Abstract
Background
Inventories of small subgraphs in biological networks have identified commonlyrecurring patterns, called motifs. The inference that these motifs have been selected for function rests on the idea that their occurrences are significantly more frequent than random.
Results
Our analysis of several large biological networks suggests, in contrast, that the frequencies of appearance of common subgraphs are similar in natural and corresponding random networks.
Conclusion
Indeed, certain topological features of biological networks give rise naturally to the common appearance of the motifs. We therefore question whether frequencies of occurrences are reasonable evidence that the structures of motifs have been selected for their functional contribution to the operation of networks.
Background
The network or directed graph description has become the preferred representation of the integrated activity of components of biological processes. The exponential growth of biological network data in the last five years has its source in recent advances in technologies such as mass spectrometry, genomescale ChiPchip experiments, yeast twohybrid assays, combinatorial reverse genetic screens, and rapid literature mining techniques [1].
The science of systems biology has the aim of understanding the functional constraints and design principles of biological networks. Alon and colleagues were the first to introduce the notion of "motifs" in biological networks [2,3]. Motifs are small patterns observed to recur throughout a network, with frequencies statistically higher than expected in random networks of similar connectivity parameters. Since the introduction of this concept, motifs have been reported in many biological networks: metabolic, signaling pathway, proteinprotein interaction, and ecological networks amongst others [26]. Moreover, the prevalence of motifs is often considered as evidence for evolutionary selection, for implementing a specific function [2,3,7]. Motifs are believed to be building blocks of the functional architecture of a biological network [3].
Consider for example the canonical set of motifs in transcription regulatory networks: Single input module (SIM), Multiple input module (MIM), and Feedforward loop (FFL) [3]. (See Figure 1. Originally, Alon and colleagues [2] proposed a dense overlapping regulon (DOR) as a motif; MIMs are special DORs that arose as a generalization of Bifan motif). Specific functions have been ascribed to each type of motif [2,711]: SIMs are commonly associated with temporal ordering of gene expression, MIMs with combinatorial gene regulation, and FFLs with filters that do not pass on transient signals [2]. These functions depend not only on the topology of the subgraph, but on the logic at nodes receiving multiple inputs. The common occurrence of these motifs, relative to corresponding randomized graphs, has been taken as evidence for their selection for function.
Figure 1. Canonical subgraph patterns in biological networks. Canonical subgraph patterns in biological networks. (a) Feedforward loop (FFL): contains a "source" (at the top), "intermediate" (bottomleft), and "target" (bottomright) nodes. (b) 3cycle: a three node directed cyclic graph, (c) Singleinput module (SIM). (d) Multipleinput module (MIM). (e) Bifan motif. SIM, MIM, and Bifan are twolayered graphs with edges from nodes in top to bottomlayer. A Bifan is a MIM with exactly 2 parent and 2 child nodes.
In this paper we investigate the role of small network subgraphs as building blocks of biological networks. We analysed several biological networks: transcription regulation networks of Saccharomyces cerevisiae under different physiological conditions, the transcription regulation network of Escherichia coli, and a neuronal signalling pathway network of the hippocampal CA1 neuron.
Contrary to previous reports, we find that commonly accepted motifs are neither over nor underrepresented in these real networks in comparison to their random formulations. We discuss how the topology of biological networks automatically predisposes them to contain a certain distribution of motifs. This suggests that the evidence for the functional significance of motifs should be reevaluated.
Methods
We use the transcription regulatory networks of Saccharomyces cerevisiae under various physiological conditions – composite, cell cycle, sporulation, diauxic shift, DNA damage, and stress response – published by Luscombe and coworkers [5]. Their largest (composite) network contains 3459 nodes and 7014 interactions (http://networks.gersteinlab.org/regulation/dynamics/index2.html webcite).
To aid comparison of our work with that of ShenOrr et al. [2], we also use their Escherichia coli transcription network containing 424 nodes and 577 interactions (http://www.weizmann.ac.il/mcb/UriAlon/Network_motifs_in_coli/ColiNet1.0/ webcite).
