Identifying interactions in the time and frequency domains in local and global networks - A Granger Causality Approach

Abstract

Background: Reverse-engineering approaches such as Bayesian network inference, ordinary differential equations (ODEs) and information theory are widely applied to deriving causal relationships among different elements such as genes, proteins, metabolites, neurons, brain areas and so on, based upon multi-dimensional spatial and temporal data. There are several well-established reverse-engineering approaches to explore causal relationships in a dynamic network, such as ordinary differential equations (ODE), Bayesian networks, information theory and Granger Causality.

Results: Here we focused on Granger causality both in the time and frequency domain and in local and global networks, and applied our approach to experimental data (genes and proteins). For a small gene network, Granger causality outperformed all the other three approaches mentioned above. A global protein network of 812 proteins was reconstructed, using a novel approach. The obtained results fitted well with known experimental findings and predicted many experimentally testable results. In addition to interactions in the time domain, interactions in the frequency domain were also recovered.

Conclusions: The results on the proteomic data and gene data confirm that Granger causality is a simple and accurate approach to recover the network structure. Our approach is general and can be easily applied to other types of temporal data.

Figure 1

Global Granger causality approach. (A) Ancestors of target node T, A₀(T) = {T₁, T₂, T₃, X₁,...,X_n, Y₁,...,Y_n}. T₁, T₂, T₃ are direct ancestors to target T. {X₁,...,X_n} connect to T through a single pathway, thus, {X₁,...,X_n} are not direct ancestors to target T. {Y₁,...,Y_n} connect to T through two distinctive pathways (B) {X₁,...,X_n} can be removed by Granger-conditioning on a single node, A₁(T) = { T₁,T₂,T₃,Y₁,...,Y_n}. (C) S is connected to T through two different paths, both {B₁,B₂} and {B₃} are sections from S to T, but {B₃} is the bottleneck. (D) There may exist other common drives to the observed nodes X and T, we assume the partial Granger causality can delete the influence of such drive and exclude such case in our analysis. (E) Histograms of the number of bottleneck for a variety of connection probability p for N = 100 and 500 simulations.

Figure 2

Conditional Granger causality approach applied on a simple linear toy model. (A) Five time series are simultaneously generated, and the length of each time series is 1000. X₂, X₃, X₄ and X₅ are shifted upward for visualization purpose. (B) For visualization purpose, all directed edges (causalities) are sorted and enumerated into the table. (C) The derived network structure by using conditional Granger causality approach. (D) The 95% confidence intervals graph for all the possible directed connections derived by conditional Granger causality. (E) Granger causality results in frequency domain.

Figure 3

Conditional Granger causality approach applied on experimental gene data. The experiment measured the expression level of 5 genes after a shift from galactose-raffinose- to glucose-containing medium. The regulatory network was inferred by using conditional Granger causality approach. Solid gray lines represent inferred interactions that are not present in the real network, or that have the wrong direction (FP false positive). PPV [Positive Predictive Value = TP/(TP+FP)] and Se [Sensitivity = TP/(TP+FN)] values show the performance of the algorithm for an unsigned directed graph. TP, true positive; FN, false negative. (A) The network structure of 5 genes derived by conditional Granger causality. (B) Gal4 and Gal80 were grouped as a single node, so that only transcriptional regulation interactions are represented. (C) Conditional Granger causality results for 5 genes. The 95% confidence intervals graph, which is constructed by using bootstrapping method, is plotted. (D) Conditional Granger causality results for a grouped genes (Gal4 and Gal80 are grouped). The 95% confidence intervals graph, which is constructed by using bootstrapping method, is plotted.

Figure 4

Conditional Granger causality approach applied on experimental protein data. The experiment measured the levels of 7 endogenously tagged proteins in individual living cells in response to a drug. (A) The time traces of 7 proteins are plotted. There are 141 time points. The time interval is 20 minutes. (B) ARIMA model is used to fit the data. We applied term-by-term differencing 3 times to the data. (C) The network structure for 7 proteins derived by using conditional Granger causality approach. (D) For visualization purpose, all directed edges (causalities) are sorted and enumerated into the table. (E) Conditional Granger causality results. The 95% confidence intervals graph, which is constructed by using method bootstrap, is plotted.

Figure 5

Conditional Granger causality in frequency domain. Conditional Granger causality was applied to experimental data in the frequency domain and power spectrum density analysis for 7 proteins (the most left column in black line). The significant causalities are shown in red lines in the figure.

Figure 6

Global Granger Causality (GGC) algorithm applied on a simple toy model. (A) The actual network structure used in toy model of global network. (B) Network structure inferred from PGC. (C) Network structure inferred from pair-wise Granger causality (solid and dashed links). By using partial Granger causality among three units, we can delete some of them (dashed links). (D) The final network structure from GGC, it is consistent with the actual relationship. (E) ROC curve summarizing the performance of the procedure on a random network with maximum non-zero coefficients. (F) ROC curve summarizing the performance of the algorithm on a random network with random non-zero coefficients. (G) True positive rate as a function of the magnitude of the non-zero coefficient.

Figure 7

Global Granger Causality algorithm applied on experimental data for global network re-construction. (A) In-, out- and total degree distributions of the large network calculated from the whole dataset. (B) For visualization purpose, the proteins are enumerated into the table (C) Direct ancestors of the protein DDX5: BC037836, C2ORF25, HMG2L1, MAPK1, RPL24 and RPS23.(D) External influences identified by the second iterative procedure, in brown ovals.

Figure 8

Global Granger Causality algorithm applied on experimental data for global network re-construction. (A) The overall mean clustering coefficient (the probability of neighbours being inter-connected) is an order of magnitude larger than the one of a random network (0.022 instead of 1/768 = 0.0013). But the network is not modular: the mean clustering coefficient with respect to degree is more or less constant. (B) Direct ancestors of RFC1, as well as their own direct ancestors. The causal link from DDX5 to RFC1 is now completely identified: an intermediate protein (SLBP) connects them.