TransMarker framework

We propose a novel framework, TransMarker, for identifying Dynamic Network Biomarkers (DNBs) from single cell time course gene expression data using gene regulatory networks. The overall framework of TransMarker is presented in Fig 1.

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Fig 1. Overview of the TransMarker framework for identifying DNBs from single cell time course data.

(A) State specific gene expression profiles are integrated with a prior regulatory network to construct attributed gene interaction graphs. (B) A set of rewired graphs is generated to represent disease progression across multiple states. (C) Topological Feature Integration incorporates both local and global structural properties into each graph. (D) GATs are applied to learn contextualized embeddings for each disease state. (E) Gromov-Wasserstein optimal transport computes alignment scores between state specific networks to identify genes with high structural shifts. (F) A union subnetwork of selected genes is used to compute a DNI, identifying DNBs based on network variability. (G) The resulting DNBs and initial feature matrices are input to a DNN for multiclass classification across disease states.

https://doi.org/10.1371/journal.pcbi.1013743.g001

The approach starts by integrating expression profiles with a prior gene interaction matrix to generate state specific attributed graphs (Fig 1A), where nodes represent genes and edges denote regulatory interactions (Fig 1B). To extract informative features, the framework incorporates both local (shortest path distances) and global (PageRank scores) structural perspectives into the interaction matrices (Fig 1c). These enriched graphs are then passed through GATs to learn contextualized embeddings for each disease state (Fig 1D).

Next, Gromov-Wasserstein (GW) optimal transport is used to compute alignment scores between states, capturing changes in gene network positions over time. Genes with high alignment shifts are selected as candidates for dynamic analysis (Fig 1E). A union network is built from these genes, and a DNI is computed to quantify structural variability. Genes with the highest DNI values are identified as DNBs (Fig 1F).

In the final step, TransMarker uses the identified DNBs along with the Initial Feature Matrices of samples and genes as input to a Deep Neural Network (DNN) for multiclass classification, evaluating the effectiveness of the biomarkers in distinguishing disease states (Fig 1G).

Performance on simulation datasets

We utilized a single cell simulation framework from the SERGIO toolkit, as introduced by Dibaeinia et al. [31]. This tool models regulatory interactions among multiple transcription factor (TF)–gene pairs based on predefined gene regulatory network (GRN) structures. SERGIO simulates cellular differentiation trajectories using stochastic differential equations and generates gene expression profiles by capturing the system at its steady state. The cell splicing step is then simulated, and several transformations are applied to approximate the actual state of the cell. Lastly, the simulator introduces technical noise, outliers, and dropout effects, and converts the simulated data into UMI count format.

In this study, we simulate gene expression profiles for 30, 50, and 100 genes within a single cell type across four disease states, resulting in three simulation datasets: D1, D2, and D3. The corresponding simulated GRNs are depicted in Fig 2A, 2B, and 2C, representing networks with 30, 50, and 100 genes, respectively. Based on these regulatory interactions, we generate 100 cells per state. For instance, Fig 2D displays the expression profiles of 30 genes over four states, with 100 simulated cells in each state, where each state captures a distinct time course progression.

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Fig 2. Simulation networks and gene expression analysis across disease states.

(A-C) Simulated GRNs for datasets D1, D2, and D3, containing 30, 50, and 100 genes, respectively, generated using the SERGIO simulator. (D) Simulated gene expression matrix for dataset D1, showing expression levels of 30 genes across four disease states, with 100 cells per state. (E) Venn diagram showing the overlap of retained edges between the four disease states of GAC after applying the PC-CMI algorithm. (F) Number of DEGs identified across the five conditions (normal and four disease states). (G) Expression profiles of selected DEGs across disease states. (H) UMAP visualizations of extracted CD8⁺ T cell barcodes, used for downstream single cell trajectory and classification analyses.

https://doi.org/10.1371/journal.pcbi.1013743.g002

To evaluate the effectiveness of the proposed framework, TransMarker was initially tested on the three simulation datasets (D1, D2, and D3), across four disease states. These datasets provide a controlled setting to assess how well the model captures dynamic gene expression patterns and identifies state discriminative features.

To ensure robustness and statistical reliability, all experiments were conducted 50 times, and the reported results represent the average performance across these runs. We trained the model on each simulated dataset and assessed its performance using multiclass classification metrics, with a particular focus on the Area Under the Receiver Operating Characteristic curve (AUROC) for each class (disease state). The ROC curves for each state in D1, D2, and D3 are shown in Fig 3A–3C, respectively. For instance, as depicted in Fig 3C, the model demonstrates high discriminative power across all four states in D3, with AUROC values of 1.00, 0.99, 0.98, and 1.00 respectively.

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Fig 3. classification results on simulated datasets and biomarkers and classification results for GAC dataset.

Each panel shows the ROC curves for four disease states in: (A) D1 with 30 genes, (B) D2 with 50 genes, and (C) D3 with 100 genes. (D) Dynamic Biomarkers for GAC. (E) ROC Curve for GAC dataset. (F) ROC curve for independent verification.

https://doi.org/10.1371/journal.pcbi.1013743.g003

A summary of classification performance across the three simulation settings is presented in Table 1, which includes Accuracy, AUROC, Area Under the Precision - Recall Curve (AUPRC), F1-score, Precision, Recall, and Specificity for each dataset. Notably, classification accuracy increases as the number of genes rises, indicating that the method benefits from a richer feature space and scales effectively with network size.

