Pathway definition and identification

All genes in one organism interact, forming a gene-network. Given the fact that some genes specifically cooperate to take over some highly specialized cellular functions, pathways are often used to represent part of the gene-network at a more detailed level. An example compensatory pathway pair is shown in Figure 1A, where the three genes shown in left and the two genes shown in right can be defined as two distinct pathways. However, it is also appropriate to group all the five genes (or plus the two master genes) into one pathway. Both definitions make sense because the first definition focuses on the functional redundancy of the two smaller pathways while the second definition focuses on the fact that all these genes are involved in one specific cellular function. Since the synthetic lethal interactions are in general considered to reveal the functional redundancy between pathways [1], [7]–[8], the pathways we pursued in this paper followed the first definition. However, the pathways we identified do not necessarily adhere to this meaning; rather, functionally distinct genes may be found to be compensatory to a multifunction pathway since we did not use the physical interaction data. This fact is discussed further in section “Multifunction effect of pathways”.

Shown in Figure 2 are several genetically redundant pathway pairs identified by our method. For example, the dynein-dynactin pathway (shown in boxes, P<1e-12, Hypergeometric test, herein and after, Figure 2A left) and protein depolymerization pathway (diamonds, P<1e-6, Figure 2A right) were found to be genetically compensatory. Genetically compensatory pathway pairs also shown in Figure 2 include histone deacetylation (diamonds, P<1e-6) and histone methylation (boxes, P<1e-6, Figure 2B), actin filament-based process (boxes, P<1e-8) and chitin metabolic process (diamonds, P<1e-11, Figure 2C), positive regulation of RNA elongation (diamonds, P<1e-8) and double strand break via single strand annealing (boxes, P<1e-14, Figure 2D), tubulin folding (diamonds, P<1e-12) and spindle checkpoint (boxes, P<1e-8, Figure 2E) and Golgi to membrane protein transport (boxes, P<1e-5) and endosome to Golgi retrograde transport (diamonds, P<1e-9, Figure 2F). Known physical interactions that were listed in Ulitsky and Shamir (2005), including protein-protein and protein-DNA interactions, were also indicated in Figure 2 using heavy blue lines.

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Figure 2. Example pathway pairs identified by our method.

Each example is shown in a pair: left: pathway 1, right: pathway 2. Heavy blue lines represent physical interactions (protein-protein interaction and/or protein-DNA interactions) between genes and red lines represent synthetic lethal interactions. Diamonds and Boxes represent genes with the dominant function for pathway 1 and pathway 2, respectively (see text for details). (A) The dynein-dynactin pathway and protein depolymerization. (B) Histone deacetylation and histone methylation. (C) Actin filament-based process and chitin metabolic process. (D) Positive regulation of RNA elongation and double-strand break repair via single-strand annealing. (E) Tubulin folding and spindle checkpoint. (F) Golgi to plasma membrane protein transport and retrograde transport, endosome to Golgi.

https://doi.org/10.1371/journal.pone.0001922.g002

By removing physical interaction data from the process of identifying genetically compensatory pathways, there was also a greater potential for spurious identifications (discussed later). Though the approach presented here is theoretically sound, predictions should make sense in the face of known physical interactions within and between pathways. Here we have detailed an example to demonstrate the utility of identifying compensatory pathways solely via genetic interaction information.

In Figure 2C, the functions of an actin filament-based process and chitin metabolism were identified as compensatory processes in this analysis. Though these functions seemed superficially disparate based on Gene Ontology [12], the potential mechanism for their interaction became apparent when the loci involved were investigated. The actin filament-based process genes are involved in various aspects of endocytosis, which require the movement of vesicles between the trans-golgi network and the cell membrane via the actin cytoskeleton (www.yeastgenome.org). The movement of vesicles within a cell depends in part on polysaccharide tags attached to the membranes of the structures [13]–[15]. One of these tags is mannan, a polymer of mannose that is important for vesicle transport, cell wall structure, and other functions dependent on protein glycosylation [14].