Additionally, we use the neuronal signalling pathway network of the hippocampal CA1 neuron published by Máayan and colleagues, containing 594 nodes and 1422 interactions [6] (http://www.mssm.edu/labs/iyengar/ webcite).
We implemented Ullmann's algorithm for subgraph isomorphism [12] to enumerate fixed sized subgraph patterns (e.g. FFL, 3cycle).
In enumerating variable sized (maximal) subgraph patterns such as SIMs and MIMs, we used our algorithms described in [13]. We note that Bifans are counted as MIMs with exactly two elements each in both parent and child sets. (See Definitions.)
To generate random networks conserving the degree sequence of the real network, we use the method described by ShenOrr et al. [2]: Starting with the same number of nodes as in an original network, nodes in the random graph are assigned a specific number of in and out"edgestubs." Randomly chosen pairs of in and outedgestubs are joined, giving rise to a random (directed) graph.
Definitions
A FFL is a set of three nodes (source, intermediate, and target) with one direct path, and another indirect path through an intermediate node, from source to target (See Figure 1(a)).
A 3cycle (3CYC) is a threenode directed cyclic graph (Figure 1(b)).
Single and multiple input modules (SIM and MIM) in a directed graph are maximal subgraphs comprising two nonempty disjoint sets (layers): and (standing for Parent and Child). By maximal we mean, for example, that each MIM is not contained in a larger MIM.
A SIM requires that contain only one node and contain at least two nodes, such that the full graph contains an edge from the parent node to every c_{i }∈ . We also require that the indegree – number of incoming edges – of every c_{i }to be strictly equal to one: within the full network, not just within the subgraph. By this definition of a SIM, no edges can exist between any c_{i}, c_{j }∈ . It follows that is the only parent of all nodes in set .
A MIM requires that both and must contain ≥ 2 nodes, that there is an edge from every p_{i }∈ to every c_{i }∈ , no edge between any p_{i}, p_{j }∈ , and no edge between any c_{i}, c_{j }∈ . A Bifan is a maximal MIM with and containing exactly 2 elements [14]. (Figure 1(e))
We note that in counting both SIMs and MIMs, we ignore selfedges.
We emphasize that we impose the criterion of maximality when enumerating SIMs and MIMs. In case of SIM, the set is maximal, whereas with MIMs both and sets are maximal.
These statements define the fundamental network motif set – FFL, SIM, and MIM – as, in a sense, "orthogonal": No subgraph can be more than one of the FFL, SIM, and MIM [13].
Results
We enumerated the occurrences of FFL, 3CYC, SIM, MIM, and Bifan subgraph patterns (see Figure 1) in:
1. the transcription networks of Saccharomyces cerevisiae (Yeast) under various physiological states [5] (see Table 1(a–f)).
2. the transcription network of Escherichia coli [2] (see Table 1(g)), and
3. the signalling pathway of hippocampal CA1 neuron [6] (see Table 1(h)).
Table 1. Frequencies of canonical subgraph patterns in biological networks
For each network, 1000 random networks were generated conserving the degree sequence of the original network. Comparisons were made between the frequencies of appearances of various patterns in the real network, and the means and dispersions of their appearances in corresponding random networks.
Table 1 presents the significance profiles of various patterns. The results show that the frequencies of various subgraph patterns are not significantly over or underrepresented in real networks when compared to their random formulations. A few outliers (where zscore > 2) appear in Table 1: FFLs in Yeast Sporulation (zscore = 2.31), 3CYCs in Yeast Stress Response (zscore = 2.47) and neuronal signalling pathway (zscore = 2.4), and Bifans in Yeast Composite (zscore = 2.05) and Cell Cycle (zscore = 2.33). Some outliers are slightly overrepresented (z > 0), and others are slightly underrepresented (z < 0). We observe no outliers with zscore ≥ 2.47.
