Mining the Modular Structure of Protein Interaction Networks

Abstract

Background

Cluster-based descriptions of biological networks have received much attention in recent years fostered by accumulated evidence of the existence of meaningful correlations between topological network clusters and biological functional modules. Several well-performing clustering algorithms exist to infer topological network partitions. However, due to respective technical idiosyncrasies they might produce dissimilar modular decompositions of a given network. In this contribution, we aimed to analyze how alternative modular descriptions could condition the outcome of follow-up network biology analysis.

Methodology

We considered a human protein interaction network and two paradigmatic cluster recognition algorithms, namely: the Clauset-Newman-Moore and the infomap procedures. We analyzed to what extent both methodologies yielded different results in terms of granularity and biological congruency. In addition, taking into account Guimera’s cartographic role characterization of network nodes, we explored how the adoption of a given clustering methodology impinged on the ability to highlight relevant network meso-scale connectivity patterns.

Results

As a case study we considered a set of aging related proteins and showed that only the high-resolution modular description provided by infomap, could unveil statistically significant associations between them and inter/intra modular cartographic features. Besides reporting novel biological insights that could be gained from the discovered associations, our contribution warns against possible technical concerns that might affect the tools used to mine for interaction patterns in network biology studies. In particular our results suggested that sub-optimal partitions from the strict point of view of their modularity levels might still be worth being analyzed when meso-scale features were to be explored in connection with external source of biological knowledge.

Introduction

One of the major challenges of systems biology is the understanding of the cellular and molecular basis of high-level biological functionality and complex phenotypes. A promising approach to address these problems relies on the characterization of cellular functionality in terms of a global description of the interwoven set of biochemical reactions that take place inside the cell. This systemic approach has received a lot of attention in recent years, fostered by the ever growing availability of massive amounts of data generated at omic-scales.

In this context, the network metaphor has appeared as an appealing framework to organize and unveil global patterns of biological relevance from the deluge of available data. It provides a systematic description language based on pairwise relationships (i.e. network links or edges) between entities of interest (i.e. network nodes or vertices). This approach allowed to uncover the role of connectivity and interaction patterns in the emergence of biological functions [1,2], to assign new functionality to non-annotated gene products [3], to propose biomarkers for several pathologies [4] to gain insights into the genotype-phenotype relationship [5–7], and to establish meaningful associations between pathological phenotypes and disruptive perturbations involving particular regions of the underlying protein interaction networks [5,8–10].

The rationale of the network-based approach is that the analysis of topological features of biological networks can unveil relevant biology. In this context, one recurrent strategy consists on the identification of central vertices according to network-based centrality indices, with the hope that meaningful biological entities could be recognized. Following this line of research several seminal studies have suggested, for instance, that hub proteins in the S. cerevisiae physical interaction network were more likely to be essential than other proteins, giving rise to the so called centrality-lethality-rule [11–15].

Modular and cluster-based descriptions of biological networks have also received much attention in recent years [16]. In this respect, a lot of effort has been paid to take advantage of meaningful correlations established between topological network clusters (which are formed from nodes which are more densely connected with each other than with their neighborhood), and Hartwell’s original idea of “biological functional modules”, defined as a group of cellular molecular components and their interactions that carry out a specific biological function [17].

The analysis of the modular structure of molecular biological networks on its own has also drawn a lot of attention as it provides a broad and global description of interaction patterns to understand the complexity of biological systems. A particular insightful use of network’s modular descriptions to unveil their organization was introduced by Guimera and Amaral [18–20]. Taking advantage of the modular organization of the network, and once disjoint network communities were recognized, they proposed to classify network nodes according to their intra and inter-module connectivity patterns into seven different universal roles [18]. To that end, they introduced two observables: the intra-cluster connectivity, Z, and the participation coefficient, P of a node (see Methods for details). While the first parameter describes the degree of a node compared to the degree of nodes that belong to the same community, the second one quantifies to what extent a node connects to different modules. Using this methodology they were able to depict highly informative ‘cartographic representations’ of several metabolic networks. Furthermore, they showed that non-hub high-participation nodes, detected in E. coli metabolic network, tend to display unusually low evolutionary rates, suggesting that relevant biology could indeed be underpinned with the proposed methodology [19].

Being a module-base scheme, an appealing factor of Guimera’s analysis is that it does not rely neither on strictly local (i.e. features involving only properties of a node and its direct neighbors) nor global network features, but rather on connectivity patterns displayed at the meso-scale level. In fact, the very notion of network community is used in order to set the meaningful scale over which the connectivity analysis is performed.

Noticeably, the identification of network modules or communities is in fact a mathematically ill-posed problem, in the sense that there is no such a thing as an a-priori objective and hypothesis-free definition of how a good cluster should be defined. This results in the co-existence of many different community recognition procedures that might produce different network partitions (see [21] for an extensive review). Moreover, it renders the question of how these methodologies would perform in terms of their ability to unveil biologically significant patterns.

Different network community detection procedures make use of qualitatively different strategies. A wide-spread used family of community detection procedures are based on the optimization of a figure-of-merit known as network modularity [22] whereas other well performing algorithms rely on more information-theoretical considerations [23,24] For instance, according to the infomap methodology clusters are defined in order to minimize the average description length of a random walk process taking place over the graph [23].

