Identifying causal regulators using Granger causality

The two-way or three-way ANOVA analysis defining time- and state-dependent gene expression signatures provides meaningful way to characterize expression changes on a global scale [18]. However, these methods on their own do not provide any information on the causal regulators driving the time-dependent gene expression behavior. To leverage the time series data more maximally towards this end, we applied Granger causality test to gene expression traits scored systematically in the fasted/fed cohort blood samples at roughly 1 hour intervals during the course of a day (Figure 2). A gene expression trait is said to be Granger causal for gene expression trait if, at previous time points, provides significantly more information on time-dependent changes in than the historical information provides on itself. In our implementation of the Granger causality test, we test this by fitting to an autoregressive model with respect to the different time points, and then testing whether extending the autoregressive model by including improves the fit (see Materials and Methods for details). If there is a statistically significant improvement testing the model fit (assessed by comparing the models using the F test), then we declare that is Granger causal for , or simply as .

Traditionally, a long time series is required to apply Granger causality test. However, it is hard to obtain a long time series of human samples collected in vivo. We have previously shown that over 80% of transcripts have significant inter-individual variances [18], which is comparable to previously reported result [19]. Thus, we can treat time series data from 40 patients as 40 independent short time series. Assuming these 40 time series have similar dynamic behavior, but with different starting points, we can combine them together to generate a virtual long time series (shown in Figure 2, and see Materials and Methods for details). Our simulation results show that the virtual long time series are as informative as long time series with similar data points (shown as Supplementary Figures S1 and S2). We constructed causal networks for the fasted and fed states by applying the Granger causality test to all gene expression trait pairs generated in the fasting/feeding cohort described in Figure 2. For gene expression traits scored in the fasting/feeding cohort, a link was inserted into the causal network if the p-value associated with the Granger causality test was less than 0.01 after multiple testing correction. The resulting fasted and fed networks were comprised of 2010 and 967 causal links (listed in Supplementary Tables S1 and S2), respectively. The corresponding false discovery rates (FDR) [20] for the causal links in the fasted and fed networks were and , respectively. Bootstrapping test results (see Materials and Methods for details) show that 80% and 90% of links in fast and fed networks have confident values above 0.5, respectively (shown in Supplementary Figure S3). Both networks were observed to exhibit the scale-free property for out-degree distributions (shown as Supplementary Figure S4). From these data it was possible to identify all expression traits supported as Granger causal for at least one other expression trait in the network (referred to here as causal regulators), and then rank order the causal regulators according to the number of genes for which they were supported as causal, shown in Table 1.

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Table 1. Top causal regulators in the fast and fed Granger causality networks.

https://doi.org/10.1371/journal.pcbi.1000671.t001

There are more causal links inferred for fast time series than for fed time series. The fasted network consists of many small subnetworks and the fed network consists of mainly two subnetworks (shown in Figures 3A and 3B). The top causal gene in the fasted network is RNF144B, a putative ubiquitin-protein ligase that plays a role in mediating p53-dependent apoptosis. Genes under RNF144B regulation including PTEN are enriched for the GO biological process of negative regulation of cellular metabolic process (p-value = 0.008). The top causal gene in the fed network is PER1, a transcription factor regulating the circadian clock, cell growth and apoptosis. The genes under PER1 regulation are enriched for genes correlated to plasma concentration of triglyceride (p-value = 0.00045) in the Icelandic Family Blood (IFB) cohort [10]. PER1's downstream genes are involved in diverse biological processes including CREB5, in circadian rhythm, PTEN and P53INP2 in apoptosis, IL1R1, IL1RAP and TLR2 all involved in inflammation response, FASN and ACSL1 in fatty acid metabolism and MVK in cholesterol biosynthesis. These results suggest that food intake interacts with circadian rhythm and the circadian rhythm has impacts on many biological processes as has been previously shown in mouse studies [21]–[22]. Further, previous research has demonstrated circadian gene (PER1, PER2, PER3 etc.) mRNA expression rhythm in human peripheral blood cells and linked that to individual's circadian phenotype [23]–[24]. Our blood causal network where PER1 is a top causal gene illustrates a potential mechanism of how the CNS control and environmental influences (e.g. external sunlight) can affect circadian rhythm gene expression which in turn regulating a host of other biological functions. More specifically, circadian rhythm genes (PER1 in particular) play important roles in cell cycle regulation and cancer processes [25]–[26]. These reports support our observations in the fed network that several genes under PER1 control are involved in apoptosis and cell cycle regulation (e.g., PTEN and P53INP2).