We employ the same random model as used in earlier related works [2,3,5,7]. While conserving the degree sequence of the original network, the edges in a random network are chosen randomly so that the resultant network is free from the pressure of "evolutionary selection" which is incident on real biological networks. However, in addition to the conservation of the degree sequence, more sophisticated random models can be generated by embedding other connectivity constraints observed in real networks, such as rules of clustering together of nodes in a neighbourhood, and pathlengths between pairs of nodes. These additional constraints will only make the random null hypothesis more stringent to refute. Nevertheless, even using the basic random model employed in our work, we fail to gather any statistical evidence that the canonical patterns appear in real networks at nonrandom frequencies.
We note that there are differences in the counts of various motifs reported by Luscombe et al. [5] and this work, even though we use the same datasets (Table 1(a–f)). Our figures supersede those reported by Luscombe et al. (see [13] for a detailed explanation).
Our reanalysis of Escherichia coli transcription network provides the most direct comparison of our results with those of Alon and coworkers (see Table 1(g)). We fail to see any statistical evidence to suggest that the canonical subgraphs appear more frequently than random. On comparing our results with those published by ShenOrr et al. [2], we find that:
1. Our definitions of fixed size subgraphs such as FFL and 3CYC are consistent with those originally defined by Alon and colleagues [2,3]. Consequently, we agree on the absolute count of these subgraph patterns in the real network. Surprisingly however, our results of appearances of FFLs in random networks greatly differ. To reconfirm our results reported in Table 1(g), we generated another set of 1000 random networks using an alternative method of random network generation – starting with the original network, over a large number of repetitions, two randomly chosen interactions are swapped. (i.e., interactions: (P1,C1), and (P2,C2) become (P1,C2), and (P2,C1)). Indeed we get similar statistical significance results using this alternative method, compared to those reported in Table 1(g).
2. Our definition of Bifan ensures that we count only those patterns where a pair of target genes are strictly regulated by a pair of transcription factors – Bifans are maximal MIMs where = = 2. We believe ShenOrr et al. [2] fail to maintain this strictness, thereby overcounting Bifans by including in their count two parent, two child subMIMs of larger maximal MIMs. (See Discussion.)
3. Similarly, our definitions and enumeration methods of SIMs and MIMs are mathematically more rigorous than those used by ShenOrr and colleagues [2]. Our counts of maximal MIMs and SIMs could be converted directly to counts of nonmaximal MIMs and SIMs (see below). We note therefore that the nonobservance of statistically significant differences between natural and randomized networks in counts of maximal MIMs and SIMs implies that there are no statistically significant differences between natural and randomized networks in counts of nonmaximal MIMs and SIMs. This comment, together with the reminder that our definitions (and counts) of FFLs and 3CYCs are identical with those of Alon et al., shows clearly that the discrepancies are not a simple effect of alternative definitions of SIMs, MIMs and Bifans.
Discussion
The observed discrepancy in occurrence frequency of FFLs and 3CYCs is a natural consequence of topological properties of networks
Occurrences of FFLs and 3CYCs in various biological networks (see Table 1) show patterns: there are a relatively large number of FFLs and relatively small number of 3CYCs. In this section we explain the topological basis for these differences in their frequencies.
First we note that random connectivity within threenode subgraphs itself favours FFLs. Consider a directed, complete – there is an edge between every pair of nodes – three node graph (3graph). Excluding bidirectional edges, for any set of 3 nodes there are 2^{3 }= 8 possible directed 3graphs. Each of these configurations is isomorphic to either a FFL or a 3CYC – any directed complete 3graph is either a FFL or 3CYC. Out of 8 possibilities, 6 form FFLs, and 2 form 3CYCs. Allowing bidirectional edges, there are an extra 19 possible configurations containing at least one bidirectional edge. Each of these possibilities gives multiple FFLs or 3CYCs or both. With or without bidirectional edges, there is a natural 3:1 bias towards forming an FFL over a 3CYC in a 3graph.