Each clustering strategy presents its own technical caveats. For example, Fortunato & Barthelemy demonstrated that a theoretical resolution limit exists for modularity-based algorithms. This leads to the systematic merge of small clusters in larger modules, even when the clusters are well defined and loosely connected to each other [25]. Many contributions, mainly developed inside the physic community, further explored this effect, proposed alternative methodologies, and established comparative studies considering ad-hoc benchmark network models [26–31]. In particular, it is now rather well estalished that the modularity function is highly degenerate and that partitions with very different resolutions can have arbitrarily similar modularities [26,32,33]. The infomap procedure on the other hand was found to be not severely affected by this resolution limit effect when benchmark networks were considered [30].

Despite these developments, modularity maximization is still one of the most popular techniques for the detection of community structure in graphs. In particular, we found that in consonance with the modularity-based community detection procedure employed by Guimera in its original series of papers, many recent cluster-based analysis of different biological problems were tackled considering slight variations of the same kind of modularity maximization guided algorithms[34–36].

In this context, we aimed to present an analysis to put a word of caution about how the “idiosyncrasy” of the considered algorithms could impact on follow-up biological analysis of real protein interaction datasets. In order to better illustrate our point we concentrated our work on two paradigmatic network community detection procedures: the modularity-based Clauset-Newman-Moore (CNM) methodology [37], and the infomap algorithm [23], and focused on two important aspects of the problem. On one hand we explored the associations that could be established between biological functional modules and the identified network communities, analyzing the biological homogeneity of these network structures. On the other, taking into account the Guimera’s cartographic role characterization we explored network meso-scale connectivity features induced by the considered network modular descriptions. In connection with this point, we studied the ability of the cluster recognition algorithms to mine connectivity patterns of a protein set of interest in order to detect biologically sensible biases in PIN topological features. In particular, we considered aging related proteins as a case study and investigated whether this complex phenotype could be linked to specific intra/inter modular connectivity pattern.

Results

CNM and infomap mined the PIN modular structure at different resolution levels

We considered the set of protein interactions recapitulated in HIPPIE, an integrated protein interaction network with experiment based quality scores (see Material and Methods for further details). The modular organization of the PIN was explored considering the CNM and infomap procedures. Both methodologies resulted in network partitions displaying similar modularity levels (Q _infomap = 0.52 and Q _infomap = 0.54). These values were much higher than the ones obtained in an ensemble of 1000 randomly rewired versions of the PIN that preserved the original degree distribution (Q _infomap-rwn = 0.255±0.001, Q _CNM-rwn = 0.313±0.001) stressing the relevance of second and higher order correlations exhibited by in the real network in connection with the emergence of the observed modular structure.

Although the partitions found by both algorithms attained similar modularity values, large differences were observed in terms of the corresponding community size distributions. For instance, whereas there were no infomap communities exceeding four hundred nodes, the CNM partition included four communities with more than a thousand nodes each.

The number of internal links, l _int, of a given cluster was a relevant magnitude to understand qualitative features of the obtained partitions, and was used as a proxy of the cluster size (see S2a Fig). In order to visualize how the network nodes were distributed among clusters, we showed in Fig 1 the cumulative cluster-size distribution function, F _c-size, as a function of l _int values. For CNM structures, an abrupt change in F _c-size (l _int ) took place for a number of internal links of order l _int ~λ≡√L (where L is the number of edges of the network). This qualitative change in F _c-size suggested the existence of a dominant size scale in the obtained modular description, as 90% of the total number of network nodes was found inside the 8 largest detected CNM communities displaying l _int > λ (red filled circles in Fig 1).

Fig 1. Analysis of cluster partitions obtained with CNM (circles) and infomap (empty squares) methodologies.

The panel shows the cumulative cluster-size function as a function of l _int. Symbol sizes were set using a scale proportional to the log-size of the corresponding cluster. Reddish filled circles were used for the 8 largest CNM communities. The horizontal line corresponds to the 10% accumulated mass level. Dashed vertical lines delimit a region of values of the order square root of total number of links in the network, [√L/2, √2L], corresponding to the natural scale found to operate in modularity optimization procedures [25, 30].

On the contrary, for infomap clusters (empty squares in Fig 1), a smooth increase of F _c-size levels was observed. In this case the network mass could be split into network sub-structures which spanned a wide range of cluster sizes and did not present any recognizable natural size scale. The results obtained for the considered PIN agreed with Lancichinetti et al general observations [30]: differently from CNM, the infomap procedure provided a network modular description of multi-resolution character.

We also found that infomap clusters were virtually included inside CNM modules, as almost 90% of infomap internal links were also internal links in CNM clusters (86% of infomap intra-cluster node pairs were preserved in the alternative CNM partition), and only 66% of CNM internal links were preserved as internal infomap links (5% of the total CNM intra-cluster pair of nodes were preserved under the infomap description). As can be seen from S2b Fig, almost the totality (~99%) of broken CNM-internal links took place in the largest CNM detected structures, and only 1% in CNM clusters of internal-link density values lower than the √L/2 level.

