Introduction

Biomarkers can provide information on pathogenic processes and pharmacological responses for a therapeutic intervention1. The identification of biomarkers for clinical diagnosis is one of several interesting topics in medical research2. Although measurements pertaining to discriminative molecules have been traditionally applied in the clinic, identifying meaningful biomarkers for clinical diagnostics based on information-rich biological data is challenging3. To identify discriminative molecules, different approaches for feature selection, such as support vector machine-recursive feature elimination (SVM-RFE)4, genetic algorithms (GAs)5 and random forests (RFs)6, have been widely applied7,8,9. These methods select features based on the feature expression values among different classes rather than the changes in the feature relationships. However, a feature is also important if it has a remarkable joint effect on others10. Since molecules interact and relate to each other, exploring changes in the relationships among molecules to obtain a comprehensive understanding of disease mechanisms has attracted increasing attention in recent years11,12,13,14. Hence, analyzing the biological data from a network perspective could be a better strategy for discovering key biomarkers and facilitating the study of disease phenotypes.

Disease development is usually studied from two aspects: static and dynamic. In clinical studies, static and dynamic (or time-series) data are applied to meet different clinical goals. Static data are used to compare changes under different conditions and to define the discriminative information. To extract information from static data, different network construction methods and network analysis techniques have been proposed. Pearson correlation coefficient (PCC) which measures associated relationships of features has been widely applied to construct the networks15,16,17, and the hubs are retained as key factors. Krumsiek et al.18,19 used the partial correlation coefficient to construct networks for biological data analyses. In metabolomics, a ratio could be designated as the pathway reaction in which one metabolite is converted into another metabolite via single or multiple reaction pathways20. Thus, Netzer et al.20 constructed a network based on the paired biomarker identifier values of the metabolite ratios. PinnacleZ21 applied mutual information to calculate the discriminative ability of the network. Graph-based iterative group analysis (GiGA)22 ranked the features in the network and identifies the informative sub-network based on the p-value calculated using the ranks of the features. KeyPathwayMiner23 applied ant colony heuristic to screen for the key sub-network. BioNet24 used the integer-linear programming approach to define the informative sub-network. Other efficient network-based methods exist, including a two-step module cover25 and condition-specific sub-networks (COSINE)26. Additional state-of-the-art methods have been summarized in a recent review paper27.

As biological processes are dynamic, the systematic exploration of the temporal responses of molecules could facilitate the extraction of potential biomarkers that indicate the onset of complex diseases28. The early diagnosis of complex diseases could prevent the qualitative deterioration of patients and improve survival rates. However, extracting potential biomarkers of complex diseases based on time-series data is a notable challenge. For example, the behavior of hepatocellular carcinoma (HCC) at early disease stages shows little apparent difference from that of precancerous cirrhosis (CIR)29. Therefore, to explore the dynamics of disease development and screen for early warning signals, some methods for analyzing time-series data have been proposed. Tai et al.30 selected important molecules using Hotelling’s T2, whereas Chen et al. calculated the composite index to identify the dynamic network biomarkers of complex diseases31,32. We also proposed a strategy for analyzing time-series data based on dynamic networks (ATSD-DN) to define the warning signal33.

In the present study, we propose a computational method that defines potential biomarkers based on differential sub-networks (PB-DSN). PB-DSN explores the changes in correlation between feature ratios among different groups to define differential sub-networks. Subsequently, the hub vertices are identified as key feature ratios to discriminate different group samples. PB-DSN can also assess changes in correlations during disease development along time points to define differential sub-networks and selects hub vertices as key information for disease phenotyping. Moreover, signals from the sub-network consisting of the edges associated with the hub vertex can be used to indicate the onset of a specific disease stage. Hence, PB-DSN can analyze both static biological data and time-series data. In this study, a static malignant tumor genomics dataset and a time-series metabolomics dataset from a rat model of DEN-induced HCC are used to validate the performance of PB-DSN.

