High-throughput experimental approaches for determining protein interactions have resulted in large-scale cellular networks for numerous organisms. Graph-theoretic analyses of these networks have been a great aid in advancing our understanding of cellular functioning and organization (review, 
). One of the most fundamental discoveries is that there is a strong relationship between the topological characteristics of cellular networks and their underlying functioning. For example, cellular networks consist of tightly clustered groups of interacting proteins, and these proteins work together as protein complexes or biological processes to achieve specific biological functions 
. An orthogonal decomposition reveals that there are recurring and over-represented topological and functional patterns within larger cellular networks, and these network motifs 
and network schemas 
can be associated with dynamic regulatory properties and shared mechanisms of functioning. Here, we revisit perhaps the most basic structure-to-function relationship that has been proposed for cellular networks—that between the number of interactions a protein has and its overall functional importance.
The importance of a gene to a cell or an organism can be quantitatively measured by considering the phenotypic effects of gene deletion or disruption. Experimental studies in the baker's yeast S. cerevisiae
have demonstrated that approximately 19% of its proteins are essential; that is, the deletion of these proteins results in cell death, even in optimal growth conditions 
. Early computational analysis of the yeast S. cerevisiae
protein-protein physical interaction network revealed a scale-free topology, where a few “hub” proteins have many interactions, and also showed that hub proteins are more likely to be essential than other proteins 
. Numerous subsequent studies have confirmed this centrality-lethality relationship, not only in yeast 
but also in other organisms 
. On the other hand, the relationship has been observed to be weak in networks consisting of interactions determined via high-throughput yeast two-hybrid experiments while stronger in other types of networks 
, and it has been proposed that, in yeast two-hybrid networks, the observed relationship is due to study bias favoring the determination of interactions of essential proteins 
. Nevertheless, the positive correlation between protein interaction degree and essentiality is generally accepted, with numerous reasons proposed in the literature to explain this relationship.
Initial work suggested that high-degree proteins may be essential due to their role in interaction network connectivity 
; however, this is unlikely to be the case as it was subsequently shown that non-essential hubs are just as important as essential hubs for maintaining connectivity, and that essentiality is better correlated with local, rather than global, measures of connectivity in protein-protein interaction networks 
. It was alternatively proposed that essentiality is a property of interactions 
. That is, there are essential protein interactions, without which an organism cannot survive, and these are randomly distributed across the network; hubs then tend to be essential as they are more likely to participate in essential interactions. However, this model implies that the probabilities that two non-interacting proteins are essential are independent of each other, and this is not the case 
. Instead, Zotenko et al. 
argued that the correlation between degree and essentiality is due to the participation of essential proteins in essential functional modules consisting of groups of densely clustered and functionally related proteins. They further showed that the essentiality of hubs that are not in these computationally extracted modules are only weakly correlated with degree 
. Indeed, it had previously been found that essential proteins tended to be densely connected to each other 
and concentrated in complexes 23
, suggesting that essentiality is a modular property rather than a property of individual proteins. Building upon this, it has been argued that essential complexes tend to be large, and thus proteins within them have a larger number of interactions, and that this explains why hubs tend to be essential 
While there is substantial evidence that essentiality is a modular property in protein-protein interaction networks, it is also clear that complexes and processes do not consist entirely of essential or non-essential proteins. Do essential proteins within an essential complex or process differ from the non-essential ones? Further, not all complexes and processes contain essential proteins. Do such essential modules have distinctive roles in cellular networks? In this paper, we aimed to discover whether, within modules, their essential and non-essential proteins differ in their interaction properties, and at a more global scale, whether essential and non-essential modules differ in their network-level properties. To accomplish this, we developed a computational framework that incorporates information about functional modules within the context of network analysis techniques. To uncover general and robust principles, we performed our analysis on three types of S. cerevisiae
protein-protein interaction networks and considered functional modules derived from protein complexes as well as Gene Ontology (GO) biological process annotations 
at different levels of resolution. Further, to address the issue of study bias, we performed our analysis on additional networks which removed interactions determined in small-scale experiments.
We began by re-examining the relationship between protein essentiality and network modularity. We hypothesized that if essentiality is a modular property, as has been proposed previously 
, then a protein's intramodular physical interaction degree should be a better predictor of a protein's essentiality than its intermodular physical interaction degree. To test this, we utilized biological process functional annotations of proteins and classified physical interactions into intraprocess interactions within processes and interprocess interactions between processes. We found that essential proteins tend to have many interactions with proteins within the same functional modules and that the intraprocess interaction degree is more correlated with essentiality than overall degree. Further, we found that the relationship between overall degree and essentiality is significantly weakened when controlling for intramodular degree, but is not as affected when controlling for intermodular degree. Thus, we show in a direct and simple manner that, for many essential proteins, their essentiality is likely to be a consequence of their participation within essential modules consisting of functionally similar proteins.
To further ascertain whether the modularity of essential proteins is due to their roles within essential protein complexes or more generally within essential biological processes, we repeated this analysis while first exclusively focusing on proteins within protein complexes and next focusing only on proteins that are not within known protein complexes. We found that most essential proteins with many intraprocess interactions in fact participate in essential protein complexes or in essential biological processes that include one or more protein complexes; that is, the modularity of protein essentiality appears to be a consequence of protein complexes, not more broadly of biological processes.
Next, we examined complexes that contain essential proteins, and compared their essential and non-essential proteins. We reasoned that if the relationship between essentiality and interaction degree for proteins within these complexes is entirely a consequence of the complexes themselves being essential, then essential and non-essential proteins within the same complex should not differ with respect to degree. On the contrary, we found that essential proteins tend to have more interactions, particularly intracomplex interactions, than their non-essential counterparts within protein complexes. That is, while essentiality appears to be a modular property, the degree of a protein is associated with essentiality within essential complexes; we hypothesize that these essential proteins may play a more important role in maintaining the structure and/or function of complexes.
Finally, we analyzed modules containing essential proteins within the context of other functional modules. We inferred significant “cross-talks” between protein complexes and biological processes and used them to build module-level networks, in which two complexes or processes are linked if they have an enriched number of physical interactions between them. Using these module-level networks, we uncovered that functional modules with essential proteins tend on average to have higher degree; that is, degree in the module-level network is positively correlated with module essentiality.
Overall, by considering proteins within the functional context of the yeast interactome, we give evidence that there is a relationship between essentiality and network topology at different levels of cellular organization: at the protein level, within protein complexes, and also more globally at the module level, with complexes and processes that are essential tending to interact with more functional groups.