Abstract
Background
A common approach for time series gene expression data analysis includes the clustering of genes with similar expression patterns throughout time. Clustered gene expression profiles point to the joint contribution of groups of genes to a particular cellular process. However, since genes belong to intricate networks, other features, besides comparable expression patterns, should provide additional information for the identification of functionally similar genes.
Results
In this study we perform gene clustering through the identification of Granger causality between and within sets of time series gene expression data. Granger causality is based on the idea that the cause of an event cannot come after its consequence.
Conclusions
This kind of analysis can be used as a complementary approach for functional clustering, wherein genes would be clustered not solely based on their expression similarity but on their topological proximity built according to the intensity of Granger causality among them.
Background
Gene network analysis of complex datasets, such as DNA microarray results, aims to identify relevant structures that help the understanding of a certain phenotype or condition. These networks comprise hundreds to thousands of genes that may interact generating intricate structures. Consequently, pinpointing genes or sets of genes that play a crucial role becomes a complicated task.
Common analyses explore genegene level relationships and generate broad networks. Although this is a valuable approach, genes might interact more intensely to a few members of the network, and the identification of these socalled subnetworks should lead to a better comprehension of the entire regulatory process.
Several in silico methodologies are available for the identification of subnetworks, or clusters, within a given dataset [15]. Most of the times, the identified clusters group genes based on similar patterns of expression in time. In a different manner, the identification of Granger causality [6] within a network allows the clustering of genes based on their topological proximity in the network [7,8]. Briefly, Granger causality [6] analysis identifies interaction in terms of temporal precedence (the cause comes before its effect) [6] and may generate a set of subnetworks within which Granger causality is intense among genes. As a result, genes are grouped depending on how close they are in terms of Granger causality. Figure 1a illustrates the clustering based on the network topological proximity while Figure 1b shows the clustering based on similar expression patterns.
Figure 1. Regulatory networks.(a) Functional clustering. Genes are clustered based on their topological proximity given by Granger causality. (b) Usual clustering. Genes are clustered based on the similarity between gene expression levels.
The concept of Granger causality [6] has been previously shown to help in the identification and interpretation of regulatory networks in time series gene expression datasets [918]. The main advantage of Granger causality analysis in the context of gene expression datasets consists in the fact that each edge of the network represents the information flow from one gene to another [19]. Nevertheless, it is necessary to point out that Granger causality is not effective causality in the Aristothelic sense because it is based on prediction and numerical calculations. Fujita et al. [2022] suggested a concept for the identification of Granger causality between groups of time series. The application was, however, limited to scenarios when clusters could be previously defined based on particular data characteristics. Here, we propose a method to define clusters by their topological proximity in the network. For this purpose we introduce an extension of the concept of functional clustering, initially proposed by [23] in neuroscience. In [23], they applied mutual information in order to group the most active brain regions. We are interested in clustering the genes by using the concept of information flow [19] between sets of time series [20]. The gene expression time series are grouped depending on the hidden structure underlying the network topology, in a way that genes which are topologically close in terms of Granger causality are clustered (Figure 1a). We use the generalization of Granger causality for sets of time series datasets proposed by [20,21] in order to define concepts of distance, degree and flow useful to determine gene sets that highly interact in terms of Granger causality. In other words, we will derive the Granger causalitybased functional clustering directly from the time series gene expression data. For this purpose, an approach that allows the identification of the optimum number of clusters for a given dataset is also presented.
Materials and Methods
Granger causality for sets of time series
Granger causality identification is a potential approach for the detection of possible interactions in a data driven framework couched in terms of temporal precedence. The main idea is that temporal precedence does not imply, but may help to identify causal relationships, since a cause never occurs after its effect.
A formal definition of Granger causality for sets of time series [20] can be given as follows.
Definition 1
[20]Granger causality for sets of time series: Suppose thatℑ_{t}is a set containing all relevant information available up to and including timepointt. LetX_{t},
where
For the linear case,
where ρ is the largest correlation calculated by Canonical Correlation Analysis (CCA).
In order to simplify both notation and concepts, only the identification of Granger causality for sets of time series in an Autoregressive process of order one is presented. Generalizations for higher orders are straightforward.
Functional clustering in terms of Granger causality
There are numerous definitions for clusters in networks in the literature [24]. A functional cluster in terms of Granger causality can be defined as a subset of genes that strongly interact among themselves but interact weakly with the rest of the network.
