protein interactions in a cell  the interactome  presents a foundational picture for biology , sitting at the lowest level of description at which it is possible to have an holistic view of a cell rather then just an isolated study of its individual components . in this article , we make a small contribution to the ongoing effort to understand the global architecture of this fundamental physical network ( 110 ) . the interactome can be represented in an abstract way as a network of nodes connected by links , where nodes stand for proteins and links for direct physical interactions between proteins . in recent years , there has been much interest in applying statistical mechanics to the study of such complex networks ( 11,12 ) . however , the validity of such an approach is always conditional on the fundamental assumptions of statistical mechanics being satisfied . for instance , many statistical measures will not be able to distinguish between a network with an intrinsic hierarchical topology and one without it ( figure 1 ) . our approach sidesteps the limitations alluded to in the preceding paragraph and focuses directly on how the interactome topology relates to two broad biological concepts . the first of these , hierarchical organization , is in this context the notion that there may exist a hierarchy in the role of proteins ( 10 ) . on one hand , there are proteins that perform very specific , local functions , relevant only within the context of a particular biological process . on the other hand , some proteins may possess a global , high level role , perhaps acting as mediators of distinct biological processes . to study the topological hierarchy in the interactome , we use the graph theoretical betweenness centrality measure ( 1315 ) ( supplementary data ) . betweenness centrality ( denoted  traffic , henceforth ) for a node is the total number of shortest paths ( between any two other nodes ) in the network that pass through that node . a high traffic value for a protein therefore correlates with that protein being topologically central in the interactome . the second of these is the concept of biological functional modularity ( 16 ) . in the context of proteins , at the extreme , this takes the form of protein machines performing specific functions in a cell ( 7 ) . more generally , it consists of an expectation that the density of protein  protein interactions will rise as we zoom into an increasingly functionally related set of proteins . to assess modularity in the interactome topology , we use the  q - modularity measure of newman ( 17 ) , which assigns a modularity score , q , to any given partition of the network into modules . the modularity q is defined as the difference between the ratio ( intra - module edges)/(total edges ) for the network in question , and the expected value for this ratio if edges in the network were randomized , subject to every node maintaining its original degree . ( 18 ) to find an interactome partition into modules that corresponds to a large q value . we now explain how to produce an interactome polar map ( figure 2 ) by combining the information contained in the modularity and traffic analyses . the position of every protein in the map is specified in terms of its radial and angular coordinates . more precisely , it is proportional to log(max traffic / protein 's traffic ) , where  max traffic is the maximum node traffic in the network ( supplementary data ) . a logarithmic scale is used due to the long tail of the traffic distribution ( 20 ) . the protein angular coordinates are assigned such that all proteins in the same module fall within the same angular range ( supplementary data ) . this way , an interactome map is created where topologically increasingly central proteins are radially increasingly closer to the center of the map , while angular sectors correspond to topological modules . to determine the circular ordering of the modules in the map , we introduce a ring ordering algorithm that , based on the interactome inter - module connectivity , attempts to place closer to each other , to the extent that it is possible , modules that are more topologically related ( supplementary data ) . we apply this analysis and discuss its implications in the context of the saccharomyces cerevisiae interactome . the interactome dataset we use is the higher confidence  filtered yeast interactome ( fyi ) ( 10 ) , consisting of interactions supported either by small - scale screens as reported in the mips database ( 21 ) or by at least two distinct methods from amongst ( i ) high - throughput yeast two - hybrid experiments ( 22,23 ) , ( ii ) computational predictions based on gene co - occurrence ( 24,25 ) , gene neighborhood ( 24,26 ) or gene fusion ( 24,25 ) , ( iii ) high - throughput affinity purification / mass spectrometric protein complex identification experiments ( 27,28 ) and ( iv ) small - scale or module - scale experimental identification of protein complexes as reported in mips ( 21 ) ( supplementary data ) . we consider only the giant connected component produced by this dataset , a network containing 1741 interactions amongst 741 proteins . our dataset choice reflects a desire to bias the data towards a thorough and accurate coverage of a limited region of the interactome as opposed to a wider , but likely shallower