Results

Application of PB-DSN in the static dataset

Many studies have explored the mechanisms of malignant tumors from the viewpoint of genomics34,35,36. A gene can signify a disease state if its expression is suppressive or augmentative under certain clinical conditions37. However, some diseases result from multifaceted gene webs that interact with each other in complex ways38.

Small, round blue cell tumors (SRBCTs) include four subtypes: neuroblastoma (NB), rhabdomyosarcoma (RMS), non-Hodgkin lymphoma (NHL) and the Ewing family of tumors (EWS). The routine histological appearances of these four tumors are similar39. These cancers are not distinguished well by light microscopy, and there is no single test that can precisely separate the different cancers39. An accurate diagnosis of the type of SRBCT is essential for providing patients with the appropriate treatment.

Training and test subsets (see supplement information) exist for four different groups of SRBCTs, including EWS, RMS, Burkitt lymphoma (BL, a subset of NHL), and NB. Detailed information about these datasets can be found in the literature39. PB-DSN is used to study genomic problems at a network level. Figure S1 shows the workflow of PB-DSN. A feature is retained if the |log(fold-change)| is greater than or equal to 3 between any two subtype groups. Eighty-one features are retained, and a total of 3240 ratios are computed to construct the networks. The network G EWS is built based on these 3240 ratios. If PCC of two ratios is greater than or equal to 0.7 in EWS, then the two ratios are linked with a red edge in G EWS . If PCC of two ratios is less than or equal to −0.7 in EWS, then the edge is green in G EWS . G RMS , G BL and G NB are constructed using the same method applied for constructing G EWS .

To define the discriminative information for separating EWS from the other three groups, in this study, an edge appearing in G EWS that has different behaviors in two of the other three networks (G RMS , G BL and G NB ) is regarded as a differential edge of EWS, and all differential edges of EWS constitute a differential sub-network of EWS (SG EWS ). The corresponding expression of the edges in SG EWS in the other three groups constitutes the sub-networks SG EWS-RMS , SG EWS-BL and SG EWS-NB . Expanding on this idea, let G = (V(G), E(G)) be a graph, V(G) is the vertex set of G and E(G) is the edge set of G. (1) SG EWS-RMS  = (V(SG EWS-RMS ), E(SG EWS-RMS )), where V(SG EWS-RMS ) = V(SG EWS ) and E(SG EWS-RMS ) = E(SG EWS ) ∩ E(G RMS ); (2) SG EWS-BL  = (V(SG EWS-BL ), E(SG EWS-BL )), where V(SG EWS-BL ) = V(SG EWS ) and E(SG EWS-BL ) = E(SG EWS ) ∩ E(G BL ); (3) SG EWS-NB  = (V(SG EWS-NB ), E(SG EWS-NB )), where V(SG EWS-NB ) = V(SG EWS ) and E(SG EWS-NB ) = E(SG EWS ) ∩ E(G NB ). The edges in SG EWS-RMS , SG EWS-BL and SG EWS-NB have the same color as the corresponding ones in G RMS , G BL and G NB , respectively.

Subsequently, the vertices in SG EWS are ranked according to their degrees in descending order. The node with the highest degree (ratio 1) is selected for further analysis. The star sub-network consisting of the edges linked to the ratio 1 in SG EWS is defined and shown in Fig. 1(a). The star sub-networks consisting of the edges linked with ratio1 in SG EWS-RMS , SG EWS-BL and SG EWS-NB are also shown in Fig. 1. These data clearly express the difference between EWS and the other three groups and reveal that the correlations of some ratios in EWS samples are significantly different from those in the other three groups. The top 5 vertices in SG EWS (see Table S1) are retained as potential biomarkers; the statistical analysis is shown in Fig. 2. The differential expression levels of ratio 1 and ratio 2 in EWS and RMS indicate that they can separate the two malignant tumor samples well. Ratio 3 is significantly decreased in NB compared with that of EWS. As the levels of ratio 1, ratio 2, and ratio 3 show significant differences between EWS and non-EWS groups, these values could be used to separate EWS from non-EWS samples. The level of ratio 4 in EWS is remarkably lower than that in RMS, thereby contributing to the discrimination of the two tumor samples. Ratio 5 increases in EWS and could be used to distinguish EWS and non-EWS samples.

