Disease and gene-disease association databases

Curated disease and disease-gene association data were retrieved (in August 2013) from the Comparative Toxicogenomics Database (CTD) [17], North Carolina State University, Raleigh, NC and Mount Desert Island Biological Laboratory, Salisbury Cove, Maine (URL: http://ctdbase.org/). The CTD disease database merges the hierarchical MESH (Medical Subject Headings) [1] and the flat OMIM (Online Mendelian Inheritance in Man) [18] databases, where OMIM diseases are either merged to the most appropriate MESH terms or are added as children of MESH diseases [19]. The gene-disease associations reported in the CTD database are either based on direct evidences or are inferred. To reduce the uncertainty in the gene-disease associations, we ignored the inferred associations in this study. Also, only the most specific human diseases (the ones with no children) were included in the network.

Protein-protein interaction database

To uncover how gene groups associated with different diseases are related to one another in the context of protein-protein interactions, a protein-protein interaction database is needed. We used ppiTrim [20] to create such a database. By processing iRefindex [21], which incorporates entries from all major protein interaction databases, ppiTrim can produce a protein-protein interaction database in a consistent way and without redundancies [20]. All required input files for ppiTrim were downloaded on June 6 2013, and the program was run on the same day to produce the protein-protein interaction network used in this paper.

Disease-protein network

The disease-protein network was created by combining the CTD gene-disease association database and the protein-protein interaction network produced by ppiTrim. Naturally, only diseases associated with proteins in the ppiTrim-produced database were included in the network. An undirected graph, consisting of 16973 nodes (2548 diseases and 14425 proteins) and 214337 edges, was created by connecting the included diseases to their associated proteins, each of which is a node in the binary interaction network produced by ppiTrim. It was found that for fourteen diseases the associated proteins were disconnected from the rest of the network. These diseases were excluded in the subsequent analysis of the results (leaving 2534 diseases) because the network cannot provide more information about them.

Information flow and disease-disease correlations

Modeling information flow by a random walk with damping, ITMProbe [14], [15] is useful in studying information flow in protein networks. Under this method, the random walk starts from one or more source nodes and either dissipates or ends at sink nodes. Source and sink nodes are also called boundary nodes, while other nodes that are neither sources nor sinks are called transient nodes. The ITMProbe program outputs the expected number of visits to each transient and sink node by random walkers originated from every source node. In this study ITMProbe was applied to the disease network described in the previous section with all diseases specified as both sources and sinks, all proteins specified as transient nodes, and with a damping factor of 0.85 (for a discussion on the effects of changing the damping factor and also the rational behind using the value 0.85 please see [15]). If we consider the flow of information starting from and ending at a given disease, we can assign a weight (proportional to the expected number of visits) to each protein (transient) node. In other words, for each disease , there is a corresponding vector of weights whose dimension equals the number of proteins in the network. Without loss of generality, we always normalize to have unit length .

The correlation between two diseases and is defined by(1)The last equality results from . For two disconnected diseases this quantity would vanish, whereas for two diseases with the same connections to the network (diseases associated with the same set of proteins) the correlation would be unity. Disconnected diseases were not included in the analyses and so disease correlations would be positive.

For later convenience, let us also define the average correlation between disease and the rest of the diseases(2)where is the number of diseases under consideration. We may also define the average pairwise disease correlation by(3)In this study, we often sort pairwise correlations and bin them. Within such a bin, the average of variable is generally denoted by , where the subscript stands for bin-averaged, with(4)where is an indicator function taking the value if is inside the bin of interest and otherwise. When there is only one bin, for all and as expected.

Enrichment analysis

In the previous section, the construction of the weight vector associated with a given disease via ITMProbe was described. Evidently, when two diseases have similar weight vectors, they are related from the perspective of protein interaction network. To provide a biological explanation of each weight vector obtained, one would need an enrichment analysis: i.e. one should find the biological terms that best describe the weight vector from an annotated term database such as GO [22] or KEGG [23]. In this study, the enrichment analysis was done using Fisher's exact test, because this is among the most used approaches.