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Table 1. Classification performance across the three simulation settings.

https://doi.org/10.1371/journal.pcbi.1013743.t001

Performance on gastric adenocarcinoma

To explore the cellular and molecular changes across different states of gastric adenocarcinoma (GAC), we utilized single cell RNA sequencing (scRNA-seq) datasets derived from multiple publicly available sources. These datasets capture the heterogeneity of the tumor microenvironment and offer a comprehensive view of the disease progression at single cell resolution.

In total, we used scRNA-seq data covering five distinct biological states: normal, chronic atrophic gastritis (CAG), intestinal metaplasia (IM), primary gastric adenocarcinoma (PGAC), and metastatic GAC. The precancerous states included CAG and IM samples, followed by malignant stages composed of primary GAC and metastatic lesions from sites such as the peritoneal cavity, ovary, and liver.

In this study, datasets GSE234129 [32] and GSE134520 [33] were used to train and test the proposed method for identifying dynamic biomarkers in GAC. An additional dataset, obtained from an institutional data portal (https://dna-discovery.stanford.edu) [34], was used for independent validation of the discovered dynamic biomarkers. After applying filtering and preprocessing steps, a total of 58,279 high quality single cells were retained for downstream analyses. The detailed description of these datasets and the computational tools used is provided in S1 Table.

To assess the effectiveness of the proposed method in real disease setting, we applied our framework to transcriptomic data of gastric adenocarcinoma covering five disease states: normal, CAG, IM, PGAC, and metastatic GAC. The model was trained on state-specific gene networks built using differentially expressed genes (DEGs) and prior regulatory knowledge. Graph embeddings from these networks were employed to identify disease-relevant dynamic network biomarkers (DNBs) and classify the disease states.

As shown in Fig 3D, our method identified 30 genes as state-associated DNBs for gastric cancer. These DNBs formed a connected regulatory subnetwork with 40 edges, based on curated interactions from RegNetwork (need the reference). Among the 30 identified genes, 19 overlapped with previously known GAC-related genes, while 11 were newly prioritized based on their differential expression profiles and topological prominence across states. This supports the model’s ability to integrate prior biological knowledge with data-driven signals to discover both known and novel candidate biomarkers. The resulting DNB network captures key regulatory transitions associated with cancer progression.

To evaluate the discriminative power of the identified DNBs, we used the learned embeddings for multiclass classification of the five disease states. The model was evaluated over 50 independent runs and achieved strong performance across evaluation metrics, with an overall accuracy of 0.8755 ± 0.0341, AUROC of 0.9230 ± 0.0385, AUPRC of 0.8871 ± 0.0413, F1-score of 0.8607 ± 0.0357, precision of 0.8688 ± 0.0556, recall of 0.8686 ± 0.0286, and specificity of 0.8808 ± 0.0531. These results demonstrate that the selected DNBs provide robust representations capable of distinguishing subtle transitions between states of GAC progression.

Further analysis of classification performance across individual states, as illustrated by the AUROC values in Fig 3E, showed high predictive accuracy across all states: Normal (0.94), CAG (0.92), IM (0.92), PGAC (0.91), and Metastasis (0.94). These consistent AUROC scores highlight the model’s ability to capture both early and late-state molecular signatures relevant to disease progression. To characterize the dynamic behavior of biomarker gene expression across disease progression, we computed and visualized the state-specific mean expression profiles of the DNBs that did not overlap with previously known, as shown in S3 Fig.

To explore the functional relevance of the 30 selected DNB genes, we conducted pathway and gene ontology (GO) enrichment analysis using Metascape (http://metascape.org/) (results provided in the S4 Fig). The most significantly enriched pathway was gastric cancer (hsa05226), reinforcing the disease specificity of the identified gene set. In addition, several other pathways known to be involved in cancer development and progression were significantly enriched, including: GO:0030335 - Positive regulation of cell migration, GO:0048568 - Embryonic organ development, GO:0048762 - Mesenchymal cell differentiation, GO:0070848 - Response to growth factor, WP5036 - Angiotensin II receptor type 1 pathway, M223 - PID Beta Catenin Nuclear Pathway.

These pathways are closely linked to oncogenic processes such as epithelial-mesenchymal transition, angiogenesis, and cellular differentiation, all of which are known to contribute to GAC progression. The enrichment of these biological themes among the selected DNBs suggests that the proposed model not only achieves accurate classification but also uncovers functionally meaningful regulators of gastric tumor development. Detailed information on the identified DNBs, including their functional summaries, is provided in S6 Table.

Independent verification

To further evaluate the robustness and generalizability of the identified DNBs, we assessed their classification performance on an independent dataset including three gastric adenocarcinoma states: Normal, IM, and PGAC. This external dataset, detailed in the Data section, was not utilized during training or model development and acts as an independent validation cohort.

