## Multi Node Properties Correlation Profile

The correlation profile of any large complex network quantifies correlations between degrees of its neighboring nodes. We have calculated correlation profiles of:

1. The protein interaction network consisting of 4475 physical interactions between 3279 yeast proteins as measured in the most comprehensive high-throughput yeast two-hybrid screen.8 A subset of this network is shown in Figure 4.

2. The transcriptional regulatory network in yeast (Fig. 2), consists of 1289 (1047 positive and 242 negative) regulations by 125 transcription factors5 within the set of 682 proteins.

3. While the regulatory network is naturally directed, the network of physical interactions among proteins in principle lacks directionality. Randomized versions of these two molecular networks were constructed by randomly rewiring their edges, while preventing "unphysical" multiple connections between a given pair of nodes, as described in the previous chapter. By construction this algorithm separately conserves the in- and out-degrees of each node. Therefore, in a randomized version of the regulatory network each protein has the same numbers of regulators and regulated proteins as in the original network. Taking in consideration the bait-prey asymmetry mentioned in,10 when generating random counterpart of the interaction network we chose to separately conserve numbers of interaction partners of the bait-hybrid and the prey-hybrid of every protein.

The topological property of the network giving rise to its correlation profile is the number edges N(Ko,K{) connecting pairs of nodes with degrees Ko and K\. To find out if in a given complex network the degrees of interacting nodes are correlated, N(Kq,Ki) should be compared to its value NXKq,K\) + ANr(K(S,K]) in a randomized network, generated by the edge rewiring algorithm. When normalized by the total number of edges E, N(Kq,K\) defines the joint probability distribution P[Kq,K\) = N(Kq,K\)IE of degrees of interacting nodes. Any correlations would manifest themselves as systematic deviations of the ratio away from 1. Statistical significance of such deviations is quantified by their Z-score where Or{Ko,K\) = ANr(Ko,K\)/N is the standard deviation of Pr(fQ),fC]) in an ensemble of randomized networks.

Figures 6 and 7 show the ratio R(Ko,K\) as measured in yeast interaction and transcription regulatory networks, respectively. In the interaction network Kq and Ki are numbers of neighbors of the two interacting proteins, while in the regulatory network Ko is the out-degree of the regulatory protein and K[—the in-degree of its regulated partner. Thus by its very construction I\Ka,K\) is symmetric for the physical interaction network but not for the regulatory network. Figures 8 and 9 plot the statistical significance Z(Ko,K/) of deviations visible in Figures 6 and 7 correspondingly. To arrive at these Z-scores 1000 randomized networks were sampled and degrees were logarithmically binned into two bins per decade.

Figure 6. Correlation profile of the protein interaction network in yeast. The ratio R(K(j,Kt)=P(K(i,K])l Pr{KoJ(\), where I\Kf),K\) is the probability that a pair of proteins with Kq and K\ interaction partners correspondingly, direcdy interact with each other in the full set of reference 8 while Pr(KoJ^\) is the same probability in a randomized version of the same network, generated by the random rewiring algorithm described in the text. Note the logarithmic scale of both axes.

Figure 6. Correlation profile of the protein interaction network in yeast. The ratio R(K(j,Kt)=P(K(i,K])l Pr{KoJ(\), where I\Kf),K\) is the probability that a pair of proteins with Kq and K\ interaction partners correspondingly, direcdy interact with each other in the full set of reference 8 while Pr(KoJ^\) is the same probability in a randomized version of the same network, generated by the random rewiring algorithm described in the text. Note the logarithmic scale of both axes.

Figure 7. Correlation profile of the transcription regulatory network in yeast. The ratio R(Kmt,Ki„) = P{Ktm*Km)/PrU(mt>Km)> where P{K0UUK,„) is the probability that a protein node with the out-degree Kmt transcriptionally regulates the protein node with the in-degree Kt„ in the transcription regulatory network obtained from the YPD database5 (Fig. 2), while P^K^pK^ is the same probability in a randomized version of the same network, generated by the random rewiring algorithm described in the text. Note the logarithmic scale of both axes.