Summing-up, all our findings were consistent with a scenario were infomap finer structures were merged into larger assemblies under the CNM description (graphical examples for this general tendency were reported in S3 Fig). Both partitions reported reconcilable descriptions of the PIN, but the community structure revealed by infomap provided a finer granularity level than the one achieved by the CNM procedure.

Network structures identified at high resolution levels presented higher biological congruency

We considered the biological homogeneity index, BHI, (see Methods) to investigate to what extent different network structures identified at different resolution levels correlated with external biological evidence. BHI values for the 8 CNM larger communities were depicted as red points in Fig 2 (CNM clusters were ordered according to decreasing size). Green triangles showed the BHI level of the infomap partition of clusters included in the respective CNM structure. For each CNM community, boxplots depicted distributions of BHI values estimated for an ensemble of 1000 random shuffling realizations of the corresponding infomap labels. The BHI levels of infomap partitions were systematically higher than the ones observed for the corresponding CNM ones (Fig 2), suggesting that the higher granularity level provided by the first algorithm resulted in a significant increase of the overall biological consistency of the detected structures. We could verify that the gain in functional coherence displayed by infomap did not come from cluster-size effects alone, as we found for all cases that more than 95% of the random label reassignments presented lower BHI levels than the value displayed by the original infomap partition. These findings supported the idea that infomap communities represented meaningful graph substructures with higher levels of biological congruence.

Fig 2. The Biological Homogeneity Index, BHI, estimated for each of the 8 CNM largest communities, is depicted as red points (CNM clusters are ordered according decreasing size).

Green triangles show the BHI level of the infomap partition of clusters included in the respective CNM structure. For each CNM community, boxplots depict main features of BHI distributions estimated for an ensemble of 1000 random shuffling realizations of the corresponding infomap labels. Noticeably, the mean BHI values of the randomized partitions agree with the BHI level obtained for the corresponding structure-less CNM clusters (red points).

Functional cartography at different resolutions

Meso-scale topological features of the PIN nodes were analyzed studying how they were distributed over the Z-P plane when the CNM and infomap procedures were alternatively considered (see Fig 3). Dashed lines in the figure delineated regions corresponding to the seven different universal roles introduced by Guimera (see Material and Methods and ref [18]). It can be appreciated from both panels that points were not homogeneously distributed over the plane. On the contrary, for both algorithms, they scattered around three local high-density regions laying on the: ultra-peripheral (Z~ -0.5, P~0), peripheral (Z~ -0.5, P~0.5), and connector (Z~ -0.5, P~0.65) areas. Moreover, the coarser resolution level achieved by the CNM algorithm resulted in a general tendency to assign lower participation coefficient values to network nodes. The infomap procedure, on the other hand, markedly populated the R4 (kinless) and R7 (kinless hub) high participation roles, whereas R5 (provincial-hub), the hub role with the lowest participation values, appeared highly depleted. A detailed quantification of all these quantities is provided in S1 Table.

Fig 3. Distributions of PIN nodes in the Z-P plane obtained when the CNM and infomap clusterization were considered are shown in left and right panels respectively.

A color-coded kernel density estimation was also depicted in the figures. Dashed lines in the figure delineate regions corresponding to the seven different universal roles.

The functional cartographical characterizations depicted in Fig 3 served to highlight inter/intra modular connectivity patterns revealed by the modular network decompositions considered in each case. The CNM detection procedure resulted in larger community structures and consequently presented less inter-cluster surfaces than the infomap methodology. In other words, ‘internal’ surfaces might appear within large CNM clusters when the infomap partition was considered (see S2b Fig), causing a number of originally intra-CNM-cluster links to become edges connecting different infomap clusters. These findings were consistent with the general increase in the participation values reported under the infomap network modular description.

The implications of these discrepancies are not properly taken into consideration in the literature. For instance, several recent studies dealing with different biological problems used modularity based optimization procedures to characterize PIN nodes in terms of topographic roles [34–36]. In all these cases, a marginal number of high-participation nodes were typically reported. However, we want to stress that the lack of identification of kinless or kinless hub nodes in these studies was not necessarily a result of an intrinsic network feature. If the infomap clustering procedure had been used for characterizing those networks, a noticeable increase in the number of high participation role nodes would have been observed (see S4 Fig). To illustrate this point, let’s consider, the two PPI-Yeast Networks used by Chang et al to investigate the party-date hub dichotomy using Z-P topological features [36]. The first one was a high-confidence yeast PIN introduced and curated by Batada [13], having a giant component of 3801 nodes, and 9742 links. The second one was another yeast PIN originally proposed by Bertin [38], having a giant component of 2233 nodes and 5750 links. We analyzed the modular structure of these networks using both, the CNM and infomap community detection algorithms. We observed that also in these cases, high participation nodes were precluded in the CNM description and that the use of the infomap clustering procedure resulted in a noticeable increase of this type of nodes. The corresponding Z-P density distributions were included as S4 Fig.

