Let G = (V, E) be a directed graph without multiple edges that represents a regulatory network, where the vertices v ∈ V denote biological entities, e.g., proteins, genes or small molecules. Causal relationships between these entities are made up of directed edges e ∈ E. A topological pattern is given by n connected vertices and the way they are connected with each other. The particular coherence which is described by a pattern is always based on all edges that exist between n vertices. The entirety of all distinct n-vertex patterns in G is then given by where is the i-th pattern consisting of n vertices. Actually, each pattern represents a set of isomorphic connected subgraphs which have the same structural properties and differ only in the participating vertices. Accordingly, a pattern comprehends a set of instances, i.e., , and each instance is a unique subgraph of G, with the subset of vertices ⊆ V and the subset of edges ⊆ E (Figure 1). The edges in are only incident to vertices in and we denote them as the intrinsic edges of the pattern instance . Other edges, e ∈ E\, do not contribute to the coherence of the vertices . Moreover, these extrinsic edges are part of the environment of which describes how the pattern instance is embedded into the network. If there are more pattern instances in G than in similar random networks, then the respective pattern is called a motif. Consequently, the entirety of n-vertex patterns in G may contain several n-vertex motifs . Then, the motif compasses its own representatives, the instances of the motif .

Following the logic of 24, we denote the topological significance of a pattern instance as how essential for all connections within a network it is. To quantify this significance we eliminate all edges of a pattern instance (i.e., its intrinsic edges ) and measure how this affects the number of connected ordered pairs of vertices in the network. An ordered pair of vertices (i, j)|i ≠ j and i, j ∈ V, is connected iff there is at least one path from vertex i to vertex j in G. Note, that the ordered pair (i, j) is different from (j, i) in a directed network. The more ordered pairs become disconnected upon the removal of all edges of a pattern instance, the higher is the topological significance of this instance for the whole network. We define the pairwise disconnectivity index of a pattern instance, Dis(), as the fraction of those initially connected pairs of vertices in a network which become disconnected if the intrinsic edges of the pattern instance are removed from the network

In Eq. 1 N is the total number of ordered pairs of vertices in a graph G = (V, E) that are connected by at least one directed path of any length. It is supposed that N > 0, i.e., there exists at least one edge in the network that links two different vertices. N' is the number of ordered pairs of vertices in the subgraph G' = (V, E') of G where E' = E/. Therefore, G' is the subgraph of G that results from removing the intrinsic edges of the pattern instance from G. The pairwise disconnectivity index of a pattern instance ranges between 0 and 1, whereas zero indicates that the removal of its intrinsic edges does not disconnect vertices within the network and one denotes the cases when no pair of vertices is connected any more.

Figure 2 illustrates how an instance of the feed-forward loop (FFL), one of the best studied network motifs 141520212223, may affect the existing communication in a network. The FFL is a three-vertex pattern that is given here by the intrinsic edges X → Y, Y → Z and X → Z. It is linked to the rest of a network by its vertices X, Y, Z where each of these can be at the start or end of an extrinsic edge (blue dotted edges). Further extrinsic edges are between other pairs of vertices in the environment of a FFL instance, e.g. the ordered pair (E₁, E₂). Whether a FFL instance can have an impact on the connection between two vertices depends on the kind of constituents of the paths that link them. If these paths consist of extrinsic edges only then the connection will not be affected upon the removal of the FFL (e.g., the pair (E₁, E₃)). Essentially, the FFL instance may be critical for those paths which include at least one intrinsic edge of the instance. However, that depends on the presence of alternative (i.e., parallel) paths between the corresponding vertices that use extrinsic edges only. For example, the pair (E₂, Y) does not critically depend on the FFL instance due to the presence of another path (E₂, X, E₃, E₄, Y) that includes no intrinsic edge of the instance. In contrast, the pair (E₂, E₆) looses its connection upon the deletion of the FFL instance though three parallel paths are connecting these vertices.

Usually, several instances of a particular pattern can be found in a network. For estimating the topological significance of the pattern itself the impact of its representatives has to be considered. We find that the average pairwise disconnectivity index of all instances of a pattern reflects this appropriately and define

as the pairwise disconnectivity index of a pattern that consists of J instances. With it Eq. 2 also states the topological significance of a randomly chosen instance of the pattern .

