Identifying the critical states and dynamic network biomarkers of cancers based on network entropy

According to our recently proposed DNB theory [1, 2], disease progression can be generally divided into three stages or states, i.e., a relatively normal (before-deterioration) state, a pre-disease (critical) state, and a disease (deteriorated) state. The normal state is a stable state with high resilience and robustness to perturbation. The pre-disease state, which is the limit of the normal state, is unstable and reversible to the normal state with appropriate intervention. The disease state, which is also a stable state with high resilience, is almost irreversible [3‐6]. Moreover, when a complex system is near the critical point, among all observed variables, there exists a dominant group defined as DNB biomolecules that satisfy the following three conditions based on the observed data [1]:

All three of the above conditions are necessary for phase transitions, and can also be approximately stated as follows: the appearance of a strongly fluctuating and highly correlated group of features/variables implies imminent transition into the disease state. Therefore, we use these three conditions to quantify the tipping point as early-warning signs of the disease, and then the identified dominant group of biomolecules consists of DNB members. These three conditions are the theoretical basis of DNB theory and have been applied to a number of analyses of disease progression and biological processes to predict critical states [2, 5].

The LNE method uses reference samples (samples from normal cells that are regarded as the background and represent healthy or relatively healthy cells) and only one disease sample to identify the tipping point with the following algorithm (Fig. 1B). Specifically, inspired by the previous study [17], the term "entropy" proposed here aims to characterize the statistical perturbation brought by each individual sample against a group of given reference samples.

[Step 1] The first step is to form a global network \({\text{N}}^{G}\) by mapping the genes to a protein–protein interaction (PPI) network downloaded from STRING (http://​string-db.​org) [18], which contains the interactions of the selected genes with a confidence level of 0.800. Isolated nodes without any links to other nodes are discarded. The PPI network is identical to a template network for each individual sample.

[Step 2] The second step is to map TCGA data to the global network \({\text{N}}^{G}\). Data for ten cancers from the TCGA database are downloaded, and then, the gene expression data are mapped to the global network \({\text{N}}^{G}\) generated in the prior step.

[Step 3] The third step is to calculate the local network entropy for each gene. For each gene \(g^{k}\), its local network \({\text{N}}^{k}\)(k = 1, 2, …, L) is extracted from the global network \({\text{N}}^{G}\)(Fig. 1B), where \(\left\{ {{\text{g}}_{1}^{k} , \ldots ,{\text{g}}_{M}^{k} } \right\}\) are the 1st-order neighbors of \(g^{k}\). L denotes the number of the local network. Then, the local entropy \({\text{E}}^{n} (k,t)\) is calculated based on n reference samples. The formula for calculating local entropy \(E^{n} (k,t)\) iswithwhere the constant M denotes the number of neighbors in the local network \({\text{N}}^{k}\) and \({\text{PCC}}^{n} \left( {g_{i}^{k} (t),g^{k} (t)} \right)\) denotes the Pearson correlation coefficient between the center gene \({\text{g}}^{k}\) and a neighbor \({\text{g}}_{i}^{k}\) based on n reference samples at time point t.

[Step 4] The fourth step is to calculate the differential entropy \(\Delta E(k,t)\). \(E^{n + 1} (k,t)\) is calculated based on n + 1 samples, mixing a single sample from an individual with n reference samples at time point t, i.e.,

Then, the differential entropy \(\Delta E^{k} (t)\) is calculated asandwhere \({\text{SD}}^{n} (g^{k} (t))\) and \({\text{SD}}^{n + 1} (g^{k} (t))\) denote the standard deviation of the gene expression for the center gene \({\text{g}}^{k}\) based on n reference samples and n + 1 mixed samples at time point t, respectively.

[Step 5] The fifth step is to calculate the global differential entropy \(\Delta E(t)\), i.e.,where constant \(Q\) denotes the number of all genes.\(\Delta E(t)\) is called the global LNE score or LNE score since it can reflect the overall effect of a single sample. According to DNB theory [1, 2], when the system is close to the tipping point, the LNE score can effectively represent the fluctuations of the network and thus serve as an early warning sign of critical deterioration. Theoretical explanation of local network entropy (LNE) has been supplemented in the page S6 of Additional file 1.

