Detecting the tipping points in a three-state model of complex diseases by temporal differential networks

The progressions of most diseases are generally divided into the following three states/stages: normal state, pre-disease state and disease state (Fig. 1a). Both the normal state and the disease state are stable or stationary with high resilience or little fluctuations, and are robust to parameter changes. For these two states, it is not easy to trigger a critical transition towards an alternative state through external perturbations [21]. Thus, these two states are modelled as two stationary Markov processes (Fig. 1b). The pre-disease state is a state defined as the limit of the normal state just before the critical phase transition. Different from the above two stable states, the pre-disease state is a time-varying state with low resilience or strong fluctuations, and sensitive to even a small parameter change [21]. However, it may reverse back to the normal state if appropriately treated, which is in contrast to the irreversible disease state. Statically, there are little state change or significant differences such as consistently high/low expressions in some single biomolecules between the normal and pre-disease states [26]. Taking the dynamically unstable nature of the pre-disease state into account, it is regarded as a time-varying Markov process with strong fluctuating dynamics. For a specific disease, the pre-disease state presents the critical stage between a relatively “healthy” stage (the normal state) and a seriously ill stage (the disease state) in which it is difficult for a system to recover to the normal state even intensive interferences involved. Thus, it is of great importance to signal the pre-disease state in order to prevent catastrophic deterioration by performing timely intervention.

The DNB strategy suggests that a dominant group of molecules or DNB module appears at the pre-disease state. In such a group, the molecules together provides an early-warning signal when the biological system is in the vicinity of the tipping point [1]. Specifically, we theoretically proved that when a biological system from a normal state approaches the tipping point of a critical transition, the DNB module appear and simultaneously satisfy the following three statistic conditions [3, 15].

The derivation of the three generic conditions is presented in Additional file 1: A1 and A2. It also should be noted that the three-state model of complex diseases is not always generalizable. Actually, when there is no abrupt deterioration phenotype during the disease progression, or a stage-wise division is not applicable, such model is improper to characterize the biological processes of diseases.

Most biological molecules perform their functions through interactions with other biomolecules, which are either in a functional module or across modules. This inter- and intra-module interconnectivity suggests that the impact of a specific genetic abnormality not only affects the activity of the gene product that carries it, but may extend along the links of a network composed by biomolecules and change the activity of other gene products [27]. Therefore, an understanding of a biomolecule’s interaction network context is essential in determining the phenotypic impact of defects that affect it.

To study the dynamical evolution of a network system during the switch of a stationary Markov process into another time-varying Markov process, we employed a differential network that integrates differential associations/edges, including differential expression variances and differential expression correlations from adjacent sampling time points (Fig. 1d), that is, to quantify the statistical significance of each differential edge (differential Pearson’s correlation coefficient (PCC) value) in a differential network [28].

The difference between this differential network and existing methods lies in that we study the differential associations/correlations of genes/proteins rather than differential expressions of genes/proteins. In particular, we assume that the dysfunction or network rewiring of those associations during disease progression is possibly related to the relevant disease processes. Using gene or protein expression to quantitatively describe the dynamic associations between biomolecules, we can identify those significantly changed associations (i.e., gene pairs rather than individual biomolecules) between two biomolecules. These are denoted as differential associations hereafter and are presumed to be highly related to the dysfunction of regulation network of interest. The temporal-ordering differential network was obtained by the following three steps (Fig. 1d).