Figure 1
figure 1

Star sub-networks based on ratio 1. (a–d) The star sub-networks consisting of the edges linked with ratio 1 in SG EWS , SG EWS-RMS , SG EWS-BL and SG EWS-NB , respectively. Ratios in the four sub-networks are same. The numbers of the connections with ratio 1 in SG EWS , SG EWS-RMS , SG EWS-BL and SG EWS-NB are 370, 10, 32 and 50.

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Figure 2
figure 2

Statistical analysis of the top 5 ratios. (a–e) The statistical analysis (the mean ± the S.E.) of ratio i (1 ≤ i ≤ 5), respectively.

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PB-DSN is compared with PinnacleZ21, GiGA22, KeyPathwayMiner23, BioNet24 and the popular statistical analysis method SVM-RFE4. We also compare PB-DSN with molecular network based on PCC (MN-PCC) which builds the networks on features instead of feature ratios and applies the same network analysis method as PB-DSN. In PinnacleZ, KeyPathwayMiner and GiGA, to reduce the irrelevant features and improve the classification performance, the upper bound of the sub-network size is \(\sqrt{N}\), where N is the number of total features in the network. PinnacleZ and GiGA select the sub-network with the best discriminative ability to discriminate the different diseases. The value of parameter l in KeyPathwayMiner is set as 0. A false-discovery rate of 10−6 is used in BioNet. In SVM-RFE, the kernel function is linear and the value of penalty factor is set as 1. In MN-PCC, if |PCC| of two features is great than, or equal to 0.7, then there is an edge between the two features. In PB-DSN τ is set as 0.7. To compare the performance of these methods, the binary logistic regression is performed. The areas under the curve (AUCs) of separated EWS and non-EWS samples are listed in Table 1. The AUCs obtained using PB-DSN, MN-PCC, PinnacleZ, GiGA, KeyPathwayMiner, BioNet, and SVM-RFE are 1.000, 0.991, 0.940, 0.953, 0.939, 0.987 and 0.789, respectively, in the training set. The corresponding AUC values in the test set are 1.000, 1.000, 0.905, 1.000, 0.786, 0.917 and 0.595, respectively.

Table 1 Comparison of the ROC analysis (AUC) for different methods in separating EWS from non-EWS.
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A similar method is also used to analyze BL vs. non-BL, RMS vs. non-RMS and NB vs. non-NB, and the results are shown in Tables 2–4. Table 2 shows that PB-DSN has a higher AUC than PinnacleZ, KeyPathwayMiner and SVM-RFE for separating BL and non-BL samples in the training set and exhibits the same performance as MN-PCC, GiGA and BioNet. In the test set, seven methods have the same AUC values. In the case of discriminating samples between RMS and non-RMS groups (see Table 3), PB-DSN has a slightly lower performance than PinnacleZ, GiGA, BioNet and SVM-RFE in the training set, but has the same performance in the test set. Compared with MN-PCC and KeyPathwayMiner, PB-DSN has a remarkable advantage for separating samples between RMS and non-RMS groups in the training and test sets. For NB vs. non-NB (see Table 4), PB-DSN has a better performance than PinnacleZ, GiGA and SVM-RFE in the training set. The AUCs of PB-DSN, MN-PCC, KeyPathwayMiner and BioNet are the same in the training set. In the test set, PB-DSN, MN-PCC, PinnacleZ, GiGA, KeyPathwayMiner and BioNet can well separate the NBs from the non-NB samples, whereas the AUCs of SVM-RFE are markedly low. The performance of PB-DSN shows better potential to identify discriminative information for the improvement of disease diagnosis.

