Motif Structure

Many real-world complex networks have been shown to be composed of well-defined building blocks called motifs. Network motifs are patterns of interconnection or subgraphs that occur in natural networks much more frequent than those in randomized networks [7], [8]. They can be thought of as simple building blocks of complex networks [8], which can provide valuable information about structural design principles of networks. First discovered in the gene regulation (transcription) network of the bacteria Escherichia coli by Alon and his team [8], [25], they have been found in many networks ranging from biochemistry to neurobiology networks, ecology, and engineering [9], [26], [27]. Study of network motifs is therefore propitious for revealing the basic building blocks of most complex networks.

Some studies have related the function of networks to the structure of their motifs. Transcription networks are among those heavily studied both theoretically and experimentally. For example, negative-autoregulation which is one of the simplest and most abundant motifs in Escherichia coli has been shown to be response-acceleration and repair system [28]. Positive-autoregulation motif is important in biomodal distribution of protein levels in cell population [29]. Feed-forward loop that is commonly found in many gene systems and organisms is important in speeding up the response time of the target gene expression following stimulus steps, pulse generation and cooperativity [30]. Dense Overlapping Regulons that occur when several regulators combinatorially control a set of genes with diverse regulatory combinations, has also been shown to be important in the function of Escherichia coli [31].

Although subgraphs of different sizes can be studied in natural networks, among them, biological networks contain three and four-node substructures far more often compared to randomized networks with similar structural properties. Many beneficial outcomes have been ensued from these observations. Often the network motifs are detected by comparing the network against a null hypothesis, that is, the number of appearance of a specific subgraph is counted in the networks and is subsequently compared with the number of appearances in properly randomized networks. The randomized networks can be constructed in various ways. However, they should at least share some common properties with the original network. For example, the randomized networks should have the same number of nodes and edges with the original network. One possible method is to build the corresponding Erdos-Renyi version for the networks [32]. A better way of constructing the randomized networks is to preserve not only their size and average degree but also their degree distribution or at least degree sequence. This can be simply done by shuffling the adjacency matrix [33]. Many of the motif detection strategies use this algorithm for constructing the randomized version of the original network under study. The motif detection algorithm can be summarized as follows [7], [8]:

1. Consider a specific subgraph i
2. Count the number of appearances of the subgraph i in the network N_i
3. Generate sufficiently large number of randomized networks with the same number of nodes and degree distribution as the original network
4. Count the number of appearances of the subgraph i in each of the randomized networks
5. Compute the average number of appearances of the subgraph i in the randomized networks <Nrand_i> and its standard deviation std(Nrand_i)
6. Compute the significance of appearances of the subgraph i as(1)
7. The networks motifs are subgraphs for which the probability P of appearing in the randomized networks an equal or greater number of times than in the original network is lower than a cutoff value (e.g. P<0.01). Thus, higher absolute values of Z-scores correspond to more significant network motifs.

Note that the Z-score of a motif can be positive or negative; positive when it is highly overrepresented in the original network as compared to randomized ones and negative when it is highly underrepresented.

It has also been proposed to normalize the Z-scores [7]. The Z-score of an specific motif may depend on the network size and it tends to be higher in larger networks [7]. Since complex networks may vary widely in size, one can take an approach that enables to compare different network's local structure. To this end, the normalized Z-scores can be calculated as(2)

The normalization emphasizes the relative significance of subgraphs rather than the absolute significance, which is important for comparison of subgraph of different sizes [7].

A motif of size k is called a k-motif. The runtime of counting process grows very fast with k. This is one of the reasons why only small k-motifs (usually three- or four-nodes) have been studied in most of the works. Different tools have been developed for the detection and analysis of network motifs such as Mfinder [34], MAVisto [35], and FANMOD [36]. In this work we used Mfinder, which uses a semi-dynamic programming algorithm in order to reduce the running time [34]. It also uses an efficient sampling algorithm that significantly reduces the running time compared to the cases where all edges are visited.

Two Biological Networks

Techniques from complex networks have been widely applied to many biological systems (e.g. see reviews [6], [13], [37]). Recent developments in designing efficient techniques in molecular biology have led to extraordinary amount of data on key cellular networks in a variety of simple organisms [8], [38], [39], [40], [41]. This allowed scholars to study networks such as protein interaction, transcriptional regulatory, and metabolic in different organisms. Networks have also been widely studied in neurosciences [42], [43], [44]. The brain networks can be studied on a micro-scale containing a number of neurons with some excitatory/inhibitory connections in-between [45], [46], [47]. However, this approach cannot be used for studying the whole-brain connectivity network. For such cases, one should use functional magnetic resonance imaging, diffusion imaging, magnetocephalography, or electroencephalography techniques to extract the large-scale functional/anatomical brain connectivity networks [48], [49], [50], [51].