We have applied our approach to the characterization of three-vertex topological patterns in transcription regulation networks from three different organisms: a bacteria (Escherichia coli) 14, a unicellular eukaryote (the yeast Saccharomyces cerevisiae) 15 and higher eukaryotes (mammals: human, mouse, rat) 2526. 3-vertex motifs were identified by means of the Z-Score as proposed by Alon and colleagues 15. This normalized value states whether the abundance of a pattern in the real network exceeds its occurrence in a number of random ensembles: that is, a positive Z-Score refers to an over-representation in the real network, whereas a negative Z-Score means under-representation. Since there is no commonly accepted threshold Z-Score value for defining motifs, we consider patterns with Z-Score > 0 as motifs and all other ones as non-motifs. For the networks of E. coli and S. cerevisiae 3-vertex motifs were already identified 1415, whereas for the mammalian transcription network this is reported for the first time. To distinguish between different motifs many of which have no commonly accepted names, we used the identification numbers (IDs) of small connected graphs as it is provided by the FANMOD software 2728. The name of a pattern instance was generated by combining a prefix E, Y or M for referring to E. coli, S. cerevisiae or mammalian, respectively, with the corresponding ID followed by the pairwise disconnectivity index rank of the instance among all instances of a given pattern.

The unusually often appearance of the same links (i.e., intrinsic edges) between genes in the pattern instances with the highest pairwise disconnectivity indices in all three networks raises the question of their contribution to the estimated significance of these pattern instances. Probably, the removal of individual intrinsic edges may already destroy the connection between many gene pairs so that their simultaneous removal is not as crucial. Otherwise they may have a significant non-additive impact taken together. However, answering this requires knowing the effect of deleting a single interaction (i.e., edge) in a network which can be accomplished in a similar way as for a pattern instance. It has been introduced as the pairwise disconnectivity index of an edge in 24 and specifies the fraction of ordered pairs becoming disconnected due to the removal of an individual edge.

As a first attempt, this fraction has been estimated for each intrinsic edge of a pattern instance in the three networks and their sum has been opposed to the pairwise disconnectivity index of the respective pattern instance. Although such kind of comparison highlights just a tendency if and how far the intrinsic edges of a pattern instance act synergistically, it is already a way that works for all kinds of patterns independent of their specific arrangement. Figure 10 illustrates this approximation for the pattern instances with the highest pairwise disconnectivity index in each network. The edge weights denote the topological significance of an edge for the corresponding network, e.g., Dis(hns → flhDC) = 0.005 for the edge from gene hns to flhDC in E. coli. Hence, the deletion of this interaction merely disconnects a half percent of all pairwise linked genes in E. coli. As expected, no effect is accomplished by removing the edge from hns to fliAZY, since there is always the alternative path via flhDC. In contrast, a relatively high pairwise disconnecivity index has been measured for the edge from flhDC to fliAZY. But still, the summarized effect of deleting these intrinsic edges separately from the E. coli network (0.049) is considerably lower as compared with the topological significance for the whole pattern instance, Dis(E.38.1) = 0.086. The same holds for the other two pattern instances in Figure 10 as well. Therewith a much stronger impact on pairwise connections between genes clearly exists due to the coherence of the intrinsic edges.

Whether this can be generalized for all pattern instances found in the three networks is shown in Figure 11. Most pattern instances in the three networks cluster near the diagonal since the joint removal of their intrinsic edges disconnects approximately the same number of gene pairs as the separate elimination of them does. However, some exceptions have been found, especially among those patterns that exhibit a high pairwise disconnectivity index per se.

A pattern instance is positioned below the diagonal dotted lines in Figure 11 due to considerable overlapping in the sets of pairwise linked genes which become disconnected upon the separate removal of the intrinsic edges of the instance. For example, consider how the vertices 1 and 5 in Figure 1A are linked. To disconnect them it is enough to delete one of the edges 1 → 2 or 2 → 5 at a time. Such kinds of dependencies seem to exist in larger scales in the analyzed networks pinpointing to lots of gene pairs that are connected in a linear chain-like manner as reflected by the pattern ID = 12 (Figure 3). There are almost no independent alternative paths between such gene pairs so that the connection between them is very sensitive upon the deletion of a single intrinsic edge. Therewith, the pattern ID = 12 is contained virtually exclusively amongst the pattern instances below the diagonal dotted lines in Figure 11.