The genes in the identified actin filament-based process pathway encode proteins for actin/myosin binding that are known to affect secretion polarity within the cell, as well as a mannan polymerase (VAN1) and a mannosyltransferase (MNN10) [16]. These mannose-processing proteins have been shown to be part of golgi-bound complexes composed of either VAN1/MNN9, or MNN10/ANP1/HOC1/MNN9/MNN11, both of which are involved in protein mannosylation [14]. This information provides a connection to the compensatory pathway involved in chitin synthesis, as HOC1 also bears a physical similarity to several other seemingly unrelated glycosyltransferases MNN1, OCH1, SUR1, and CHS1 (a chitin synthase) proteins, indicating the presence of necessary functional components [17] in proteins of both pathways, and revealing the possibility of redundant functions performed by those proteins. Since chitin is a polymer of N-acetylglucosamine, which bears structural similarity to mannose, and is also a critical component of the cell wall, it is not surprising that a protein or complex could be involved in the transport or polymerization of both molecules. N-acetylglucosamine is used to link mannan to yeast proteins [14]–[15], indicating the necessity of the mannan producing complex to recognize and bind sugars to the molecule that is the building block for chitin. It is also possible that the molecular outputs of both complexes (mannan and chitin) are compensatory. Indeed, evidence of overlap between chitin and mannan glycosylation exists as mutations in the MNN10 complex alter chitin levels in yeast cells [16], [18]. This indicates that either the complexes themselves are partially redundant, or that cell wall stress resulting from missing cell wall mannan can be relieved by the increased production of chitin. Given these similarities in sugar and protein structures and functions between mannan and chitin-based biological processes, it was not surprising (and expected) that they were identified as compensatory processes in this analysis.

Summary statistics of compensatory pathway pairs

Our search generated 2,590 pathway pairs (7.7±3.1 genes per pair), which cover 5,284 (56% of 9,376) synthetic lethal interaction pairs involving 689 yeast genes. Since we did not require the physical interaction to support our pathway models, it was expected that the pathways we identified to be larger than that identified by integrating genetic and physical interaction data [1], [8]. To test this hypothesis, we compared the sizes of the pathways from different methods. As shown in Figure 3 upper panel, the pathways (redundancy removed, see Methods) identified by our method were significantly (P<1e-16, Wilcox rank sum test) larger than that of Kelley and Ideker [1], whereas the sizes of the pathways in Kelley and Ideker [1] was significantly (P<1e-6, Wilcox rank sum test) larger than that of Ulitsky and Shamir [8]. In addition, our investigation suggested that this result is not an artifact of parameters specific to our method (Figure S1 and Text S1). A close inspection revealed that many pathways (40%; 110/280) identified by Ulitsky and Shamir [8] had one pathway member of size 2 (however, we also noted that Ulitsky and Shamir [8] had identified 5 pathways with more than 50 genes and the largest one had 201 genes). This result implied that we can potentially increase the size of gene sets at the cost of losing the physical interaction support. However, the pathways we identified can sometimes be smaller than the congruence score method by Ye et al [7]. For example, BIK1 was absent from the dynein-dynactin checkpoint pathway, as shown in Figure 2A, and it was identified by the congruence score method. It should be noted that pathway size distribution is not a measure of accuracy of our method. In fact, the size of a pathway could be very arbitrary if we take a gene-network perspective, where a pathway is used to highlight a functionally homogenous sub-network (see section “Pathway definition and identification”). However, we noted that in functional genomics a practical problem is that many genes are not functionally annotated, thus it may be helpful to provide biologists reasonably larger lists of genes showing compensatory interaction patterns. In fact, the example in Figure 2C discussed previously suggests the value of providing a larger pathway.

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Figure 3. Summary statistics of the pathway pairs we identified.

Upper panel: Distribution of the sizes of the pathways of our analysis (GI only; box) and that of Ulitsky and Shamir ([8]; circle) and Kelley and Ideker ([1]; star). Bottom panel: The distribution of the completeness of the between-pathway synthetic lethal interactions in our identified pathway pairs.

https://doi.org/10.1371/journal.pone.0001922.g003

We next asked how the synthetic lethal interactions were depleted from either pathway of each pathway pair and how the synthetic lethal interactions were enriched between pathways in each pair. Since we imposed a stringent constraint on the within-pathway synthetic lethal interactions (α = 0.01; see Method), we found that only one of the pathways we identified have within-pathway synthetic lethal interactions. Thus we proceeded to determine how the synthetic lethal interactions were enriched between pathways (defined as completeness d₁₂, which represents the proportion of all potential synthetic lethal interactions that have been observed; see Equation (1)). According to the assumptions in Figure 1A, if all possible pairs of genes between the two compensatory pathways are connected by synthetic lethal interactions, they have a completeness of 100%. In practice, two compensatory pathways may have a completeness less than 100%, due to either lack of experiments or biological reasons. As can be seen from Figure 3 bottom panel, the pathway pairs we identified had a typical completeness of between-pathway synthetic lethal interactions at the range of [0.75, 0.85], which may have indication to the false-negative rate of the synthetic lethal interaction data (Text S2). Noticeably, some pathway pairs we identified had completeness d₁₂ = 100%.