According to the discussion above, the progression of a network system through a normal state is described by a stationary Markov process. Therefore, detecting the outset of a pre-disease state is equivalent to identifying the end of this stationary Markov process, or the switching point from a stationary Markov process to a time-varying Markov process. It is supposed that each time point \(t = T (T > 2)\) is a candidate transition point, or equivalently, the switching point of a stationary Markov process. Then we need to determine whether the observable samples or differential network derived at this candidate point is not coincident with that in the preceding stationary Markov process. Therefore, an index I-score is proposed for measuring the probability of how likely a candidate time point is the ending point of a stationary Markov process. The approach contains the following two steps. First, utilizing an observable network sequence O _T − 1 = {DN ₂, DN ₃, …, DN _T − 1}, i.e., T − 2 temporal-ordering differential networks based on samples from the preceding time points t = 1, 2,…, T − 1, we trained and obtained a hidden Markov model (HMM) \(\varTheta_{T - 1} \left( {O_{T - 1} } \right) = \left( {A,B,\pi } \right)\), with the subscript T − 1 of \(\varTheta\) denoting that \(\varTheta\) is constructed from the training samples/networks up to t = T − 1; parameter A represents a state transition matrix; B stands for an observation matrix; and π denotes the initial probabilities. The training of \(\varTheta\) follows the Baum-Welch procedures, an unsupervised learning method, which is presented in Additional file 1: A3. Second, we test if a candidate point is belong to the switching period, by calculating the I-score of the form:where \(\varTheta_{T - 1} (O_{T - 1} )\; = \;(A,\,B,\,\pi )\) is the trained HMM. O _T = {DN ₂, DN ₃, …, DN _T} is a sequence of temporal-ordering differential networks. {s ₁, s ₂, …, s _T − 1, s _T} denotes a state sequence where s _i is the system state at the i-th time point. S _normal and S _pre are both hidden (unobserved) states. S _normal stands for the normal state, and S _pre represents for the pre-disease state. On one hand, if the differential network {DN _T} is derived in the normal stage, or equivalently the testing point t = T, at which DN _T is built, is still within the stationary Markov process characterized by HMM \(\varTheta_{T - 1}\), then the I-score I(T) is supposed to be identical or similar to I(T − 1) (Fig. 1c). Thus the I-score curve progresses steadily and remains low value if the system is in the normal state. On the other hand, once the observation {DN _T} is from the pre-disease state, or the system transits into a time-varying Markov process at testing time point t = T, I(T) rises rapidly, suggesting that the observation o _T = {DN _T} is highly inconsistent with the HMM \(\varTheta_{T - 1}\) constructed from the prior information, i.e., the differential network series {DN ₂, DN ₃, …, DN _T − 1}. Clearly the inconsistent observation or distinct differential networks only appear during the switching period between the normal stage described by a stationary Markov process and the pre-disease stage described by a time-varying Markov process. Therefore, the boost of I(T) signals the onset of the pre-disease stage, or the impending of catastrophic transition. The specific algorithm for calculating I-score is presented in Additional file 1: A3.

We applied the I-score scheme to three time-course datasets (GSE2565, GSE1393 and GSE13009) downloaded from the NCBI GEO database (http://​www.​ncbi.​nlm.​nih.​gov/​geo). For all these genomic data, we discarded the probes without corresponding NCBI Entrez gene symbol. For each gene mapped by multiple probes, the average value was employed as the gene expression. The procedure of building a molecular interaction network was as follows. First, the biomolecular association networks for Homo sapiens and Mus musculus were downloaded from several public databases, e.g., protein–protein interactions from STRING (http://​string-db.​org), and transcriptional regulations from TRED (rulai.cshl.edu/cgi-bin/TRED/tred.cgi?process = home). We integrated these linkage information together without redundancy into a whole molecular interaction network including 65,625 functional linkages in 11,451 molecules for Homo sapiens, and 37,950 linkages in 6683 molecules for Mus musculus. Second, the genes from each microarray dataset were mapped to the integrated network to extract the related linkages. The molecular network was used for the initial construction of correlation networks and consequent analysis. Finally, our main results were visualized by Cytoscape (http://​www.​cytoscape.​org) in the post-processing step. The functional analysis including gene ontology and pathway enrichment were based on GO database (http://​www.​geneontology.​org/​page/​go-database) and KEGG mapper tool (http://​www.​genome.​jp/​kegg/​tool/​map\_​pathway2.​html).

For real datasets, to reduce computational complexity, the differential network was partitioned into many local networks. Each local network contained a centre node (a differentially-expressed gene) and all of its first-order neighbours based on the network structure, such as a mapped STRING network. The I-scores for the local networks were then calculated one by one, thus generating a weighted average score. Specifically, suppose there are k subnetworks partitioned from a differential network,where n _i denotes the number of nodes in the i-th local network and I _i stands for the local I-score of this subnetwork.

In Additional file 1: C, data description and processing procedures were presented and discussed in detail.

In order to validate the proposed computational method and I-score, we employed a theoretical model of an eight-node regulatory network (Additional file 1: Figure S1) to illustrate the identification of a critical stage when the system approaches a tipping point. Such model of regulatory network which is of Michaelis–Menten form, is usually employed for studying genetic regulatory activities such as transcription, translation, diffusion, and translocation processes [29]. Detailed description of the network characterized by a set of eight stochastic differential equations, was provided in Additional file 1B. Based on a sequence of 23 sets of parameters, a dataset was generated for numerical simulation from the network.

In Fig. 2a and b, we demonstrated the generation of differential network based on time-course data. There were a sequence of specific correlation networks constructed (Fig. 2a), based on which consequent differential networks were built (Fig. 2b). Taking the differential networks as observable samples, we carried out the numerical experiment. It can be seen that a sharp increase of the I-score, i.e., probability of being a critical transition point, indicated the coming tipping point at a bifurcation parameter value q = 0 (Fig. 2b).

