The relationships uncovered by each method form networks whose structures are method-dependent (Figure 1a). Phylogenetic profile links, which we will refer to as 'phylo links', are transitive; that is, two genes linked to a common gene are also linked to each other. An example is the fully connected clique formed by flagella motor proteins (also shown in Pelligrini et al. [14]); that is, the proteins are generally either all present or all absent in a given genome and therefore have identical phylogenetic patterns. Phylo networks are characterized by the number and size distribution of their cliques. The occurrence of fully connected cliques follows from the definition that requires perfectly correlated patterns of occurrence to assign a link. A more permissive phylogenetic profiling method would result in reduced transitivity.

Gene networks uncovered by conserved chromosomal proximity links, which we will refer to as 'chromo links', typically exhibit a chain structure. For example, one of the long chains in the network of Helicobacter pylori genes (Figure 1b) corresponds to the chromosomal region containing highly conserved ribosomal genes. This network formation results from the fact that the conservation of gene order tends to involve more than two genes. As a particular conserved group tends to be arranged in an order conserved among organisms, linear concatenation dominates the network.

The domain fusion links, which we will refer to as 'fusion links', have complicated network relations, including the appearance of a major cluster. Even for small networks, as in Figure 1c, we find a few nodes that have a large number of links, and a large number that have few links [25,26,27]. We discuss this power-law behavior below.

In summary, the three methods for constructing networks all produce distinctive, non-random structures: branched but highly cliquish for phylo networks; generally branched structures for fusion networks; and linearly concatenated structures for chromo networks. By combining these three representations on the same grid of genes, we generate a largely non-overlapping map of functional linkages that provides more information than any of the three maps alone. The combined networks have a number of important functional and structural properties.

When the three networks are superimposed, four new types of links are formed combinatorially (Figure 2). The combined network of fusion, chromo and phylo links for a given genome (henceforth, combined network) captures between 30 and 80% of the genes in a genome and 57% on average, whereas the chromo, fusion and phylo networks independently capture 48%, 29% and 19% of the genes on average. As these numbers suggest, the graphs overlap significantly in terms of nodes (Table 1). We found, as did Huynen et al. [28] in Mycoplasma genitalium, that of the three methods the chromo networks have the greatest coverage. The fusion and phylo networks share 72% and 75% of their nodes with chromo networks; that is, 72% and 75% of the nodes found in fusion and phylo networks are also found in chromo networks. Functional links, on the other hand, tend to be complementary. Only 20% and 14% of the links in fusion and phylo networks, respectively, are found in chromo networks. The least overlapping of all are the phylo and fusion graphs, with only 41% of the nodes in a phylo net found in a fusion net and 6% of the links in a phylo net found in a fusion net. These results indicate that although the three methods capture overlapping sets of genes, and conserved chromosomal proximity captures more than the other two methods combined, the links generated by the different methods individually show much less overlap than the nodes.

A fundamental question is the quality of the links in terms of a direct functional correlation between the linked members. This quality can be estimated by reference to databases that classify genes into broad functional categories (clusters of orthologous groups (COGs) [12]) or biological pathways (Kyoto Encyclopedia of Genes and Genomes (KEGG) [18]). In particular, 72%, 68% and 64% of the fusion, chromo and phylo links are in the same COGs functional category, respectively [19,22,24].

We find, as did Marcotte et al. [29], that the links corroborated by multiple methods are of exceedingly good quality. However, as discussed above, this intersecting set corresponds to a small fraction of the total links (approximately 10%). Thus, in our study of combined networks, we construct the union, instead of the intersection, of the links generated by each method.

To evaluate the significance of a network we first produced 100 shuffled versions of this network (see Materials and methods). For each of these we calculate the percentage of linked pairs whose members are present in the same functional category (according to COGs) or pathway (according to KEGG) and then estimate the average and standard deviation for the population of random networks. We find a statistically significant difference between the observed networks of all types (fusion, chromo, phylo and combined), and the shuffled networks (p = 2.2e-16 in a t-test). On average, 33 standard deviations of separation distinguish an observed network from its 100 shuffled networks. As can be seen in Figure 3, the observed networks have high functional correlation centered around 60% agreement with COGs and 70% with KEGG, whereas the shuffled networks form a tight distant cluster with a low background functional correlation - a noise level in the range of 10 to 20%.

