The genetic interactions used in this study are taken from published experiments detecting cell growth defects through screening a deletion of interest (query gene) against ~5000 viable yeast single-deletion strains (target genes) 3456. As only ~150 queries have been reported, the current network covers ~2% of the entire map. Therefore, many observations will be re-assessed after completion of the map. Specifically, the entire observed genetic network is expected to be symmetric when query and target genes are reversed. To account for the symmetric property of the entire genetic network, we have constructed both an asymmetric genetic network that includes all currently available high-throughput genetic interactions and the symmetric genetic network that covers interactions only between genes that have been used as queries (Fig. 1A). The graph of the symmetric genetic network is shown in Fig. 1B. Each node represents a gene, and each edge represents the genetic interaction between two connected genes. The edges are considered undirected, and we do not distinguish between edges that were detected in one or both directions. High connectivity in the symmetric genetic network (Fig. 1B) reflects that query genes were selected based on related functionality 6.

Previous analysis has suggested that shared genetic interaction partners correlate with physical interactions 6. Quantitative measures for partner sharing in physical interaction networks has been defined as Mutual Clustering Coefficients (MCC) 14. Here we use the negative Log₁₀ of the P-value of the hypergeometric MCC as a quantitative measure of neighbor sharing in the genetic interaction network, and for convenience term it the congruence score 25. Higher scores indicate that two genes share more genetic interaction partners than expected by chance. A genetic congruence network is then derived from introducing non-directed edges between congruent genes, using the congruence score to provide an edge weight (Fig. 1C). Asymmetric and symmetric congruence networks have been constructed from the corresponding genetic networks, respectively. A P-value of 0.01 for shared genetic interaction partners after correcting for multiple testing corresponds to a congruence score of 8 for the congruence network derived from the asymmetric genetic interactions and a congruence score of 6 for the network derived from the symmetric genetic interactions.

The protein interaction network we used is derived from ~45,000 protein-protein interactions compiled from the large-scale yeast two-hybrid and mass spectrometry analyses 78910. Each interaction has been assigned with a confidence score that corresponds to the network edge weight. The confidence score represents the probability that two proteins interact with each other 12.

The size and global topological measures for genetic, congruence, and protein networks are provided (Table 1). The average degree is the number of edges per node, and the clustering coefficient measures the interconnectivity around a node. Interestingly, average degrees nearly halve but clustering coefficients double from genetic networks to congruence networks. The values for the protein network are in between those for genetic and congruence networks. These suggest that the congruence network tends to be highly clustered. We quantitatively demonstrate with the following results on path lengths and local motifs that the inferred congruence links from shared patterns of genetic interactions have greater relevance to protein interactions than underlying direct genetic interactions.

Conventional analysis shows genetically interacting genes encode proteins in the same complex more often than would be expected by chance 6. Because physical associations and genetic interactions each report on functional similarity, we might naively expect that physical and genetic links should be correlated. However, it has also been recognized that the number of genetic interaction pairs having direct physical interaction is a small fraction of the total number of genetic interaction pairs (~1%) 6. Therefore, given currently known genetic and protein interactions and their overlap, the majority of genetic interactions do not connect physical partners.

To quantitatively study the global relationships between genetic and physical interactions, we calculated the shortest path length for any two genes in the genetic interaction network and the shortest path length for corresponding gene products in the protein interaction network, and then compared these two path lengths. Our results reveal that most protein pairs are distributed 3–4 links apart in the protein interaction network, regardless of whether there is a genetic interaction between the gene pair (Fig. 2A). This indicates that characteristic path lengths in genetic and physical interaction networks are incommensurate. Results are similar using symmetric and asymmetric genetic networks. These observations support the concept that genetically interacting pairs usually have no direct physical interactions. If we define pathways by the context of physical interactions and assume genes with physical interactions function in a single pathway and without physical interactions act in parallel pathways, then our results suggest that genetically interacting genes are more likely to belong to parallel compensating pathways. Other groups have used similar reasoning to identify components of pairs of complementary pathways from joint analysis of physical and genetic interactions 22.