Although there are different implementations of Fisher's exact test, we chose to use Saddlesum [24] because it is an in-house tool and it has already integrated the two term databases (GO and KEGG) of interest for the analyses. For each weight vector, the cutoff (minimum weight included in the analysis) for Fisher's exact test was chosen to be 0.001 times the mean of the ten largest weights. This variable cutoff was chosen to, in the case of more uniform vectors, prevent including too many genes in the analysis that could result in false term associations. Terms with E-values less than 0.01 were considered significant. For each disease, we used this approach to find the corresponding GO (only terms in the “biological processes” family) and KEGG terms, providing yet another way to compare different diseases.

Clustering

To elucidate the relationships among diseases, a two-stage clustering procedure was employed. At the first stage, the diseases, each associated with a weight vector, were clustered probabilistically based on their vectors' correlations with cluster centers' vectors. An enrichment was then done for each of the final cluster centers. After the first stage, however, many cluster centers were found to be associated with similar, or even identical, sets of GO/KEGG terms. In terms of biological inferences, it was therefore necessary to do the second stage of clustering to group cluster centers based on the similarity of biological terms associated with them.

Although many algorithms are already available for clustering vectors, they usually result in non-overlapping clusters. Many diseases, however, belong to more than one family. Therefore, it is important to have a clustering method that produces overlapping disease classes. Thus, we introduced a probabilistic algorithm in which each disease was iteratively assigned a probability for belonging to a particular cluster. Each cluster was characterized by its center, a vector containing a set of weights for all proteins in the network. For a disease, the probability of being in a cluster was defined to be proportional to the cosine similarity between the weight vectors associated with the disease and the cluster center. The center vectors were initially chosen to coincide with the diseases' weight vectors. In other words, at first the number of clusters was the same as that of the diseases. The initial probabilities were computed using these initial centers. In each iteration the new positions (vectors) of the cluster centers were calculated using(5)where denotes the probability of disease to be in cluster , is the vector associated to the disease provided by ITMProbe, and the vector representing cluster is always normalized to have unit length. The probabilities were then recomputed and used in the next iteration. Since, in this approach, the cluster centers are basically the average of the disease weights, after each iteration they would move closer to each other. For this reason before each iteration the cosine similarities between cluster centers were calculated and the centers that were closer than a cutoff were combined. Specifically, if two centers and , containing and diseases respectively, had the highest cosine similarity and if their similarity was more than the cutoff, the two centers were merged to arrive at a new center vector that carries with it diseases. This procedure was continued until there was no pair of centers with a similarity bigger than the cutoff, which was set to where is the standard deviation of the diseases' pairwise correlations (see the Result section). The rational behind this choice is that our model cannot distinguish between two vectors if they are closer than this cutoff. It is worth noting that this process of elimination was performed even before the first iteration, because there were a number of diseases (initial cluster centers) whose correlations was more than the cutoff.

If this iterative method were continued long enough all cluster centers would be combined and there would be only one cluster. The goal of clustering was to group only highly correlated diseases, and so the iterations had to be stopped at an appropriate point. Two observations help us to find this point. First, it is desirable to express the diseases in terms of the lowest number of parameters possible, meaning a lower number of clusters. Second, a disease should be associated with fewest number of clusters possible. This is especially true for diseases that have low correlations with all other diseases (see the Results section). Such a disease should mainly belong to only one cluster (consisting only of that disease). Combining these competing observations, we stopped the iterations when the following quantity was minimized:(6)where denotes averaging over all diseases, is the number of clusters, , and (see Eq. (2)) is the average correlation between disease and all other diseases. For disease , is the participation ratio [14] and varies between (when the disease belongs to only one cluster) and (when the disease is associated with all clusters with the same probability). Therefore, a large means the disease is mainly associated with only a few clusters. In the denominator of eq. (6), the contribution of diseases that are far from all others is larger due to the presence of . This would make it unlikely for these diseases to be in more than one cluster when the iteration is stopped. The quantity went through a minimum after 10 steps (See Fig. S1 in the first section of Supporting Information S1) at which point the iteration was stopped and the final probabilities were calculated. This procedure resulted in 1707 clusters.