Table 2 Comparison of the ROC analysis (AUC) for different methods in separating BL from non-BL.
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Table 3 Comparison of the ROC analysis (AUC) for different methods in separating RMS from non-RMS.
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Table 4 Comparison of the ROC analysis (AUC) for different methods in separating NB from non-NB.
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Application of PB-DSN in the time-series dataset

Metabolomics, a powerful platform in systems biology used to study changes in holistic low-molecular-weight metabolites (≤1500 Da), plays a significant role in different fields of life science40,41,42. The dynamics of metabolite concentrations reflect physiological and pathological disturbances, and studying cancer from the perspective of cell-reprogrammed metabolism can provide insights into the process of carcinogenesis28,43. Thus, metabolomics studies have been successfully employed in some cases to screen for biomarkers of malignant tumors44,45,46. HCC is one of the major diseases with serious effects in humans. The early and precise diagnosis of HCC is crucial for ensuring that patients receive the appropriate treatment. However, due to the rapid development and early metastasis of HCC47, it is difficult to improve the performance of HCC diagnosis and, in particular, to distinguish small malignant HCCs from precancerous CIR samples. Although some traditional tumor markers (i.e., α-fetoprotein) are effective for HCC discrimination, the poor sensitivity of these molecules suggests that they are far from ideal47,48. Thus, developing efficient methods for the extraction of new biomarkers that signal HCC onset is urgently needed.

The metabolomics training set used in this study contains control and model groups and has been reported in a previous study28. Week 0 was defined as the starting time point of animal experiment. The collection of time-series sera set was conducted from week 8 to week 20 once every 2 weeks. The model group contains three stages: week 8 (hepatitis (H) stage, S 1), week 10–14 (CIR stage, S 2–S 4) and week 16–20 (HCC stage, S 5–S 7). S 1, S 4, and S 7 are the typical time points of the corresponding liver disease stages, whereas S 2 and S 5 are the first time points of the corresponding liver diseases. If a variable has missing values in a group, we replace these values with the minimum nonzero value in that group at the same time point. A |log(fold-change)| greater than or equal to 1 is used to filter the non-informative features, and seventeen features are selected based on the typical time points in three sub-problems (H vs. CIR, H vs. HCC and CIR vs. HCC). In total, 136 metabolite ratios are computed based on these 17 metabolites to construct the networks.

To screen the prospective information of HCC, PB-DSN focuses on S 5, which is the starting time point of HCC, and extracts differential edges to infer the differential sub-network of S 5 (SG 5). The differential edges are those that appear in the network of S 5 but have different behaviors in most (2/3 in this study) of other networks in this time-series dataset at S t (1 ≤ t ≤ 4). In SG 5, the hub vertex (N,N-dimethylglycine/threonic acid) having the largest degree and its associated nodes are selected for further analysis. Figure 3 shows the dynamics of the correlation between N,N-dimethylglycine/threonic acid and its associated nodes in disease initiation and progression. We observed that the correlations of feature ratios change with the development of liver disease. Differences in the correlations of ratios between the starting time point of HCC and the stages prior to HCC could represent the onset of HCC. Therefore, changes in the correlations between N,N-dimethylglycine/threonic acid and its associated nodes could, at the network level, be critical information signaling the emergence of disease deterioration during liver disease development. The statistical result of N,N-dimethylglycine/threonic acid, as shown in Fig. 3(h), indicates that N,N-dimethylglycine/threonic acid is significantly different between HCC and non-HCC groups; thus, N,N-dimethylglycine/threonic acid exhibits potential for distinguishing HCC samples from non-HCC samples.

Figure 3
figure 3

Star sub-networks based on N,N-dimethylglycine/threonic acid and box plot. (a–g) The star sub-networks consisting of the edges linked with N,N-dimethylglycine/threonic acid during disease development. Ratios in the seven sub-networks are same. The numbers of the connections with N,N-dimethylglycine/threonic acid in SG i (1 ≤ i ≤ 7) are 4, 5, 4, 18, 36, 4 and 4. (h) The box plot of N,N-dimethylglycine/threonic acid.