We asked whether the other view of genetic interactions, i.e. genetic congruence, might yield improved concordance with physical interactions. We first explored the relationship between pair-wise genetic congruence versus direct physical interaction. High-throughput physical interaction data sets are known to include many false-positives, which can confound analysis. Confidence scores have been developed to reflect the probability that a physical interaction is a true-positive 12. We observed that protein interaction confidence increases with the congruence score (Fig. 2B). Above the congruence score of 8 and 6, which corresponds to the network P-value of 0.01 for the asymmetric and symmetric networks respectively, all protein pairs exhibit high confidence interactions with confidence score greater than ~0.8. This implies that genetic congruence acts as an indication of high-confidence protein interactions. It is notable that information from a purely genetic experiment correlates well with information from a purely proteomic experiment. We also used receiver operating characteristic (ROC) curves to assess the relationship of congruence scores and physical interactions. ROC curves for asymmetric and symmetric congruence scores both climb rapidly away from the origin with high true positive rates and low false positive rates [see additional file 1, supp. fig. S2]. According to the area under the curve, the congruence score from the symmetric network performs better than the score from the asymmetric network, but at the cost of making fewer predictions. This is in agreement with the result from Fig. 2B that congruence scores of the symmetric network predict higher confidence physical interactions as compared with those of asymmetric network. The reason for the differences may be due to biased selection of query genes, as the symmetric network only contains query genes and all query genes were selected from a few related biological processes 6.

We further investigated the pair-wise congruence in the context of the protein interaction network. Our results show that the shortest path of physical interactions between congruent pairs decreases from ~3.6 links to 1 link (direct physical interaction) with increasing of congruence score (Fig. 2C). The path length transition begins when the congruence score increases beyond 8 and 6 for asymmetric and symmetric congruence networks, respectively. Once the score reaches 21 and 20 for asymmetric and symmetric networks, the congruent gene encoded proteins coincide with known direct physical interactions (4 pairs with congruence score = 21 in the asymmetric network and 1 pair with congruence score = 20 in the symmetric network).

Finally, to explore the connection between the congruence network and the protein network, we computed the highest score path for any two genes in the congruence network. Edge weights are in the range of 0 and 1 generated by applying a sigmoid function to the congruence scores (see Methods). The higher the path score, the higher probability two genes share similar genetic interaction partners. When comparing the highest path score in the congruence network with the shortest path length in the protein interaction network, we observed that the physical distance decreases monotonically from the average path length ~3.6 links to 1 (direct physical interaction) as the highest path score increases in both asymmetric and symmetric congruence networks (Fig. 2D). Therefore, transitive genetic congruence is commensurate with physical distances, which is similar to direct genetic congruence (Fig. 2C).

Network motifs represent significantly recurrent patterns of simple interactions in complex networks 17. Comparison of local structures in the network can help reveal the connections among superficially unrelated biological or social networks 18. Additionally, the local structure of the network contributes to the understanding of global organization of the network 16. To contrast the local structure of three types of networks, we counted the abundance of non-directed triads and tetrads in genetic, congruence, and protein networks. The random networks used to detect tetrads were generated to preserve the same triad counts as the real network 18.

More significantly, we can determine network transitivity through the observation of whether a transitive or intransitive motif is enriched or depleted in the network. Transitivity is a common network property that interactions of A-B and B-C imply elevated probability of interaction of A-C. We developed a characteristic, termed the motif transitivity score (MTS), as a quantification of the motif transitivity [see Methods and additional file 1, supp. table S1]. The positive values indicate transitive motifs while the negatives represent intransitive motifs. The network transitivity has been quantified by the clustering coefficient before 2627, which is closely related to the motif transitivity score defined here. We have found good agreement between motif enrichments (Fig. 3A) and average clustering coefficients (Table 1), i.e. congruence and protein networks are more clustered compared with the genetic network.

When using the asymmetric genetic and congruence networks for comparison with the physical network, the pattern of enriched motifs (the relative motif ratio) is significantly correlated for congruence and protein interaction networks (Pearson correlation coefficient R = 0.76, P-value = 0.03), and these are anti-correlated with the enriched motifs for direct genetic interactions (R = -0.66, P-value = 0.08; R = -0.69, P-value = 0.06, respectively) (Fig. 3A). This is consistent with the above global distance analysis, supporting significant overlap between congruence and protein networks.