To exhibit the distinct dynamics of the system between the normal state and the pre-disease state, we illustrate the underlying mechanisms of the I-score in Fig. 2c, that is, the occurrence-frequency of differential edges changes along the progression of the system. It is clear that when the system is far away from the tipping point, few edges are in a differential network, while when the system approaches the tipping point, considerably more differential edges are expected to appear in a differential network. Through this numerical experiment, it is clear that our computational approach is model-free and works even we do not know which variable or subnetwork is important in detecting the early-warning signal. Thus I-score and differential networks are capable of exploiting the high-dimensional information and thus distinguishing pre-disease samples from normal samples. The numerical experiment validates that the I-score is reliable and accurate in signalling the critical/pre-disease stage.

We present the simulation and calculation details in Additional file 1: B.

We further identified the pre-disease state through I-score in three real time-course datasets, i.e., the gene expression profiling dataset (GSE2565) generated from a mouse experiment, in which pulmonary edema was triggered by inhalation of carbonyl chloride [30]; the microarray dataset (GSE1393) of acute corneal trauma after a chemical burn with silver nitrate was also obtained from a mouse experiment, which is carried out for examining the effect of stimulation stress produced by chemical burn, in comparison with the non-stimulated control animals [31]; and the genomic dataset (GSE13009) derived from an study on human breast cancer cells, in which heregulin induced sustained signal activity in MCF-7 cells and triggered an irreversible cell phenotype change to differentiation [32]. The application on the lung injury dataset is set as a concrete demonstration. A stepwise algorithm is also presented in Additional file 1: C.

In the experiment of phosgene-induced mice lung injury (GSE2565), two groups of CD-1 male mice, each comprising six mice, were exposed to either air (as the control group) or phosgene (as the case group). Lung tissues were collected and processed from the two groups of mice at nine sampling points, i.e., 0, 0.5, 1, 4, 8, 12, 24, 48, and 72 h after exposure, based on which two series of gene expression data were produced. By comparing the gene expression levels between phosgene- and air-exposed mice, the experiments studied the oxidative detriments on lung tissue caused by phosgene exposure [30]. Identifying the critical point follows a procedural scheme. The differential network was constructed at each sampling point on the basis of differential-expression genes screened by the P value from a paired Student’s t test between case and control samples. According to the I-score scheme, except for the first sampling time point (0 h) which is always assumed as in a normal stage, we take each time point as a candidate critical point, that is, the candidate ending point of the stationary Markov process representing for the normal stage. The I-score was then calculated based on the testing differential network derived at each candidate critical point (Fig. 3a). Among the eight I-score curves in Fig. 3a, the red curve represents the inconsistency probability for testing the candidate time point at 8 h, and the seven blue ones are for testing other candidate time points. The drastic increase of I-score as well as the largest value both appeared in the red curve, indicating the maximum likelihood of 8 h being the critical point. It is also clear that there was a drastic increase of I-score around 4 h, which reached a peak at 8 h for the red curve, signalling the imminent critical transition around 4–8 h, and thus indicating the pre-disease stage just before the deterioration into serious lung pulmonary injury. As mentioned in the Methods section, the I-score for the real dataset was an average of local scores based on local differential networks, each of which contain a centre node (a differentially-expressed gene) and all of its first-order neighbours based on STRING network structure. To illustrate the relevance of these local differential networks, the local I-scores at three time points (1, 8, and 48 h) from case and control samples respectively are presented in Fig. 3b, where each local network is labelled by its centre gene. Obviously, the local I-score from case samples demonstrate the sensitivity and significance at the identified transition point (8 h). Furthermore, Fig. 3c illustrates the dynamical changes of the whole mouse molecular network from 0.5 to 48 h, where a drastic change in terms of expression variations and network structure of a group of genes located in the lower-right corner can be observed around 8 h. In Fig. 3c, the temporal alteration of the network is exhibited through differential variations and correlations, based on which the differential network construction are constructed. The detailed calculating procedure are presented in Additional file 1: C.

Clearly, from Fig. 3a–c, when deterioration is impending, the abrupt change in I-score implies a warning signal at the 4–8 h period. Conversely, the I-score is sensitive in the pre-disease state, since there is no obvious indicative sign or dynamics either in normal state or disease state. Reminding that the I-score signal is yielded from the collective behavioral dynamics of members in the differential network, the pre-disease state is hardly detected by any single biomolecules. Therefore, it may make the identification and management of high-risk cases more effective by applying the I-score scheme to the detection of pre-disease stage.