Global properties of biological networks also favour FFLs over 3CYCs. Most biological networks, such as those used in our study, are scalefree [15]. In scalefree networks, the connectivity of nodes follows the power law: the probability of a node having k neighbours is P(k) ~ k^{γ}. Only a few nodes in such a network are highlyconnected (and form hubs), while most nodes are sparsely connected [15].
We asked how many of the FFLs in various networks contain hubs among their nodes. (We consider as hubs the top 10% of nodes in the network that are highlyconnected, having more than 10 neighbours.) Table 2 contains the percentages of FFLs enumerated in various networks, having n = {0, 1, 2, 3} nodes as hubs. A large majority of the FFLs contain at least one hub; most common being the FFLs with hubs at two of their nodes. In the Yeast composite network, 961 of 997 FFLs have at least one common sourceintermediate edge between them. These 961 FFLs can be grouped into 114 clusters (containing distinct sourceintermediate edges) revealing that connected hubs often share many common children, automatically giving rise to FFLs. We believe that the principle of preferential attachment predisposes a biological network to have connected hubs that have shared children. This gives a network its robustness to random node failure [15].
Table 2. Percentage of FFLs in various networks having exactly n of its nodes as hubs
We also observe that there is an imbalance between indegree and outdegree around hubs – there are significantly more outgoing edges than incoming edges. We have seen above that FFLs are naturally favoured over 3CYCs in 3graphs. The imbalances between in and outdegree around the hubs further enhances the formation of FFLs. Consider a hub with m incoming edges and n outgoing edges. With a random addition of an edge between any pair of (m + n) nodes adjacent to this hub, the probability of forming an FFL in this system is: while that of forming a cycle is: . Then, , which is symmetric in m and n. If there is a large disparity between m and n (i.e., m ≪ n, or m ≫ n), then one of the terms or dominates, resulting in . For example, when m = 2 and n = 20, P_{FFL }= 0.91, and P_{3CYC }= 0.09. This shows the odds against the formation of a 3CYC in networks with structures typical of biological networks.
There have been suggestions that 3CYC is an "antimotif" – a motif that is selected against in many biological networks [14]. But, as described above, the suppression of 3CYCs is an expected consequence of topological properties of biological networks.
These properties are sufficient to account for the observed profiles of FFLs and 3CYCs.
Assemblies of motifs
Kashtan and colleagues [16] observed that regulatory networks contain multioutput FFL generalizations (see Figure 2(a)) in frequencies much higher than multiinput (Figure 2(d)) and multiintermediate (Figure 2(f)) generalisations. (These authors also suggested that multioutput FFLs were selected to achieve some information processing role [16].)
Figure 2. SelfAssemblies of two FFLs. Various possible selfassemblies of two FFLs sharing a common edge.
We, in contrast, observe that the varied frequencies of assemblies of multiple FFLs are a consequence of the occurrence of FFLs around hubs. Figure 2 shows all possible assemblies involving two FFLs sharing a common edge. In Table 3 we enumerate the occurrences of each such assembly in various networks. Clearly, the multioutput assembly of two FFLs abounds over other possibilities, simply because a large number of FFLs share a common sourceintermediate edge.
Table 3. Number of occurrences of various assemblies shown in Figure 2
Thus the numbers of multioutput FFLs grow combinatorially with the number of FFLs sharing a common sourceintermediate edge. The count of (k<n)output assembly of FFLs, where n is the number of FFLs sharing two common (source and intermediate) nodes, is expected to increase as ^{n}C_{k}. For example, 5 FFLs having a common sourceintermediate edge (see Figure 3) will give rise to 10 nonredundant bioutput FFLs. Table 4 shows the statistical significance of finding biouput FFLs in various real networks used in this work, by comparing the occurrences with those observed in their corresponding random networks. Statistically, their frequencies are not significantly greater than in random networks.