In this work, we have considered two biological networks: protein structure network [7], and human brain functional network extracted through functional magnetic resonance imaging [24]. Figure 1 shows their structure by representing the nodes and edges connecting them. Their properties including, size, average degree, standard deviation of the degrees, average path length and clustering coefficient is represented in Table 1. We used Mfinder to determine the significance of all three- and four-nodes subgraphs of these networks. In order to obtain a high level of accuracy, we set the parameters of random network generation algorithm and counting motifs in the tool as follows [34]

• Number of random networks = 10000
• Uniqueness threshold is ignored
• No threshold on mfactor to use when counting motifs
• No threshold on Z-score to use when counting motifs
• Default values were considered for other parameters, including switching method for generating random networks.

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Figure 1. Topology of sample biological networks.

(a) Protein structure network [7] and (b) human brain functional network extracted through functional magnetic resonance imaging [24].

https://doi.org/10.1371/journal.pone.0020512.g001

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Table 1. Characteristics of considered biological networks.

https://doi.org/10.1371/journal.pone.0020512.t001

Table 2 summarizes the set of three- and four-node motifs with their corresponding normalized and non-normalized Z-scores in the networks. As we can see motif #7 — a four-node motif with five edges — has the highest positive Z-score, and thus, is the most significance motif structure in both of the networks and can be considered as the dominant motif. On the other hand, motif#1 has the highest negative Z-score in both of the networks, and thus, is the most significant anti-motif in the set of three- and four-node subgraphs. There is a significant direct correlation between the Z-scores of the motifs in these two networks (r = 0.9328, P<0.001; Pearson linear correlation and r = 0.9286, P<0.0025; Spearman rank correlation). This indicates the similarity of these two networks in the structure of their building blocks, i.e. #2, #5, #7, and #8, have always positive Z-score, i.e. they are significantly more abundant in these networks as compared to random networks. As the clustering coefficient of the real networks is relatively large (see Table 1), it seems natural that the subgraphs that include a triangle structure have a positive Z-score. In some sense, the Z-score of motifs #5, #7 and #8 seems strongly dependent on the Z-score of motif #2. The negative Z-score of motif #1 seems also correlated to the positive Z-score of motif #2. Subgraph #1 and #4 (motif #6 that has small Z-score and is not a significant motif) has always negative Z-score meaning that they are anti-motifs appearing much less in the original networks as compared to random ones.

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Table 2. Significance profiles of motifs in the networks.

https://doi.org/10.1371/journal.pone.0020512.t002

Random and Systematic Failures in the Edges

Random or systematic failures can occur in some of the networks' components, i.e. nodes and edges. For example in protein-protein interaction network, while attacking nodes may correspond to breakdown of polypeptides by appropriate enzymes, attacking edges of the network can be interpreted as preventing physical interaction between two polypeptides in order to prevent carrying out their biological functions. In this work we considered failures in the edges and investigated its influence on the profile of the motif structure of the networks. Failures in the networks are of two types, in general: random failures that are called errors or systematic failures that are called attacks.

Let define some preliminary metrics of graph theory. Consider an undirected and unweighted network with adjacency matrix A = (a_ij), i, j = 1, …, N, where N is the size of the network. Let denote the edges between the node i and the node j by e_ij. The degree of the node i can be obtained as(3)

Edge betweenness centrality (load) is a centrality measure of an edge in a graph, which counts the number of shortest paths passing through the edge. The betweenness centrality L_ij of the edge e_ij between nodes i and j that is defined by [52](4)where Γ_pu is the number of shortest paths from nodes p to u in the graph and Γ_pu(e_ij) is the number of these shortest paths making use of e_ij. The betweenness centrality of an edge is indeed the load of shortest paths using that edge, i.e. the larger the betweenness centrality of an edge is the more its significance in the formation of the shortest paths in the network is.

In a topological space and complex network analysis, closeness is a basic and important concept. In graph theory closeness is the inverse of the sum of the shortest distances between each node in the network. In other worlds, the closeness centrality C_i of node i is defined as(5)where d(i,j) is the length of the shortest path between the nodes i and j. Indeed, the closeness centrality of node i is the inverse of the average shortest path from i to other nodes in the network.

We considered different failure strategies in the networks. In order to choose candidate edges for removal four strategies were considered as follows:

1. Random failure: at each step, one edge was randomly chosen and removed from the network.
2. Systematic failure based on the node degrees: at each step, the quantity k_ik_j was calculated for each edge e_ij, and then, the edge with the maximum amount of k_ik_j was removed from the network. If some edges have the same value of k_ik_j, one of them was removed randomly.
3. Systematic failure based on the edge betweenness centrality: at each step, the quantity L_ij was calculated for each edge e_ij, and then, the edge with the maximum amount of L_ij was removed from the network.
4. Systematic failure based on the node closeness centrality: at each step, the quantity C_iC_j was calculated for each edge e_ij, and then, the edge with the maximum amount of C_iC_j was removed from the network.