The enriched motifs in asymmetric congruence and protein networks are all transitive, including triad2 (triangle motif) and tetrad6. The triangle motif is the most significantly enriched motif, suggesting the transitive property of congruence interactions and physical interactions. This result is in agreement with our observation in the previous section that the transitive congruence is correlated with short physical distance (Fig. 2D). The asymmetric genetic interaction network, however, consists of both intransitive motif tetrad4 (square motif) and transitive motif tetrad6, with the square motif as the most enriched structure.

The detection of intransitive motifs in the asymmetric genetic network may be due to the artifact that the interactions have not yet been tested. It does not necessarily mean that these interactions do not exist. To overcome this bias, we repeated motif-finding procedure using the symmetric genetic network and corresponding congruence network (Fig. 3A). The pattern of enriched motifs is still significantly correlated for symmetric genetic congruence and protein interaction networks (Pearson correlation coefficient R = 0.73, P-value = 0.04), but these are insignificantly correlated with those for the symmetric genetic network (R = 0.29, P-value = 0.49; R = 0.10, P-value = 0.82, respectively). The enriched motifs in the symmetric congruence network remain the same as for the asymmetric congruence network, i.e. all transitive motifs, triad2 (triangle motif) and tetrad6.

A final concern is that the transitive motifs arise from the mathematical process of generating congruence scores: if genes A and B share synthetic lethal partners, and B and C share partners, then A and C may have an increased probability of sharing partners. To address the question, we followed the following protocol [see additional file 1, supp. fig. S3]: (1) Randomize the genetic interaction network. (2) Calculate congruence scores for gene pairs in the randomized network. (3) Set a threshold and calculate motif enrichment for the random congruence network. We repeated this process 100 times for both the symmetric and the asymmetric genetic interaction networks. The typical extreme value for the maximum congruence score observed was 5 for the symmetric network and 6 for the asymmetric. Thus, applying the same cutoff for congruence scores as in the actual network, 6 for symmetric and 8 for symmetric, typically rejects all the congruence edges in the randomized network. We reduced the thresholds to retain the same number of congruence edges as in the actual network, with mean values of 1.8 (symmetric) and 3.2 (asymmetric) over the 100 randomizations. The average clustering coefficient is significantly smaller in the random networks than the actual network: 0.23 vs. 0.84 (random vs. actual symmetric, P-value 10^-402), and 0.12 vs. 0.73 (random vs. actual asymmetric, P-value 10^-933). Although the transitive motif triad2 (triangle) is enriched in the random congruence network relative to a fully random network, the motif count is far below that observed in the actual congruence network [see additional file 1, supp. table S3]. Other patterns of motif enrichment are quite different: tetrad4 (square motif, intransitive) is enriched in the random congruence network and depleted in the actual network, and tetrad6 (4-clique, transitive) is enriched in the actual network but not in the random network [see additional file 1, supp. fig. S4]. The transitive motifs in the congruence network are therefore enriched significantly beyond what would be expected based solely on the method of defining congruence edges.

Both transitive and intransitive motifs are still detected in the symmetric genetic interaction network. However, the types are different from those in the asymmetric genetic network. The transitive triangle motif becomes the most enriched structure in the symmetric genetic network, in agreement with a previous study that genetic interaction partners of a gene have an increased likelihood to interact with each other 24. One source of the triangle motif could be the requirement for any two of three pathways for viability. Notably, the square motif is still highly enriched in the symmetric genetic network despite the abundance of the triangles, indicating that the square motif will remain enriched when the complete genetic interaction network is determined.

The view from recent studies indicates that high clustering is a generic feature of biological networks, as exemplified by protein interaction and protein domain networks 13. However, we find that the genetic interaction network has both transitive and intransitive motifs. The coexistence of triangle and square motifs in the genetic network suggests two scenarios for genetic interactions between pathway components. In one scenario, genetic interactions between two pathways generate a square motif. Each edge crosses between the pathways, and genes at opposite corners are in the same pathway. In the second scenario, any two of three pathways are required for viability. Genetic interactions cross between all three pathways, generating the triangle motif.