A usual approach for network clustering when the structure of the graph is known is the spectral clustering proposed by [25]. However, in biological data, the structure of the regulatory network is usually unknown.
In order to overcome this limitation, we developed a framework to cluster genes by their topological proximity using the time series gene expression information. We developed concepts of distance and degree for sets of time series based on Granger causality, and combined them to the modified spectral clustering algorithm. The procedures are detailed below.
Functional clustering
Given a set of time series
Let G = (V,E) be an undirected graph with vertex set V = {v_{1},…,v_{p}}(where each vertex represents one time series) and weighted edges set E. In the following we assume that the graph G is weighted, that is each edge between two vertices v_{i} and v_{j} carries a nonnegative weight w_{ij} ≥ 0. The weighted adjacency matrix of the graph is the matrix W = w_{ij}; i,j = 1,…,p. If w_{ij} = 0, this means that the vertices v_{i} and v_{j} are not connected by an edge. As G is undirected, we require w_{ij} = w_{ji}. Therefore, in terms of Granger causality, w_{ij} can be set as the distance between two time series
Definition 2
Distance between two (sets of) time series
Notice that
Moreover, notice that the CCA is the Pearson correlation after dimension reduction,
therefore,
Another necessary concept is the idea of degree of a time series
Definition 3
Degree of
where indegree and outdegree are respectively
Notice that indegree and outdegree represent the total information flow that “enters” and “leaves” the vertex v_{i}, respectively. Therefore, the degree of vertex v_{i} contains the total information flow passing through vertex v_{i}.
Without loss of generality, it is possible to extend the concept of degree of a vertex
v_{i} (time series
Definition 4
Degree of subnetwork
where indegree and outdegree are respectively
Now, by using the definitions of distance and degrees for time series and sets of time series in terms of Granger causality, it is possible to develop a spectral clusteringbased algorithm to identify subnetworks (set of time series that are highly connected within sets and poorly connected between sets) in the regulatory networks. The algorithm based on spectral clustering [25] is as follows:
Input: The p time series (
Step 1: Let W be the (p × p) symmetric weighted adjacency matrix where
Step 2: Compute the nonnormalized (p × p) Laplacian matrix L as (Mohar, 1991)
where D is the (p × p) diagonal matrix with the degrees d_{1},…,d_{p}(
Step 3: Compute the first k eigenvectors {e_{1},…,e_{k}} (corresponding to the k largest eigenvalues) of L.
Step 4: Let U ∈ ℜ^{p×k} be the matrix containing the vectors {e_{1},…,e_{k}} as columns.
Step 5: For i = 1,…,p, let y_{i} ∈ ℜ^{k} be the vector corresponding to the ith row of U.
Step 6: Cluster the points (y_{i})_{i=1,…,p} ∈ ℜ^{k} with the kmeans algorithm into clusters {X_{1},…,X_{k}}. For kmeans, one may select a large number of initial values to achieve (or to be closer) the global optimum configuration. In our simulations, we generated 100 different initial values.
Output: Subnetworks {X_{1},…,X_{k}}.
Notice that this clustering approach does not infer the entire structure of the network.
Estimation of the number of clusters
The method presented so far describes a framework for clustering genes (time series) using their topological proximity in terms of Granger causality.
Now, the challenge consists in determining the optimum number of subnetworks k. The choice of the number of subnetworks k is often difficult depending on what the researcher is interested in. In our specific problem, one is interested in identifying the clusters presenting dense connectivity within a cluster and sparse connectivity between clusters.
In order to determine the most appropriate number of clusters in this specific context, we used a variant of the silhouette method [28].
Let us first define the cluster index s(i) in the case of dissimilarities. Take any time series
After calculating a(i) and b(i), the cluster index s(i) can be obtained by combining them as follows:
Indeed, from the above definition we easily see that −1 ≤ s(i) ≤ 1 for each time series
The average cluster index s(i) of a subnetwork is a measure of how tightly grouped all the genes in the subnetwork are. Thus, the average cluster index s(i) of the entire dataset is a measure of how appropriately the genes have been clustered in a topological point of view and in terms of Granger causality.