The methods described here uncover a large number of functional systems such as citrate cycle, purine metabolism, and fructose and mannose metabolism. To provide two specific examples, the highlighted nodes in the network shown in Figure 2 correspond to genes that encode proteins involved in oxidative phosphorylation and phenylalanine, tyrosine and tryptophan biosynthesis. We find that all but four of the 35 H. pylori genes annotated as participating in oxidative phosphorylation are present as six clusters in the combined network for this organism (Figure 4a). The clusters found by the three methods reflect the functions subserved by different groups of oxidative phosphorylation genes. In particular, the six clusters shown in Figure 4a, ranging in size from 1 to 12 are found in the network corresponding to five of the six receptor complexes that act in oxidative phosphorylation: NADH dehydrogenase, ATP synthase, succinate dehydrogenase, cytochrome bc, and cytochrome d.

As can be gleaned from Figure 4a, seven of the genes involved in ATP synthase (HP1131 to HP1137) form a conserved cluster of genes (detected in almost all known bacterial genomes). The chromo links among HP0828, HP1212 and HP1137:HP1136 (the last two are paralogs), are based on conserved proximity in other genomes. Six of the genes also have identical phylogenetic profiles, and the profile is unique to these six genes. The link between genes HP1135 and HP1137:HP1136 is strengthened because it is established by both chromosomal proximity and fusion in two Mycobacterium genomes. The five (of six) H. pylori genes that form the cytochrome bc and c complexes are linked. Essentially, the string of links is based upon two separate chromosomal sections: HP0144, HP0145 and HP0147; and HP1539 and HP1540. Within these two sections, two links are further strengthened by phylo links. The two sections are linked by a fusion link between two of the genes (HP1539 and HP0147). Three genes (HP0191, HP0192 and HP0193) compose the fumarate reductase complex and are connected by two chromo links. Chromosomal links build a skeleton of links that unite all the genes involved in NADH dehydrogenase. This cluster is further supported, however, by numerous phylo and fusion links. Two links are supported by all three methods. Two genes, HP1010 and HP1420, involved in oxidative phosphorylation are not connected to other genes involved in this functional system.

Figure 4b shows the functional links among the H. pylori genes involved in phenylalanine, tyrosine and tryptophan biosynthesis. Genes HP0402, HP0403 and HP0774 correspond to tRNA synthetase domains of phenylalanine and tyrosine linked by fusion events elsewhere as well as the proximity between HP0402 and HP0403 (HP0402 corresponds to two nodes, one for each of its domains, see Materials and methods). HP1249, HP1277, HP0663 and HP0401 share the same phylogenetic profile. HP0401, HP1249, HP0157 and HP0283 are found fused together as the yeast Ar01 gene and are thus all fusion linked. The proximity of HP1277 and HP1278, tryptophan synthase α and β chain respectively, is conserved in other genomes. These two genes are also found fused together in some genomes. The relationship between HP1279 (represented twice for each of its domains), HP1281 and HP1282 is enforced by phylo, chromo and fusion links for each pair of the three genes, with the exception of a lack of a fusion link between HP1279 and HP1282. HP0154 and HP0672 are present in the network but are not linked to other genes known to act in this pathway.

Overall, in E. coli, 26 clusters of seven or more genes corresponding to distinct KEGG pathways are identified in the network; where a cluster is a minimally connected subgraph such as those shown in Figure 4. The ability to reconstruct pathways depends on an integration of the three methods. In other words, the clusters shown would become fragmented if the methods were used individually. To illustrate this, for each of the 26 E. coli clusters we ask what fraction is obtained by the individual methods alone (Figure 5). Although some of the pathway clusters can be completely recovered by one of the methods - for example, the ribosomal system can be completely accounted for by chromo links - most can only be found by the integration of the methods. As an example, in the cluster in the combined network of 10 genes involved in the citrate cycle, 30%, 60% and 30% can be associated by only the fusion, chromo and phylo links, respectively.