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The vertices in SG 5 are ranked in descending order based on their degrees, and the top 5 ratios (see Table S2) are selected for the subsequent statistical analysis. Among the 5 ratios, the levels of 3 metabolite ratios showed significant differences between the model and age-matched groups at any time point (Table S3); thus, these metabolite ratios (N,N-dimethylglycine/mucic acid, N,N-dimethylglycine/threonic acid and betaine/mucic acid) contribute to separating the samples between control and model groups. Figure 4(a–c) show the metabolic trajectory of these 3 ratios along the time points in the training set. The significant differences of these 3 metabolite ratios are shown between HCC and non-HCC groups. Thus, we find that when the levels of these 3 metabolite ratios in non-HCC samples significantly decrease, HCC occurs. The AUCs of the 3 ratios used to discriminate HCC from non-HCC groups are 0.954, 0.923, and 0.939 in the training set, respectively (Fig. 4(d)). The detailed results of statistical analysis shown in Tables S4–S6 suggest that the levels of the 3 metabolite ratios exhibit significant differences between any time point in the HCC stage and any time point in the non-HCC stage, further indicating the ability to discriminate between HCC and non-HCC samples. Notably, for N,N-dimethylglycine/threonic acid, significant differences are also observed between the H stage and any time point in the CIR stage. The significantly decreasing level of N,N-dimethylglycine/threonic acid at different disease stages could suggest its potential for a more complete presentation of liver disease development.

Figure 4
figure 4

Statistical analysis of the 3 metabolite ratios. (a–c) The metabolic trajectories (the mean ± the S.E.) of N,N-dimethylglycine/threonic acid, N,N-dimethylglycine/mucic acid and betaine/mucic acid in the training set. (d,e) The ROC curves of these 3 metabolite ratios in the training and test sets.

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In the present study, the external test set (see supplement information) contains 36 sera from 6 model rats monitored at 6 time points (i.e., S 1–S 6). Histological examinations to validate HCC reveal that S 1–S 4 are the pre-cancer stage, whereas S 5–S 6 are the HCC stage. The AUCs of the 3 metabolite ratios in the test set are 0.948, 0.903, and 0.865, respectively, for the separation of HCC and non-HCC samples (Fig. 4(e)).

To evaluate the performance of PB-DSN, we compared this method with multivariate empirical Bayes statistics (MEBA)30, ATSD-DN33, MN-PCC, PinnacleZ21, GiGA22, KeyPathwayMiner23 and BioNet24. In ATSD-DN, τ is set as 0.7, and the ratio of imidazole-4-acetic acid/trimethylamine N-oxide is selected. Trimethylamine N-oxide is selected by MN-PCC. The top ratio in PB-DSN is N,N-dimethylglycine/threonic acid. In MEBA, based on Hotelling’s T2, top 3 features are selected to discriminate different diseases. PinnacleZ, GiGA, KeyPathwayMiner and BioNet select the sub-network with the best discriminative ability.

The comparison results shown in Table 5 indicate that in the training set, the performance of PB-DSN for separating disease and normal groups is better than those of ATSD-DN, MN-PCC and PinnacleZ. The AUC obtained using PB-DSN for discriminating HCC and non-HCC samples is 0.954, which is higher than the AUCs of 0.808, 0.820, 0.884, 0.815, 0.900 and 0.934 obtained by ATSD-DN, MN-PCC, PinnacleZ, GiGA, KeyPathwayMiner and BioNet, respectively. The performance of PB-DSN is only 0.002 lower than that of MEBA. In the test set, the AUC obtained using PB-DSN is better than those of other methods except ATSD-DN and KeyPathwayMiner. To discriminate H from the CIR samples, the AUCs obtained for the training set by PB-DSN, MEBA, ATSD-DN, MN-PCC, PinnacleZ, GiGA, KeyPathwayMiner and BioNet are 0.966, 1.000, 0.776, 0.619, 0.912, 0.687, 0.966 and 0.959, respectively, and the corresponding AUCs in the test set are 0.972, 0.917, 0.870, 0.741, 0.787, 0.750, 0.907 and 0.889, respectively. In our previous study28, a ratio of creatine/betaine was identified. The AUCs of creatine/betaine in the training and test sets are shown in Table 5. In all cases, the AUCs of N,N-dimethylglycine/threonic acid selected by PB-DSN are superior to those of creatine/betaine. The better performance of PB-DSN for analyzing the time-series dataset illustrates the potential of this method to define prospective signals that indicates the onset of HCC, thereby improving the precise diagnoses of different liver diseases.