Enrichment analysis was then performed for the cluster centers with the same parameters as before, except that the maximum number of genes involved was limited to where is the average number of genes included in the enrichment analysis per disease and std denotes standard deviation. This limit was imposed because averaging weight vectors results in a vector of more uniformly distributed weights and so the number of included genes could reach much higher values than before. The E-values corresponding to the terms found by the enrichment analysis could be used to create a new vector for each disease or cluster center with term association. For this purpose the union of all significant terms associated with all centers was determined. For cluster the th component of was then defined as if and zero otherwise. Here is the E-value corresponding to the th term when querying SaddleSum using . The vectors were then used for the second stage clustering.

In the second stage, the clusters obtained in the first stage were separated into two groups depending on whether or not they had been associated with biological terms. The first (with term associations) and second group had and members respectively. For the first group, the set of vectors were then defined and were clustered using a distance-based hierarchical approach. The distance between and was defined as , and the cutoff was chosen to be . To avoid over-clustering, we used a more stringent similarity measure here than the one used before for comparing the terms assigned to different diseases. For disease the probability of being in the th new cluster, containing clusters from the first group, was defined as where is the probability of disease belonging to the th cluster in the first group after the first stage of clustering.

We also used Cfinder [25], [26], a program for clustering nodes of a graph, to create overlapping clusters of diseases. The advantage of Cfinde over similar algorithms is that, like the clustering method explained here, it produces overlapping clusters. To use Cfinder, we first created a disease network by connecting each pair of diseases with an edge weighted by their correlation. Using this method, we obtained similar results to those from our clustering algorithm described above. For a detailed description of the procedure and the results please see Supporting Information S1.

Evaluating the accuracy of p-values

The accuracy of p-values reported (by Saddlesum) for the original (not averaged) weight vectors has already been evaluated by Stojmirovic and Yu [24]. The cluster centers, however, are weighted averages of all weight vectors. To investigate whether or not averaging affects the accuracy of the p-values (and consequently the E-values) reported by the enrichment analysis, we took a similar approach to that of Stojmirovic and Yu [24] to calculate the “empirical” p-values and to compare them to the reported ones. Briefly, the gene list was shuffled 672 times and, for each gene list, enrichment analysis was performed for all cluster centers and the reported p-values were recorded. This number (672) was chosen to have approximately weight-term matches, i.e. , where is the number of clusters, is the total number of GO/KEGG terms, and is the number of randomized gene lists. The empirical p-value corresponding to the cutoff value was then defined as , where is the total number of reported p-values (for all cluster centers and all gene lists) that are smaller than or equal to . The results, given in Fig. S2 and section 2 of Supporting Information S1, showed that the reported p-values were indeed accurate.

Statistics of disease-disease correlations

Based on the proteins that the two diseases were connected to, disease pairs were classified into three categories. If both diseases in a pair were connected to the same set of proteins, the pair was assigned to category (1). Members of this category, by definition, had the largest correlation possible (unity) and were equivalent in our study. The number of “independent” (not equivalent to any other) diseases was 1962. If, on the other hand, the two diseases shared some (but not all) associated proteins, the pair was classified in category (2). Category (3) consisted of the rest of the disease pairs (the ones with no shared connections to the protein network). Category (1), (2), and (3) had 978, 5243, and 3203090 pairs respectively.

Using eq. (1), the (total number of pairs from all three categories) pairwise correlations were calculated. The median, mean and standard deviation of were , , and 0.02 respectively. These statistics indicate that although the correlations were generally very small, there was a large number of outliers (disease pairs with high correlations compared to the mean). Obviously, disease pairs in category (1) had the largest correlation possible (). The members of the second category had overall larger correlations (with a mean value of 0.13, a standard deviation of 0.23, and a median of 0.03). However, some pairs in this class had low correlations (with a minimum of ). In fact 179 disease pairs in the third category (pairs with no common gene associations) had higher correlations than the median correlation of pairs in category (2), indicating that having some shared network connections does not necessarily translate to high correlations, although 95.9% of disease pairs in the second category had larger than average correlations. The maximum correlation of the third category, containing 99.8% of the disease pairs, was 0.237.