Meso-scale connectivity patterns of aging-related proteins

In this section we aimed to investigate to what extent the resolution of the considered modular description could condition the finding of significant and non-trivial correlations between complex high-level phenotypes and PIN’s meso-scale connectivity patterns. We focused our attention on a set of gene products related to aging: the aging related genes (ARG). Aging is a complex process associated to several complex diseases, that is affected by both, environmental and genetic factors [39]. A lot of effort has been devoted to characterize the genetic basis of aging and resulted in the identification of genes that: are able to modulate the aging process (e.g. gene mutants that increase maximum lifespan in model organisms or linked to human longevity) [40], display transcriptional changes that correlate with age [41], or show specific DNA methylation patterns [42]. Integrative network-based methodologies have already been employed to provide a system-level understanding of aging [40,43,44]. In particular, Xue et al have considered a network model of aging integrating a PPI network with gene expression data [43]. They defined network modules analyzing correlation patterns of gene transcriptional profiles and, in the same spirit of the present contribution, found that aging genes were unevenly distributed in their aging-network. Interestingly, they reported that module interfaces—loosely defined as vertices presenting first neighbors located in different modules- had 2–3 fold enrichment in aging associated genes over that the module’s cores. In connection with this last finding, we reasoned that the participation feature analyzed in the present contribution is particularly well suited to provide a further quantitative topological description of aging-related genes in the context of PIN analysis.

Aging genes tended to be at the interfaces of high-resolution clusters

As was already shown in Fig 3 and S1 Table, the use of infomap gave rise to a noticeable increase in protein vertex participation values with respect the CNM-based characterization. This effect was particularly evident for ARG nodes. Participation-based ROC curves calculated for ARG genes (AUC_infomap = 0.76 and AUC_CNM = 0.65) displayed (see S5 Fig) statistically significant differences between high and low granularity modular descriptions (p_v = 2.2 10^-16, deLong’s test). This finding suggested that, when estimated at the finer resolution level provided by infomap communities, ARG genes were actually boosted toward much higher relative participation levels. Therefore, the participation feature estimated using infomap cluster’s definition could better bring out the same tendency reported in [43] regarding aging-related genes to be located at the interfaces of network-communities.

Aging genes displayed specific topographical roles

We then explored whether aging-related genes were biased to display specific roles over the network. Results, reported in Table 1 (ARG-dataset column) showed that provincial-hub and connector-hub roles exhibited the strongest enrichment in ARG when the CNM methodology was adopted. On the other hand kinless and kinless-hub categories were significantly enriched when infomap methodology was considered. Under this last analysis alternative, a 64% larger set of ARG were involved in enriched topographic categories and more extreme significance signal levels were achieved.

Table 1. Summary of Fisher statistical association test between the ARG set and cartographic role assignments.

		Cartography		ARG dataset		ARG' dataset
		Role	N	ARG	pv	ARG'	pv	pv'
CNM	Non Hubs	R1	3884	47	1	47	1	1
		R2	2964	119	0.00989	119	4.60E-05	0.488
		R3	909	47	0.00413	46	0.000382	0.121
		R4	13	1	1	1	0.447	0.111
	Hubs	R5	58	15	1.72E-09	8	0.000584	0.417
		R6	170	32	2.19E-15	12	0.00693	0.998
		R7	2	0	1	0	1	1
Infomap	Non Hubs	R1	3162	26	1	26	1	1
		R2	2347	51	1	51	1.00E+00	1
		R3	1610	65	0.13	65	0.0558	0.286
		R4	635	67	4.95E-18	67	6.67E-21	0
	Hubs	R5	12	1	9.86E-01	1	0.418	0.622
		R6	102	11	2.32E-03	7	5.07E-02	0.98
		R7	132	40	1.00E-27	16	4.37E-06	0.352

Summary of Fisher statistical association test between the ARG set and cartographic role assignments considering the CNM (first 7 rows) and infomap (last 7 rows) modular descriptions are shown in the first 4 columns. Results of the corresponding bootstrap control tests are shown in the last three table columns. The number of network’s nodes, ARG nodes, and ARG’ nodes, assigned to a given role are displayed in columns: N, ARG, and ARG’ respectively. Fdr–adjusted Fisher enrichment p-values are reported in ARG- pv and ARG’-pv columns, where fdr-adjusted bootstrap p-values (see Methods) are shown in column pv’. Cartographic role abbreviation: Ultra peripheral (R1), Peripheral (R2), Connector (R3), Kinless (R4), Provincial Hubs (R5), Connector Hubs (R6), Kinless Hubs (R7).

The participation feature highlighted non-trivial connectivity patters of the aging gene set

Previous studies have shown that aging genes had high degree values in PINs, and argued that they played a central role in the cell connecting functionally diverse cellular processes [44–49]. This high-degree bias was also observed in our case (see S6 Fig) and partially explained the enrichment found for aging genes in connection with hub roles (provincial-hub and connector-hub CNM categories, and infomap kinless-hub role, see Table 1, ARG column). Interestingly we found that aging genes nodes were also enriched in the infomap-kinless category. Even though this last result could in principle be linked to the high degree levels observed for ARG’s, we argue that more subtle meso-scale connectivity patterns (kinless nodes are characterized by presenting links more or less homogeneously distributed among different clusters) could also be a prominent feature of this protein set. The observed kinless role enrichment could support the functional connectivity hypothesis for aging genes without necessarily relaying on high degree levels of the corresponding network nodes.