Table 5 Comparison of the ROC analysis (AUC) in metabolomics data.
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Discussion

The precise, early diagnosis of malignant tumors can better facilitate appropriate treatments and improve the survival rates of patients. However, due to the complex factors of individual differences, epigenetics and environmental effects, identifying efficient biomarkers remains a challenge. Molecules interact with each other in networks or pathways to implement biological functions49. In contrast to the molecule level, deregulation at the pathway level is more critical to carcinogenesis50. Thus, discovering biomarkers from a network-based perspective can provide a more efficient strategy to characterize disease phenotypes.

To define discriminative information for classifying different disease groups in the static dataset and to identify the warning signals of disease in the time-series dataset, PB-DSN examines changes in the relationships among ratios and extracts differential edges to infer differential sub-networks.

To identify discriminative information in the static dataset about a specific group, the differential edges with different behaviors between the network of the specific group and most of other networks are extracted. Based on these differential edges, the differential sub-network, which reflects the discriminative information between the specific group and other groups, is constructed. The vertices with large degree in the differential sub-network contain crucial information, as their relationships with many other feature ratios (i.e., adjacent vertices) have changed and, thus, the correlations of the feature ratios are much close in the specific group.

Exploring the important prospective information about the onset of a specific physiological or pathological stage is crucial. To define prospective information about the disease severity and phenotype based on the time-series dataset, PB-DSN focuses on a specific time point (e.g., the starting time point of HCC) and traces the changes in relationships of the ratios from the beginning of the assessment to the specific time point. Subsequently, PB-DSN extracts the differential edges that have different behaviors between the network of the specific time point and most other networks of the earlier time points. Based on these differential edges, a differential sub-network, which reflects the discriminative information between the specific time point and the before time points, is constructed. The vertices with the large degree in the differential sub-network contain crucial prospective information about the onset of a specific physiological or pathological phenomenon, as their relationships with many other feature ratios (i.e., adjacent vertices) have changed; thus, the correlation of feature ratios is much close at the specific time point.

Based on the comparisons in the analysis of static datasets, PB-DSN outperforms MN-PCC, PinnacleZ, GiGA, KeyPathwayMiner, BioNet and SVM-RFE. Hence, by studying the differences of the feature ratio correlations among different groups, PB-DSN can be used to mine important discriminative information.

In the time-series metabolomics data, three metabolite ratios, N,N-dimethylglycine/threonic acid, N,N-dimethylglycine/mucic acid and betaine/mucic acid, are defined. Increased N,N-dimethylglycine is considered an important indicator for a shift of homocysteine remethylation towards the betaine-homocysteine-methyltransferase reaction in liver CIR51. Betaine is an important methyl donor that plays a significant role in hepatic methyl balance28. These two compounds are closely associated with homocysteine remethylation. Moreover, threonic acid is a product of ascorbic acid oxidation52 and is thus associated with systemic oxidative stress in patients. In the present study, mucic acid is decreased in the model group compared with controls, which likely indicates that low levels of mucic acid are associated with the onset of liver disease. The combinations of N,N-dimethylglycine/threonic acid, N,N-dimethylglycine/mucic acid and betaine/mucic acid as biomarkers may improve the diagnosis of HCC development. These three ratios can promote the discrimination of metabolic differences, as the combination patterns indicate that different physiological perspectives are considered based on different metabolic pathways rather than traditional individual features or single pathway-derived metabolites. Larger HCC patient cohorts are needed to validate these results in future studies.