Interestingly, the average correlation between a disease and all others varies dramatically. In fact the lowest average correlation () was more than five orders of magnitude smaller than the largest, suggesting that some diseases have relatively low correlations with all other diseases. For example, 286 diseases have correlations less than with all other diseases.

Interpretations of correlations

To investigate what high correlation between two diseases could imply, a number of quantities were calculated. First, we calculated, for disease pairs with correlations in a certain interval, the probability of being siblings (having the same parents in the CTD/MESH database). To achieve this, the disease pairs were sorted according their correlations and were divided into bins. The probability (see eq. (4)) of finding a sibling pair in each bin was then defined as the ratio of the number of sibling pairs to the total number of disease pairs in that bin. The number of disease pairs in each bin was 1000.

Second, for each disease pair, we calculated the similarity between the associated enriched terms, which was defined as the ratio of the number of significant terms shared between the two diseases to the total number of identified significant terms. In other words, for two diseases associated with exactly the same GO and KEGG terms would be unity, whereas for diseases with no common terms it would vanish. If one or both diseases in a pair did not have any term associated with them, was not defined. To find the distribution of disease pairs with , the pairs were partitioned into bins as described in the previous paragraph. In each bin, the average probability for two diseases in a pair to hit the same GO/KEGG terms was then defined as (see eq. (4)).

As expected, the enrichment analysis did not find any significant biological terms associated with some of the diseases, implying that, for some disease pairs, calculation of was not possible. In this study Saddlesum was able to find significant GO/KEGG terms for about 60% of the diseases (1530 out of 2534). For diseases with significant term hits, the average numbers of identified GO and KEGG terms were 34.7 and 5.2, and the standard deviations were 52.2 and 8.2. The large spread of the number of hits was due to the fact that some diseases, with a large number () of connections to the network, had hundreds of GO and tens of KEGG terms associated with them. Overall 3182/203 unique GO/KEGG terms were hit by the enrichment analysis.

Interestingly, there was a significant difference between the percentage of pairs with undefined when disease pairs with low and high correlations are considered. For example, was undefined for 23% of disease pairs with correlations greater than , as opposed to 64% for pairs with correlations smaller than . This can be understood through the fact that the percentage () of diseases that had been successfully assigned one or more GO/KEGG terms by Saddlesum was smaller for diseases with very low average correlations. This behavior is shown in Fig. 1. After sorting into ascending order and placing them in bins each containing diseases, we computed the average in a bin and, in the same bin, the number of diseases that had one or more GO/KEGG term hits. In Fig. 1, is plotted versus the average per bin and the aforementioned behavior is clearly displayed.

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Figure 1. The relation between the results of enrichment analysis and the average correlation .

The percentage of diseases for which GO/KEGG terms were identified by Saddlesum as a function of average correlation . To facilitate the calculation, we sorted all s in ascending order and placed them into bins each containing diseases. The percentage is then measured by the number of diseases with GO/KEGG term hit(s) per bin. For very low average correlations is significantly lower.

https://doi.org/10.1371/journal.pone.0110936.g001

Since was undefined for a large number of disease pairs, the pairs were divided into two sets: with defined (first set) and undefined (second set) term similarities. For the first set (with defined ), Fig. 2 (A) illustrates the behavior of (in green), (in blue), and (in red), where () is for pairs with (). The figure clearly shows, when , a rise in the probability of a disease pair to have common biological associations as correlation increases. The figure also indicates, when , that disease pairs with higher correlations are more likely to be siblings if they have . However, the siblings without shared terms have almost a flat (correlation-independent) distribution, although the percentage of such pairs is very small (about 0.5%). One possible explanation for these results is that the increase in the percentage of siblings in highly-correlated diseases is in fact due to an increase in the percentage of the pairs with . In other words, in high correlation regime, most of the siblings are a subset of disease pairs with shared GO/KEGG terms. Figure 2 (B) shows how varies with correlation for the second set of disease pairs (the ones with undefined term similarities).