In order to further characterize non-trivial connectivity patterns of the analyzed gene set we decided to de-convolve the degree signal from the participation values, performing a bootstrap analysis for the cartographic role enrichment calculation (see Methods). Due to data scarcity of network nodes of high degree levels we considered for this analysis a reduced aging-related gene set (ARG’) obtained discarding the top-10% most connected vertices of the original ARG set (see Methods and S7 Fig for details). Interestingly, we found that only the infomap-kinless category enrichment was significant under the bootstrap analysis (see Table 1, ARG’-dataset column). Hence, these results showed that the high resolution level of the infomap community structure allowed highlighting the single non-trivial cartographic role enrichment that could not be explained by the effect of the aging gene-set degree distribution.

Suboptimal performance of alternative topological descriptors to characterize the aging gene set

We further wanted to examine whether other network features were also able to provide non-trivial evidence to distinguish aging related genes from the rest of the considered protein interaction dataset. In particular, we analyzed the performance of these indicators in connection with their ability to bring out mid/poorly connected ARG genes, i.e. unimportant and non-central nodes from the point of view of their degree level. We thus focused our attention on a subset of PIN by removing the 10% of genes with highest degree values (i.e. removing from the analysis nodes with k>18), and examined the use of infomap-participation to bring out this subset of ARG genes from PIN data. We compared its performance with: the participation feature estimated at a broader resolution (CNM-participation), the node degree, and two alternative measures of a node’s information-flow related capabilities: betweenness and bridging centrality. This last feature quantified to what extent a node was located between well-connected regions (see Material and Methods and [50] for further details).

Fig 4 shows ROC curves obtained for the considered features calculated over the analyzed degree-bounded gene-set. Noticeably, along the false-positive-rate (i.e. 1-specificity) range spanned by infomap-kinless ARG genes (1-specificity values in the interval [0,0.045]) the infomap-participation presented the largest sensitivity among the considered descriptors. In particular, regardless of its absolute performance as a topological predictor, the infomap-participation performed better than the CNM-participation feature (i.e. a participation characterization estimated at a broader resolution), and also better than the degree, betweenness and bridging centralities. This observation agreed with the significant and non-trivial link we found between the infomap-kinless category and this group of genes. Moreover, this result suggested that the infomap-participation feature provided the most effective topological alternative, among the considered ones, to bring out this particular gene-set from protein interaction data (in particular, more effective than other information-flow related quantities like node betweenness or bridging centrality).

Fig 4. ROC curves for ARG genes based on node’s infomap-participation, CNM-participation, degree, bridge-centrality and betweenness feature levels are shown as solid black, orange dashed, blue dotted-dashed, dotted-dashed red and green dashed lines respectively.

Only nodes with mid/low connectivities (i.e. network nodes with degree values lower than the 90% percentile of the entire degree distribution) were considered. The horizontal dotted line depicts the maximum sensitivity level achieved infomap-kinless ARG gene with the lowest infomap-participation value.

Discussion

In our analysis of the modular structure of a human PIN we found that both, CNM and infomap clusterization algorithms, produced high-quality network partitions in terms of achieved modularity levels. However, significant differences arose in terms of the granularity of each description. In particular we verified that the largest structures detected by CNM were further broken up in smaller clusters according to the infomap network modular description.

In concordance with these findings, we observed for the high-granularity network partition a general increase in the number of nodes with high-participation roles. Internal surfaces appeared within large CNM clusters when the infomap partition was considered, causing a number of originally intra-CNM-cluster links to become edges connecting different infomap clusters. Importantly, the same behavior was observed when already published data was re-analyzed with the infomap prescription (S4 Fig). This finding certainly relativized Guimera’s original claim that non-hub kinless nodes were not supposed to be found in real-world networks [18]. We have shown instead that this could eventually arise only as a consequence of the employed community detection methodology, better than reflecting an intrinsic feature of the analyzed network. Furthermore, in our work we found that the observed discrepancies in modular descriptions had non-trivial counterparts in the biological coherence of the detected network structures (infomap structures presented greater levels of biological coherence), and secondly in the kind of connectivity patterns each algorithm was able to unveil for the analysis of a considered protein-set of interest.

Studying topological network features of proteins related to aging we observed that they could be significantly linked to low and mid participation hub-roles according to the CNM partition (Table 1). However, should the infomap partition be taken into consideration, the same gene-set would have been found to be significantly enriched in high-participation roles (kinless and kinless-hub categories) instead. Noteworthy, for neither hub category we could rule out that the observed enrichment could have arisen from the particular degree distribution exhibited by the corresponding network nodes, as degree-aware random samples showed associations of similar statistical significance levels than the originally observed one. Only the non-hub high participation kinless role detected within the infomap description, proved to be non-trivially connected to the aging related gene set. This meant that the corresponding association was particularly supported by inter and intra modular connectivity patterns of the network nodes.

As we already mentioned, previous studies have shown that aging genes had high degree values in PINs and it has also been suggested that their higher connectivity was an indicator of a central role of these genes in cellular functioning [44–49]. Additionally, they might also connect functionally diverse cellular components [46]. This is important because aging is a complex process, that involves alterations in many biological processes, including genomic instability, telomere remodeling, epigenetic alterations, loss of proteostasis, metabolic alterations, mitochondrial dysfunction, etc. [51]. However, it is important to keep in mind that the gene's degree in a PIN is a measure of centrality that does not completely account for the node’s level of connectivity at the meso-scale. On the other hand the gene's participation coefficient is a measure of its level of connectivity to different modules, and (given the functional-topological connection) to different biological processes. In particular when considering the network modular description provided by infomap, our findings favor the hypothesis of aging genes as central players in the communication among different biological processes. In the same line that the study of Xue et al [43], our results suggested that being associated to kinless, high infomap-participation nodes, (i.e. nodes mostly located at infomap-cluster’s interfaces) these proteins could serve for coordination and/or information flow purposes between modules of specific biological functionality.