To further validate the identified critical point and elucidate the local differential networks with apparent turnover features, we presented 12 pairs of local networks with dynamically significant changes in I-scores around the identified critical time point at 8 h (Fig. 4). These local networks were mapped to an integrated STRING network. Each turnover gene (node) or turnover regulation (edge) expressed lowly (or highly) before the tipping point and highly (or lowly) afterward. The mass turnover features of these local networks demonstrate the biological significance of the identified critical point. The graph-related information is shown in Table 1, from which it can be seen that the structures of these networks changes drastically between normal and disease states. The ratios of the turnover neighbours and edges overwhelm those of the background 28.6% (turnover neighbours) and 18.1% (turnover edges), i.e., the ratio of the turnover nodes/edges in the whole STRING molecular network.

Briefly, these analysis implied that during the first 8 h after inhaling phosgene, the main pathological process of the case group resulted in an enhancement of bronchoalveolar lavage fluid protein levels, triggered pulmonary edema, and eventually increased death rates [30]. The severe phosgene-induced acute lung injury was around 8 h and lasted until 12 h after exposure. As the phosgene exposure continued, 50–60% mortality occurred after 12 h and 60–70% mortality was observed after 24 h [30]. Indicative signals of severe injury through I-score scheme are shown in Fig. 3a, illustrating that the onset of pre-disease state is around 8 h, and the system steps into the disease state (pulmonary edema) after 12 h. Our prediction based on the I-score was in agreement with the observations of the experiment. To better show the significance of the results, the validation I-score was calculated based on a leave-one-out bootstrap procedure in Additional file 1: Figure S3. Based on each set of re-sampled data, the similar increasing trend of I-score curve around 4–8 h also demonstrated the consistence of the approach.

We then exhibited the application of I-score for acute corneal trauma. In Fig. 5a, each I-score curve is calculated from an HMM that obtained based on a set of differential networks yielded before a candidate critical time point. The red curve in Fig. 5a shows a drastic increase of I-score around 1–3 h and reach the peak at the 8-h sampling point, which implies that 8 h is a tipping point with the highest probability. To demonstrate the system progression at the network level, four consequent whole gene regulatory network were illustrated on the basis of the case data of acute corneal trauma in Fig. 5b, from which the network structure also showed a significant change at 8 h. In the original experiment, the heat shock genes were upregulated beginning at the 8 h time point, indicating the start of a stress response [31]. The changes in gene expression attained two peaks at 8 and 72 h, and then declined to low levels of activity at later time points. In line with the actual observation, our analysis successfully identified the critical state before the emergence of strong stress response.

The third application of I-score on the heregulin-induced breast cancer (GSE13009) is demonstrated in Additional file 1: Figure S4 and related contents placed in Additional file 1: C.

As one of the most important and common chemical industry compounds, the phosgene gas and its inhalation damage receives extensive attention [33]. Some pathogenic and genomic mechanisms of the pulmonary injury activated by phosgene exposure have been studied [30]. From our analysis, a significant change in the average I-score of the local differential networks occurred from 4 to 8 h. Analysing the genes from local differential networks with dynamically significant change in I-score (Fig. 4), pathway enrichment and gene ontology functional analysis suggested that the genes in these local networks are highly related to the mechanism of disease progression [30, 34]. Additionally, some well-known genes involved in apoptosis or related to the inflammatory response were included, such as JUN (local network 8), NOTCH2 (local network 12), MYC (local network 1), IL1B (local network 7), and PTGS2 (local network 5) (Fig. 4), which may reduce the number of injured cells.

Moreover, some genes in the identified local differential networks were consistent with the dysfunction in glutathione metabolism (such as ASNS and GCLC) and the inflammatory immune response associated with chemokine signalling pathway (such as IL1B), which may relate to the abnormality of antioxidant reactions (such as PTGS2) caused by continually inhaling of phosgene. Thus, resulted from oxidant reaction of carbonyl chloride and the decrease of antioxidant enzyme concentrations, some protein-modified signalling pathways, including the Wnt signalling pathway and the mitogen-activated protein kinase (MAPK) signalling pathway, were disordered and thus may affect pro-inflammatory cytokines and other inflammatory mediators (such as NFKB1) [35].

The enrichment analysis also suggested that some signalling pathways may also be involved or affected in the phosgene-induced pathogenic processes, such as TGF-beta signalling pathway and hormone-mediated signalling pathway, which associated with the cell repair and reproduction.

Besides, some biological processes were also suggested to be highly relevant to lung injury through GO functional analysis, i.e., the differentially high/low expression of genes in the identified local differential networks associated with abnormal alterations in primary metabolic processes, which indicates the denaturation of lipids, proteins, and nucleic acids that may have been oxidized by phosgene [30, 34].

The functional analyses for acute corneal trauma and heregulin-induced breast cancer are shown in Additional file 1: C.