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Figure 2. The probabilities of having common term associations or being siblings.

(A) The probabilities of finding a pair of diseases with (1) common GO/KEGG terms (red), (2) the same parents and common associations (blue), and (3) the same parents without shared biological terms (green) are shown. Here only pairs with a defined term similarity are considered. (B) For pairs with undefined (pairs with at least one member not associated with any biological terms), the distribution of siblings is plotted as a function of correlation. (C) and (D) show similar quantities to (A) and (B) respectively, when the biological term associations are directly retrieved from the KEGG DISEASE database.

https://doi.org/10.1371/journal.pone.0110936.g002

Using only the terms from the manually curated disease-pathway associations [23] in the KEGG DISEASE database (downloaded on December, 12 2013) in place of the terms retrieved from enrichment analyses, term similarities () and the probabilities , , , and were recalculated. In this KEGG database, one or more OMIM diseases are associated with one or more KEGG pathways. The CTD disease database [17] was used to find the equivalent MESH diseases that were used in our study. The results are shown in Figs. 2 (C) and (D). Overall, the trends observed in these figures are very similar to those shown in 2 (A) and (B): i.e. increase in / with correlation (for ) and correlation-independence in for pairs with defined , and a uniform and then increasing for pairs with undefined term similarities.

We have also computed the degree of overlap between the enrichment analysis and KEGG DISEASE database. For each disease , out of the total number of annotated KEGG pathway associations we calculated the number of associations that were also reported significant by enrichment analysis. The ratio , for disease , measures the agreement between the curated pathway assignment and the enrichment analysis. There were 490 diseases that had been annotated in the KEGG DISEASE database and also had term hits using the enrichment analysis. The average value of for these diseases was found to be , indicating on average 48% of the annotated terms for each disease in the KEGG DISEASE database were also found using our enrichment analysis.

Comparison with MimMiner and human disease network

MimMiner [13] uses a text mining approach to calculate a pairwise disease similarity score that, like correlation defined here, ranges from 0 to 1. However, since these two measures have very different distributions and are defined based on different concepts, they cannot be directly compared. For this reason we adopted the procedure described in the third section and Fig. S3 of Supporting Information S1 to find equivalent cutoff correlations and MimMiner scores, and respectively, and to compare the two methods.

In the absence of a true gold standard, we used disease-pathway associations reported by KEEG DISEASE databse [23] to compare the retrieval agreement with KEGG DISEASE from MimMiner and that from our model. This was done by comparing the effectiveness of the two methods to identify disease pairs that are associated with the same biological pathways as annotated in KEGG DISEASE. To make a fair comparison, only disease pairs with defined term similarities (see the “Interpretations of correlations” section) and with available MimMiner scores were included in this analysis (336610 pairs, about 10% of all pairs). By ranking the disease pairs based on either their correlations or their MimMiner scores, two lists of pairs were created. For each list, the weighted number of disease pairs with common associated biological terms that were among the first (highest ranking) pairs was calculated as , where is the term similarity between diseases of pair using KEGG DISEASE as the standard. The function provides a measure for comparing the two methods: a faster rise in would mean a larger number of pairs with high term similarity have been ranked higher than the others, indicating a better agreement with KEGG DISEASE.

The results (MimMiner in red, our method in blue) are shown in the inset of Fig. 3 (A). The green curve shows the weighted number of disease pairs identified (ranked higher than ) by our method, but missed (ranked lower) by MimMiner. Similar trends are observed for both methods, but a better performance (faster rise in ) for MimMiner is indicated. This finding is expected, because MimMiner is based on mining the literature, which is also the source of the manually curated data in the KEGG DISEASE database. However, an important observation is that the two methods do not find the same pairs, especially in terms of less apparent relationships. To see this feature, we first excluded the disease pairs that were obvious candidates for being related, i.e. sibling diseases and pairs with common gene associations (3847 pairs were excluded leaving 332763). We then recomputed the blue and the green curves, shown in Fig. 3 (A). The closeness between these two curves indicates that for non-apparent relationships, the disease pairs identified by our method are largely missed by MimMiner. In Fig. 3 (A), about 87% of pairs ranked higher than (equivalent to a correlation of and a MimMiner score of 0.41) by the method presented here were missed by MimMiner.