We also determined the statistical significance of the pathway pairs we identified. To achieve this goal, we generated 10,000 random networks by crossing pairs of edges as done in Kelley and Ideker [1] and Milo et al [19]. For each pathway pair we identified (with completeness d₁, d₂ and d₁₂; see Equation (1)), we counted the chances of observing this pathway pair to have d₁⁰≤d₁, d₂⁰≤d₂ and d₁₂⁰≥d₁₂, where the superscript 0 means the score was under permutation. As it turned out, all pathway pairs we identified had P-value less than 1e-4. This result suggests that the constraint β = 0.75 of Equation (1) was rather strict. In fact, we found from our permutation study that given d₁⁰≤0.01and d₂⁰≤0.01, d₁₂⁰ had a mean of 0.09 and a standard deviation of 0.06 (Figure S2 and Text S2), which further suggests that our threshold β = 0.75 of Equation (1) was very strict.

Physical interactions in the discovered pathways

Biologically, we expected the enrichment of physical interactions within pathways we identified, which was an important component in Kelley and Ideker [1] and Ulitsky and Shamir [8]. To study the enrichment of physical interactions within the pathways identified by our method, we downloaded the physical interaction data collected by Ulitsky and Shamir [8], which included 67,856 physical interaction pairs. For each pathway pair identified, we calculated the completeness scores d₁, d₂, d₁₂ as defined in Equation (1) using the physical interaction data to measure the enrichment of physical interactions within pathway 1, within pathway 2, and between pathway 1 and pathway 2, respectively. Similar to the idea of Kelly and Ideker [1], we expected completeness scores d₁ and d₂ to be large. On the other hand, we hypothesized that physical interactions will be less enriched between compensatory pathway pairs, i.e., d₁₂ to be relatively small, according to Figure 1A. The distribution of completeness was shown in Figure 4. As can be seen, there were significantly (P-value<10⁻¹⁶, Wilcox rank sum test, one-sided) more within-pathway physical interactions than between-pathway physical interactions, suggesting that our identified pathway pairs do have biological meanings.

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Figure 4. Physical interactions are enriched in pathways we identified and depleted from between-pathway gene pairs.

https://doi.org/10.1371/journal.pone.0001922.g004

Functional homogeneity of discovered pathways

We next sought to study how different cellular functions were enriched in the pathways we identified. The functional annotation from Gene Ontology [12] was used for the evaluation. For each pathway we identified, a hypergeometric distribution was used to test the enrichment of genes annotated with a specific Gene Ontology concept. Since there were 1720 Gene Ontology biological process concepts being screened (see Methods), the −log10 of the smallest P-value of all the 1720 P-values (not corrected for multiple testing) was defined as the functional homogeneity score of our pathways. Thus, a higher homogeneity score implied a better grouping of genes into pathways. To provide a comparison, we also ran the same enrichment test on the pathways (redundancy removed, see Methods) identified by Kelley and Ideker [1] and Ulitsky and Shamir [8]. As expected, the pathways identified by our method (Figure 5, upper panel) had lower functional homogeneity than those identified with the aid of physical interaction data (Figure 5, bottom panel, stars and circles).

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Figure 5. The distribution of the number of identified pathways as a function of the functional homogeneity.

Upper panel: the functional homogeneity of all the pathways from each pathway pair identified by our method. Although a considerable percentage of pathways identified by our method had high homogeneity scores, many of them showed a low functional homogeneity due to multifunction effects. Bottom panel: at least one of the two pathways of each pathway pair identified by our method had high homogeneity. Note that the functional homogeneity score was still significantly lower than that of the pathways identified using physical interaction data.