TWhen the model was applied to this dataset, it exhibited strong generalization performance, achieving an accuracy of 0.8206 ± 0.0576, AUROC of 0.8738 ± 0.0429, AUPRC of 0.8091 ± 0.0473, F1-score of 0.7445 ± 0.0636, precision of 0.7365 ± 0.0621, recall of 0.8003 ± 0.0741, and specificity of 0.8168 ± 0.0944. These results emphasize the discriminative ability of the discovered DNBs in an unseen dataset and highlight their potential applicability across different clinical cohorts.

To further investigate state-specific classification effectiveness, we calculated AUROC values separately for each class. As shown in Fig 3F, the model achieved AUROC scores of 0.84 for Normal, 0.87 for IM, and 0.91 for PGAC. These findings confirm that the proposed framework can accurately identify key transitions even in datasets with limited state granularity, reinforcing the biological and predictive relevance of the selected DNBs across real-world gastric cancer datasets.

Ablation experiments

To quantify the contributions of key components in the proposed TransMarker framework, we conducted a series of ablation experiments examining the effects of (i) topological feature integration, and (ii) rewiring strategies. The results are summarized in Fig 4, which collectively shows the impact of these elements on model performance and provides a comprehensive evaluation of metrics under different training conditions.

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Fig 4. Ablation studies and performance evaluation under varying training conditions.

(A) Performance comparison under four topological feature integration settings: no features (None), only global features (w/o local), only local features (w/o global), and both local and global features (Full). The full model achieves the best results across ACC, AUROC, and AUPRC metrics, confirming the complementary value of integrating both feature types. (B) Comparison of different network rewiring strategies: no rewiring, correlation-based rewiring, our proposed regulatory rewiring, and rewiring with random noise injection (10%, 20%, and 30%). Our method outperforms others, demonstrating both the importance of state-specific regulatory rewiring and the robustness of the approach under noisy conditions. (C–D) Sensitivity analysis of the balancing coefficient α that controls the contribution of local versus global topological features. Optimal performance is achieved at α=0.3, highlighting the importance of a balanced integration. (E–F) Evaluation of model performance with varying training data sizes (10% to 50%). AUROC box plots (E) and full performance metrics (F) show that larger training sets yield more stable and accurate predictions. (G) Evaluation of model performance with different embedding dimensionalities (32 to 512) based on ACC, AUROC, and AUPRC. (H) Performance of the model under different dropout rates. (I) Classification performance using varying proportions (1% to 30%) of top-ranked genes; optimal performance is achieved using the top 15%.

https://doi.org/10.1371/journal.pcbi.1013743.g004

The first set of ablation experiments evaluates the contribution of local and global topological features to the final performance. As shown in Fig 4A, we tested four configurations: (1) using neither local nor global topological features (None), (2) using only global features (w/o local), (3) using only local features (w/o global), and (4) using both local and global features (our full model). The full model achieved the best performance across ACC, AUROC, and AUPRC evaluation metrics, confirming the complementary value of incorporating both local and global topological signals. Notably, excluding local features led to a performance drop, with accuracy of 0.796, AUROC of 0.841, and AUPRC of 0.784 ranking second overall. Omitting both feature types resulted in the most pronounced decline, underscoring the critical role of structural context in node representation learning. These findings validate the design of the topological integration module and emphasize that both local connectivity patterns and global graph structures are essential for accurate modeling of disease progression.

More detailed experiments investigate the sensitivity of the model to the balancing coefficient α, which governs the relative contribution of local versus global topological features. As illustrated in Fig 4C and 4D, optimal performance was obtained at α = 0.3, indicating that a slight emphasis on global features over local ones yields the best results.

Performance declined as α deviated from this optimum. Notably, when α approached 0.9, heavily favoring local features, model performance deteriorated significantly. This trend is consistent with the earlier observation (from Fig 4A) that global features contribute more substantially than local ones to predictive accuracy. These results confirm that the integration balance between local and global features is a key hyperparameter that must be carefully tuned to optimize performance. A detailed table of results across all evaluation metrics for the full range of α values is provided in the S2 Table.

The second group of experiments, visualized in Fig 4B, explores the effect of different network rewiring strategies on model performance. We compared four distinct conditions. In the first, no rewiring was applied, meaning all states shared the same regulatory network structure. The second condition involved correlation-based network construction, where state-specific co-expression networks were generated using Pearson correlation coefficients with a resampling-based thresholding strategy. The third condition implemented our proposed method, which constructs state-specific regulatory networks by integrating prior regulatory knowledge with dynamic expression signals. Lastly, to assess robustness to noise, we conducted additional experiments in which 10%, 20%, and 30% randomly added edges were injected into the regulatory network, simulating scenarios with incomplete or noisy prior knowledge.

The results demonstrate that our proposed rewiring method achieved the best classification performance across all evaluation metrics. In contrast, the no-rewiring condition produced the weakest performance, emphasizing the importance of modeling state-specific network dynamics. While correlation-based rewiring offered some improvement over the non-rewired setup, its performance was still lower, with AUROC dropping to approximately 0.80 and both accuracy and AUPRC falling below 0.80. Notably, the model maintained strong predictive capability under all three noise-injection scenarios, with accuracy above 0.81, AUROC above 0.84, and AUPRC above 0.82. These findings highlight the robustness of our approach and its potential applicability even in settings with noisy or incomplete regulatory information.