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Figure 3. Comparison with MimMiner.

(A) The inset figure shows the number () of weighted disease pairs with shared KEGG pathways that were ranked higher than by MimMiner (in red) and or by our method (in blue). Also shown in the inset (in green) is the weighted number of pairs with common term associations missed (ranked lower) by MimMiner, but identified (ranked higher) by our model. In the main panel, the same quantities corresponding to the proposed method are plotted after exclusion of obvious candidates for being related. The closeness between the blue and green curves indicates that the non-apparent candidates found by our method are largely missed by MimMiner. Displayed in panel (B) is the inverse of average normalized rank versus the term similarity cutoff. At large similarity cutoff, the higher the average normalized rank (the smaller and thus the larger ) the better the agreement between the quality scores (cosine similarity or the MimMiner score) and the KEGG annotation.

https://doi.org/10.1371/journal.pone.0110936.g003

Given the fact that our method and MimMiner effectively find different pairs, one may wish to look at the quality of retrieval using a different measure other than . In Supporting Information S1, we have described how to find the cutoff cosine similarity and MimMiner score, above which there exists an apparent positive correlation between the similarity/score and the pair relatedness. Denote the number of disease pairs with similarity/score above the cutoff by /. We defined the normalized rank as rank divided by or by depending on whether cosine similarity or MimMiner score was in use. For a given cutoff term similarity , we first found among or disease pairs with term similarity larger than , and then computed their average normalized rank when all pairs were ranked either by correlation (cosine similarity) or by MimMiner score. Evidently, for large cutoff , a larger indicates a better retrieval fidelity. Using this measure, results shown in Fig. 3 (B), our methods seems to provide higher retrieval fidelity. One should bear in mind, however, comparisons of such should always be taken with a grain of salt due to the difficulty in constructing a gold standard and totally impartial datasets.

It is also worth noting that for a very large subset (90%) of disease pairs investigated in this study, KEGG term similarities or MimMiner scores were not defined. However, many of these disease pairs, or the ones with or with a MimMiner score of zero, may in fact be related. For example, 5090 (97%) of disease pairs in the second category and 838 (86%) of the members of the first category of pairs, which have common gene associations and are more likely to be related, were in this subset. Even diseases with no gene associations or the ones that have been classified in totally different families could share biological pathways or phenotypic similarity. On the one hand, many pairs with documented relationships may not have yet been annotated by KEGG DISEASE or scored by MimMiner, or may have been reported as being not related. Table 1 lists ten example pairs, with correlations much larger than , from all three categories of disease pairs. On the other hand, it is likely that some disease-disease relationships have not yet been discovered. One should keep in mind that the members of the majority of pairs with significant correlations in our study have no obvious relationships (do not share genes and are not siblings), and also their possible relationships have not yet been experimentally verified. From a practical point of view, however, these pairs are more interesting, because they suggest unknown and non-trivial relationships that, if verified, could add to our knowledge about the causes and possibly cures of certain diseases.

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Table 1. Examples of relationships between diseases that have undefined (or zero) term similarity and undefined (or zero) MimMiner score, and are from different disease families.

https://doi.org/10.1371/journal.pone.0110936.t001

From the perspective of finding “related” disease pairs with zero MimMiner score, Li and Patra [16] found non-apparent related pairs while combining a phenotype similarity network (created using MimMiner similarity scores), a gene-phenotype network and a protein interaction network. For out of the pairs, support information for relatedness was provided by Li and Patra [16]. The relatedness evidence for of the pairs is founded on that the member diseases are classified in the same disease class. Using our method, we have found disease pairs with zero MimMiner scores, each of which has its member diseases classified under the same disease family according to MESH. We have also found disease pairs with zero MimMiner scores, each of which has its member diseases share at least one biological pathway according to KEGG DISEASE database.