PB-DSN is compared with MEBA30, ATSD-DN33, MN-PCC, PinnacleZ21, GiGA22, KeyPathwayMiner23 and BioNet24 in the analysis of time-series data. The performance of PB-DSN is better than those of other methods, except MEBA for discriminating between non-HCC and HCC samples in the training set. In the test set, the biomarker performance indicated by PB-DSN is also efficient. In the case of separating samples between H and CIR groups, PB-DSN only has the slightly lower performance than that of MEBA in the training set, but has the best performance in the test set. The biomarkers identified by PB-DSN are effective in discriminating diseases from normal control samples. The better performance shows that compared with other methods, PB-DSN is more advantageous for extracting prospective information to facilitate the diagnosis of HCC.

Moreover, for PB-DSN and some compared methods, we made experiments to show the influence of the parameters on the effectiveness. Different parameter settings are tested and the corresponding performances are given in Tables S7–S14.

In summary, PB-DSN analyzes biological data from networks to define discriminative information and prospective signals of complex diseases. The application of PB-DSN in the two malignant tumor datasets shows that this method has the potential to effectively define discriminative information from a static dataset and to identify the prospective signals from a time-series dataset. Moreover, the network analysis method of PB-DSN can be applied in molecular networks, which also have the effective performance in some cases.

Methods

Let \(D=\{{S}_{t}|1\le t\le {N}_{s}\}\) represent the data set, where S t is a data subset of the tth disease group in a static problem or a data subset of the tth time point in the dynamic problem and N s represents the number of the various disease groups or the number of time points. Let \(F=\{{f}_{1},\,{f}_{2},\,\ldots ,\,{f}_{m}\}\) represent the feature set, where m represents the number of features. Since a ratio can be designated as the pathway reaction in which one metabolite is converted into another metabolite via single or multiple reaction pathways, the relationship of these ratios was explored to construct the network used to analyze the metabolomics data20. Furthermore, ratios of gene expression levels were also studied in genomics53,54. Hence, PB-DSN also studies feature ratios to analyze the biological data.

In PB-DSN, the feature ratio that means the ratio of feature concentration or expression level is defined as r ij (t) = f it /f jt , where f it is f i at S t (1 ≤ i < j ≤ m, 1 ≤ t ≤ N s ).

Network construction

The Pearson correlation coefficient of two feature ratios r x (t) and r y (t) in group (or at time point) S t (1 ≤ t ≤ N s ) is defined as

$$PCC({r}_{x}(t),{r}_{y}(t))=\frac{1}{{n}_{t}-1}\sum _{k=1}^{{n}_{t}}(\frac{{r}_{x}(t,k)-{\mu }_{{r}_{x}(t)}}{{\sigma }_{{r}_{x}(t)}})(\frac{{r}_{y}(t,k)-{\mu }_{{r}_{y}(t)}}{{\sigma }_{{r}_{y}(t)}}),$$
(1)

where r x (t, k) and r y (t, k) are the values of ratio r x (t) and r y (t) in the kth sample at S t , \({\mu }_{{r}_{x}(t)}\) and \({\mu }_{{r}_{y}(t)}\) are the means of ratio r x (t) and r y (t), \({\sigma }_{{r}_{x}(t)}\) and \({\sigma }_{{r}_{y}(t)}\) are the standard deviation, and n t is the number of the samples at S t . The Pearson correlation coefficient describes the relationships between variables in a phenomenological form. When two variables occur adjacently in a pathway or are derived from a common precursor, the correlation coefficient is positive, and when one variable is used to directly or indirectly generate the other, the correlation coefficient is negtive55. Large |PCC(r x (t), r y (t))| suggests that the two corresponding feature ratios are closely related to each other at S t . Hence, the network G t is built based on the Pearson correlation coefficient for depicting the relationships among feature ratios at S t . Let each feature ratio represent a vertex in the network, and when the Pearson correlation coefficient of the two feature ratios |PCC(r x (t), r y (t))| ≥ τ, there is an edge between the two corresponding feature ratios r x (t) and r y (t) in G t . Since the Pearson correlation coefficient, which represents the different relationships of the two ratios, may be positive or negative, the edge is colored red for PCC(r x (t), r y (t)) ≥ τ and green for PCC(r x (t), r y (t)) ≤ -τ.

Defining the differential sub-network

In a complex biological system, the relationship of the feature ratios in different physiological or pathological phenomena may be different. Thus, the correlation difference among the different sample groups or along the time points could reflect different physiological or pathological changes.