Table 4. Frequencies of BiFFL assembly in various networks
Figure 3. Example of FFLs sharing two hub nodes. Example of FFLs sharing two hub nodes that are connected.
On SIMs, MIMs and Bifans
SIMs and MIMs are variable sized subgraphs. Alon and colleagues [2] defined the dense overlapping regulon (DOR) as a twolayered subgraph with not necessarily complete connections between them. MIMs are special DORs, a concept that arose as a generalization of the Bifan (Figure 1(e)) subgraph. These Bifans were observed to be present in large numbers in biological networks. However, some investigators fail to impose the criterion of maximality while counting MIMs. This can lead to significant inflation of counts [2,5]. Note that this applies equally to natural graphs and random ones (Hence we emphasize that the differences between our results and those of Alon et al. are not explicable solely on the basis of alternative definitions of some of the motifs).
A maximal MIM with m parents and n children contains [2^{m } (m + 1)] × [2^{n } (n + 1)]  1 easily enumerable nonmaximal "subMIMs". Our definition of a Bifan ensures that we are only counting (maximal) MIMs that contain 2 parents and 2 children. Counting subMIMs as Bifans will combinatorially increase their counts, as each maximal MIM will contribute to ^{m}C_{2 }× ^{n}C_{2 }Bifans. For example, the Yeast composite network contains a large MIM containing 2 parents and 119 children. This alone contributes to 7021 nonmaximal Bifans. The same consistency is maintained when counting SIMs. The list of subgraphs occurrences in various networks used in this paper can be downloaded from http://hollywood.bx.psu.edu/networks/analysis/ webcite.
The natural appearance of bipartite graphs in dense general graphs has received some attention in graph theory [17]. It has also been demonstrated, using Ramsey theory [18], that bipartite cliques appear in sufficiently dense bipartite graphs [19,20]. MIMs are bipartite cliques. Biological networks contain regions in which dense bipartite graphs naturally appear, and hence giving rise to bipartite cliques. This in itself speaks against the notion of evolutionary selection of MIMs [2].
Evidence for selection of motifs?
Analysis of natural networks shows that several commonly observed subgraphs identified as motifs do not appear at frequencies significantly greater than in corresponding random graphs. Instead, their frequency of occurrence is the result of the smallworld character of many biological networks, and of the associated degree distribution.
What does this imply about the idea that motifs have been selected, by evolution, for function? The statement that motifs are selected for function has two possible interpretations, not necessarily incompatible:
1. It might be asserted that the general type of motif – for instance FFL rather than 3cycle – is selected because of a general propriety to serve a particular function (For example, Alon et al. [1] pointed out that a FFL with AND logic at the output node can function as a filter rejecting transient stimuli).
2. Or it might be asserted that individual FFLs (or 3cycles) within a network play specific functional roles at specific points.
Statistics of frequency of occurrences of specific motifs, and the comparison of observed frequencies in natural networks relative to random networks, do not – no matter what numerical results emerge – provide evidence for or against assertions of type 2. If any individual subgraph at some node plays an essential functional role in a network, it could be selected – whether it is a commonlyoccurring subgraph or not. Conversely, an observation of significantly nonrandom occurrence frequencies of motifs would suggest the action of positive or negative selection, acting at the level of assertions of type 1 or type 2. Indeed it seems inescapable that if assertions of type 1 are true, then at least some assertions of type 2 must also be true, but not vice versa.
Our results suggest that there is no evidence for type 1 assertions.
Conclusion
We have analysed several biological networks. Our results suggest that there is no evidence suggesting selection for or against subgraph patterns such as FFL, 3CYC, SIM, MIM, Bifan. We have shown that, in contrast to the need to invoke selection to explain the structure of observed networks, it is the topological properties of networks that automatically favour the observed frequency profiles of various subgraph patterns.
Authors' contributions
Both the authors contributed equally to the planning and execution of this study; both authors contributed to the draft, and have read and approved the final manuscript.
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