and more error prone , sampling . note that when defining whether two proteins interact , a binary description is being imposed on what ideally would be characterized in terms of an affinity constant . experimentally , effectively the aim has been for a cut - off that is high enough to exclude indiscriminate low - affinity interactions , such as those that occur between a general protein and proteasomal , ribosomal , or heat shock proteins , since in principle these are less informative interactions ( 29,30 ) . hence , such interactions are largely absent from datasets . with the vast majority of the proteins at the periphery of the map and well - defined modules connected through a handful of more central proteins , the yeast interactome polar map next , we discuss how this interactome architecture fits in with the biology of the cell . an examination of the mips database functional ( biological process ) annotation of the proteins ( 21 ) demonstrates that the topological modules make very good sense as biological functional modules . on average 88.8% of the proteins in a module share a similar function based on their mips classification ( 21 ) ( supplementary data ) , confirming earlier studies connecting topology and function ( 3140 ) . part of the mismatch between the topological modules and current protein functional annotations will likely vanish , once more complete and accurate interactome datasets become available . however , it is the disparities not due to dataset limitations that are truly interesting , for those are instances where the functional modules based on the interactome topology are not the ones we currently assign in functional classification schema . in view of ( i ) the good overall match found and ( ii ) the foundational role of the interactome in the cell , we propose that the interactome topology represents a fundamental source for the division of proteins into functional groups . as such , its modularity analysis provides a rigorous alternative to the currently subjective functional annotation present in protein databases . in accordance , we name the topological modules found so as to reflect their perceived biological roles ( figure 2 ) . the average degree of essential ( 41,42 ) proteins in the dataset is 5.7 , while that of non - essential proteins is 3.9 , a difference that is too large to be attributable to chance alone ( 43 ) ( supplementary data ) . although physically knocking out a gene associated with a high or a low degree protein , say of degree 2 or 10 respectively , may be considered equivalent , from a mathematical network perspective it is not . in one case , it involves deleting 1 node and 2 links , in the other it involves deleting 1 node and 10 links . note that such an explanation would not involve ascribing any out of the ordinary , higher - level role to hubs , the large degree proteins . alternatively , the observed higher average degree of essential proteins could still stem from lingering systematic biases in the fyi dataset . while the translation of the interactome modular topology into biological functional modularity is straightforward , this is not the case for the interactome topological hierarchical organization . a fundamental question that the traffic analysis gives rise to is how the interactome topological hierarchical organization phenomenologically expresses itself . do more hierarchically central proteins in fact perform a higher - level , coordinating role in the cell ? at present , these are open biological questions . comparing proteins of equal degree , we find no significant correlation between a protein 's traffic and its essentiality ( 41,42 ) ( supplementary data ) , something not too surprising , as knocking out a key protein that renders an essential functional module inoperant is plausibly more damaging than knocking out a protein that mediates two distinct processes that nonetheless can still function independently . an intriguing , though at the moment still unsupported hypothesis is that , if a protein is disrupted , its traffic level correlates with the likelihood of causing non - lethal side effects in multiple areas of the biology of the cell . speculating further , perhaps our representation of the interactome can provide clues as to where those side effects may arise  a matter of critical importance in drug development . a different possibility is that , for some of these module - connecting proteins , interacting with multiple modules is not a sign of a role in coordinating the functionality of the modules , but rather just a result of the protein being independently used in those modules . in opposition to a true functional widget ) , in reference to how , analogously , a mechanical bolt can be used in multiple functional modules of a human engineered machine , while playing no role in coordinating their functionality . finally , note that a priori the observed 47 inter - module interactions are particularly susceptible to be false - positives , because a false - positive interaction between two random unrelated proteins is likely to result in an inter - module , high traffic interaction . however , significantly , 45 out of these 47 inter - module interactions belong to the set of interactions supported by small - scale targeted experiments and arguably it is not very likely that a false - positive interaction of the type just described would go unnoticed in a targeted experiment and further make it into the peer - reviewed literature . of the remaining 2 inter - module interactions , one ( 22 ) high - throughput yeast two - hybrid datasets as well as in the mips dataset of protein complexes identified via small- or module - scale experiments ( 21 ) , while the other is reported in the gavin et al . ( 27 ) high - throughput protein complex identification study and again in the uetz et al . ( 22 ) yeast two - hybrid dataset . out of the 45 supported by small - scale experiments , four are also reported in a high - throughput yeast two - hybrid study ( 22,23 ) , three in a high - throughput protein complex identification study ( 27,28 ) and two in the mips dataset of protein complexes identified via small or module - scale experiments ( 21 ) . whatever biological role central proteins turn out to play , we submit that they call for further experimental investigation , given their unique topological placement in the interactome . having hierarchically classified the proteins with the traffic measure , we are now in a position to consider network degree distribution related questions without falling prey to previously noted statistical problems ( figure 1 ) . of particular interest is how degree changes as one moves hierarchically across the interactome ( 14 ) ( figure 3a ) . surprisingly , nodes of different degree are rather homogeneously hierarchically spread across the interactome : note the large spread between the green , red and blue curves relative to their small positive slopes ; or , for a more quantifiable attribute , how the 10% of nodes with the largest degree in the periphery of the interactome have a significantly larger degree than the average degree at the center of the interactome . thus , the interactome is not hierarchically stratified by degree . in particular , the interactome has a non - hub - centric hierarchical organization . further , note that should the inter - module protein interactions indeed represent a form of functional coordination , then this coordination is apparently occurring overwhelmingly in a pair - wise fashion : out of the 76 proteins that possess links to modules outside their own , only 4 connect their home module to more than one other module ( taf25 and srp1 have links to four other modules , nup1 and clb2 to two other modules ) . the other 72 proteins connect their home module to a single other module ( supplementary data ) . it is also noteworthy that , amongst these 76 connecting proteins , the higher degree ones in general belong clearly in their assigned home modules ( specifically , for connecting proteins of degree 4 or higher , let us exclude the two interactions that each of these proteins must have by default to connect it to its home module and to one linked module ; then , 95.3% of the remaining interactions of connecting proteins are with the protein 's respective home module . this pairwise - coordination is in sharp contrast with the picture of a hub protein connecting and mediating multiple modules ( 11,43,44 ) . the non - hub - centric organization runs contrary to a number of network growth models that have been proposed to explain the topology of the interactome ( 11,4548 ) . the models are based on evolution by stochastic gene duplication and divergence ( 49,50 ) . amongst other reproduced statistics , the models are able to generate the power - law degree distribution observed in early interactome datasets ( 51 ) . since these models do not make an appeal to evolutionary selection pressures , a major conclusion taken from their success was that natural selection is not required to reproduce the global structure of the interactome ; instead , stochastic gene duplication and divergence suffices to give rise to that topology ( 45,47 ) . however , here we report that these gene duplication models lead to hierarchically hub - centric networks . in figure 3b , we show data pertaining to an interactome built using the model of pastor - satorras et al . ( 45 ) . by comparison with the same plot for the yeast interactome , this time the network is clearly stratified by degree , with the larger degree nodes concentrated at the hierarchical center of the network . now it is the average degree at the center of the network that is significantly larger than the average degree of the 10% of nodes with largest degree at the periphery . the models of vzquez ( 46 ) ( slightly different implementation of the gene duplication and divergence process ) and of wagner ( 47 ) ( emphasizing a continuous divergence in the form of gain and loss of interactions amongst existing proteins ) produce similar hub - centric networks ( supplementary data ) . in summary , there are at least three possible explanations for the non - hub - centric hierarchy we observe in yeast : ( i ) it is a spurious effect associated with the limitations of existing data and the interactome is in fact hub - centric ; ( ii ) there is some crucial feature of the gene duplication and divergence mechanics that is not understood and/or is not captured by current models or ( iii ) the non - hub - centric hierarchy is in fact shaped by natural selection pressures on the global interactome structure . our study is based on the present - day knowledge