To determine the optimal training data proportion, we experimented with varying training set sizes ranging from 10% to 50%; the results are shown in Fig 4E (AUROC box plot), and Fig 4F (plots of all evaluation metrics) and S3 Table.

We evaluated the impact of different embedding dimensionalities (ranging from 32 to 512) on model performance using ACC, AUROC, and AUPRC metrics, as shown in Fig 4G. Fig 4H presents the performance across various dropout rates. Additionally, Fig 4I illustrates the classification performance when using different proportions (1% to 30%) of top-ranked genes, with 15% achieving the best results. Further robustness analyses are provided in the Supporting Information. Specifically, S2 Text and S7 Fig examine the robustness of TransMarker to the choice and completeness of the prior knowledge network, while S3 Text and S8 Fig analyze the influence of the entropic regularization parameter (ε) in the Gromov–Wasserstein alignment step.

Case study on esophageal squamous cell carcinoma

To assess the applicability and generalizability of our proposed method beyond gastric adenocarcinoma, we performed one more case study on Esophageal Squamous Cell Carcinoma (ESCC) using transcriptomic data (GSE199654) [10] encompassing six pathological states: Normal, High-Grade Intraepithelial Neoplasia (HGIN), Inflammation (INF), Stage I, Stage II, and Stage III. This experiment aimed to determine whether the framework could successfully identify DNBs and distinguish disease progression states in a different cancer type.

Applying the same pipeline, we identified a set of 28 genes as state-associated DNBs for ESCC. Among these, 14 genes overlapped with prior knowledge, while the remaining 14 were newly prioritized differentially expressed genes (DEGs), indicating the model’s capacity to integrate known biology with data-driven discoveries. The corresponding DNB network is visualized in Fig 5A. Functional enrichment analysis of these genes further supports their involvement in cancer-related pathways, with detailed results provided in the S5 Fig.

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Fig 5. Case study on Esophageal Squamous Cell Carcinoma (ESCC).

(A) State specific DNB subnetwork identified from GSE199654 across six ESCC states, including 28 genes (14 known, 14 novel). (B) State wise AUROC for multiclass classification using DNBs.

https://doi.org/10.1371/journal.pcbi.1013743.g005

The identified DNBs were employed for multiclass classification of the six ESCC states, yielding an overall ACC of 0.8451, AUROC of 0.9025, and AUPRC of 0.8602, indicating high classification performance. State-wise AUROC values, shown in Fig 5B, ranged from 0.85 to 0.94, demonstrating consistent predictive performance across both early and advanced disease states.

These results confirm the robustness and transferability of our method, highlighting its potential for broad application across different cancer types and progression models.

Comparison with highly related methods

To rigorously evaluate the performance and generalizability of TransMarker in identifying disease-relevant DNBs, we compared it with five state-of-the-art node ranking methods specifically developed for multilayer networks, 13 widely used node ranking strategies based on classical centrality measures in complex networks, and two gene ranking frameworks: RL-GenRisk [21], which applies reinforcement learning to prioritize cancer genes in renal carcinoma, and DyNDG [20], which models gene expression dynamics through time-series multilayer networks for leukemia. These comparisons provide a comprehensive benchmark across both static and dynamic network analysis paradigms.

We constructed state-specific gene networks and extracted top-ranked genes using each method, selecting the same number of top genes as determined by our method’s optimal selection threshold identified through preliminary experiments. The resulting gene subsets were then assessed through multiclass classification across disease states, and performance was evaluated across multiple standard metrics. To ensure a fair comparison, all methods, including ours, were run 50 times, and results were averaged across runs.

The node ranking methods for multilayer networks include multilayer entropy [35], ElementRank [36], Versatility Centrality [37], Versatility Degree Centrality [38], and Eigenvector Multicentrality [39]. Each of these methods captures distinct aspects of node importance across network layers and has been applied in various multilayer network analysis tasks. However, none of them is tailored for dynamic disease progression modeling or state-specific biomarker prioritization as addressed in our study. Each of these multilayer node-ranking methods captures distinct aspects of node importance across complex multilayer topologies. Specifically, Multilayer Entropy evaluates nodes based on structural information entropy, emphasizing those bridging different layers. ElementRank extends the PageRank algorithm by weighting layer-specific importance to reflect both local and global influence. Versatility Centrality employs a tensor-based approach to aggregate eigenvector centrality across layers, highlighting nodes consistently influential throughout the network. Versatility Degree Centrality integrates degree information from multiple layers to identify nodes with stable cross-layer connectivity. Eigenvector Multicentrality generalizes eigenvector centrality using a tensor model that simultaneously accounts for intra-layer and inter-layer propagation effects. These methods offer complementary perspectives on multilayer structure; however, unlike TransMarker, they do not explicitly capture dynamic regulatory rewiring or state-specific transitions using optimal transport principles, which are central to our framework. Details of the compared multilayer ranking methods, performance metrics, and the practical significance of evaluation criteria (ACC, AUROC, AUPRC, etc.) in the context of state-specific biomarker identification are provided in the S1 Text.