Figure 7. Correlation profile of the transcription regulatory network in yeast. The ratio R(Kmt,Ki„) = P{Ktm*Km)/PrU(mt>Km)> where P{K0UUK,„) is the probability that a protein node with the out-degree Kmt transcriptionally regulates the protein node with the in-degree Kt„ in the transcription regulatory network obtained from the YPD database5 (Fig. 2), while P^K^pK^ is the same probability in a randomized version of the same network, generated by the random rewiring algorithm described in the text. Note the logarithmic scale of both axes.

The combination of R- and Z-profiles reveals the regions on the Kq - K\ plane, where connections between proteins in the real network are significandy enhanced or suppressed, compared to the null model. In particular, the blue/green region in the upper right corner of Figures 6-9 reflects the reduced likelihood that two hubs are direcdy linked to each other, while

Figure 8. Statistical significance of correlations present in the protein interaction network in yeast. The Z-score of correlations 7{Kf»K\) = (P(Kt)tK\) - PJJQs,Kl))/or(Ko,K-i), where P(Kq,K{) is the probability that a pair of proteins with Ka and K\ interaction partners correspondingly, dirertly interact with each other in the full set of reference 8 while Pr(Ko,K\) is the same probability in a randomized version of the same network, generated by the random rewiring algorithm described in the text, and ar(A^,A'1) is the standard deviation of Pr(IQy,K\) measured in 1000 realizations of a randomized network. Note the logarithmic scale of both axes.

Figure 8. Statistical significance of correlations present in the protein interaction network in yeast. The Z-score of correlations 7{Kf»K\) = (P(Kt)tK\) - PJJQs,Kl))/or(Ko,K-i), where P(Kq,K{) is the probability that a pair of proteins with Ka and K\ interaction partners correspondingly, dirertly interact with each other in the full set of reference 8 while Pr(Ko,K\) is the same probability in a randomized version of the same network, generated by the random rewiring algorithm described in the text, and ar(A^,A'1) is the standard deviation of Pr(IQy,K\) measured in 1000 realizations of a randomized network. Note the logarithmic scale of both axes.

Figure 9. Statistical significance of correlations present in the transcription regulatory network in yeast. The ratio Z(KouvKJ = (P(K0UVKJ-P^K^/a^ uvKin)' where is the probability that a protein node with the out-degree A"oul transcriptionally regulates the protein node with the in-degree Km in the network from the YPD database,5 while Pr(^ouD-^in) is the same probability in a randomized version of the same network, generated by the random rewiring algorithm described in the text, and <Jr(KoavKin) 's the standard deviation of Pr(Kout>-^in) measured in 1000 realizations of a randomized network. Note the logarithmic scale of both axes.

Figure 9. Statistical significance of correlations present in the transcription regulatory network in yeast. The ratio Z(KouvKJ = (P(K0UVKJ-P^K^/a^ uvKin)' where is the probability that a protein node with the out-degree A"oul transcriptionally regulates the protein node with the in-degree Km in the network from the YPD database,5 while Pr(^ouD-^in) is the same probability in a randomized version of the same network, generated by the random rewiring algorithm described in the text, and <Jr(KoavKin) 's the standard deviation of Pr(Kout>-^in) measured in 1000 realizations of a randomized network. Note the logarithmic scale of both axes.

red regions in the upper left and the lower right corners of these figures reflect the tendency of hubs to associate with nodes of low degree. One should also note a prominent feature on the diagonal of the Figure 6 and 8 corresponding to an enhanced affinity of proteins with between 4 and 9 physical interaction partners towards each other. This feature can be tentatively attributed to members of multi-protein complexes interacting with other proteins from the same complex. The above range of degrees thus correspond to a typical number of direct interaction partners of a protein in a multi-protein complex. When we studied pairs of interacting proteins in this range of degrees we found 39 of such pairs to belong to the same complex in the recent high-throughput study of yeast protein complexes.19 This is about 4 times more than one would expect to find by pure chance alone.

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