To further answer the question whether the enriched triangles and squares overlap with each other or are excluded from each other, we compared the members of triangle and square motifs in the symmetric genetic network (Fig. 3B). Results show that one-node sharing is the dominant scenario (76%) for triangles and squares. Assuming three pathways for the triangle motif and two pathways for the square motif, the one-node sharing case defines four parallel pathways with one shared by the square and triangle. Two-node sharing accounts for 22% of total possibilities, and suggests three parallel pathways with two shared by the triangle and square. Only 2% of total cases are the complete overlap of the triangle and square, which is in an agreement with our observation that tetrad5 is not an enriched motif in the symmetric genetic network (Fig. 3A).

Because the completed genetic interaction map will necessarily be symmetric (except for false-positives or false-negatives), the enriched motifs in the symmetric genetic network are more relevant than the enriched motifs in the asymmetric genetic network.

Correct interpretation of the relationship between genetic and protein interactions enables interesting biological predictions. As we have demonstrated in previous sections, genetic congruence and protein networks are similarly organized with corresponding distances and motifs. Then, we would expect that two genes closer in the congruence network have higher tendency to physically interact with each other, reside within one protein complex, and involve in similar biological process.

To validate this prediction generally, we plotted protein complex membership versus the distance in the genetic network and the path score in the congruence network (Fig. 4A). The probability of co-residence in a protein complex increases with the congruence path score, and scores greater than 0.7 indicate same protein complex membership. On the other hand, gene products binned by distance in the genetic interaction network have uniformly low probability of protein complex co-residence.

Physical interactions usually suggest functional association. Accordingly, we asked whether congruence also indicates functional connection besides physical connection. As an initial validation, we found that genes close in the congruence network share similar functional annotations recorded in the database of Gene Ontology (GO)28, i.e. biological process and molecular function (Fig. 4B). Moreover, the functional similarity is consistently higher for gene pairs based on path score in the congruence network than based on distance in the genetic network.

An example of congruence coinciding with protein interaction and function association is the prefoldin complex, which includes PAC10, GIM3, GIM4, GIM5, and YKE2. These five genes are clustered in the congruence network and the average path score between any two members of this complex is 0.51 (Fig. 1C). They are all chaperone proteins forming a complex, which promotes efficient protein folding 2930.

The genetic interaction dataset is derived from a recent high throughput study in budding yeast 6. The interaction is detected by cell growth defect through introducing a deletion of interest (query gene) into all viable yeast single-deletion strains (target gene). Interactions derived from 6 essential query genes, including MYO2, SCC1, CDC2, CDC7, CDC42, and CDC45 were removed in our study because phenotypes exhibited by conditional alleles of essential genes may include loss of function, unregulated function, and gain of function, while null alleles of nonessential genes are by definition solely loss of function mutations. Results and conclusions do not change, however, when these 6 essential genes are included in the analysis.

We constructed two types of genetic networks. The asymmetric genetic network includes currently available high throughput genetic interactions, i.e. 3799 genetic interactions between 126 non-essential query genes and 982 target genes. The symmetric genetic network only contains interactions between query genes, i.e. 813 genetic interactions between 108 non-essential query genes and 104 target genes that have been used as queries.

Genetic interactions from the high throughput study 6 were reported as an interaction between the query gene and the target gene. A randomized network was generated by keeping the query gene list unchanged, randomly matching one of the target genes according to the probability of each target gene shown in the interaction list with replacement. Duplicate query-target pairs and self-interaction pairs, which are not possible in the experimental networks, were rejected during randomization. Results depict the average over 1000 randomizations.

The congruence score was defined as -log₁₀ [hypergeometric P - value (x ≥ k_obs], and hypergeometric P−value (x≥kobs)=∑x=kmin⁡(m,n)C(m,x)C(t−m,n−x)/C(t,n) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@6AB8@, where two target genes having m and n genetic interaction partners share x partners from a list of t query genes, and C(j,k) is the binomial coefficient j!/k!(j-k)! 25. Related measures have been used to analyze protein interaction networks to predict protein-protein interactions 14. The congruence score is calculated for every target gene pair in the symmetric and asymmetric genetic networks. The symmetric and asymmetric congruence networks are derived from the corresponding genetic networks, respectively. The distribution of network size over different congruence scores is provided [see additional file 1, supp. fig. S1]. The congruence score of 8 (122 nodes with 267 edges) for asymmetric congruence network corresponds to the network P-value of 0.01 after correction for multiple testing of per-link P-value 0.01/982² = 10^-8. Similarly, the congruence score of 6 (61 nodes with 146 edges) is the cutoff value for the symmetric congruence network.

