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,
Institute of Molecular Biotechnology, Beutenbergstrasse 11, D-07745 Jena, Germany
¶ Department of Biochemistry, National University of Ireland, University Road, Galway, Ireland
|| Department of Surgery, Children’s Hospital, Harvard Medical School, Boston, Massachusetts 02115
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ABSTRACT |
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 TOP ABSTRACT EXPERIMENTAL PROCEDURESRESULTSDISCUSSIONREFERENCES |
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EXPERIMENTAL PROCEDURES |
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TOPABSTRACTEXPERIMENTAL PROCEDURESRESULTSDISCUSSIONREFERENCES |
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i=1s/2 (is) - (s/2s)/2 if s is even and Ns = i=1(s-1)/2 (is) if s is odd. It follows the simple exact result Ns = 2Ns-1 + 1 = 2s-1 - 1. If all possible dissociations of a given complex occur on average with equal probability, it follows the exponential decay of the average lifetime 
s
of a complex with s proteins: s 
Ns-1. If the number Ss of all complexes of size s, Ss = ins xs,i (ns = n0 exp(-as) is the number of different complexes of size s (Fig. 2), and xs,i is the number of complexes of species i (consists of exactly the same type of s proteins)), is proportional to s, it follows by using the above equations for Ns and s: Ss/Ss - 1 0.5. If for each complex size xs,i is normally distributed around xs,i, it follows by using the equations for Ss and ns: xs,i exp(-as)/(xs -1,i exp(-a(s -1))) 0.5 and therefore with the experimentally determined characteristic number of different protein complexes a = acompl.size (Fig. 2): xs,i 
0.6xs -1,i. This finding suggests that the mean number of one type of protein complexes of a given size decreases by 40% if the size is increased by one.
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Construction of Protein Complex Interaction Networks (CN) (Fig. 1B)—
In this network graph, one node represents a whole protein complex, and two complexes are connected if both contain one (or more) same protein, as proposed (8).
Measures to Characterize Protein Networks—
i) Connectance C = 2(number of actual links in the network)/(n(n -1)), n is the number of nodes (in the PN, for instance, n is the protein number); ii) diameter D of the largest cluster: D is the number of links in the shortest path between two nodes, averaged over all pairs of nodes; iii) clustering index cc = ici/n with ci = 2ki/(ki(ki - 1)) (ki is the number of connections between the ni neighbors of node i) (11).
PN and CN are Small-World Networks—
Small-world networks are highly clustered (like regular networks), but have nevertheless a small network diameter (like random networks) (11). Both requirements are fulfilled by the PN and CN as can be seen in Tables I and II.
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The clustering index of random matrices equals their connectance C, whereas the clustering index of regular matrices is somewhat below one.
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RESULTS |
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TOPABSTRACTEXPERIMENTAL PROCEDURESRESULTSDISCUSSIONREFERENCES |
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exp(-as) (Fig. 2), for the TAP data astonishing exactly, whereas the HMS-PCI complexes follow this function up to s = 15 but have some more different large complexes (a is a constant). For TAP and HMS-PCI we find a characteristic number s* = 1/a = 7.3 and 6.4, respectively. Despite similar average number the HMS-PCI data exhibit more different larger complexes (step in the cumulative graph in Fig. 2, around s = 15). The exponential decay of the number of different protein complexes with size s may have implications on the underlying dynamics of complex formation and dissociation. We propose here a simple model that considers the observed "destabilizing effect" when a given complex grows by one subunit. With increasing size s of a complex (i.e. containing s proteins), it has Ns possible ways of dissociation, where Ns = 2s- 1 - 1 (for details, see "Experimental Procedures"). Assuming that Ss, the number of all complexes (of all species) of size s is proportional to the average life time, which in turn is inversely proportional to Ns, it follows that Ss/Ss - 1 = 0.5. Because Ss is related to the observed exponential decay f(s) with the characteristic number s*, we can estimate that the average number of complexes with a given composition decreases by 40% when the complex size increases by one additional subunit (see "Experimental Procedures"). It should now be possible to test this quantitative prediction experimentally in order to validate the physical interpretation of the complex size distribution.