of the yeast interactome , which is still rather deficient ( 5258 ) . we minimized false - positives using the higher confidence yeast fyi dataset as the source for our study . the good correspondence between the topological modular breakdown of the interactome and the known functional annotation of proteins corroborates that false - positives are not an overriding problem in this dataset . nonetheless , we repeated the interactome analysis using a dataset of interactions reported in the mips database that are validated through small - scale screenings , the most reliable source of data ( 55 ) ( giant component : 392 proteins , 675 interactions . the correspondence between topological and functional modules ( now on average 92% of the proteins in a module shared a similar function based on their mips classification ) , as well as the reported non - hub - centric hierarchy were again supported by this small - scale dataset ( supplementary data ) . regarding the limited coverage of our datasets [ fyi 12% coverage , small - scale 7% coverage , assuming 6000 proteins in yeast ( 23 ) ] , it is of note that the doubling in size of the network , going from the small - scale to the fyi dataset , did not dilute the observed non - hub - centric topology nor the pair - wise inter - module connectivity pattern . in fact , the fyi dataset produces an even slightly less hub - centric interactome than the small - scale dataset ( supplementary data ) . likewise , the pairwise inter - module connectivity pattern is no less present in the fyi than in the small - scale dataset ( where out of the 47 proteins with inter - module links , four connect their home module to more than one other module . the observation that higher degree connecting proteins are in general strongly attached to their home modules is equally confirmed in both datasets ( the earlier mentioned 95.3% of home module interactions for the fyi higher degree connecting proteins , now becomes 95.7% in the small - scale dataset ) . for instance , regarding the typical pairwise coordination , the possibility that this observation is only the result of a high - number of false - negatives for inter - module protein interactions can not be ruled out . note , for example , how in the map there are no interactions between the translation initiation module and the ribosomal subunit modules , even though such interactions must certainly exist . ultimately , only the generation of more accurate and comprehensive interactome datasets can unequivocally confirm or disprove some of the results and hypotheses put forward in this article ( 52,53 ) . now we would also like to highlight its potential as a framework for exploiting the wealth of interactome data . first , one may zoom into a module of interest in the interactome and locally repeat the analysis , producing a single module polar map ( figures in supplementary data ) . such a local map provides one with a starting point to discuss the biology of the process under study , interpret and design experiments , and generate new biological hypotheses . second , we note that the entire proteome is rarely , if ever , evenly expressed by the cell ( 49 ) . therefore , perhaps the interactome is best viewed as a potential network at the cell 's disposal , with different parts of it being turned on and off to different degrees , as biologically required . integrating mrna expression data with the interactome polarmap ( 59 ) ( figure 4 ) , the interactome represents an elementary abstraction of the multitude of complex biochemical interactions taking place in the rich physiological environment of the cell . in the trade - off between simplicity and realism , arguably some facts may be beneficially incorporated in future interactome models : for instance , protein interactions vary within a continuum of binding affinities and post - translational modifications , as well as allosteric interactions effectively change the possible binding partners of a protein , to name a few of the more prominent omissions at present ( 49 ) . we end by noting that the organization of a network through a procedure akin to the one used in this paper may also be of relevance to the problem of network motif finding ( 60 ) , as certain motifs may turn out to occur sparsely overall and yet be statistically significant in specific regions of the network ( e.g. only in the high traffic central area , or only in a particular module ) . hidden intrinsic hierarchy in networks . in this example sketch , network ( a ) has two hierarchical levels : it consists of a linear string of nodes at the top level that connect to a lower level set of subgraphs possessing scale - free degree distributions , but that are otherwise random . network ( b ) has only one hierarchical level , also with a scale - free degree distribution and random in other respects . arguably , for many applications the difference between topology ( a ) and topology ( b ) is of relevance . yet , an analysis based on common measures such as degree distribution ( 11,12 ) , clustering coefficient as a function of degree ( 11,44 ) , or degree correlation measures ( 61 ) , to name a few , would indicate networks ( a ) and ( b ) to be topologically identical . this is due to the much larger number of nodes at the lower hierarchical level statistically overwhelming , and