We used 47,783 protein-protein interactions with confidence scores 12 compiled from the large-scale two-hybrid data sets of protein-protein interactions 910 and mass spectrometry analysis of protein complexes 78. The distribution of network size over different confidence scores is provided [see additional file 1, supp. fig. S1].

The shortest path distance was counted for any two nodes in the un-weighted genetic interaction and protein interaction networks. The shortest path length is the sum of lengths of individual linkage.

The SEEDY algorithm 31 was used to compute highest score path distance for the weighted genetic congruence and protein interaction networks. The highest score path is the path with the maximal value of the product of edge weights. Disconnected components are ignored for both shortest path and highest score path calculations.

The edge weight for the protein network is the confidence score (in the range of 0 and 1) 12. The edge weight for the genetic congruence network is derived from a sigmoid function w=e(s−a)/b1+e(s−a)/b MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciGacaGaaeqabaqabeGadaaakeaacqWG3bWDcqGH9aqpdaWcaaqaaiabdwgaLnaaCaaaleqabaGaeiikaGIaem4CamNaeyOeI0IaemyyaeMaeiykaKIaei4la8IaemOyaigaaaGcbaGaeGymaeJaey4kaSIaemyzau2aaWbaaSqabeaacqGGOaakcqWGZbWCcqGHsislcqWGHbqycqGGPaqkcqGGVaWlcqWGIbGyaaaaaaaa@432F@ (in the range of 0 and 1), where s is the congruence score, a and b are parameters. The rationale of introducing the above sigmoid function is derived from the probability distribution of Pr(true positive|s) = Pr(protein interaction|s) as genes sharing genetic interaction partners usually exhibit physical association 6. The parameters a = 15.9 and b = 1.6 are the best-fit values for the sigmoid function to form a smoothed interpolation of Pr(protein interaction|s) for the asymmetric congruence network [see additional file 1, supp. fig. S6]. Results were not sensitive to the choice of parameter values [see additional file 1, supp. fig. S7]. Similarly, a = 17.7 and b = 3.4 are the best-fit values for the symmetric congruence network.

We used the mfinder1.1 – network motifs detection tool http://www.weizmann.ac.il/mcb/UriAlon/groupNetworkMotifSW.html to count non-directed triad and tetrad motifs in genetic interaction, genetic congruence, and protein interaction networks. Both symmetric and asymmetric genetic networks were used for motif searching. Motifs were also counted for the symmetric congruence network with cutoff value of 6, the asymmetric congruence network with cutoff value of 8, and the protein network with confidence score greater than 0.5 12. Motif results are insensitive to the threshold values for congruence and protein networks [see additional file 1, supp. table S2]. The Metropolis algorithm was used to conserve the number of triads in random networks for tetrad motif counting. The relative motif ratio (RMR) was calculated to represent the abundance of each motif relative to random networks in which each node has the same number of edges as the corresponding node in the real network. The formula for RMR is defined as RMR=Δi/(∑Δi2)1/2, Δi=Nreali−<Nrandi>Nreali+<Nrandi>+ε, and ε=4 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@7985@. The criteria taken for enriched motifs are N_realZscore > 2, N_real/N_rand > 1.1, Uniqueness ≥ 4 where Uniqueness is the number of times a motif appears in the network with completely disjoint groups of nodes 1718.

To quantify the motif transitivity, we give the definition of motif transitivity score (MTS) as MTS=3×number of 'Δ' - number of 'V'3×number of 'Δ' + number of 'V' MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@7A19@, where 'Δ' is a group of 3 vertices each of which is connected to the other two, and 'V' is a group of 3 vertices only one of which is connected to the other two. The 'Δ' and 'V' are mutually exclusive subgroups in the MTS calculation. The factor of 3 accounts for the fact that each 'Δ' is equivalent to three 'V'. This formula quantifies the motif transitivity in the range from -1 to 1, and is insensitive to the motif size. The MTS is 1 for a fully connected motif, and is -1 for a motif without the triangle. The values of MTS for triads and tetrads are listed [see additional file 1, supp. table S1].