New functional information on particular genes can be predicted on the basis of the combined networks. Naturally, the most fundamental unit of prediction in the network corresponds to a link between two nodes. For example, M. jannaschii genes MJ1313 and MJ1407, whose functions are completely uncharacterized, are linked by fusion, chromo and phylo links. From this superlink we can confidently conclude that, although the actual functions of the genes in question remain elusive, a functional link probably exists between the two genes and they need to be experimentally studied together.

By superimposing known pathway information onto the combined networks, functional predictions can be made on the basis of a gene's location in the network. For example, Figure 6 shows a fraction of the Thermotoga maritima network. TM0885 and TM1367 have no characterized function, but their position in the network suggests that they have a role in energy production. These two genes form a four-gene clique of phylo links with TM0397 (a ferrodoxin-like domain) and TM0034 (an uncharacterized protein with a putative Fe-S center from the COGs database). The context of the completely uncharacterized genes (TM0885 and TM1367) within the network further strengthens the inference of their functional association with their neighbors (Figure 6). TM0397 is fusion linked to TM0396, a dehydrogenase with a Fe-S cluster, which is in turn fusion linked to an oxidoreductase protein, TM1640. From this network locus of TM0885 and TM1367, we predict for them a role in energy production.

A striking property that emerges when the networks are combined is the formation of a giant cluster. This occurs for all but three genomes (Mycoplasma pneumoniae, Mycoplasma genitalium, and Ureaplasma urealyticum) perhaps because the number of links in these short genomes is very small. Although the number of nodes in the combined networks range from 400 to 1,600 (Figure 7a), depending on the genome, the percentage of nodes contained in a genome's largest cluster is relatively invariant, ranging from 80 to 90% (Figure 7b). These observations suggest that the properties of the combined networks have stabilized, and their general characteristics, as described below, should be relatively invariant against further increases in the number of functional links discovered. We note that random networks generated using the same number of nodes and edges as the chromo, fusion or combined networks, also form giant clusters, as is theoretically expected. In contrast to the very large clusters we find in chromo and fusion links, the largest cluster of phylogenetic links contains 19 ortholog families. This is not surprising because, by definition, all genes in a phylo cluster must have the same phylogenetic profile. Thus, the probability of finding a cluster with a very large number of genes is low.

In general, many paths of different lengths connect each pair of nodes in the giant cluster. A useful measure of the global characteristics of the cluster is the minimum path length between each pair of nodes. It seems intuitively plausible that such minimum paths are the most biologically relevant of all paths connecting a pair. For each genome, the set of shortest path lengths has a Gaussian distribution. Its average is referred to as the characteristic path length. The characteristic path length averaged over the 43 genomes that have a giant cluster is 7, and the standard deviation about this average is 3. In other words, on average, seven comparative genomic links separate any two genes in a giant cluster. This is in contrast to a characteristic path distance of 5 ± 1 for random networks simulated with the same number of nodes and edges. Thus, the characteristic path distances of the combined networks are slightly larger than those of random networks but much smaller than the characteristic path of uniform lattice networks.

An example of a minimum path is shown in Figure 8. The E. coli gene tyrA of the phenylalanine, tyrosine and tryptophan biosynthesis pathway is separated from the aspS gene of the aminoacyl-tRNA biosynthesis and alanine and aspartate metabolism pathways by five links. This shortest path proceeds by way of four histidine metabolism genes. Each link along this path relates genes with a common functional pathway. However, as many genes are mapped to multiple functional pathways, paths in the network typically traverse many pathways. Snel et al. have proposed that such 'linker' genes be used to mark the boundaries between functional modules of genes in such networks [30].

Although global properties appear to be nearly random, local properties are not. In particular we expect local properties to be structured as each biological pathway that subserves a particular function invariably has several members, and the relationships between these members are likely to emerge through the comparative genomics methods applied here. One descriptive characteristic of a local environment is the clustering coefficient: the average probability that two genes linked to a common gene are also linked to each other [2]. This number is 1 for systems that are fully transitive (for example, networks constructed using phylogenetic profiling as applied here; see Figure 1) and becomes very small for random networks.