The proposed method was also compared to the Human Disease Network (HDN), introduced in the pioneering work of Goh et al. [3]. In this method the disease network is created by linking diseases that have common gene associations. The method proposed here, however, links the diseases based on their correlations, i.e. the diseases are linked if they have significant correlations (larger than the cutoff , as obtained in Supporting Information S1). Interestingly, in our study the minimum correlation between diseases with shared gene associations was . In other words, the links of a disease network created by the method of Goh et al. would be a small subset of those of our disease network. To find out if the additional disease-disease relations suggested by the proposed method are supported by the available experimental data, once again KEGG DISEASE database was used. We considered only diseases annotated by KEGG (1272 out of 2534 included diseases) and created three disease networks by linking the diseases using three different connectivity measures, i.e. having shared gene associations, high correlation, and having common pathway associations as annotated by KEGG DISEASE. The total number of links between the diseases in the three networks were 527, 14202, and 45577 respectively. The number of coinciding links between the KEGG-based and the correlation-based networks was 2988, as opposed to 389 when comparing KEGG and HDN networks. In other words, 2599 pairwise disease relations predicted by our method and missed by HDN are supported by the KEGG DISEASE database. Both methods however failed to predict the relationships between a large number of diseases that, according to KEGG DISEASE, have shared biological pathways. On the other hand there are many diseases that have high correlations, but are not reported by KEGG as having common pathway associations. As discussed before, these are not necessarily false positives. The KEGG database does not yet contain many literature-supported relationships (see Table 1 for some examples), but more importantly, there might be many disease relations that have not yet been discovered. An important aspect of the proposed method is that it suggests disease relationships that should be experimentally verified.

Effect of clustering

Based on the hypothesis that highly correlated diseases are more likely to have common pathways, one can use correlation-based clustering to increase the number of hits when searching for biological terms associated with the diseases. Assuming that all diseases in a given cluster share some pathways/processes, one can increase the chance of finding these pathways/processes by weighted averaging of the weight vectors assigned by ITMProbe to the diseases in the cluster. The rational behind this method is that each vector may be contaminated with “noise” and that the “signal” could be amplified by averaging. To accommodate the scenario that a disease might belong to several families, we used a probabilistic clustering method (see Methods) that allowed overlapping clusters and assigned a probability to each disease for being in a particular cluster.

Our iterative approach resulted in 1707 clusters. Enrichment analysis was run for all cluster centers obtained in this stage and found significant hits for 1301 clusters with an average of 70.9/7.5 GO/KEGG terms per cluster, which was higher than the average number of terms found for the diseases. The probabilities of belonging to different clusters were calculated for each disease and were used to determine the percentage of diseases with term hits, defined by(7)with being an indicator function taking value when cluster has a term hit and otherwise. Interestingly, the number of such diseases showed an increase from 60% (when enrichment was directly performed for the diseases) to 85%. For the diseases that had term hits using both methods (direct and through clustering) the term similarity , was calculated using(8)with being the number of diseases that have significant term hits, being the set of terms associated with the th disease, being the set of those assigned to the cluster , and denoting the number of members in the set . We found . This seems to indicate that more than of the terms associated with the diseases were dropped upon merging to clusters and some information might have been lost in the process. What is really important, however, is whether terms of small number of annotated genes are preserved, as these terms are most specific and usually most informative. Upon examining the distribution of minimum GO/KEGG term size (number of annotated genes for that term) when running SaddleSum using diseases directly and using cluster centers, we find that the most informative terms are largely kept in the process. The distribution of the minimum term size is shown in Fig. 4.

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Figure 4. The effect of clustering on the minimum term size.

The minimum term size distribution of (A) GO and (B) KEGG terms reported by SaddleSum enrichment analyses when using disease weight vectors directly (red curves) and when using cluster center vectors (blue curves). Not only the most informative (smallest size) terms are preserved during clustering, the clustering procedure seems to shift the minimum term size distribution towards the small end, indicating the likelihood of providing even more specific terms when weight vectors are grouped under the proposed clustering procedure.

https://doi.org/10.1371/journal.pone.0110936.g004

To illustrate how clustering through weight vectors may increase the likelihood of associating a disease with terms, we examine the late-onset Parkinson's disease (OMIM:168600). This disease was not associated with any terms when enrichment analysis was directly performed for the disease. After clustering, however, the top four clusters, ranked by their probabilities of including the Parkinson's disease, were associated with the Parkinson's disease pathway. Specifically, the term hits (with E-values smaller than 1e-4) for the cluster with the highest probability (13%) are listed in Table 2, which include Parkinson's disease, Alzheimer's disease and other neurological processes.