Table 2 summarizes the performance comparison on the GAC dataset. Our method consistently outperformed all others across all metrics. Among the competing methods, Versatility Degree Centrality achieved the highest AUROC after our method (0.8693 ± 0.0122), followed by ElementRank (0.8519 ± 0.0425). Notably, the performance gap in terms of AUPRC and F1-score was substantial, demonstrating the superior discriminative capacity of our DNBs. Methods such as Eigenvector Multicentrality and Versatility showed weaker performance, with accuracy below 0.5 and unstable precision-recall balance.

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Table 2. Quantitative performance comparison of our method and five multilayer network ranking baselines on the GAC dataset across multiple evaluation metrics.

https://doi.org/10.1371/journal.pcbi.1013743.t002

To further assess the distinctiveness and potential biological relevance of the genes prioritized by our approach, we compared the DNBs’ genes selected by our method to those obtained from five multilayer network ranking methods in GAC using a Venn diagram analysis. As illustrated in Fig 6A, 19 of the 30 genes identified by TransMarker were not shared with any of the other methods, underscoring our method’s ability to discover distinct molecular features that may play critical roles in the GAC progression process. Classification experiments using the expression profiles of these 30 genes across GAC states demonstrated strong discriminative performance, suggesting that the genes prioritized by our method are highly informative for distinguishing disease states.

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Fig 6. Comparison of Gene Discovery and Performance Evaluation with Baseline Methods.

(A) Venn diagram comparing our method and five multilayer network baselines, highlighting 19 unique genes identified only by our approach, suggesting its potential to uncover novel GAC-related candidates. (B) Performance comparison between our method and 13 classical centrality-based ranking algorithms on the GAC dataset. (C) Performance comparison between our method and two gene ranking frameworks.

https://doi.org/10.1371/journal.pcbi.1013743.g006

Among the 19 uniquely identified genes, 7 (GSPT1, TFF3, KLF4, ITK, GNB2L1, FLI1, and GNG5) were not included in the KEGG gastric cancer pathway, indicating that they have not been previously recognized as canonical GAC-related genes. However, several of these genes have been implicated in tumor biology through independent studies. For instance, TFF3 has been associated with epithelial regeneration and cancer metastasis [40], and KLF4 is known for its dual role in tumor suppression and promotion depending on the cellular context [41]. GSPT1, a translation termination factor, has been reported to influence cell cycle regulation and proliferation in cancer [42]. The identification of these genes highlights our method’s potential to uncover underexplored biomarkers that may contribute to GAC development or progression.

We also compared our method against 13 widely used node ranking strategies based on classical centrality measures in complex networks. These methods—including degree, betweenness, diffusion, PageRank, and eigenvector-based approaches—are commonly applied to biological networks to prioritize key nodes based on their topological properties. Each method captures a different aspect of network structure, such as node connectivity, shortest paths, local clustering, or influence propagation. The full list of methods and their mathematical definitions are provided in S5 Table. Fig 6B presents the comparative results of these 13 methods and our approach, evaluated using ACC, AUROC, and AUPRC on the GAC dataset. As shown, our method significantly outperforms all baselines across all three metrics, while most existing methods perform notably worse, with the best-performing alternative (PageRank) achieving only AUROC = 0.804 and AUPRC = 0.549. These results demonstrate the advantage of integrating both topological and biological signals in our framework. The comparison protocol follows the same evaluation strategy used in earlier benchmarking. Additional details of the performance metrics for all 14 methods are provided in S4 Table, and the overlap of selected genes is visualized in S6 Fig using an UpSet diagram.

To complement the benchmarking against classical and multilayer node ranking methods, we further evaluated our framework against two recent learning-based gene prioritization strategies, including RL-GenRisk and DyNDG—which have been designed for cancer-related gene identification in time series or multi-state contexts. As shown in Fig 6C, our method (TransMarker) substantially outperforms both RL-GenRisk and DyNDG across all seven evaluation metrics, including accuracy, AUROC, AUPRC, F1-score, Precision, Recall, and Specificity. For instance, TransMarker achieves the highest AUROC (0.923 ± 0.0385) and AUPRC (0.8871 ± 0.0413), reflecting strong discriminative power and robustness in identifying state-relevant genes. In contrast, RL-GenRisk and DyNDG exhibit considerably lower AUPRC scores (0.790 and 0.692, respectively), suggesting limited precision-recall balance in multiclass classification. Similarly, TransMarker shows higher recall (0.8686 ± 0.0286) and F1-score (0.8607 ± 0.0357), indicating superior sensitivity and overall predictive performance. These results further demonstrate that our method not only captures the dynamic nature of disease progression more effectively than static or semi-dynamic models, but also generalizes well across diverse evaluation criteria. Taken together, the consistent advantage across all comparisons supports the robustness, adaptability, and clinical relevance of our framework for state-specific biomarker discovery in complex diseases such as GAC.