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Figure 3. Fasted and fed Granger causality networks.

(A) the global view of the fast network; (B) the global view of the fed network; (C) the zoom-in view of the subnetwork around PER1 (black box) in (B). There are more links in the fast networks. The fast network consists of many small subnetworks and the fed network consists of two large subnetworks and a few small ones. Nodes in red are obesity causal genes.

https://doi.org/10.1371/journal.pcbi.1000671.g003

380 human genes are cataloged as obesity causal genes in the human obesity map [27]. In recent years, many large genome-wide association studies (GWAS) have convincingly identified a number of genes causing human obesity. 34 genes including FTO were replicated in many populations [28]–[31] Taking consideration of these two sources, there are 409 obesity causal genes, and 246 of them were expressed in our blood data set. When the obesity causal genes were overlapped with the fasted and fed networks, 7 genes (ADA, BBS5, CBL, CCND3, FASN, FTO and SCARB1) overlapped with PER1's downstream genes in the fed network (Fisher's Exact Test p-value = 0.037) (shown in Figure 3C). It has been shown that circadian rhythm links to metabolic processes in mouse [32]–[33]. For instance, mutations in mouse genes involving circadian rhythm regulation, such as Clock, can lead to obesity [34]. Our results provide evidence that human obesity causal genes are under circadian rhythm control in a peripheral tissue like blood.

Connecting different biological processes using dynamic Bayesian networks (DBN)

Constructing DBN using the model described in Figure 1A, requires a large amount of data and computational resources. However, when the intra-slice structure (the SBN) is known, then there is a dramatically reduced demand for large amounts of both data and computational resources. A large dataset of profiled peripheral blood samples (IFB) is already described and available [10]. The fasting feeding study group and the IFB cohort are derived from the same population both in terms of geological location and genetic background, therefore the static networks based on these two studies are assumed to be similar. The IFB data set consists of both gene expression measured in the fasting state and genotype data. Previously, we demonstrated that Bayesian networks constructed by integrating gene expression data and genotype data were of high quality [12]–[13],[35]. To match for gender, data from 455 males in the IFB cohort was used to construct a static Bayesian network which consisted of 7310 nodes (genes) and 11047 links (see Method Section for details). The static Bayesian network was fixed as the intra-slice network in the DBN model shown in Figure 1A, and then the time series data (fast or fed) were used to construct inter-slice connections.

The fasted and fed DBNs consisted of 1125 and 1290 inter-slice links (listed in Supplementary Tables S3 and S4), respectively. Among them, 846 (75%) and 936 (73%) were self links. 404 self links are common between the fasted and fed DBNs. The genes under self control (with self links in DBNs) are enriched for cis expression quantitative traits (cis eQTLs) in blood (enrichment p-values =  and for the fasted and fed DBNs, respectively).

One important goal for utilizing time series data is to study the dynamic changes in molecular networks. Under static condition, many biological processes may be disconnected or loosely connected, whereas under a perturbation, these processes will change coordinately. 409 obesity causal genes mentioned above were collected from two resources, namely the human obesity map [27] and recent GWAS data [28]–[29],[36]. 138 out of the 409 genes are included in the DBNs. These 138 genes were used as seeds to construct obesity related sub-networks for fast and fed DBN and the SBN as previously described [13]. The fasted and fed subnetworks were compared with the subnetworks constructed from the SBN. The largest change was from the fed subnetwork, where three segmented subnetworks in the SBN were connected in the fed DBN by two inter-slice links (shown as red in Figure 4). CDCA7, a transcription regulator for the cell cycle, is found in the center of the connected subnetworks. It connects genes involved in lipid metabolism such as NPC1, FABP5 and APOE to the large subnetwork on the left which consists of inflammatory response genes such as STAT3, STAT5, GPR109A, TNF, NTSR1, ORM1 and IL1RN. This suggests that the expression of genes involved in either inflammatory response or lipid metabolism change coordinately in response to food intake. It is also worth noting that the circadian rhythm regulator PER1 is in the subnetwork on the left, which consists of many genes involved in inflammatory response pathways. As well, in the fed DBN, both cell cycle regulation and lipid metabolism processes are linked to the circadian rhythm.

