross-validation was implemented to enhance the reliability of the model. For the two core classification metrics (AUC and F1) obtained from the 5-fold cross-validation, one-sample t-tests and coefficient of variation analyses were conducted separately to statistically assess the stability and significance of the model performance. with detailed results available in Supplementary Files S8, S9, and S10. All comparative experiments of different methods were performed on a computer equipped with a 24-core CPU (Intel Xeon Platinum 8260 CPU @ 2.30 GHz), and model training was carried out using a single RTX 3090 GPU.

To address the issue in the original partitioning scheme where the test set only contained small chromosomes and thus failed to fully reflect the complexity of genome-wide chromatin organization, we further optimized the dataset partitioning strategy as a supplement: the training set consisted of chromosomes with odd numbers (specifically [1, 3, 4, 6, 8, 10, 12, 14, 16, 18, 20]), the validation set included chromosomes with even numbers ([5, 7, 9, 11, 13, 15, 17, 35]), and the test set was adjusted to chromosomes 2, 20, and 22. By incorporating the longer chromosome 2, this adjustment enables the test set to cover the diversity of chromosome lengths, thereby more comprehensively reflecting the generalization ability of the model at the genome-wide level. with detailed results available in Supplementary Files S6 and S42 - S47.

To rigorously validate the generalization ability of the HyperDeepTAD model, we designed a stringent independent validation protocol: the model was exclusively trained on chromosomes 1–19 of the GM12878 cell line, with chromosomes 20–22 completely excluded from the training process; the trained model was then applied to chromosomes 20–22 of the K562 cell line for performance evaluation. This protocol enabled dual independent validation across cell types (GM12878-K562) and chromosomes, completely eliminating the risk of data leakage and thus truly reflecting the model’s generalization performance on unseen cell types and chromosomal regions (see Supplementary Files S49-S58 for detailed results). To better demonstrate the model’s generalization ability, we conducted simulation experiments using chromosome 22 as an example, with detailed results provided in Supplementary Files S59–S62.

Hyperparameter settings

To maximize model performance and clarify the contribution of each core architectural component, this study conducted systematic hyperparameter tuning and ablation experiments. The experimental design focused on the four core modules of the model: Dynamic 1D Convolution, Bidirectional Gated Recurrent Unit, residual connections, and hierarchical attention mechanism. Multiple controlled experiments were performed to quantify the necessity of each component, while iterative optimization of training-related hyperparameters was carried out to enhance the model’s training stability and cross-dataset generalization ability.Ablation experiments specifically investigated the impact of the number of DynamicConv1D layers on model performance and verified the functional necessity of each core module by selectively removing one or more components. The number of DynamicConv1D layers was set to three gradient groups (1 layer, 2 layers, 3 layers), with all other module configurations held constant. The results showed that two layers of dynamic convolution was the optimal configuration: too few layers resulted in insufficient feature extraction capacity, failing to fully capture the deep semantic information of input data; too many layers induced overfitting and increased computational overhead, both of which led to significant performance degradation. Furthermore, the introduction of residual connections effectively improved model accuracy and F1-score by mitigating the vanishing gradient problem in deep network training and ensuring the integrity of feature transmission. The hierarchical attention mechanism enabled the model to adaptively focus on key feature information, making a significant positive contribution to F1-score improvement. Removing any single module caused varying degrees of performance decline, confirming the necessity of the synergistic effect of the four core modules. The model achieved optimal performance when all modules were retained and two layers of DynamicConv1D were used.

Hyperparameter tuning experiments further focused on learning rate and the key parameters of the Adaptive Focal Loss function. In learning rate comparison experiments, multiple gradient learning rates were used for iterative training. The results indicated that a smaller learning rate yielded better AUC and accuracy, whereas an excessively high learning rate caused training oscillations, unstable convergence, and ultimately performance degradation, confirming that the model converges more sufficiently under a smaller learning rate. In the parameter tuning of the Adaptive Focal Loss function, grid search identified the optimal parameter combination as Gamma = 2.0, Alpha = 0.5, Delta = 0.03, at which the model achieved the best overall performance. Parameter sensitivity analysis revealed that increasing or decreasing Gamma led to over-focusing or under-focusing on hard-to-classify samples, respectively, affecting the balance of learning between positive and negative samples. Adjustments to Alpha needed to match the distribution of positive and negative samples in the dataset, as its core role was to balance the loss weights of different classes. A reasonable Delta value ensured numerical stability during loss calculation and prevented extreme values from interfering with the training process. These results further validated the effectiveness of the Adaptive Focal Loss function in balancing positive-negative sample distribution and focusing on hard-to-classify samples.

Based on the optimal configuration determined by the above experiments, the overall workflow of this study is as follows: First, the first dynamic 1D convolution module is used to extract basic features of the input data. Then, it is connected to the second dynamic 1D convolution module to further deepen feature mining and semantic information capture. To suppress overfitting, a dropout layer is added after the convolution layers for regularization. Next, the features extracted by convolution are fed into the bidirectional gated recurrent unit layer, where bidirectional modeling of sequential data is performed to enhance the temporal correlation and contextual relevance of the features. After that, residual connections are used to fuse the output of the BiGRU layer with the feature maps extracted by the convolution layers. Meanwhile, layer normalization is applied to standardize the feature distribution, thereby improving the training efficiency and stability of the model. The fused features are adaptively weighted and optimized via the hierarchical attention mechanism, which accurately focuses on the key feature information for the task and filters out redundant noise. The weighted features are then input into the fully connected layers to complete the integration and abstraction of deep features. Finally, a dense layer activated by Sigmoid outputs the binary classification results.During model training, the Adam optimizer and the adaptive Focal Loss function with optimally configured parameters are adopted. Accuracy, AUC, recall, precision, and F1-score are selected as the core evaluation metrics to comprehensively assess the classification performance and generalization ability of the model. The detailed settings of all hyperparameters are available in Supplementary Files S3, S4, S5 and S7.

To elucidate the decision-making mechanism of our model, we performed SHAP value visualization analysis. Results in Supplementary Files S12 and S13 demonstrated that the model primarily predicts TAD boundaries based on variations in local interaction strength, rather than relying on global signals. This finding not only validates the biological rationality of the model’s decision-making logic but also provides clear targets for subsequent experimental validation of key regulatory loci at TAD boundaries.

Authors’ contributions

JWL and KHL participated in the design of the method and the analysis of the experimental results. KHL and RPF provided important support in data collection and preprocessing. JWL and FG provided technical support during model training, hyperparameter adjustment, and experiments. All authors have read and approved the final manuscript for publication.

Funding

This research was supported by the National Natural Science Foundation of China (Grant No. 62372156).

Data availability

The source code can be obtained from GitHub at https://github.com/liukaihua0213-source/HyperDeepTAD, https://github.com/Mr-liu213/HyperDeepTAD.

Declarations

Ethics approval and consent to participate

Not applicable.

Consent for publication

Not applicable.

Competing interests

The authors declare no competing interests.