The concurrent elimination of the intrinsic edges of a pattern instance located above the diagonal dotted lines breaks also pairwise connections between genes that are not so easily assailable as described above. At least two paths between such genes exist, each using a unique combination of intrinsic edges. Thus, they cannot be affected by eliminating a single intrinsic edge only. For example, in Figure 2 there are three paths linking vertex E₂ with E₆: The first one includes the intrinsic edge X → Z. The second consists of the intrinsic edges X → Y and Y → Z whereas the third path contains only the edge Y → Z. However, no matter which of the intrinsic edges is deleted, the vertex pair (E₂, E₆) remains untouched since at least one of the three paths is still present. Their connection is disrupted only if the whole pattern is deleted. Such dependencies can be observed in Figure 11 for few pattern instances in E. coli, but increasingly in the other two networks. This trend is most distinctive in the mammalian network. Besides the pattern instances with a high pairwise disconnectivity index, a considerable number of motif instances appear in the lower left corner of the plot for the mammalian network (Figure 11, red triangles): their intrinsic edges have an extremely small or even no impact at all on pairwise connections between genes. But as motif instances, they are a bottleneck for linking many gene pairs.

The work presented here describes a method that has been proved to be suitable for evaluating the role of topological patterns within a network. This holds true regardless of the size and complexity of these patterns. The method assesses the significance of a pattern depending on the contribution of its instances, i.e. connected subgraphs, for the connectivity of a network. The approach is based on the technique described previously in 24, which estimates the necessity of a network element (e.g., a vertex or an edge) for sustaining the communication ability between connected pairs of vertices in a network. This is accomplished in a similar way as wet experiments in a lab: a gene (corresponding to a vertex in a graph) is knocked out and the effect of this removal is observed in the considered context. The same may be applied to a reaction (an edge in the graph), when a gene has been mutated and the encoded product (vertex) is still present, but unable to undergo a certain reaction.

In this work, we have proposed to proceed likewise for pattern instances, but disturbing the interactions between the involved vertices rather than eliminating the vertices themselves. Consequently, only the causal links between these vertices are destroyed and therewith the respective pattern is removed in a minimally invasive way. This is conducted without making any a priori assumptions on the analyzed network and its properties. In contrast to the attempt made in 18, we destroy the coherence between the edges of only one single pattern instance at a time, leaving the remainder of the network intact. On the one hand, different impacts on the network connectivity exerted by the various instances of a pattern can thus be discovered. On the other hand, the topological role of a pattern can be determined more realistically since an overrating is avoided.

We exemplarily applied the method developed and proposed here to the analysis of transcriptional regulation networks of three very distinct taxa (E. coli, S. cerevisiae and mammals, i. e. human, mouse, and rat); for simplicity, we focused here on 3-vertex topological patterns in these networks, but the method can easily be adopted to the analysis more complex and larger patterns. A first check of which of the thirteen possible 3-vertex patterns are present in these networks at all revealed that all of them can be found in the mammalian network, the S. cerevisiae network contains seven and that of E. coli only four of them. Moreover, these latter four patterns are shared by all three networks. Amongst them, only the "feed-forward loop" is statistically over-represented and, thus, could be considered as a "motif" (Figure 3).

As to be expected, the abundance of a pattern decreases with its complexity: Thus, 3-vertex patterns with two edges occur much more frequently than those with three edges, etc. The order of the abundance is almost the same in all three transcription networks. It is of interest that the network patterns "coupled feedback loop" (Figure 3, ID = 78) and "3-vertex-circuit" (Figure 3, ID = 140) do not exist in the networks of E. coli and S. cerevisiae and are clearly under-represented in the network of mammals (Figure 3), although they are widespread in signaling circuits of various bacterial and eukaryotic organisms 4041424344. We assume that this is an intrinsic property of transcriptional networks and cannot be explained by the incompleteness of the underlying knowledge, since other patterns of similar complexity (e.g., the mentioned feed-forward loop) are not consistently under-represented among these three networks.