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Figure 4. Changes in the obesity causal genes subnetworks between the fed DBN and the SBN.

Nodes in red represent obesity causal genes. Edges in black are links in the SBN and Edge in red are inter-slice links in the fed DBN. Three subnetworks in the BN are connected in the fed DBN. The circadian rhythm regulator PER1 is also in the subnetwork.

https://doi.org/10.1371/journal.pcbi.1000671.g004

Time series data

40 healthy participants from an Icelandic company were recruited to participate in a randomized, two-arm, cross-over study to examine the effects of fasting and feeding on human blood gene expression [18], shown in Figure 2. For the first period of the study the 40 participants were randomized to two treatment groups, with 20 individuals making up each group. All participants began fasting at 9pm the night before the first period of the study. The first treatment group comprised the fasted arm of the study for the first period, where participants continued to fast through the day for the duration of the study (participants were only allowed to drink water during this time). The second treatment group comprised the fed arm of the study for the first period, where participants were fed a standard meal in the morning and then fasted through the rest of the day for the duration of the study. The second period of the study was carried out one week later from the start of the first period. The protocol for the second period of the study was identical to the first period, except those in the fasted arm for the first period were switched to the fed arm, and those in the fed arm for the first period were switched to the fasted arm. Figure 2 shows the schematic for the experimental design.

A total of 560 peripheral blood samples were collected from the 40 participants at 7 time points for each period of the study. Significant inter-individual variation has been noted in human blood gene expression profiles [43]. Previous analyses carried out on this data set detailed the inter-individual variation and overall expression differences between the fasted and fed conditions [18]. In the present study we focus mainly on using temporal information to infer causal relationship by applying a Granger causality test and a dynamic Bayesian network so that possible causal drivers of dynamic changes can be identified from the causal networks. To correct for the individual differences in gene expression we referenced each individual expression profile to the corresponding individual profile at time point 0. This reduced the effective number of time points to 6 for this study.

Constructing causal networks using Granger's causality test

The time series based causality test was proposed by Wiener [44] as the notion that, if the prediction of one time series could be improved by incorporating the knowledge if a second one, then the second series has a causal influence on the first. Granger was the first to formalize the idea in the context of linear regression model [17], so that time series based causality test is generally referred as Granger causality test. There is a variety of models for testing Granger causality, such as multivariate autoregressive model (MVAR) and bivariate autoregressive model (BVAR). If coefficients in the regression model do not change depending on time, the model is referred as a stationary model. Otherwise it is referred a non-stationary model. The simplest model is stationary bivector autoregressive model. Even though comparing to MVAR, BVAR tends to infer many indirected links, the causal directions of these inferred links follow causal information flows [45]. To remove potential in-direct links, for each gene, we only keep one causal link pointing to it, which has the most significant p-value in the BVAR model.

Traditionally, Granger causality test is applied to long time series. However, it is hard to collect long time course data from human samples. Our data consists of many short time series from multiple individuals. There are several theoretical studies related to combining multiple time series in a general regression frame work, including for instance that of Berhane & Thomas [37] and Guerrero & Pena [46]. Berhane & Thomas [37] proposed to use a mixed-effects model to combine time series from different locations, while Guerrero & Pena [46] outlined a weighted least squares approach. In both approaches, some constraints were applied after a number of assumptions were made.