Single-cell RNA-seq data processing and cell type identification

To identify major immune cell types from the raw single-cell RNA-seq data, we implemented a systematic processing pipeline using the Seurat R package (v4.0) [43]. The raw count matrix was initially loaded into R and used to create a Seurat object with a minimum feature threshold of 200 genes per cell. Cells with fewer than 200 detected genes, more than 6,500 detected genes (to exclude potential doublets), or mitochondrial gene expression exceeding 15

Next, genes expressed in fewer than three cells were excluded to reduce noise and retain biologically relevant features. The filtered dataset was normalized, and 2,000 highly variable genes were identified. These variable genes were then used for downstream scaling and dimensionality reduction via principal component analysis (PCA).

For clustering, we constructed a shared nearest neighbor (SNN) graph and applied Louvain clustering with a resolution of 0.5, followed by UMAP for two-dimensional visualization of cell clusters. Known marker genes were utilized to aid in the identification of immune cell types.

Differential expression analysis was performed to identify marker genes for each cluster. Finally, the barcodes of CD8+ T cells were extracted and saved for further downstream analyses, with their UMAP visualizations presented in Fig 2H and S1 Fig.

Differentially expressed genes identification and state-specific gene regulatory network construction

To delineate dynamic regulatory changes during gastric adenocarcinoma progression, we performed state-wise differential expression analysis and constructed corresponding gene regulatory networks. The detailed process for state-specific network construction is depicted in Fig 1A and 1B.

Differential expression analysis was carried out using the edgeR package in R. For each disease state, the matrix was compared to the normal state, and differentially expressed genes (DEGs) were identified using thresholds of and p-value <0.05. Genes not consistently present in both conditions or showing missing or constant expression were excluded before modeling. The union of all DEGs identified across states was compiled, and expression matrices for each state were filtered to retain only the DEGs.

To enhance biological relevance, 150 gastric cancer-related genes from the KEGG [44] pathway database were added to the DEG set. This combined gene set was mapped onto a prior regulatory network sourced from RegNetwork [45], which integrates gene regulatory relationships from over 25 sources. The resulting state-independent background network comprised 2,363 genes and 6,798 directed regulatory interactions.

We then applied the PC-CMI algorithm [46] to eliminate indirect or spurious associations from the background network using expression data for each condition. This yielded five condition-specific gene regulatory networks (normal state and the four disease states), where nodes represent genes and directed edges represent condition-specific regulatory relationships supported by both prior knowledge and conditional mutual information. Details of the identified DEGs, edges across different states, and expression profiles of selected genes are shown in Fig 2E, 2F and 2G.

The proposed method

Our model constructs a set of state-specific gene regulatory interaction graphs where denotes the disease state or time point. Each graph shares the same node set (representing genes), with each node for , while the edge set E^t captures state-dependent gene-gene relationships. By modeling these networks across states, the dynamic nature of disease progression is reflected in the evolving topologies. A graph attention network (GAT) is applied to each to generate gene embeddings that integrate both topological features and node attribute information. These embeddings are then used to analyze expression and connectivity dynamics across states, facilitating the identification of dynamic network biomarkers (DNBs). Finally, these DNBs are employed to train a deep neural network for multi-class classification of disease states.

Topological feature integration.

To enhance node representation with structural information, we integrate both local and global topological features derived from each state-specific graph . This integration produces a structural matrix that captures multi-scale relational properties among genes. Specifically, two types of topological descriptors are utilized: Local Topological Profile and Global Topological Profile.

To construct a unified representation that incorporates both local and global perspectives, we compute a convex combination of the two similarity matrices:

where is a tunable hyperparameter controlling the balance between local and global influences. This combined structural matrix is then employed as an adjacency matrix to enhance topological learning within the graph attention networks.

Local topological profile.

The local topological structure of each graph is captured using the shortest path distances between nodes. For each pair of nodes and , the shortest path length is calculated. To transform these distances into a similarity measure, we apply an exponential decay function:

If two nodes are disconnected in the graph, a large constant value is assigned to represent an infinite distance. The resulting matrix is then row-normalized to ensure comparability across nodes. This descriptor emphasizes local neighborhood proximity and captures how easily one gene can influence another through direct or short-path interactions.

Global topological profile.

To capture the global structural importance of genes within each state-specific interaction network, we construct a Global Topological Profile. Unlike local structure, which emphasizes neighborhood proximity, the global profile reflects each gene’s influence across the entire graph.

To quantify the global importance of each gene in the network, we compute PageRank scores for the graph . These scores estimate the long-range influence of each node throughout the graph. The global structural similarity between nodes and is then defined as the product of their PageRank scores:

This formulation encodes pairwise global similarities such that genes with higher joint importance receive stronger weights. The resulting matrix is also row-normalized and captures global interaction potential by emphasizing nodes with central roles in the network’s structure ().

State specific graph embedding module.

Our proposed model projects gene expression profiles along with the knowledge-based interaction matrix to a low-dimensional space. The objective is to optimize the embedding space to learn state-specific molecular representations from gene interaction networks, facilitating the identification of biomarkers linked to disease progression.