Estimation of the number of clusters in biological data
In order to estimate the most appropriate number of subnetworks present in the data set, we estimate the average cluster index s of the entire dataset for each number of clusters k. When the number of identified subnetworks is equal or lower than the adequate number of subnetworks, the cluster index values are very similar. However, when the number of identified subnetworks becomes higher than the adequate number of subnetworks, the cluster index value s decreases abruptly. This is due to the fact that one of the highly connected subnetworks is split into two new subnetworks. Notice that these two new subnetworks present high connectivity between them because they are in fact, only one subnetwork. In order to illustrate this event, see Figure 2 for an example. In Figure 2a, genes in cluster 1 are highly interconnected. Now, suppose that one wants to increase the number of clusters by splitting cluster 1 into two clusters namely clusters 1 and 5 (Figure 2c). Notice that clusters 1 and 5 are highly connected between them. If the number of clusters is higher than the adequate number of clusters (four, in our case), the value s decreases substantially, since the Granger causality between clusters increases and the within cluster decreases. The breakpoint where the value s decreases abruptly can be used to determine the adequate number of subnetworks. In fact, this can be visually identified by analyzing the breakpoint at the plot similarly to the standard elbow method used in kmeans. However, if one wants to determine the breakpoint in an objective manner, this can be done by adjusting two linear regressions, one with the first q dots and another with the remaining dots, thus identifying the breakpoint (the value q) that minimizes the sum of squared errors (Figure 3).
Figure 2. (a) The representation of network with four clusters; (b) The network obtained by applying the proposed method with four clusters (k=4); (c) The representation of the network when the number of clusters is set to five; (d) The network obtained by applying the proposed method with five clusters (k=5). The solid edges represent Granger causality. Notice that the structure of the “true” network ((a) and (c)) is not observed and it can only be estimated.
Figure 3. The optimum number of subnetworks is indicated by the breakpoint in the graph. The breakpoint appears when the number of subnetworks is greater than the adequate number of subnetworks. The breakpoint selection criterion is based on two linear regressions that best fit the data.
Network construction
The network connecting clusters is constructed following procedures previously described [20,21]. Briefly, after Classification Expectation Maximization (CEM) [29] Principal Component Analysis (PCA) is used to remove redundancy and to extract the eigentime series from each cluster. PCA allows us to keep only the most significant components leading to variability in the dataset, thus reducing the number of variables for subsequent processing. In this study, we retained only components accounting for more than 5% of the temporal variance in each cluster [22]. The eigentime series are then clustered as described in the section Functional clustering and the network can be inferred by applying the method proposed by [20,21].
The Granger causality between cluster is identified by:
where
Then, test
Simulations
Four sets of Monte Carlo simulations were carried out in order to evaluate the proposed
approach under controlled conditions. The first scenario represents four subnetworks
without Granger causality between them (Figure
4a). The second scenario consists of four subnetworks constituting a cyclic graph
(Figure
4b). The third scenario presents a feedback loop between subnetworks A and B (Figure
4c). The fourth scenario is composed of a network with one subnetwork (subnetwork
D) that only receives Granger causality and one subnetwork (subnetwork A) that only
sends Granger causality (Figure
4d). Since biological data usually possess several highly correlated genes (genes which
hold the same information from a statistical stand point), we constructed 10 highly
correlated time series for each
Figure 4. (a) Four independent subnetworks without Granger causality between them; (b) Four subnetworks in a cyclic graph; (c) Feedback loop between subnetworks A and B; (d) A network between subnetworks, where subnetworks A only sends Granger causality and D only receives Granger Causality.
For each scenario, time series lengths varied: 50, 75, 1000 and 200 time points. The number of repetitions for each scenario is 1,000. The synthetic gene expression time series data in subnetworks A, B, C and D were generated by the following equations described below.
Simulation 1:
Simulation 2:
Simulation 3:
Simulation 4:
where β = 0.6, γ = 0.3, ε_{i,t}∼N(0,Σ) with
and
for i = 1,…,20.
Actual biological data
In order to illustrate an application of the proposed approach, a dataset collected by [30] was used. The work presents whole genome gene expression data during the cell division cycle of a human cancer cell line (HeLa) characterized using cDNA microarrays. The dataset contains three complete cell cycles of ∼16 hours each, with a total of 48 time points distributed at intervals of one hour. The full dataset is available at: http://genomewww.stanford.edu/HumanCellCycle/HeLa/ webcite.