Our approach is a simplified version of the Berhane & Thomas approach [37]. Instead of using community-specific slopes, we assumed response slopes for individuals are similar. Further, in order to reduce individual specific variation which could affect the response slope, an individual's gene expression data were normalized according to its own expression data at the first time point. Our simulation study shows that causal relationship can be accurately inferred by combining these short time series.

Simulation of short time series

Under first order stationary BVAR model, a set of data was simulated for causal relationship as following:(1)There are independent time series of length , , . All coefficients and noises follow normal distributions as(2)The initial conditions are draw from an uniform distribution with mean 0. 1000 independent time series were simulated, and each series consists of 240 time point (shown as Supplementary Figure 1).

The test of Granger causality under BVAR model can be carried out by comparing the full model(3)with the autoregressive model(4)The significance of the Granger causality test (full model explains more variance than the autoregressive model) is then measured by F-test statistics(5)where and are sum of squared residuals of full model and autoregressive model, respectively; and is the length of the time series.

For the 1000 time series simulated above, the p-values of Granger causality are estimated as Eq. 5. If only partial time points are used, then the power to detect Granger causality decreases (shown in Supplementary Figure 2). It is worth to note when the same number of time points are used, it is more likely to inferred correct causality if the interval between time points is shorter.

If only 6 time points are used, no Granger causality test is significant if considering the time series independently. If assuming and are similar, then these short series can be combined together to infer Granger causality, and the Eq. 3 can be modified as(6)where and are sum of squared residuals of full model and autoregression model, respectively. For example, a virtual time series by combining the first 6 time points of randomly selected 40 time series is as informative as a long time series with the same time points.

To estimate the false positive rate, we permuted the assignment of 1000 time series generated above (for example was assigned as where ) so that the autoregressive assumption was valid. For each permutated data set, we followed the same procedure mentioned above to calculate p-values for the Granger causality test. At different p-value cutoffs, we calculated the recall (positive rate) and the false positive rate (shown in the Supplementary Figure 2).

It is of note that choosing the optimal time lag length in the autoregressive (AR) model normally requires comparing model residuals and statistics at different p-value thresholds. However, because of the small sample size (40) and limited number of time points (6), we restricted our analyses here using AR models with only first order time dependency, similar to what has been done in previous studies [5]–[6]. Similarly, we assumed the Granger causal relations were stationary from time point 1 to 6. That is, we were mainly interested in the mean α and β values in Eq.(1), which represent the averaged Granger causality between genes from time point 1 to 6.

Bootstrapping test

A bootstrapping procedure of re-sampling individuals with replacement, was used. At each time, one subject (along the associated data at 6 time points) was sampled from a pool of 40 individuals. A bootstrapped data set consisted of 40 sampled individuals (40×6 data points). The same Granger causality test outlined above was applied to the re-sampled data. The bootstrapping procedure was performed 100 times. The link confident value is the percentage of a link's p-values above a multiple testing corrected threshold in the results of the 100 bootstrapping tests.

Reconstructing the static Bayesian network

455 male samples in IFB cohort [10] was used in reconstruction of the static Bayesian network. A set of informative genes were identified as follows: (1) a gene expressed in the blood (with mean log intensity >−1.5), (2) the variation of the mean log ratio was larger than 1.23. Of the 23720 genes represented on the microarray, 7310 were selected for inclusion in the network reconstruction process as previously described [12],[35]. One thousand Bayesian networks were reconstructed using different random seeds to start the reconstruction process. From the resulting set of 1000 networks generated by this process, edges that appeared in greater than 30% of the networks were used to define a consensus network.

Reconstructing dynamic Bayesian networks

For a two-slice dynamic Bayesian network represented in Figure 1A, it can be decomposed as , where is the parent set of . The static Bayesian network reconstructed above was used as the intra-slice network. The intra-slice network is fixed and is not refined in the process of reconstructing dynamic Bayesian networks. Thus, only inter-slice links () are added or removed during the reconstruction process. Similar to the static Bayesian network reconstruction process, 1000 networks were reconstructed using different seeds and the Bayesian information criterion (BIC) score [47] was used for the optimization. Edges appeared in 30% of the 1000 structures are included in the final network.