The Protein-Protein Interaction Network—
To map the physical protein complexes onto a protein-protein interaction graph that represents the topology of the protein network (PN), we have to extract the interaction information from the complex data. In contrast to the yeast-two hybrid data, where the elementary experimental finding is a pair that directly translates into a link in the interaction graph, the protein complex data allow various definitions of "interaction" to build a PN (see "Experimental Procedures"). However, the context of a given complex might enable inherently weak, direct physical interactions to take place, which would not be found in isolation or in other complexes, e.g. due to the presence of a scaffolding protein in that complex. For instance, while bait A might not be able to pull out protein B, bait C might pull out a complex that includes A and B that may or may not have direct physical contact. To embrace these scenarios of indirect and scaffold-protein-mediated interactions, we use here a "large PN" definition that counts "functional interactions" between all proteins participating in a complex (Fig. 1C), as it also was suggested (1). Our "medium PN" corresponds to the "spoke" model, while the "large PN" corresponds to the "matrix" model in a previous study (10). We used the latter, most encompassing PN definition for further analysis, because we are interested in the observed complexes as entities rather than in the interactions (Table I, Figs. 2 and 3). However, similar results with respect to the major network topology characteristics were obtained with the "small" and "medium PN" definition (Table II).
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To measure the extent of modular organization in the large PN graph, we calculated the clustering coefficient cc which quantifies for a given network the extent of formation of subnetworks (clusters) that are highly interconnected inter se. Column 2 of Table I shows that cc of PN is much higher than the clustering coefficient in corresponding random networks with the same connectance C = 2I/(n(n -1)), because ccrandomCrandom = CPN. This quantifies the high modular organization of the cellular protein interaction network.
The "Null Model"—
Because in the large PN definition all proteins in a complex are considered to interact with each other, the complex as a physical module will necessarily give rise to a maximally connected cluster (or clique) in the network graph. We thus asked whether the partitioning of the proteome into complexes of the observed size distribution alone explains the high clustering. To answer this question, we simulated the simplest model, called the "null model." We generated 455 and 487 complexes with the exponential size distribution corresponding to the observed complex size distribution in the TAP and HMS-PCI data, respectively. "Proteins" were randomly drawn (without removing them) from a pool with ng proteins (ng is chosen to obtain the same number of "proteins" as in the experimental PN: ngTAP = 1,450, ngHMS-PCI = 1,700; a given protein can be assigned to more than one complex, but no two same proteins can occur in one complex). Then an interaction graph is extracted as defined above according to the "large PN" scheme, and the topology is analyzed. For TAP, this "null model" yields an exponential distribution for connectivity k that is similar to the observed one (Fig. 2), although the total number of interactions I in the simulation is higher than in the TAP data (Table I). In contrast, for HMS-PCI the null model yields a distribution of k that is clearly steeper than in the observed data, and, consistently, the number of interactions I in the data are higher than predicted by the model (Table I). This is in line with the notion that the HMS-PCI data contain more larger complexes than a pure exponential size distribution (as assumed for the "null" models), as e.g. shown by the TAP data, would allow.
Interestingly, in both cases the simulated cc was significantly smaller than that in the corresponding experimental PN (ccnull = 0.49 versus ccexp = 0.73 for TAP and ccnull = 0.54 versus ccexp = 0.70 for HMS-PCI). Thus, the PN are strongly clustered, to an extent that cannot be accounted for by the physical arrangement of proteins into complexes that represent cliques in the interaction graph. In other words, the high cc value of the PN must be due to "higher-level" interactions between the physical complexes.
The Protein Complex Interaction Network—
Because the complexes detected by mass spectroscopy are by definition independent entities, such an apparent link between complexes in the network graphs must correspond to the sharing of the same protein by different complexes (Fig. 1B). We thus analyzed the topology of the "complex-complex interaction network" (CN). Fig. 3 shows that the connectivity distribution of the CN again follows an exponential decay. A comparison of columns 6 and 8 in Table I shows that the simulation gave rise to analogous results as for the PN: the null model yielded more interactions than observed for TAP and fewer for HMS-PCI. The clustering coefficients for both ccexp and ccnull are much higher than the cc values of the corresponding random networks (Table I). Again, as for the PN, in the CN the clustering coefficients of the experimentally determined networks, ccexp (0.52 and 0.54 for TAP and HMS-PCI, respectively), are still considerably higher than the simulated one, ccnull (0.30 and 0.28, respectively). The difference ccexp - ccnull is nearly the same for the two network types, PN and CN.