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Table 2. Terms associated with the cluster with the highest probability to include the Parkinson's disease.

https://doi.org/10.1371/journal.pone.0110936.t002

Retinitis Pigmentosa is an eye disease, with many different types, which is characterized by progressive retinal degeneration. As a second example, we examined the cluster and term associations for type 7 of this disease (MESH:C564284), which had no term hit before clustering. The disease was in multiple clusters (with relatively high probabilities 10%) that were associated with the phototransduction pathway. Given in Table 3 are the terms associated with the cluster with the highest probability (10%), which are related to phototransduction, detection of light and response to light. The phototransduction pathway, along with the Retinal metabolism (KEGG:hsa00830) and Spliceosome (hsa03040) pathways, has been indeed annotated to be related to this disease by the KEGG DISEASE database. Figure 5 visualizes the clusters that contain Parkinson's disease and Retinitis Pigmentosa 7.

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Figure 5. Two example clusters.

The clusters that include Parkinson's disease (OMIM:168600) and Retinitis pigmentosa 7 (MESH:C564284) are shown in panels (A) and (B) respectively. In each case, only diseases with membership probabilities larger than 5% are shown. The size of each node (circle) is proportional to the probability of membership of that node in the cluster. For a disease pair, the thickness of the line linking the diseases is proportional to , where is the correlation between the two diseases and is the minimum correlation between all diseases shown in each cluster. The names and IDs of the members of each cluster are also given. Diseases whose names are written in the same color (other than black) have exactly the same gene associations and so are equivalent in our study. Equivalent diseases are represented by one node in the figure. For example, the node identified by C566637 in panel (B) represents the four diseases whose names are in green, i.e. C535804, C566637, C565827, and C562479.

https://doi.org/10.1371/journal.pone.0110936.g005

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Table 3. Terms associated with the cluster with the highest probability to include the Retinitis Pigmentosa type 7.

https://doi.org/10.1371/journal.pone.0110936.t003

As a third example, the associations of the Knobloch syndrome (MESH:C537209) were also investigated. This is another eye disease that is characterized by different abnormalities, including cataracts, dislocated lenses, vitreoretinal degeneration, and retinal detachment [27]. Unlike the other two examples, this disease is primarily a member of one cluster with a probability of 62%, and the second highest probability was much smaller (3%). According to the KEGG DISEASE database, the pathways involved in this disease are focal adhesion (KEGG:hsa04510), ECM-receptor interaction (KEGG:hsa04512), and cell adhesion molecules (KEGG:hsa04514). Focal adhesion, and ECM-receptor interaction were in fact among the KEGG terms that were found to be associated with the cluster. Due to the relatively large number of associated terms with this cluster, the whole list is not given.

The first two examples indicate that there was a high degree of overlap between the GO/KEGG terms associated with different clusters. In other words, several clusters have phototransduction or Parkinson's disease pathways associated with them. For this reason a second round of clustering was performed, which was based on term similarity rather than disease correlations through weight vectors, and the probabilities of belonging to each new cluster was calculated. The second stage clustering reduced the number of clusters with term associations to 217, which substantially reduced the term overlap. For example, the highest new membership probability () for Parkinson's disease became 31%, and the new cluster, as expected, was associated with the Parkinson's disease pathway. Similarly, after the second stage clustering, Retinitis Pigmentosa type 7 was primarily in one cluster with the probability of 44% (which is associated with phototransduction) and all other probabilities were less than 15%. On other hand, for the Knobloch syndrome only increased modestly to 73%. The large reduction of the number of clusters is consistent with the view that many diseases share common modules or biological pathways.