Given the gene expression matrix , where N represents the number of genes and M denotes the number of samples, along with an associated adjacency matrix representing the gene topological network at disease state t, our model aims to learn a mapping function f that generates low-dimensional representations for state t:

where θ represents the model parameters, and d is the dimension of the latent space. These embeddings are utilized to uncover meaningful molecular patterns and identify state-specific biomarkers associated with disease progression.

Our model comprises two layers of Graph Attention Networks (GATs), applied independently to each state-specific graph. In the first GAT layer, we employ multi-head attention, allowing the model to capture diverse contextual relationships by aggregating information from multiple perspectives.

Let denote the initial feature vector of gene i, and be the weight matrix for the k-th attention head. The attention coefficient for the interaction between gene i and gene j is computed as:

where is the neighborhood of gene i as defined by the state-specific network A^(t); LeakyReLU is a non-linear activation function. In the first layer, the output for node i is:

where denotes concatenation, σ is the ELU activation function, and K is the number of attention heads.

In contrast, the second GAT layer uses a single attention head to project the intermediate features into a unified embedding space. The layer can be formulated as:

Since our model is unsupervised, it does not rely on ground-truth labels. Instead, we design a contrastive objective to enforce embedding similarity between connected nodes. Specifically, we minimize the average Euclidean distance between the embeddings of node pairs connected in the adjacency matrix A^(t), encouraging the model to preserve the network’s structural topology in the latent space.

Alternatively, this can be seen as minimizing a reconstruction loss based on the similarity of node embeddings. The likelihood of an edge between node i and node j is computed as:

where are the final embeddings of genes i and j, and σ is the sigmoid function. The binary cross-entropy loss between the reconstructed adjacency matrix and the original A^(t) is:

The model is trained using the Adam optimizer, which adapts the learning rate for each parameter based on the first and second moments of the gradients, promoting faster and more stable convergence. We apply early stopping based on validation loss to prevent overfitting.

By training the GAT independently on each state-specific network, the model captures expression-informed topological embeddings that reflect key biological variations across disease progression. These embeddings are then analyzed to uncover potential biomarkers, focusing on genes whose network roles and expression profiles change significantly over time.

Gromov-Wasserstein optimal transport for disease progression alignment.

To capture dynamic changes in gene-level structural relationships across disease progression, we employed Gromov-Wasserstein (GW) optimal transport, a method designed to match data distributions from different domains by exploiting their internal structural similarities, to align gene interaction embeddings between consecutive disease states. Unlike classical optimal transport, which assumes direct correspondence between elements, GW optimal transport (GWOT) identifies alignments based solely on the relational structure within each domain. Given that embeddings at each state encode structural gene-level features within the interaction network, GWOT enables alignment across states without requiring pointwise correspondences.

Let denote the pairwise dissimilarity matrices for two disease states, constructed from embedding representations of biological samples. GWOT seeks a probabilistic transport plan that minimizes the discrepancy between the intra-domain structures by solving the following optimization problem:

where p and q are probability distributions over the source and target domains, respectively.

The resulting transport matrix Γ can be interpreted as a soft matching between samples across different disease states. An entry indicates the degree to which a sample in one state corresponds to a sample in the other, based on the similarity of their internal neighborhood structures.

To enhance computational efficiency and obtain smoother solutions, an entropy regularization term can be introduced. The entropic GWOT formulation adds a negative entropy term to the objective:

Here, the regularization parameter balances structure preservation and entropy maximization. Smaller values of ε encourage sparser alignments, while larger values yield smoother transport plans. This formulation helps avoid poor local optima and improves alignment robustness.

Overall, GWOT provides a principled approach for discovering relationships between biological samples across multiple disease stages based on their intrinsic topological organization, without requiring prior alignment information.

To quantify the extent of structural variation experienced by each gene across states, we computed a cumulative alignment score. Specifically, for each gene, we summarized the standard deviations of its transport plan row vectors across all states transitions:

This cumulative alignment score reflects how consistently or variably a gene’s relational context shifts as the disease progresses, enabling the identification of genes undergoing significant structural changes across states.

Identification of biomarker candidates.

To identify dynamic biomarker modules linked to disease progression, structurally variable biomarker candidates are initially selected based on changes in their local network structure across states. Union subnetworks are then built using these genes, and their connected components are extracted. The dynamic behavior of each component is quantified by calculating the DNI across states, and its variability is evaluated using the standard deviation of DNI values. Components are subsequently ranked according to DNI variability across states, and the one with the highest standard deviation is chosen as the most dynamically disrupted subnetwork, representing a disease-relevant biomarker module.

The computed alignment scores are stored in a vector , where R_i represents the cumulative alignment score of gene . An unsupervised filtering strategy is applied to identify structurally variable genes, utilizing the elbow method.

Let be the alignment scores sorted in descending order, and let π be the permutation indices such that . The elbow point, indicating a significant shift in the slope of the ranking curve, is detected using the KneeLocator algorithm[47]. This point determines a threshold that separates highly variable genes from more stable ones. If the elbow is located at index δ, the set of structurally variable genes is defined as:

Genes ranked above the elbow are regarded as structurally variable across states. In cases where the elbow point is ambiguous or not well-defined, we adopt an empirically derived proportion determined through systematic examination to select the top-ranked genes. The final set captures genes with dynamic topological behavior, serving as candidates for downstream disease progression analysis.