Ten unrelated, clinical tumor datasets (kidney renal clear cell carcinoma (KIRC), lung squamous cell carcinoma (LUSC), stomach adenocarcinoma (STAD), liver hepatocellular carcinoma (LIHC), lung adenocarcinoma (LUAD), esophageal carcinoma (ESCA), colon adenocarcinoma (COAD), rectum adenocarcinoma (READ), thyroid carcinoma (THCA), and kidney renal papillary cell carcinoma (KIRP)) were downloaded from The Cancer Genome Atlas (TCGA) database (GDC) (cancer.gov https://​portal.​gdc.​cancer.​gov/​). These datasets included RNA-seq data from tumor and tumor-adjacent samples and clinical information. The tumor samples were categorized into different stages according to the clinical (stage) information, samples lacking stage information were discarded, and the complete clinical staging information were provided in the Additional file 2: Table S1.

The molecular global template network was built with the following steps. First, the protein–protein interaction networks for Homo sapiens were downloaded from STRING (http://​string-db.​org). Second, the genes from each microarray dataset were mapped to the integrated network to construct the molecular network for subsequent analysis. Finally, the subsequent analysis results were visualized with Cytoscape (www.​cytoscape.​org).

The enrichment analysis was performed based on the KEGG Mapper tool (KEGG Mapper Color) and the DAVID Functional Annotation Tool (DAVID: Functional Annotation Tools (ncifcrf.gov). All statistical analyses were performed with R software v4.0.3 (R: The R Project for Statistical Computing (r-project.org)). The differential expression and Cox survival analyses were performed with the R DESeq2 package and R Survival package.

We used the LNE algorithm to identify the critical transition points of ten cancers in datasets that were downloaded from TCGA. The samples were categorized into different stages according to clinical information, and the tumor adjacent (TA) samples were used as reference samples for each tumor. For example, the KIRC samples were categorized into four stages, i.e., stages I, II, III, and IV. The LUSC samples were categorized into seven stages, i.e., stages IA, IB, IIA, IIB, IIIA, IIIB, and IV. The LNE score was then calculated for each single sample following the proposed algorithm, and the average LNE score curves of each stage are shown in Fig. 2A–J. Based on LNE theory, the transition point in the LNE score curve is the critical state during disease progression (Fig. 1A). To validate the identified critical state, the prognoses of before-transition and after-transition samples were determined and compared through Kaplan–Meier (log-rank) survival analysis (Fig. 2K–T). Regardless of specific biological and pathological differences, the stage II-III is a special state correlated with the lymph node metastasis, while the stage III-IV stands for a period associated with distant metastasis [19]. Therefore, we consider that stage II and stage III represent the critical point before lymph node metastasis and the critical stage before distant metastasis, respectively.

For KIRC, there was a drastic increase in the LNE score before stage III (transition point; Fig. 2A), suggesting the upcoming abrupt critical transition into the disease state (stage IV); that is, the stage at which the tumor migrates to form distant metastasis at stage III [20]. Furthermore, survival analysis was applied to compare survival curves for samples before and after the transition point (stage III) by log-rank tests to validate the identified critical state. The survival curves showed a significant difference (p < 0.0001) between stage I ~ III and stage IV KIRC samples according to clinical information (Fig. 2K). In addition, the survival time of patients after stage III was significantly shorter than that before stage III.

For LUSC, the average LNE score abruptly increased at stage IIB (Fig. 2B), indicating an upcoming critical transition after stage IIB; that is, after stage IIB, lymph node metastasis and tumor invasion of the visceral pericardial surface would occur [21]. Moreover, the survival curves of samples taken before and after the transition point (stage IIB) were significantly different (p = 0.0063; Fig. 2L). Furthermore, patients with diseases in stages after the identified critical transition had shorter survival times than those with diseases in stages before the transition.

For STAD, the average LNE score abruptly increased at stage IIIA (Fig. 2C), indicating an upcoming critical transition after stage IIIA; that is, the cancer would spread to the serosal layer of the stomach wall (stage IIIB) and ultimately cause distant metastasis (stage IV) [22]. Moreover, the survival curves of samples taken before and after the transition point (Stage IIIA) were significantly different (p = 0.0036; Fig. 2M). Furthermore, patients with diseases in stages after the identified critical state had shorter survival times than those with diseases in stages before the transition.

For LIHC, the average LNE score abruptly increased at stage II (Fig. 2D), suggesting an upcoming critical transition after stage II. A literature search showed that direct invasion of adjacent organs occurs at stage III [23]. Moreover, the survival curves of samples taken before and after the transition point (Stage II) were significantly different (p < 0.0001; Fig. 2N). Furthermore, patients with diseases in stages after the identified critical state had shorter survival times than those with diseases in stages before the transition.

The results of the same above-mentioned method for the other six cancers are shown in Fig. 2. Moreover, a method (Additional file 1: Fig. S5) was applied to validate the identified critical state, and the results were provided in the page S12 of Additional file 1. As demonstrated in these graphs, LNE is an effective method for identifying the critical state of cancers.