therefore hiding , the top level structure . the map is constructed , in an unsupervised manner , based solely on protein  protein interaction data . the module captions ( blue text boxes ) were manually chosen , a posteriori , to reflect the biological role of each module . the map suggests a  coordinated - functionality architecture for the interactome , arguably an ideal framework for the cell to physically implement the concept of distinct , yet coordinated , biological functional modules . this would be a pairwise - coordination , as inter - module physical interactions occur in a pairwise fashion : of the 76 proteins that possess inter - modular connections , only four connect their module to more than a single other module ( taf25 in module # 21 and srp1 in module # 13 have links to four other modules , while nup1 in module # 13 and clb2 in module # 1 have links to two other modules ) . protein interaction dataset ( 10 ) , consisting of interactions validated either through small - scale experiments or through at least two distinct procedures . the giant connected component , shown here , consists of 741 proteins and 1752 protein  protein interactions . a non - hub - centric hierarchical organization . in the yeast interactome ( a ) , the degree distribution does not change greatly as one moves from the periphery to the center of the figure 2 polar map ( i.e , as one moves from a low to a high traffic region ) . in contrast , gene duplication and divergence interactome stochastic growth models [ 4547 ] produce hub - centric interactomes , where the average degree markedly increases with traffic and the hierarchical center of the interactome is therefore dominated by hubs . ( a ) analysis for the yeast interactome giant connected component , based on the fyi data set ( 741 proteins , 1752 interactions ) . red curve  the average degree for the set of nodes whose log ( traffic ) value falls within 0.3 of the log ( traffic ) value indicated in the x - axis . that is , a log ( traffic ) bin of size 0.6 is continuously slid along the log ( traffic ) axis and the average degree for all the nodes that fall within the bin is calculated . the last bin also includes all nodes with a log ( traffic ) value larger than the range shown in the figure . the bins cover the entire data set , with every bin containing at least 26 nodes . green curve  similar to the red curve , except this time the degree average is done only over the 10% largest degree nodes in the bin . blue curve  similar to green curve , but this time averaging over the 10% lowest degree nodes in the bin . the average degree in the highest traffic bin ( rightmost data point in the red curve ) is only 0.42 times the average degree of the 10% largest degree nodes in the lowest traffic bin ( leftmost data point in the green curve ) . ( b ) corresponding plots for the giant component of an interactome evolved under the gene duplication and divergence stochastic growth model of pastor - satorras et al . . the average degree in the highest traffic bin is now 4.6 times larger than the average degree of the 10% largest degree nodes in the lowest traffic bin . similar results were achieved under multiple trials , model parameters and gene duplication growth models ( supplementary data . ) . overlay of 20 min post heat shock mrna expression data ( 62 ) upon the interactome polar map . 8 ( proteins connected to other modules in the global map have names in blue ) . 62 ) , there is a sharp decline in expression of mrna for ribosomal proteins . other modules with a significant proportion of constituent proteins showing a decline are the translation initiation and core rna polymerase modules . these observations are consistent with the repression of genes involved in rna and protein synthesis upon heat shock ( 62 ) . modules with a significant proportion of constituent proteins showing an increase in expression are the small cell cycle checkpoint control module and parts of the signal transduction and cell cycle progression modules . overlay of mrna expression data upon the single module maps of the latter two modules shows clear over expression in the protein chaperone submodule of cellcycle progression ( supplementary fig . 9 ) , and into the phosph - cyclin and cell growth / morphogenesis submodules of signal transduction ( above ) . the increased expression of genes in these submodules supports the notion that cell growth is checked upon heat shock .
the architecture of the network of protein  protein physical interactions in saccharomyces cerevisiae is exposed through the combination of two complementary theoretical network measures , betweenness centrality and  q - modularity. the yeast interactome is characterized by well - defined topological modules connected via a small number of inter - module protein interactions . 
 should such topological inter - module connections turn out to constitute a form of functional coordination between the modules , we speculate that this coordination is occurring typically in a pairwise fashion , rather than by way of high - degree hub proteins responsible for coordinating multiple modules . 
 the unique non - hub - centric hierarchical organization of the interactome is not reproduced by gene duplication - and - divergence stochastic growth models that disregard global selective pressures .