https://doi.org/10.1371/journal.pone.0001922.g005

However, an interesting observation was that in each pathway pair identified by our method, at least one of the two pathways had high functional homogeneity. To show this fact, we took the pathway with a smaller P-value from each pathway pair identified by our method and drew the distribution of their functional homogeneity (redundancy removed, see Methods). As shown in Figure 5 bottom panel (boxes), these pathways showed enhanced functional homogeneity compared to Figure 5 upper panel (although still lower than that identified using physical interaction support; Figure 5, bottom panel, stars and circles). This phenomenon was actually a result of the multifunction effects of the pathways. To explain, suppose a pathway A has compensatory effects with pathways B and C. Then with a high probability our method will group genes from pathway A into one set and genes from pathways B and C into another set, resulting in one genuine pathway corresponding to A with a high functional homogeneity score and another spurious pathway corresponding to B and C, with a low functional homogeneity score. Thus, at the cost of reducing the accuracy of one of the two pathways in each identified pathway pair, it was still possible to gain much insight of the gene functions using synthetic lethal interaction data alone.

Network of compensatory biological functions

The above result showed that our method can group genes into pathway pairs, and in most cases at least one pathway from each pathway pair had specific biological functions. Since synthetic lethal interaction data often predict functionally compensatory pathways [1], [7]–[8], we next determined how different cellular functions compensated each other. To make the analysis strict, we selected the pathways pairs where each member pathway had a functional homogeneity score larger than 5.24 (or P-value 5.8e-6, which corresponds to the bonferroni-corrected P-value 0.01) and more than 30% of the member genes in each pathway had the same function as the one annotated to the pathway. This requirement resulted in 89 non-redundant pairs of Gene Ontology concepts connected by extensive synthetic lethal interactions (Table S1). As shown in Figure 6, the Gene Ontology concepts could be arranged in a network. For example, the genes involved in Golgi to plasma membrane protein transport were found to be synthetic lethal with the genes involved in intra-Golgi vesicle-mediated transport; retrograde transport, endosome to Golgi; and vesicle-mediated transport.

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Figure 6. Network of compensatory biological functions revealed by synthetic lethal interaction data.

Each node represents a Gene Ontology concept. Each edge represents the existence of extensive synthetic lethal interactions between gene-pairs from the two connected Gene Ontology concept pairs.

https://doi.org/10.1371/journal.pone.0001922.g006

Interestingly, the obtained network showed considerable un-evenness toward several Gene Ontology concepts, such as tubulin folding and histone exchange. For example, the histone exchange was found to be connected to many other functions, such as transcription, DNA dependent; negative regulation of meiosis; chromatin silencing; tubulin folding; etc. Another hub node was the positive regulation of RNA elongation, which was also connected to many functions, mostly involving DNA repair biology. The underlying biological implications await additional exploration.

Multifunction effects of pathways

Since we deliberately excluded the physical interaction data when searching for compensatory pathway pairs (see Methods), it was not surprising that sometimes we discovered pathways containing members that obviously deviated from the most common function of these pathways (Figure 2). At least three possibilities can explain this phenomenon. First, false positives might exist in the synthetic lethal interaction data set. Thus, an obviously unrelated gene may be incorporated into one pathway. However, the current synthetic lethal interaction data is believed to contain only few false positives [5]. Thus this explanation is unlikely true. Second, the “strange member” might be a new member of that cellular function. In other words, the Gene Ontology data contains false positives and false negatives. The existence of physical interactions between this new gene and other genes known to belong to that function will increase the probability of this explanation. Third, it might be simply due to multifunction (Figure 1C). In this case, the multiple functions contained in one identified pathway may suggest that its collaborator pathway participates in more than one cellular function.

The third consideration was identified in this analysis. In Figure 2E, the prefoldin complex (GIM3, GIM4, GIM5, YKE2, PAC10) was found to be compensatory to the spindle checkpoint complex (BUB1, BUB2, BUB3, MAD1, MAD2, BMH1). However, the VID21 gene was found to be grouped with prefoldin complex, which had synthetic lethal interaction with spindle checkpoint complex except BMH1. VID21 is a component of the NuA4 histone acetyltransferase complex. Thus, this fact suggested that the spindle checkpoint complex may also participate in a function in collaboration with histone modification. In this sense, we determined that the spindle check point was a multifunction pathway.

The multifunction effect of pathways is better illustrated using a network representation in Figure 6. The fact that a pathway with a given function may be connected to many other pathways, each with a distinct function, suggests the multifunction effects of a pathway. In this sense, the “hub” pathways such as those involved in positive regulation of RNA elongation and histone exchange are good examples of pathways with multifunction effects.