A paradigmatic example of an infomap-kinless protein related to aging is sirtuin 1. SIRT1 and other members of the sirtuin family (SIRT3 and SIRT6) contribute to healthy aging in mammals [51]. In particular, the association of SIRT1 with aging has been proposed based on its role in several processes such as genomic stability, metabolic efficiency, mitochondrial biogenesis, proteostasis and inflammatory responses related to aging [51]. The proteins encoded by the CDKN2A gene provide another interesting example of aging related gene-products associated to infomap- kinless nodes. This gene gives rise to several isoforms known to function as inhibitors of CDK4 kinase, such as p16 and p19. The levels of both p16 and p19 are correlated to the chronological age of tissues in humans and animal models. More interestingly, the CDKN2A gene locus was found to be associated to several of age-associated diseases in a meta-analysis of GWAS [52]. Based on these evidences, the CDKN2A gene is regarded as the best documented gene that control human aging and is associated to age-related diseases.

A final remark is in order here. Even though we have found that the CNM partition presented a slight modularity gain when compared with infomap, the higher granularity of the later modular description allowed us to highlight network structures of more biological congruence and statistically significant associations between the cartographic role classification scheme and the analyzed aging related protein-set. A major drawback of the CNM partition was the existence of extremely large structures for which no clear associations with Hartwell’s functional modules could be established. These findings pointed out that the biological significance of a partition obtained through an optimization procedure of a pure topological figure-of-merit should not be taken for granted. Of course this result did not mean that modularity optimization is a flaw methodology per-se. Other modularity optimization heuristics exist apart from the CNM procedure that could produce partitions at different resolution levels [21,33,53,54]. However, our results suggested that sub-optimal partitions from the strict point of view of their modularity levels might still be worth being analyzed when meso-scale features were to be explored in connection with external source of biological knowledge.

Conclusions

In this manuscript we addressed in a systematic manner how two alternative modular descriptions of a biological network could condition the outcome of follow-up network biology analysis. In particular we analyzed the use of two paradigmatic and well-known community recognition algorithms, namely the CNM and infomap procedures, and thoroughly characterized their performance in terms of the granularity of the corresponding inferred network partitions and the biological homogeneity displayed by the detected network structures.

We observed that the infomap partition resulted in a keener description of the network’s modular structure than the CNM prescription. Noticeably, we found that infomap clusters not only corresponded to congruent structures from the topological perspective, but also displayed higher levels of biological homogeneity. Discrepancies in the cluster resolution level displayed by each algorithm had also impinged on the specific kind of meso-scale connectivity patterns each methodology was able to unveil. In this regard, we presented a thoughtful analysis of differences arising in the significant statistical associations that could be established between intra/inter modular connectivity patterns, and specific protein sets related to complex phenotypes like aging.

In this paper we did not aim to present an exhaustive review of existing clustering procedures, but to raise a word of caution regarding the technical tools usually considered in network biology analysis. It is important to keep in mind that meso-scale descriptions of complex networks are not unique and that results one gets depend on the description you choose. At this respect our work illustrates the following apparently trivial but often disregarded consideration: optimal partitions from a strict topological point of view do not always provide the best modular description to highlight biologically relevant patterns. Other features like the resulting partition granularity are worth to be considered as well.

Material and Methods

We considered the high-confidence version of the HIPPIE protein interaction network [55]. The v1.5 version of this network (downloaded on April 2012) included 31068 interactions among 8277 proteins. We focused our analysis on the giant component of this graph, comprising 8000 nodes and 30835 edges, that we dubbed PIN for future reference. An analysis of several network topological features is included as Supplementary Material (see Text A in S1 Text). In our analysis, we also considered a curated database of genes associated with the human aging phenotype provided by GenAge [56]. The downloaded dataset (October 2013) comprised 298 genes, and 261 of them could be mapped to PIN.

Network topological features

In our work, we took advantage of a handful of topological network features. First, we considered the simplest local node-centrality measure, i.e. the degree of a node, defined as the node’s number of direct neighbors. A second local centrality measure considered in this work was the clustering coefficient of a node [57]. It is defined as the ratio between the actual number of connections between two neighbors and the number of all possible connections of this kind, and it specifies the probability that two randomly selected neighbors of the node of interest were connected to each other, and. In addition, we also made use of the node betweenness concept, a global centrality measure defined as the number of shortest paths among all network vertices pairs that traversed across the considered node [58]

The bridging centrality, BC, is another interesting topological feature devised to explore information-flow related capabilities of a given node [50]. It is a measurement of the extent how well a node is located between well connected regions and it is calculated as the product of two factors: the node’s betweenness, be, and the bridging coefficient, bc, of a node v:

$$BC\left(v\right) = be\left(v\right)*\stackrel{bc\left(v\right)}{\overbrace{\left(\begin{array}{l}1/k(v)\hfill \\ \overline{{\sum }_{i\in N(v)}1/k(i)}\hfill \end{array}\right)}}$$

where k(v) is the degree of a node v and N(v) is the direct neighbor sub-graph of node. The bc(v) factor then determines the extent of how well the node is located between high degree nodes, assessing the local bridging characteristics in the neighborhood.