Above, we use the numerical algorithm LNE to identify the critical state. Subsequently, we wanted to validate the identified critical states at the network level. For each sample, the top 5% of genes with the highest LNE score were chosen as the LNE genes. The common LNE genes of samples in the identified critical state were regarded as DNBs for consequent functional and biological analyses and may participate in key associations among biomolecules related to tumor deterioration during disease progression. First, these common LNE genes were mapped to the PPI network to further study the dynamic evolution of the network and to identify changes in cancer-related molecules.

For KIRC, the dynamic changes of gene’s LNE score in its PPI regulation network across all 4 stages are shown in Fig. 3A. An obvious change in the network structure occurred at stage III, indicating an upcoming critical transition, in line with the experimental results above. Furthermore, previous studies fully support that the critical state was correctly identified.

For LUSC, the dynamic changes of gene’s LNE score in the PPI regulation network across all 7 stages are shown in Fig. 3B. A drastic change in the PPI network occurred in stage IIB, indicating an upcoming critical transition, in line with the experimental results above.

For STAD, the dynamic changes of gene’s LNE score in the PPI regulation network across all 7 stages are shown in Fig. 3C. A drastic change in the PPI network occurred in stage IIIA, indicating an upcoming critical transition, in line with the experimental results above.

For LIHC, the dynamic changes of gene’s LNE score in its PPI regulation network across all 4 stages are shown in Fig. 3D. A drastic change in the PPI network occurred in stage II, indicating an upcoming critical transition, in line with the experimental results above.

The results in Fig. 3 also reflects some additional information in the network point of view. First, the evolution of the LNE gene group (i.e., the top 5% genes with the highest LNE values) can signal the critical transition at the network level, that is, for the subnetwork composed of the LNE genes, a significant change in its topological structure occurs when the system approaches the critical state. Second, the LNE genes composed a connected subgraph when they were mapped to the protein–protein interaction (PPI) network, which can’t be implied by neither the LNE definition nor its mean value. Third, the changes of the network structure across all stages provide a clue of the dynamical evolution of the gene network during the disease progression, which may help to understand the underlying mechanisms of the associative relationships among the LNE genes.

Furthermore, results of the dynamic network changes for the other six cancers are presented in Additional file 1: Fig. S2, and also in line with the experimental results above.

LNE genes can be divided into two types of biomarkers: optimistic LNE (O-LNE) biomarkers and pessimistic LNE (P-LNE) biomarkers. Specifically, from a statistical perspective, those samples identified as O-LNE biomarkers tend to have an association with good prognosis, while those identified as P-LNE biomarkers tend to have an association with poor prognosis. More details about the identification method are provided in Fig. S1 and the page S1 of Additional file 1.

The predicted prognoses of samples with optimistic LNE biomarkers in their LNE genes were more favorable than those of other samples; that is, the survival time of these samples was expected to be longer than that of other samples. In contrast, the predicted prognoses of samples with pessimistic LNE biomarkers in their LNE genes were less favorable than those of other samples; that is, the survival time of these samples was expected to be shorter than that of other samples.

For KIRC, the survival times of samples with the O-LNE biomarkers CLIP4 (p = 0.0002; Fig. 4A) and PGD (p = 0.00093; Fig. 4A) were significantly longer than those of samples without these O-LNE biomarkers according to survival curve analysis (Fig. 4). The survival times of samples with the P-LNE biomarkers CDCP1 (p = 0.0023; Fig. 4A) and EPB41 (p = 0.0081; Fig. 4A) were significantly shorter than those of samples without these P-LNE biomarkers.

For LUSC, the prognoses of samples with the O-LNE biomarkers ACP1 (p = 0.019; Fig. 4B) and ERH (p = 0.00025; Fig. 4B) tended to be significantly better than those of other samples; i.e., the survival time was expected to be longer. The prognoses of samples with the P-LNE biomarkers AHNAK (p = 0.0083; Fig. 4B) and PLAU (p = 0.011; Fig. 4B) tended to be significantly worse than those of other samples; i.e., the survival time was expected to be longer.

For STAD, those samples with the O-LNE biomarkers CATSPERB (p = 0.026; Fig. 4C) and ZNF350 (p = 0.026; Fig. 4C) tended to have significantly better prognoses than other samples. Those samples with the P-LNE biomarkers ACE2 (p = 0.039; Fig. 4C) and SRI (p = 0.012; Fig. 4C) tended to have significantly worse prognoses than other samples.