Not surprisingly, the combined networks for all 43 genomes show a significant amount of local clustering - or 'cliquishness' - when compared to random graphs. The chromosomal proximity and fusion networks both have a similar clustering coefficient of 0.24 and 0.25 respectively, as calculated for the giant component of the networks. When shuffling the networks (see Materials and methods) the clustering coefficients for these random graphs are 0.02 and 0.06 for chromo and fusion networks, respectively. The clustering coefficient increases to 0.36 for the combined network (0.02 for the shuffled networks), reflecting the high coefficient of the fully transitive phylogenetic graphs. The combination of high clustering coefficients and random-like characteristic path distance places these biological networks among the well studied class of so-called 'small-world' networks, to which social nets often conform [2,31].

Quantitative measures of the local (clustering coefficient) and global (characteristic path distance) properties of networks allow us to analyze the similarities and differences among all of the networks. Figure 7c shows the properties of the network mapped onto a two-dimensional space defined by the characteristic path distance and clustering coefficient. We find that networks of the same type (phylo, chromo, fusion or combined) cluster together, demonstrating the universality of network structures of each method. The combined networks have a characteristic path that is roughly the average of the fusion and chromo networks and a clustering coefficient that is greater than both.

Furthermore, the networks appear to be scale-free in terms of the number of links per node (degree): a few nodes have many connections while most have few connections. As shown in Figure 7d, the distribution of links per node for the chromo, fusion and combined networks differs markedly from those of random graphs with the same number of nodes and edges (Figure 7d). While the power law distribution (linear regression on a log-log scale) holds well for the fusion and phylo graphs, the number of nodes of degree 1 in the chromo graph is less than would be expected from a power law, and is, indeed, not significantly larger than those of degree 2. The explanation lies in the operon organization of genes on bacterial and archaeal chromosomes, making two the typical number of chromo links of a conserved proximity of a gene with other genes - one on each end (see also chromo links in Figures 1 and 2).

We stress that the comparative genomic links do not necessarily correspond to direct protein-protein interactions, and thus are not expected to be detected by yeast two-hybrid methods. Links made in silico have been shown to be indicative of broader functional relationships [19,22]. In a recent study, Rain et al. identified over 1,200 interactions using a high-throughput application of the yeast two-hybrid method using approximately 15% of the H. pylori genes as 'baits' [32]. Of the 1,200 interacting pairs detected, only 17 correspond to links also found in our comparative genomic network. We note that low overlap between sets of links is even observed between high-throughput datasets [33,34] and may be explained by a lack of saturation of the complete functional relationship space, a high false-positive rate, and/or a bias towards a certain type of link [35].

Uncovering functional links between genes is a major step towards deducing the function of individual genes. Here we describe the properties of networks generated by the combination of three comparative genomics methods for 43 microbial genomes representing the three domains of life. We find that a giant 'small-world' cluster consistently includes 80% to 90% of the nodes in these networks, so that the average minimum path between any two genes is small, but local cliquishness is frequent relative to random networks. This structure for a society of genes, along with the observation that local order corresponds to genes of the same functional pathways, supports the notion that the network of relationships among a cell's genes is a set of highly cliquish functional systems interlinked by genes that are common to multiple pathways.

We used three previously published methods for detecting functional links between proteins by comparative genomics. In this section we describe the implementation of these methods. As a pair of proteins can be coded as distinct genes in some genomes but fused as a multidomain protein in others (the basis for the fusion method), a comparative genomics study must adopt protein domains (instead of whole proteins) as the unit of analysis. Thus, multidomained proteins are split into domains (as they are at present in the COGs database, which clusters orthologous protein domains that are present in three or more lineages) and these, in turn, form the units of the comparative genomics links generated by the three methods. All the links are available in the Predictome database [36] and as additional data files (see Additional data files). These methods all depend upon detecting orthologs across genomes and, consequently, each node in any network corresponds to a cluster of orthologous groups [12,37] - a COG - with a representative sequence in that genome. Thus, the term 'gene' which, for brevity, is used throughout as indicative of a single node in the network, may in some cases be represented by multiple nodes.

Two different methods for generating random networks were used. Both methods compare random networks with observed networks having the same number of nodes and edges.