Data

We downloaded the synthetic lethal interaction data from BIOGRID ([6]; version 2.0.31). It contains 9376 non-redundant synthetic lethal genetic interactions, involving 2348 yeast genes. Note that in this paper we focused on the synthetic lethal genetic interactions. Other genetic interactions, such as 7,233 synthetic growth defect and synthetic rescue interactions were not included.

The 68172 protein-protein and protein-DNA interaction data covering 6814 yeast genes was obtained from Ulitsky and Shamir [8]. The functional annotation data of yeast genes is downloaded from SGD [24], as of August 2007. We parsed the Gene Ontology [12] data structure and mapped yeast genes to all ontology nodes and the resulting 1720 biological process nodes, each with no more than 500 genes, are selected to perform functional analysis.

Network visualization

Network figures were created using Cytoscape [25].

Algorithm

By viewing genes as nodes and synthetic lethal interactions between genes as edges, the synthetic lethal interaction data can be represented as a network. Our goal was to identify approximately complete bipartite graphs within the synthetic lethal interaction network which satisfied the following criteria: 1) there were no or few edges within each sub-network; 2) there was an abundance of edges connecting the sub-network pairs; 3) the sub-networks contained at least four genes. Mathematically, we denote the original network as G(V, E), where V is the set of yeast genes and E is the set of edges connecting yeast genes: each element e_ij in E takes value 1 if there is synthetic lethal interaction between genes i and j and 0 otherwise. Our goal was to find all node set pairs (V1, V2) that satisfied(1)where I(•) is the identity function, |•| is a function on sets, counting the number of elements in it. α and β are any pre-chosen numbers so that α is close to 0 and β is close to 1. Here we set α = 0.01, β = 0.75 (Text S3 and Text S2).

Another consideration was the balance between the sizes of node set pair (V1, V2). Due to the formidability of the searching space, we wanted to first focus on the sub-network pairs with similar sizes. Taking the above issues into consideration, we use the following objective function to search for candidate node set pairs:(2)where δ is a tuning parameter to control the penalty to the size differences between node sets V1 and V2 and abs(•) is the absolute value. A larger δ gives more penalties to the size differences between set V1 and V2. Here we set δ = 1.5 (Figure S3, Figure S4 and Text S4).

To implement the above objective, we proposed the following heuristic algorithm.

1. Input: (1) G(V, E), the network;
   (2) α = 0.01, β = 0.75, T = 0.4, parameters in Equation (2).
2. Output: R = (G₁, G₂, …, G_m), superset of m sub-networks G_i(V1,V2,E′) to be discovered and returned.

1: R = Φ. C = 0.

2: repeat

3:  Set V1 = Φ, V2 = Φ. S(V1, V2, E′) is the sub-network pair we will discover.

4:  Randomly select an edge e_ij = 1 from E. Set V1 = {i}, V2 = {j}.

5:  while there are nodes that can be added to S do

6:   Find the set of all nodes in G that are connected to S: V′ = {v | v∈V and e_vk = 1, for some k∈V1∪V2}

7:   if V′ is empty then

8:    break

9:   for all v∈V′ do

10:    Calculate score(v) = max(score(V1∪v, V2), score(V1, V2∪v))

11:    set score(v) = 0 if score(v)<T

12:   end for

13:   if score(v) = 0 for all v∈V′

    break

14:   sample a node v randomly from V′ according to vector score(v), v∈V′

15:   add v to V1 if score(V1∪v, V2)> = score(V1, V2∪v); to V2 otherwise

16:  end while

17:  add S into R if |V1|>3, |V2|>3, and S∉R

18:  C = C+1

19: until C>C^*

Our algorithm randomly chose an edge from the whole set of edges and initialized the sub-network pair V1 and V2. It then enumerated all other nodes and assigned them score according to Equation (2). We then sampled a node from these candidates according to the probability proportional to their scores. Note that a candidate with a score less than our threshold (T = 0.4) was not sampled. The sampled node was added into the sub-network according to its score. A pre-defined number C* ( = 10000 in this work) of searches are performed.

Redundancy

Similar to Kelley and Ideker [1], repeat pathway pairs were removed. Specifically, if pathway pair A and pathway pair B shared more than 50% synthetic lethal interaction edges, the smaller pathway pair was removed. For summary statistics and function enrichment analysis of the identified pathways, if pathway A and B shared more than 50% member genes, the smaller pathway was removed. The same procedure applies to pathways from Kelley and Ideker [1] and Ulitsky and Shamir [8].