The higher clustering in both the experimental PN and CN in comparison to the "null model" indicate that the latter does not fully determine the PN and CN topology. The finding that even at the higher-level of the CN the experimental cluster coefficient, ccexp, is considerably higher than the simulated one, ccnull, appears to point to a kind of "super-clustering." In fact, protein complexes are not random aggregates of subunits but represent functional entities that perform specific cellular functions. Moreover, as recently suggested, complexes that perform similar cellular roles and belong to the same functional group (such as cell cycle, mRNA metabolism, transcription, etc.) extensively share proteins (8, 9, 12, 13). Gavin et al. (8) proposed 9 and Mewes et al. (12) 11 of such functional groups.
The "Stage One Model"—
To account for the bias introduced by the sharing of proteins between functionally related complexes, we extended the "null model" to a "stage one model." Herein, the pool of ng proteins (ng1TAP = 1,650, ng1HMS-PCI = 1,800) from which the complex subunits are drawn is now divided into gr functional groups of equal size. With a high probability pr we took the "proteins" for a given complex from the same group to capture the finding that a complex with a certain cellular function contains mostly proteins that have been assigned to the same functional category. The connectivity distribution of the corresponding PN extracted from the "stage one model" with gr = 10, pr = 0.9 is shown in Fig. 2 (crosses). In the case of the PN, when compared with the null model the new model only slightly changes the distribution of k by shifting the weight to the tail. In the case of the CN, the stage one model increases the decay of the exponential distribution of connectivity in CN as compared with the null model (Fig. 3) and strongly decreases I (Table I). This finding suggests that the increased promiscuity of complexes is not associated with an increase of new links between previously unconnected complexes, but instead results from the increase of number of links between already connected complexes, reflecting the sharing of multiple proteins.
With gr = 10, pr = 0.9, the cluster coefficient for the stage one model, ccone, is only slightly (but significantly) higher than ccnull but still fails to produce the observed high value of ccexp of PN and CN. However, with higher values of pr and gr the clustering coefficient increases; for pr = 0.99 and gr = 100 the clustering coefficient reaches the experimental values, ccone ccexp. Taking into account the combination of "functional" and spatial cellular compartmentalization, gr = 100 may not be an overestimation, because Ho et al. discriminated 34 functional and 15 spatial groups (9) and Schwikowski et al. discussed 42 functional and 9 spatial groups (13).
Comparison with Other Protein Interaction Networks—
As mentioned above, in contrast to the CN, the PN depend on the assumed definition for protein interactions (see "Experimental Procedures"). Table II shows that the PNs corresponding to our "small" and "medium" definition do also belong to the class of small-world networks. The clustering coefficients cc of these PNs are more than one magnitude higher than that of the corresponding random networks cccrn. Note that the clustering coefficient of random networks equals the connectance of these networks: cccrn = C. However, it is remarkable that the TAP networks have a higher cc than the HMS-PCI networks.
For the sake of comparison, we also add the analysis of three other protein interaction data sets: i) Y2H data for yeast protein interactions, as analyzed in (5) (data is available at www.nd.edu/
networks/cell), ii) the carefully curated yeast protein interaction data set discussed in Ref. 2 (data is available at dip.doe-mbi.ucla.edu), and iii) protein interactions of the human signal transduction network of the TRANSPATH data base (data is available at www.transpath.de). Again, these PNs have much higher clustering coefficients than their corresponding random matrices (Table II).
It has been claimed that protein networks are of the scale-free type (5), i.e. the distribution of the number of connections per protein should follow a power-law. In contrast, we have shown that both the distributions for the PN ("large" definition, Fig. 2), and the CN (Fig. 3) clearly follow an exponential law p(k) exp(-ak). For all the networks analyzed in Table II, the corresponding distributions are between a power-law and a pure exponential law: all these connectivity distributions follow a stretched exponential distribution: p(k) exp(-akb), with b < 1.