In order to evaluate our proposed approach, we chose to analyze the same gene set examined in Figure 5 of [10], which comprised a set of 50 genes.
Figure 5. The optimum number of subnetworks in the actual biological data is indicated by the breakpoint in the graph (the optimum number in this case is three).
Results
Simulated data
In order to study the properties of the proposed functional clustering method and to check its consistency, we performed four simulations with distinct network characteristics in terms of structure and Granger causality.
Table 1 describes the frequency that each number of clusters was identified as optimal in each simulation and time series length. Notice that the accuracy of the method in identifying the correct number of clusters clearly converges to 100% as the time series length increases (the correct number of clusters is four for all the scenarios). The same result was obtained with varying numbers of subnetworks or when Granger causality within clusters increased, demonstrating the consistency of the method. Moreover, both the cluster indices value and the respective standard deviation for each simulation and time series length are described. The average cluster index value was calculated by using the value at the breakpoint as described in Figure 3 in 1,000 repetitions. By analyzing Table 1, it is possible to verify that the longer the time series length, the smaller are the standard deviations and the greater is the silhouette width demonstrating that the method is consistent.
Table 1. Frequency of the selected number of clusters for each scenario and time series length
Table 2 describes the average of the frequency (in percentage) the time series were correctly clustered for each scenario and each time series length given the correct number of clusters. It is important to point out that the number of correctly classified time series increases as the time series length increases.
Table 2. Average of the percentage of correctly clustered timeseries in 1,000 repetitions given the correct number of clusters
Table 3 represents the frequency (in percentage) each edge of the simulated network was identified when the estimated number of clusters were correctly identified as four. The correctly identified edges are in bold. Since the pvalue threshold was set to 0.05, it is expected to identify ≈5% of false positive edges where there is indeed no Granger causality. In fact, where there is no Granger causality, the rate of false positives was controlled to 5%, and where there is Granger causality, the number of identified edges is clearly higher than where there is no Granger causality.
Table 3. Percentage of edges with time series length equals to 50/75/100/200 when the estimated number of clusters were correctly identified as four
Biological data
By applying the method described in section Functional clustering to the biological dataset, the optimum number of subnetworks was identified as three. Notice in Figure 5 that there is a clear breakpoint when the number of clusters is three.
Once clusters were obtained, the clustercluster network (Figure 6) was modeled by applying the method described in [20,21]. Two of the depicted clusters, clusters one and two, provide interesting material for biological interpretation. Genes belonging to cluster two highlight expected interconnections in cell cycle regulation. For instance, aberrant activation of signal transcription factors NFκB or STAT3, and alterations in p53 status, have each been reported to affect cell survival individually. The presence of the three genes in the same cluster is in agreement with a recent study which examined the hypothesis that alterations in a signal network involving NFκB, STAT3 and p53 could modulate expression of proapoptotic BAX and antiapoptotic BCLXL proteins, promoting cell survival [31]. The authors show that overexpression of p53 together with inhibition of NFκB or STAT3 induced greater increase in the BAX/BCLXL ratio than modulation of these transcription factors individually. As discussed earlier in this paper, this is a situation in which similar patterns of gene expression are not sufficient to comprehend the biological process.
Figure 6. The network obtained with three (k=3) subnetworks. Solid arrows are significant Granger causality with pvalue < 0.05 and dashed arrow is significant Granger causality with pvalue < s0.10. The circles represent the clusters.
In [10], a network depicting Granger interaction among genes from this same gene dataset was presented. The authors analyzed the network in the context of tumor progression and identified genegene connections associated with NFκB, p53, and STAT3. Here, cluster 1 groups not only NFκB, p53, and STAT3, but also the functionally associated gene BCLXL, NFκB regulator A20 and targets IAP and iκBα. The presence of NFκB and fibroblast growth factors (FGFs) and receptors (FGFRs) in the same cluster is also in agreement with the previous work. Members of the FGF family and NFκB have been shown to interact in various contexts and, despite distinct roles, are involved in cell proliferation, migration and survival [32,33].