Meso-scale network topological features

Two topological features, introduced by Guimera and Amaral, were central to our analysis: the intra-cluster connectivity, Z, and the participation coefficient, P of a node [19]:

$$Z_i=\frac{k_i-\overline{k_{C_i}}}{{\sigma }_{k_{C_i}}} , P_i=1-\sum _{C=1}^M{\left(\frac{k_{ic}}{k_i}\right)}^2$$

where k_i is the degree of the node-i, $\overline{k_{C_i}}$is the mean degree of nodes in module C_i, σ_Kci is the standard deviation of the nodes degree in that cluster, k_iC is the number of connections of node i to members of cluster C, and M is the total number of communities in the network. These quantities can be considered meso-scale descriptors as they explicitly depend on the network’s modular structure. Zi measures how well-connected node-i is with respect to the other nodes in the module. On the other hand, the participation coefficient P_i is close to 1 if node-i links are uniformly distributed among all the modules.

Network null models

To explore local and global structural properties of PIN we considered two different network null models: an Erdos-Renyi (ER) [59], and a fully rewired version (RW) of the real network [60]. Both control graphs preserved the number of nodes and edges of the original network. The ER graph had the same link density than the original network but, as edges were assigned randomly, it presented no correlations of any order. On the other hand, the RW model preserved the original degree distribution, but lacked second and higher order correlations that might exist in the real graph. All network-related calculations were performed using the R statistical framework (v2.15.1) [61] and the igraph library (v0.6–2) [62].

Modularity

The modularity parameter, Q, was originally introduced by Mark Newman in order to assess for the possible modular structure of a complex network [22 and references therein]. For a given network partition into disjoint communities, Q is defined (up to a multiplicative constant) as the number of edges falling within groups minus the expected number in an equivalent null model network. It is customary to consider a so called configurational model for this control network, where edges are placed at random between network vertices displaying the same degree distribution as the original graph. In this case, the modularity Q can be expressed as:

$$Q = \frac{1}{2L}{{\displaystyle \sum }}_{ij}^N\left(A_{ij}-\frac{k_i{k}_j}{2L}\right)\delta \left(C_i,C_j\right)$$

In the above equation, k _i is the degree of node-i, L is the total number of network edges, N the total number of nodes, A _ij is the adjacency matrix of the network (A _ij = 1 if there is a link between nodes i and j, and zero otherwise). C _i identifies the cluster that includes node-i, and δ(C _i,C _j) is a delta function (i.e. δ(C _i,C _j) = 1 if node-i and node-j belong to the same cluster, and zero otherwise). Note that the expected number of edges between vertices i and j if edges are placed at random is k _i k _j /2L.

Modular network description

We considered two well-established network community recognition methodologies: the Clauset-Newman-Moore (CNM) modularity optimization algorithm [37], and the infomap procedure [23]. A brief description of the optimization criteria at use by each algorithm was included as Supporting Infotmation (Text B in S1 Text). A thorough analysis and performance comparison of both algorithms can be found in [27, 29, 30].

Universal topographical roles

Guimera and Amaral presented a heuristic definition of seven universal topographic roles, based on within-community degrees, Z _i, and participation coefficient, P _i, node values [18]. They first classified nodes presenting Z ≥ 2.5 as module hubs (note that Guimera’s hub categorization, which is considered in the present manuscript, is not equivalent to the more commonly used hub notion exclusively based on the idea of nodes presenting large degree values). The hub/non-hub node classification was further refined taking into account the participation coefficient values. Non-hub nodes could then be assigned to one of the following roles:

• Ultra-peripheral (role R1): Nodes with all their links within their module (P ≈ 0)

• Peripheral (role R2): Nodes with most links within their module (0 <P < 0.625)

• Connector (role R3):Nodes with many links pointing to other modules (0.625 <P < 0.8)

• Kinless (role R4):Nodes with links homogeneously distributed among several modules (P> 0.8)

Similarly, hub nodes were further classified as:

• Provincial hub (role R5): Hub nodes with the vast majority of links within their module (P < 0.3)

• Connector hub (role R6): Hub nodes with many links pointing to other modules (0.3<P< 0.75)

• Kinless hub (role R7): Hub nodes with links homogeneously distributed among several modules (P> 0.75)

A detailed justification of this classification scheme can be found in [18] and [19]. A schematic representation of the corresponding role-induced partitioning of the Z-P plane is included in S8 Fig.

Biological homogeneity index (BHI) of network partitions

The BHI figure-of-merit measures the degree a given partition embodies biologically meaningful clusters, using a reference set of functional classes [63]. It basically quantifies whether genes placed in the same cluster belong to the same functional class. In our case, we relayed on biological knowledge embedded in Gene Ontology protein annotations. A brief description of the considered BHI was included as Text C in S1 text file.