All networks studied here appear to be rather robust against the elimination of a randomly chosen pattern instance. Therewith, the various 3-vertex patterns in these networks display a low topological significance on average. Mostly, the overall majority of the instances of a pattern have a rather small effect on the existing pairwise connections between genes, in most cases even less than 1% of all pairwise connections are affected.

Also the motifs among the 3-vertex patterns examined did not exhibit a generally higher importance for the connectivity of the whole network than non-motif patterns, as one might have expected. This is, however, in agreement with previous studies on the evolutionary and functional assessment of motifs in the regulatory networks of different yeasts, which have provided evidence that motifs are not subject to any particular evolutionary pressure to preserve the corresponding interaction pattern 4546. No simple relationships have been found between evolutionary conservation and over-representation of network patterns, on the one hand, and their functional enrichment, on the other hand, in the yeast regulatory network 42. In accordance with these observations, our results indicate that there is no positive correlation between the abundance (i.e., over-representation) of a network pattern and its topological significance. Thus, focusing on motifs exclusively rather than searching for important pattern instances in general would have lead to a completely different and deceptive picture.

In spite of the generally low impact of all types of patterns (including motifs) found in the analyzed networks, a few pattern instances cause a significant perturbation upon their removal. This trend is manifested in the heterogeneous distribution of the pairwise disconnectivity index among all the instances of a pattern (Figs. 3, 4, 5). Topologically, this may originate from the way how a pattern instance is embedded, i.e., its particular position within the whole context of the respective network. Biologically, such heterogeneity might be caused by the influence of the genes in the network that are forming a pattern instance. In the networks, the topologically most significant pattern instances consist preferably of genes that provide basic functions for the organism. Interestingly, most of these instances belong to one of the patterns that are shared by the three networks, which may emphasize the importance of these patterns. Furthermore, such instances may indicate locations within the networks rendering them vulnerable upon a targeted removal.

Among the pattern instances that are of particular importance for the network connectivity, motif instances again do not play a predominant role over instances from non-over-represented patterns. In the mammalian network, most of the outliers even belong to the non-motif patterns. Altogether, our data support the view that far not all instances of any pattern (motif or not), but only few of them may play specific functional roles 47 and thereby exhibit a strong impact on pairwise connections between genes in transcription networks.

In all the networks analyzed here, a limited number of genes repeatedly appears in the pattern instances displaying the highest topological significance. For example, in E. coli the gene flhDC is part of all pattern instances that disconnect at least 4% of the gene pairs, preferably together with the genes fliAZY or ompR_envZ. Similar observations can be made in S. cerevisiae for the genes MCM1, SIN3, SNF2_SWI1 and SWI5. Likewise in the mammalian network, the interaction between the genes c-myc and PAX3 participates in many of the pattern instances with a high pairwise disconnectivity index. Altogether, the common occurrence of genes and interactions between them underlines the key importance of these constituents for the corresponding organism. All these genes are engaged in important processes and at least in E. coli and S. cerevisiae they are crucial for linking a significant number of gene pairs 24. Hence, their damage can be lethal for the respective organism. Furthermore, these pattern instances are not located in different regions of a network. They are connected with each other and seem to form a bigger pattern cluster that controls a lot of pairwise connections between genes in these networks.

In many cases, the intrinsic edges of a pattern instance contribute to its pairwise disconnectivity index in a synergistic manner, i.e., the simultaneous removal of the respective edges exerts a much higher than merely additive effect (Figure 11). Although the approach we used for this purpose is a conservative approximation, it shows a principal tendency in these networks. More exact computations of this feature may be desirable but developing suitable algorithms for this, which have to take into account the particular characteristics of every pattern separately, was beyond the scope of this paper. However, we find that our approach was adequate to disclose clearly that the intrinsic edges of certain pattern instances display synergistic effects. This is the case for the pattern instances with the highest pairwise disconnectivity index in each of the three networks. Some other candidates have been found in E. coli and increasingly more in S. cerevisiae and mammals. This trend goes along exactly with the increasing density of the networks (1.2 edges per vertex in the E. coli network, 1.6 in S. cerevisiae and 2.3 in Mammals). The reason for this is on the hand: a more densely connected network provides a higher average vertex degree and thereby offers more alternative paths between pairs of vertices. These paths need not to share a similar set of edges, i.e., the connection of a pair is becoming more robust requiring more edges to be removed in order to disconnect it.