Definition 1. Let D = {S t | 1 ≤ t ≤ N s } represent the data set, where S t is a data subset of the tth disease group in a static problem. Let G t represent the network at S t . If e ∈ V(G t ) has different behaviors (i.e., disappears or has a different color) in most of the other networks, then e is a “differential edge” at S t . The sub-network, SG t , consisting of all the differential edges at S t is called the differential sub-network at S t in the static problem.

Definition 2. Let D = {S t | 1 ≤ t ≤ N s } represent the data set, where S t is a data subset of the tth time point in the dynamic problems. Let G t represent the network at S t . If e ∈ V(G t ) has different behaviors (i.e., disappears or has a different color) in most of the networks G p (1 ≤ p < t), then e is a “differential signal edge” of S t . The sub-network, SG t , consisting of all the differential signal edges of S t is called the differential sub-network at S t in time-series problems.

Hence, the differential sub-network at S t in a static problem contains information to discriminate group S t from other groups based on the changes in the relationships between the ratios. Moreover, the differential sub-network of S t in a dynamic problem contains information that could signal the onset of the specific physiological or pathological stage at S t , which contains certain information that is markedly different from that of the previous time point.

PB-DSN applies topological structure analysis to select the most important ratios from the differential sub-network. This method ranks the nodes of the differential sub-network at S t in a descending order according to their degrees, and the top k ≥ 1 nodes are selected.

The compared methods

SVM-RFE

This method selects the important feature subset by sequential backward elimination4. In each iteration, SVM-RFE measures the weights of the features based on the contribution to the hyperplane, and the features with the lowest weights are removed from the current feature subset. During the recursive procession, the feature subset with the best classification performance is retained as the selected feature subset.

MEBA

This method is based on multivariate empirical Bayes statistics and uses Hotelling’s T2 to measure the importance of features in a systematic time dimension30. To reduce the false positive and false negative results, MEBA applies the time-course mean profiles to evaluate treatment differences.

MN-PCC

PCC is applied to measure the relationship of the molecules and construct the networks. Subsequently, the same network analysis method as PB-DSN is applied to identify the key factors.

ATSD-DN

This method constructs networks based on the non-overlapping ratios of the feature ratios33. Two network analysis techniques, dynamic concentration analysis and topological structure analysis, are performed to identify the informative feature ratios. Among the feature ratios selected by both two techniques, the ratio with the highest AUC is considered a potential biomarker.

PinnacleZ

This method uses mutual information to define the discriminative ability of a sub-network21. To search a sub-network, PinnacleZ starts from each node in the network and performs a greedy search. During the search, PinnacleZ iteratively adds a node that is associated with other nodes in the current sub-network and can yield a maximal score increase in the sub-network. This procedure is continued until the improvement rate of the score is less than or equal to 0.05.

GiGA

This method selects the discriminative sub-network based on p-values22. First, GiGA assigns a rank to each feature based on expression changes and identifies the local minimum (i.e., the node with a lower rank than all direct neighbors in the network). Subsequently, each local minimum is viewed as a seed and is iteratively extended. In each iteration, the neighboring node with the smallest rank is added. After n steps, a sub-network with n nodes with a maximum rank m is built. Based on p-value, the sub-network is scored as

$$p=\underset{i=0}{\overset{n-1}{{\rm{\Pi }}}}\frac{m-i}{N-i},$$
(2)

where N is the number of total nodes in the network.

KeyPathwayMiner

The goal of this method is also to define the informative sub-network23. A strategy called Global Node Exceptions is used in KeyPathwayMiner. Moreover, ant colony heuristic is used for finding the solution to subnet problem.

BioNet

BioNet is an effective method that can compute provably optimal or sub-optimal solutions to screen for the maximal-scoring sub-network24. First, BioNet calculates the maximum likelihood score for each features based on the beta uniform mixture distribution of the p-values. Subsequently, integer-linear programming is applied to define the optimal sub-network in reasonable computation time.