Degree control for role enrichment estimation

A bootstrapping procedure was devised to control the node’s degree distribution confounding factor for the role enrichment analysis. For each enrichment test, we considered an ensemble of 1000 control random gene-sets having the same degree distribution than the genes under study. A p-value level was assigned according to the number of random realizations displaying the same or larger effects (over/under representation significance) than the ones observed in the original data. Each random realization was built blindly selecting genes from pools of given degree levels in order to conform the degree distribution displayed by the original gene set (see Text D in S1 Text for details).

Supporting Information

S1 Fig. Topological features of HIPPIE network.

(A) node degree distribution for HIPPIE network (gray circles), and for the respective Erdos-Renyi random network (yelow triangles). Nodes of the HIPPIE network (gray circles), Rewired Network (red diamonds) and Erdos-Renyi network (yellow triangles) were displayed over the Degree-Betweenness and the Degree-Clustering Coefficient planes in panels (B) and (C) respectively.

(TIF)

S2 Fig. Modular structure of HIPPIE Network.

(a) Number of internal links, l_int, as a function of cluster sizes. The dotted and dashed lines depict the expected relationships for fully connected cliques and linear structures respectively, and are included for reference purposes. (b) Fraction of internal CNM cluster’s links that did not appear as internal links in the infomap modular description (i.e. fraction of broken links, fBL). Circle sizes are proportional to each CNM cluster log-size.

(TIF)

S3 Fig. Examples of module comparison in HN.

CNM clusters with low internal-link density and more than ten nodes. infomap communities were depicted using different colors. Gray colored nodes belonged to infomap clusters not-totally included in the displayed CNM structure. Two scenarios can be recognized. For cases (a)-(d) a rather good agreement between the alternative modular descriptions was observed. However, for the cases illustrated in panels (e)-(j) internal structure not resolved by the CNM procedure was indeed highlighted by the infomap prescription.

(TIF)

S4 Fig. Cartographic description of already published high confidence PPI networks.

Z-P planes for Batada yeast PPI network (panels A-B) and Bertin yeast PPI Network (panels C-D). Left and right panels correspond to CNM-based and infomap-based cartographical descriptions respectively. An overall increasing behavior in node participation levels can be observed when the infomap cluster recognition procedure was considered.

(TIF)

S5 Fig. Participation based analysis of aging related genes.

Participation-based ROC curves estimated for ARG genes for CNM and infomap modular descriptions are shown with dashed brown and continuous black lines respectively. The vertical dotted line represents the 90% specificity level. Statistically significant differences between total AUCs are observed (AUC-IFM = 0.76; AUC-CNM = 0.65; pvalue< e-16, deLong’s test), suggesting that the resolution level provided by infomap enhanced the detection of the topological bias displayed by ARG genes toward high-participation levels.

(TIF)

S6 Fig. Evidence of high-degree bias in aging related gene set.

The degree distribution of aging genes, and the whole HIPPIE network remaining nodes are shown in the left and right boxplots respectively. Significant differences (pv< e-16, Wilcoxon test) were observed between both degree distributions.

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S7 Fig. Control of degree-aware random sampling employed in the bootstrap analysis of aging related gene set.

Degree distributions for selected quantiles of 1000 control random realizations are displayed as boxplot. Blue circles depict ARG degree values for the respective quantiles. It can be observed that the top-10% of ARG with highest degree levels, lay outside the inter-quartile levels of their corresponding control random samples.

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S8 Fig. Role-specific regions in the Z-P plane according to Guimera and Amaral topographical role classification scheme.

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S1 Table. Distribution of cartographic role assignments.

Distribution of cartographic role assignments according to the Infomap and CNM descriptions. Cartographic role abbreviation: Ultra peripheral (R1), Peripheral (R2), Connector (R3), Kinless (R4), Provincial Hubs (R5), Connector Hubs (R6), Kinless Hubs (R7). The coarser resolution level achieved by the CNM algorithm resulted in a general tendency to assign lower participation coefficient values to network nodes. For instance 68% of infomap-connector vertices were assigned to lower participation roles (59% peripheral, 9% ultra-peripheral) when the CNM procedure was considered. More strikingly, the majority (94%) of the 635 infomap-kinless nodes were re-classified as: CNM-connectors (45%), CNM-peripheral (48%), and CNM-ultra-peripheral nodes (1%). Finally, it can also be observed that nodes originally assigned to hub-like roles when infomap procedure was employed were not only affected by this lowering effect in the participation, but in addition, almost 50% of them were also reassigned to non-hub roles when CNM was considered.

(DOCX)

S1 Text. Supporting Information Text.

(DOCX)

Data Availability

Data are from third parties. The HIPPIE network (v1.5) was downloaded from: http://cbdm.mdc-berlin.de/tools/hippie/information.php (April 2012). The curated database of genes associated with the human aging phenotype was downloaded from http://genomics.senescence.info/genes/ (October 2013).

Funding Statement

CONICET (grant PIP0087), UBACyT (grant 20020110200314), ISCIII-FEDER (PI13/00082 and CP10/00524), IMI JU (grant agreements n° [115002] (eTOX) and n° [115191] (Open PHACTS)], resources of which are composed of financial contribution from the EU's FP7 (FP7/2007–2013) and EFPIA companies’ in kind contribution). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.