Detecting early-warning signals for sudden deterioration of complex diseases by dynamical network biomarkers

Abstract

Considerable evidence suggests that during the progression of complex diseases, the deteriorations are not necessarily smooth but are abrupt, and may cause a critical transition from one state to another at a tipping point. Here, we develop a model-free method to detect early-warning signals of such critical transitions, even with only a small number of samples. Specifically, we theoretically derive an index based on a dynamical network biomarker (DNB) that serves as a general early-warning signal indicating an imminent bifurcation or sudden deterioration before the critical transition occurs. Based on theoretical analyses, we show that predicting a sudden transition from small samples is achievable provided that there are a large number of measurements for each sample, e.g., high-throughput data. We employ microarray data of three diseases to demonstrate the effectiveness of our method. The relevance of DNBs with the diseases was also validated by related experimental data and functional analysis.

Figure 1. Schematic illustration of the dynamical features of disease progression from a normal state to a disease state through a pre-disease state.

(a) Deterioration progress of disease. (b) The normal state is a steady state or a minimum of a potential function, representing a relatively healthy stage. (c) The pre-disease state is situated immediately before the tipping point and is the limit of the normal state but with a lower recovery rate from small perturbations. At this stage, the system is sensitive to external stimuli and still reversible to the normal state when appropriately interfered with, but a small change in the parameters of the system may suffice to drive the system into collapse, which often implies a large phase transition to the disease state. (d) The disease state is the other stable state or a minimum of the potential function, where the disease has seriously deteriorated and thus the system is usually irreversible to the normal state. (e)–(g) The three states are schematically represented by a molecular network where the correlations and deviations of different species are described by the thickness of edges and the colors of nodes respectively. When the system approaches the pre-disease state, the deviations of (z₁, z₂, z₃) increase drastically, and the correlations among (z₁, z₂, z₃) also increase drastically whereas their correlations with other nodes (z₄, z₅, z₆) decrease drastically ((e) and (f)). We call (z₁, z₂, z₃) as the dominant group or the DNB. (g) At the disease stage, the system settles down in another steady state, i.e., the disease state, with lowered deviations and correlations for the DNB. (h) Graphs show an example of the dynamical fluctuations of the molecular concentrations for the DNB in the pre-disease state, which dynamically change with strong temporal deviations but are closely (positively or negatively) correlated.

Figure 2. Numerical validation of theoretical results.

(a) A five-gene model for a DNB and an early-warning signal. The network model and detailed background are described in Supplementary Information B. The tipping point is at P = 0 in the theoretical model, at which the system undergoes a critical transition or a bifurcation detected by z₁ and z₂. (b)–(c) When the system approaches the tipping point (P = 0), z₁ and z₂ become closely correlated with increasingly strong deviations from P = 0.4 to P = 0.01. (d)–(e) Figures show the curves of SDs and PCCs for the variables against the parameter P, which clearly indicate the tendency of z₁ and z₂, i.e., their fluctuations (SD(z₁) and SD(z₂)) and correlation (|PCC(z₁, z₂)|) increase drastically whereas their correlations with other nodes (|PCC(z₁, z₃)|), |PCC(z₁, z₄)|, |PCC(z₁, z₅)|, |PCC(z₂, z₃)|, |PCC(z₂, z₄)|, and |PCC(z₂, z₅)|) decrease drastically when the system approaches the tipping point, which satisfies all three criteria for the DNB. (f) The curve shows the clear tendency of the composite index near the tipping point for the DNB composed of (z₁, z₂), which can be used as the early-warning signal for predicting the imminent change in the concerned system.

Figure 3. Detecting early-warning signals for complex diseases.

Detecting early-warning signals for diseases from two sets of high-throughput experimental data for the lung injury with carbonyl chloride inhalation exposure, i.e., acute lung injury ((a), (b), (c), and (d)), and the hepatic lesion by chronic hepatitis B, i.e., HBV induced liver cancer ((e), (f), (g), and (h)). In each sampling period, there are 2–5 samples for gene expressions. (a) and (e) represent the mean SDs in the DNB (i.e., SD_d in Eq.(2)), (b) and (f) are mean PCCs in the DNB (i.e., PCC_d in Eq.(2)), (c) and (g) represent the PCCs between the DNB and other molecules (i.e., OPCCs, or PCC_o in Eq.(2)), and (d) and (h) represent the composite index/in Eq.(2). The dotted green line indicates the period of the pre-disease state. Both cases demonstrate strong and significant early-warning signals before the diseases are eventually deteriorated. The results of these diseases show the effectiveness of our method to detect the early-warning signals using a small number of samples.

Figure 4. DNBs for two complex diseases.

(a) and (c) show expression profiles of the DNB genes and other genes (randomly selected genes with two times the size of the DNB) for the acute lung injury and the HBV induced liver cancer, respectively, which also indicate that genes in each DNB during the pre-disease period are correlated with strong deviation. Each horizontal part boxed by lines is the DNB, and each vertical part boxed by lines is the pre-disease period in (a) or (c) for the respective disease. The profiles for entire genes are described in Supplementary Information F. (b) and (d) show the identified DNBs for the acute lung injury and the HBV induced liver cancer, respectively. The DNB and whole mouse network are linked by the documented functional interactions from various databases (see Methods). Genes in each DNB are indicated in red and some of their nearest neighbors are indicated by grey nodes in (b) and (d). For acute lung injury, we also show the dynamic evolution of the network structure for the identified DNB (220 genes and 1167 links) and the whole mouse network (3452 genes and 9238 links) including the DNB. (e) The DNB at 0.5 h. (f) The DNB at 4 h. (g) The DNB at 8 h (the pre-disease state). (h) The DNB at 24 h. (i) The whole mouse network at 0.5 h. (j) The whole mouse network at 4 h. (k) The whole mouse network at 8 h (the pre-disease state). (l) The whole mouse network at 24 h. The dynamic evolution of the DNB for total 8 time points is shown in Supplementary Fig. S8 and the corresponding dynamics of the whole mouse network is shown in Supplementary Fig. S9.