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DISCUSSION |
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TOPABSTRACTEXPERIMENTAL PROCEDURESRESULTSDISCUSSIONREFERENCES |
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In contrast to others (5, 14, 15), we cannot affirm that protein networks are of the scale-free type. We find that the distributions of the number of connections per protein clearly follow an exponential law, or a stretched exponential law. This may have implications for the evolution of protein networks because scale-free networks need some preferential attachment to arise; without preferential attachment exponential networks emerge (16). The protein complex networks that do not rely on special definitions also show the exponential connectivity distribution. Our null model reveals that this is mainly due to the exponential distribution of the number of different protein complexes of a given size. However, in order to explain the high clustering, both in PNs and CNs, we had to expand the null model. Although the pr and gr values are somewhat arbitrary, as is the ontological classification of proteins into functional groups, the stage one model reveals an interesting property of protein complexes: The ingredient to be added to the minimal null model to reproduce the high clustering coefficients observed in the PN and the CN is the massive overlap of protein subunit usage by the complexes, caused by the use of highly similar combinations of proteins in complexes with similar cellular roles. This extensive promiscuity between complexes is what gives rise to high clustering coefficients in the network topology, and thus to the impression of modularity.
We have shown that the statistical properties of the TAP and HMS-PCI data slightly differ in some details, especially the HMS-PCI data contain more large complexes. This may be due to the fact that the HMS-PCI data were obtained by overexpressing the tagged protein, which could have resulted in increased chance of pulling out weakly interacting proteins. Furthermore, in the TAP data the complexes are purified in a two-step procedure, in contrast to the one-step procedure used by HMS-PCI, which could also contribute to a finding of weaker interactions by HMS-PCI. Interestingly, the perhaps weaker interactions mainly occur in larger protein complexes. Knowledge-based analysis of the two data sets, TAP and HMS-PCI, showed that in HMS-PCI the bait often copurified complexes of independent origin, resulting in larger complexes. In contrast, TAP yielded single complexes in most cases. These complexes often consist of only core complexes with some already biochemically and immunologically characterized subunits or auxillary proteins missing (data not shown). Consider, for example, the biochemically and immunologically purified complex replication factor C (RFC), which has 5 subunits called RFC1 through RFC5. In TAP, the bait protein RFC2 pulled down RFC3 and RFC4 and one additional protein, EFD1 (supplemental data of Ref. 8). The same bait protein, RFC2, also copurified RFC3 and RFC4 in HMS-PCI, but pulled down 13 additional proteins as well (supplemental data of Ref. 9). These questions must be examined more carefully in future studies. However, with our simple null and stage one model we can reproduce the main features of the underlying protein networks. Further refinements of these models can be done for more consistent future data.
Our results show that the graph-theoretical analysis of clustering and modularity in the topology of protein interaction networks needs to take into account the observed physical modules (complexes) and their particular organization into functional modules and higher-order (complex-complex) networks by the shared usage of proteins. These aspects are lost in the usual graph representation of protein networks. The importance of the analysis of physical protein complexes is also underscored by the demonstration of the exponential distribution of the number of different complexes of a given size that reflects the physicochemical dynamics of complex formation and dissociation. This has been shown already for the exponential distribution of the number of domains in proteins (17). Higher quality, exhaustive protein complex data in the near future will allow one to translate the topological maps of biochemical networks that contain potential interactions into complexes defined by actual physical interactions.
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ACKNOWLEDGMENTS |
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FOOTNOTES |
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Published, MCP Papers in Press, May 23, 2003, DOI 10.1074/mcp.M300005-MCP200
1 The abbreviations used are: TAP, tandem affinity purification; HMS-PCI, high-throughput mass-spectrometric protein complex identification; PN, protein-protein interaction network; CN, protein complex interaction network. 
* This work was supported by Grant 0312704E of the Bundesministerium für Bildung und Forschung and Grant SFB 604 of the Deutsche Forschungsgemeinschaft.
To whom correspondence should be addressed: Institute of Molecular Biotechnology, Beutenbergstrasse 11, D-07745 Jena, Germany. Tel.: 49-3641-656208; Fax: 49-3641-656191; E-mail: wilhelm{at}imb-jena.de
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REFERENCES |
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TOPABSTRACTEXPERIMENTAL PROCEDURESRESULTSDISCUSSIONREFERENCES |
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(2002) Protein interactions. Two methods for assessment of the reliability of high throughput observations.
Mol. Cell. Proteomics
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