Extraction of candidate components.

To identify biomarker modules undergoing dynamic structural changes across disease states, the analysis quantifies the variability of gene-gene interactions over time. This is achieved by computing the DNI, which captures abrupt topological changes within subnetworks.

A union network is constructed by merging all state-specific subnetworks induced from the previously selected structurally variable genes (). The union network is defined as:

where Here, denotes the set of edges in state t restricted to nodes in .

From , connected components are extracted:

where each . These components serve as candidate gene modules. For each component C_j, its corresponding state specific subgraph is defined at each state t as:

where . These subgraphs are then used in subsequent steps to quantify the dynamic behavior of each component across states.

Quantifying dynamic behavior via DNI variability.

To evaluate the structural and functional dynamics of each gene module during disease progression, we introduce a novel Dynamic Network Instability (DNI) metric. This metric measures the temporal variability of a module by assessing changes in both alignment uncertainty and network cohesion across states. For a connected component C_j, the DNI is defined as:

where is the state-specific dynamic score of component C_j, calculated as:

The Graph Cohesion Score (GCS) is a composite term defined as:

where is the sum of alignment scores for nodes in C_j, reflecting the heterogeneity of node alignment within the component.

is the cohesion density, defined as the edge density of the component:

with n_t and m_t representing the number of nodes and edges in the component , respectively.

This formulation ensures that modules with high internal connectivity and strong overall alignment yield lower GCS values, leading to higher scores. All components are ranked by their DNI variability:

The selected C^* represents the most temporally dynamic module, exhibiting the highest variation in structure across states.

Implementation details

In the embedding process, the GAT model comprised two attention layers with a dropout rate of 0.6, using 64 hidden units and 2 attention heads in the first layer, followed by a 62-dimensional output layer. The model was trained using the Adam optimizer [48] with a learning rate of 0.001 and weight decay of 5e-4. A contrastive loss function was employed to enforce similarity between embeddings of connected nodes. Early stopping was implemented to terminate training if the loss did not improve for 50 consecutive epochs. Parameter choices were further refined through experiments on dropout rates and embedding sizes.

Experiments were also conducted to identify the optimal proportion of top-ranked genes for downstream classification. By systematically evaluating multiple candidate percentages, we determined that selecting the top 15% of genes consistently delivered the best classification performance across datasets. This empirically determined selection ratio was subsequently adopted for all methods to ensure fair and consistent comparison.

To assess the state-discriminative power of the identified biomarker genes, a multiclass classification model was implemented using an MLP. Expression values across all disease states were concatenated to form the input feature matrix. One-hot encoded labels were assigned to each sample based on its state. The dataset was divided into 70% training, 10% validation, and 20% test sets, and class weights were computed to address any class imbalance.

The MLP architecture consisted of three fully connected layers with 128, 64, and 32 units, respectively, each using ReLU activation. The final output layer comprised 5 units with a Softmax activation function to represent the five states. The model was trained using categorical cross-entropy loss and evaluated using multiple performance metrics. Early stopping was applied based on validation AUROC, with a patience of 50 epochs, and the best-performing model weights were retained for final evaluation.

Complexity analysis

The proposed framework combines topological learning, state specific embedding, optimal transport-based alignment, and biomarker module identification across disease states. Its overall time complexity is primarily determined by three core computational components: (1) Topological Feature Integration, (2) State-specific GAT-based Embedding, and (3) GWOT for dynamic alignment.

In the Topological Feature Integration step, for each state-specific network , computing the Local Topological Profile requires all-pairs shortest path distances, with a time complexity of for a sparse graph with N nodes and |E^(t)| edges. The Global Topological Profile is derived via PageRank, which has an average-case complexity of per iteration and typically converges in a small number of steps. The convex combination and normalization of similarity matrices add negligible overhead , resulting in an overall time complexity per state of .

The State-Specific Graph Embedding Module consists of two layers of GATs. In the first layer, multi-head attention is applied with K heads, each aggregating features from neighboring nodes. Given d as the embedding dimension and M as the input feature dimension, the per-layer complexity is . Since the second layer uses a single head and operates on the concatenated output, the total embedding complexity per state becomes . This step is repeated for each time point , and since each graph is processed independently, the embedding step is inherently parallelizable.

In the GWOT module, the alignment between each pair of consecutive states involves solving a quadratic optimization over the dissimilarity matrices of size , resulting in a theoretical complexity of . However, by introducing entropic regularization and using Sinkhorn-like iterative solvers, the practical complexity is significantly reduced to approximately per pair of states. For T–1 consecutive state transitions, the cumulative cost becomes .

The Biomarker Identification and Component Extraction steps are dominated by sorting operations and connected component extraction. Sorting the cumulative alignment scores takes , and connected component detection from union subnetworks is , which is negligible compared to earlier steps.

In summary, the overall time complexity of the proposed model is:

This expression reflects the model’s dependence on the number of disease states T, the network size N, the number of edges |E^(t)|, and embedding dimensionality d. Importantly, both the embedding and alignment components are highly parallelizable, making the model efficient and scalable for large-scale, state-resolved biological networks.