It should be noted that the observations reported here have been made for the networks as they are known at present. In particular the mammalian network may still suffer from incomplete knowledge. However, our method can be used for monitoring changes in such networks obtained from updated pathway databases like TRANSPATH^® 25 in the future. We see our results as the beginning of a large work which may consider the analysis of increasingly larger patterns including more than 3 vertices. More regulatory networks of various types (e.g., signal transduction networks, protein-protein interaction networks, gene expression networks) from different organism must be considered and tested in this regard in future as well. First attempts with signaling networks have confirmed the basic conclusions drawn here in spite of small characteristic differences in some details. Thus, we feel that the basic trends reported here will hold true for the more complete transcriptional as well as for other types of networks that will come up in future with increased reliability of high-throughput approaches and their systematic application.

On the other side, our method provides for the first time the possibility to assess the impact of patterns and motifs in general as well as individual pattern instances onto the overall connectivity of a graph. It is therefore suitable to identify bottlenecks in a biological network, which may be particularly important for the normal function of a cell, and may be top candidates to investigate disease mechanisms related to these functions. Since it identifies individual components in a network (vertices, edges, or pattern instances), it works independently of any a priori knowledge about the statistical over- or under-representation of certain network features. Though our approach was developed for the analysis of biological regulatory networks, it seems to be suitable for the analysis of other networks regardless of the particular nature of processes they represent (e.g., ecological, social, technical networks).

Literature-based databases of experimentally verified direct relationships for Escherichia coli 14 and Saccharomyces cerevisiae 15 have been used where E. coli V1.1 and S. cerevisiae V1.3 are available at http://www.weizmann.ac.il/mcb/UriAlon. The mammalian network of transcription factor genes (human, mouse, rat) was retrieved from the TRANSPATH^® Professional database (release 8.3, made in 2007) on signal transduction 25 and TRANSFAC^® Professional database (release 11.3, made in 2007) on eukaryotic cis-acting regulatory DNA elements and trans-acting factors 26. The network describes the causal relationships between genes that are coding for transcription factors, based on the regulation of these genes from transcription factors. However, the transcription factors themselves are not part of the network, i.e., the interaction chain "gene A codes for transcription factor A regulates gene B" has been summarized to: "gene A → gene B", which is a commonly used technique when inferring gene regulatory networks. Furthermore, genes are represented at the level of "ortholog abstraction", at which all species-specific data (human, mouse, rat) that refer to mammalian genes have been summarized to corresponding generic entries.

Selected genes (vertices) in the yeast and mammalian transcription networks were checked for their viability using the BIOBASE Knowledge Library™ http://www.biobase.de and the Saccharomyces Genome Database (Stanford Genomic Resources 48).

The networks were scanned for 3-vertex topological patterns using the FANMOD software with default settings 2728. The statistical significance of the network motifs was evaluated by means of the Z-Score 15, Z = (M_real - M_rand)/SD, where M_real and M_rand are the numbers of appearance of the motif in the real network and the randomized networks, respectively. SD is the standard deviation. The sign of edges (such as 'positive' for activation or 'negative' for inhibition) is not considered.

The pairwise disconnectivity index was calculated using the DiVa software 49. The statistical analysis was accomplished with R 50.

For estimating the impact of a single intrinsic edge on the existing pairwise connections between genes we have applied the pairwise disconnectivity index on an edge as defined in 24. In this manner it states the fraction of those ordered pairs of vertices that have been disconnected upon the removal of an edge, i.e., . Similar to Eq. 1, N is the number of linked ordered pairs of vertices in a network and we assume N > 0. The term N' stands for the number of connected ordered pairs of vertices in the network we obtain when deleting the edge e. Hence, Dis(e) = 0 the edge e is not crucial for linking at least of vertex pair. In contrast, Dis(e) = 1 if no vertex pairs remains connected.