Even though MCL1 and P21 play important roles in cell survival, and BAI1 is transcriptionally regulated by P53, the analysis run here clustered them separately from P53 containing cluster. This result suggests that, in the context of this dataset, their interaction is stronger with genes such as cJUN, also functionally related to cell survival, protooncogene MET and tumor suppressor MASPIN, for instance. Also worth noticing is the interaction of this cluster with the two members of cluster 3: FGF5 and FOP. Like the other members of FGF family grouped in cluster 2, FGF5 is involved in cell survival activities, while FOP was originally discovered as a fusion partner with FGFR1 in oncoproteins that give raise to stem cell myeloproliferative disorders. It would be interesting to identify specific details regarding the intensity and direction of the information flow within this cluster for a clearer understanding of their relationship in the context of cell cycle progression.
Discussions
Fujita et al. [20,21] suggested both a concept of Granger causality for sets of time series and a method for its identification with a statistical test to control the rate of false positives. Although this method is useful for the identification of Granger causality between sets of time series in Bioinformatics and Neuroscience [22], the application was limited to predefined clusters, i.e., the time series composing each cluster needed to be previously known. We developed an objective method to define clusters based on the intuitive concept that a gene cluster should interact more intensely in terms of Granger causality within itself than with neighboring clusters.
Krishna et al. [34] proposed a Granger causality clustering method based on the structure of a pairwise network. Their method consists in identifying pairwise Granger causality between gene expression time series and then, by applying the method proposed by Bader and Hogue (2003), to detect dense regions in the network. The difference between their approach and ours is that they take into account the number of edges, and the density of the network which is given by the number of estimated edges divided by the total number of possible edges. The presence of an edge is determined by the pvalue’s threshold. Notice that depending on the threshold, the results can change. In our framework, we take into account the weight of Granger causality between sets of time series in order to identify how close two sets are. Consequently, it is possible to obtain a notion of distance between two clusters based on Granger causality, i.e., a continuous measure (distance in terms of Granger causality) instead of a discrete measure (presence or absence of an edge). Moreover, by using the concept of Granger causality between sets of time series proposed by [20], the concept of density of a network can be easily defined in terms of Granger causality instead of a density based on the number of edges as proposed by [34].
A disadvantage of our method is that it cannot be applied for very large datasets. The larger is the number of time series (genes), or the higher the order of the autoregressive process to be analyzed, the higher the chance to generate noninvertible covariance matrices in the calculation of distance (definition 2) and degree (definition 4) between clusters. We believe that this drawback can be overcome through sparse canonical correlation analysis [35], recently proposed in the literature. However, this topic deserves further studies before it can be used in both clustering and identification of Granger causality between sets of time series, since penalized methods relying on L1 penalization [35] or kernel [36] may present biased estimators.
We only analyzed the autoregressive process of order one because gene expression time series data, possibly due to experimental limitations, are typically not large. However, if one is interested in analyzing greater orders, one minus the maximum canonical correlation analysis value among all the tested autoregressive orders can be used as the distance measure between two time series.
The clustering algorithm used here is based on the wellknown spectral clustering. Although results were satisfactory, other graph clustering methods may be used. The normalized cuts algorithm proposed by [37], for instance, presents better results in non Gaussian data sets.
Finally, which biological process underlie time series datasets correlation, remains a difficult question to be answered. Studies suggest that correlated genes may belong to common pathways or present the same biological function. However, it is also known that methods based exclusively on correlation cannot reconstruct entire gene networks. Further studies in the field of systems biology might be able to answer this question in the future.
Conclusions
We propose a time series clustering approach based on Granger causality and a method to determine the number of clusters that best fit the data. This method consists of (1) the definition of degree and distance, usually used in graph theory but now generalized for time series data analysis in terms of Granger causality; (2) a clustering algorithm based on spectral clustering and (3) a criterion to determine the number of clusters. We demonstrate, by simulations, that our approach is consistent even when the number of genes is greater than the time series’ length.
We believe that this approach can be useful to understand how gene expression time series relate to each other, and therefore help in the functional interpretation of data.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
AF has made substantial contributions to the conception and design of the study, analysis and interpretation of data. KK, AGP and JRS contributed to the analysis and interpretation of mathematical results. PS contributed to the analysis and interpretation of biological data. AF and PS have been involved in drafting of the manuscript. SM directed the work. All authors read approved the final manuscript.
Acknowledgements
The supercomputing resource was provided by Human Genome Center (Univ. of Tokyo). This work was supported by FAPESP and CNPq  Brazil and RIKEN  Japan.
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