For LIHC, those samples with the O-LNE biomarkers ATP2B4 (p = 0.014; Fig. 4D) and IRF6 (p = 0.03; Fig. 4D) tended to have a significantly more optimistic prognosis than other samples. Those samples with the P-LNE biomarkers TSEN34 (p < 0.0001; Fig. 4D) and ENO1 (p < 0.0001; Fig. 4D) tended to have a significantly worse prognosis than other samples.

Furthermore, these novel biomarkers may play significant roles in essential biological processes during critical deterioration. As shown in Table 1, in KIRC, CLIP4 was identified as an O-LNE biomarker; CLIP4 is involved in regulating the expression of several tumor-associated genes, and its expression is considered to stimulate tumor metastasis [13]. In LUSC, FGF11 was identified as an O-LNE biomarker; and FGF11 may be involved in the stabilization of capillary-like tube structures associated with angiogenesis and may act as a modulator of hypoxia-induced pathological processes such as tumorigenesis [14]. In STAD, ACE2 was identified as a P-LNE biomarker, and ACE2 can affect macrophage expression of tumor necrosis factor (TNF-α) [15]. In LIHC, TTK was identified as a P-LNE biomarker, and TTK alone or in combination with other therapeutics may selectively kill tumor cells [16].

The O-LNE and P-LNE biomarkers of the ten cancers are provided in Additional file 3: Table S2. The results imply that O-LNE and P-LNE biomarkers are effective in determining patient prognosis from a statistical perspective and may participate in essential biological processes. Moreover, these biomarkers are mainly non-differentially expressed genes and are therefore usually missed by traditional studies. For example, only KIF2C and MYBL2 were differentially expressed genes in LIHC (Additional file 3: Table S2), and GRB7 was differentially expressed in THCA (Additional file 3: Table S2). Hence, we present a method to identify novel biomarkers, drug targets and key regulators.

For cancers affecting the same system of the human body, common LNE genes identified in the critical state may participate in the same biological processes during tumor deterioration. In this study, the cancers were categorized into 3 groups based on the system affected, i.e., the respiratory system (LUSC and LUAD), urinary system (KIRP and KIRC), and digestive system (STAD, READ, LIHC, ESCA, and COAD).

For each of the 3 systems, there were not only many overlapping the signaling genes in different cancers but also tight functional relationships among them (Fig. 5A, 5D, and 5G). For each system described above, KEGG pathway enrichment analysis of the common signaling genes among the different cancers was performed. For the 3 systems of the human body, as presented in Fig. 5B, 5E, and 5H, the common signaling genes were significantly enriched in pathways related to the process of cancer development. For the respiratory system (LUSC and LUAD), the common LNE genes were enriched in cancer-related pathways, including the PI3K-Akt signaling pathway, pathways in cancer, and the cell cycle pathway (Fig. 5B). Activation of the PI3K-Akt pathway and pathways in cancer has been reported to be involved in the development and progression of lung cancer [28]. In melanocytes driven by lung cancer cell-derived exosomes, the cell cycle pathway may contribute to tumor progression more than any other pathway [29]. For the urinary system (KIRP and KIRC) (Fig. 5E), the common LNE genes were enriched in pathways associated with cancer, such as the PI3K-Akt signaling pathway, focal adhesion pathway, and MAPK signaling pathway. Suppressing the PI3K-Akt signaling pathway can inhibit cell proliferation and induce apoptosis [30]. Activating the MAPK signaling pathway can promote the proliferation and invasion of renal cell carcinoma cells [31]. For the digestive system (STAD, READ, LIHC, ESCA, and COAD), the common LNE genes were enriched in cancer-related pathways, including the PI3K-Akt signaling pathway, microRNAs in cancer, and proteoglycans in cancer (Fig. 5H). The PI3K-Akt signaling pathway has been shown to drive tumor progression and regulate metastasis in multiple cancer cells [32, 33]. MicroRNAs are critical factors in cancer biology [34]. Proteoglycans can perform multiple functions in cancer and normal angiogenesis and are directly related to cancer [35, 36]. Although unique pathways were present among these 3 different systems, some common pathways were present as well, such as the PI3K-Akt signaling pathway. Moreover, we found that the regulatory patterns of the PI3K-Akt signaling pathway for these 3 systems were quite similar. Specifically, upstream regulators, such as factors in the extracellular matrix (ECM) (common signaling genes), activate the downstream factors ITGA/ITGB (1st-order DEGs), which subsequently activate the key downstream molecules PI3KA and AKT together with other common signaling genes and 1st-order neighboring DEGs; this cascade may trigger tumor development (Fig. 5C, F and I). In general, the synergy of common signaling genes and their 1st-order neighboring DEGs may have biological significance in tumor-related biological processes.