Introduction

Recent advances in the mathematical theory of random graphs (Watts 1999; Watts and Strogatz 1998) have led to an explosion of research concerned with the architecture of biological networks (Albert et al. 2000; Barabasi and Albert 1999; Bhalla and lyengar 1999; Jeong et al. 2000, 2001; Wagner 2000, Wagner 2001; Wagner and Fell 2001; Watts 1999). This research has shown that the topology of many biological networks, including metabolic networks (Jeong et al. 2000; Wagner and Fell 2001) and protein-interaction networks (Jeong et al. 2001; Wagner 2001), share two important features. First, they have a small diameter, L, defined as the shortest path between network nodes, averaged over all nodes. Second, the frequency, P(D), of nodes with D immediate neighbors is a broad-tailed distribution consistent with a power-law, i.e., P(D) ~ D −γ, with a constant γ characteristic of the network (Jeong et al. 2000; Wagner and Fell 2001).

The discovery of power-laws and small-worldness has given rise to an intriguing hypothesis relating large-scale network structure to mutational robustness (Albert et al. 2000). This hypothesis rests on the observation that random node removal leaves the diameter of networks with power-law connectivity largely unchanged. In a biological network, node removal corresponds to synthetic-null or gene-knockout mutations. In contrast to networks with power-law connectivity, more homogeneous random networks respond to node removal with a rapidly increasing diameter (Albert et al. 2000). This striking structural stability has led to the suggestion that we observe networks with power-law connectivity in cells because of their robustness to random node removal (Albert et al. 2000; Jeong et al. 2000). However, power-law networks are extremely dependent on the small number of highly connected nodes: removal of these can lead to huge changes in network architecture and diameter.

Do small network diameters matter to cells? A possible advantage of small mean path lengths in metabolic networks stems from the importance of minimizing transition times between metabolic states in response to environmental changes (Edwards and Palsson 2000). Metabolic networks with small diameters thus might adjust more rapidly to environmental change. Answering this question by direct experimentation, however, is currently impossible, for doing so would require comparing biological networks of different large-scale structure in vivo. Absent direct experimental tests showing whether genetic network diameter matters to organisms, one can still test key predictions of this hypothesis. One such prediction is that highly connected proteins should be more important to the cell and, thus, be subject to severe selective and evolutionary constraints. One way to test this prediction is to study the rate of evolution of proteins in these networks. We examine this rate in two genetic networks: the Saccharomyces cerevisiae protein-interaction network (Ito et al. 2001; Uetz et al. 2000; Wagner 2001) and the Escherichia coli metabolic network (Wagner and Fell 2001).

Several recent papers have studied the relationship between a gene product’s position in protein-interaction networks and its rate of evolution. Using 164 genes in the yeast protein-interaction network and their Caenorhabditis elegans orthologs, Fraser and coauthors (2002) found a weak negative correlation between protein distance and the degree of connectedness of network proteins. Two recent analyses used the more closely related yeast S. pombe as a reference taxon but different protein-interaction data to construct networks (Fraser et al. 2003; Jordan et al. 2003a). Fraser and coauthors again found a significant, though weak, negative correlation between network degree and rate of evolution. Jordan and coauthors, however, found no significant correlation in either their original analysis or their later corrected analysis (Jordan et al. 2003b). Below, we attempt to resolve this controversy using several reference taxa, including a very recent ancestor of S. cerevisiae, S. paradoxus (Kellis et al. 2003), as well as multiple types of protein-interaction data.

Materials and Methods

Constructing Cellular Networks

Wagner and Fell (2001) represented the E. coli core metabolic network as a graph in which two proteins are connected if a reaction product of one is a substrate for the other. In other words, proteins are connected if they share at least one metabolite. We define the connectedness, D, of a protein in this network as the number of proteins it is connected to. The resulting metabolic network is comprised of 237 distinct proteins with a range of connectedness, D, from 1 to 46 (Wagner and Fell 2001).

In our analysis of the Saccharomyces cerevisiae protein-interaction network, our primary dataset (hereafter Pooled Data I) consists of three independent, large-scale datasets on pairwise protein interactions, two of which were generated using the yeast two-hybrid assay (Ito et al. 2001; Uetz et al. 2000). The third dataset was generated using various non-two hybrid techniques (Mewes et al. 1999). In our graph representation of this network, nodes (proteins) are connected to other proteins if they interact in vivo according to the experimental data. We define the connectedness, D, of a protein as its number of protein interaction partners. The protein-interaction data we use contain enormous amounts of experimental noise (Ito et al. 2001; Uetz et al. 2000; von Mering et al. 2002). However, the relevant feature of the network, the distribution of the number of interaction partners per protein, is robust. It is identical in all three datasets, and highly connected proteins in one dataset are also highly connected in the others (Wagner 2002). We are thus confident that the broad-tailed connectivity distribution is not an artifact of experimental technique.

For comparative purposes, we also study the filtered interaction dataset of von Mering and coauthors (2002; hereafter Pooled Data II), where we have accepted only interactions annotated with “medium” or “high” confidence by these authors. This is a dataset very similar to that of Fraser and coauthors (2003) and includes information on pairwise protein interactions, as well as information on the composition of purified protein complexes (Gavin et al. 2002; Ho et al., 2002). We also pooled two datasets from the high-throughput spectrometric identification of purified protein complexes (Gavin et al. 2002; Ho et al., 2002) for a third dataset, which is a subset of Pooled Data II. Self-interactions were excluded from the data of Gavin and coauthors, and we used the filtered data presented by Ho and coauthors.

To compare the degree distributions of the protein complex data and Pooled Data II datasets to that of Pooled Data I, we used a maximum likelihood approach to fit each dataset to the continuous power-law distribution described by the probability density function:

To determine the significance of the differences in the power-law exponent b seen between datasets, we employed a likelihood ratio test. The test is based on comparing the likelihood of Pooled Data II and protein complex data under the constraint that these two datasets have the same degree distribution as Pooled Data I, to the maximum likelihood estimate of the degree distribution for the two datasets. In terms of Eq. (1), we estimate the maximum likelihood value of b for Pooled Data I, referred to hereafter as b I. We then calculate the likelihood L c of observing the Pooled Data II and the protein complex data with b fixed at b I. This constrained likelihood will never be greater than the likelihood L of observing either dataset when b is allowed to take on its maximum likelihood value. The ratio of the likelihoods L/L c between the unconstrained (L) and the constrained (L c) models is a measure of whether Pooled Data II and the protein complex data follow a different degree distribution than does Pooled Data I. To gauge the statistical significance of L/L c, we first simulate 10,000 sample degree distributions where b = b I. Each of these simulations has the same number of data points as either Pooled Data II or the protein complex data but follows the degree distribution of Pooled Data I. We then calculate L/L c for these simulated datasets (because the simulations have finite size, the maximum likelihood estimate [MLE] for b in a simulation will almost never be exactly b I). Performing this calculation for all 10,000 simulations yields a distribution of L/L c, which indicates how much L may differ from L c due only to chance. If L/L c for our real data does not fall within the range of values seen in the simulations, then the two datasets have significantly different distributions.

We discuss results in detail for Pooled Data I. However, we note that our results are qualitatively identical when separately considering each of the datasets contained within Pooled Data I.

Estimating Evolutionary Rate

We were interested in whether the connectivity (number of interaction partners) of a gene’s product affects that gene’s rate of sequence evolution. We used two related approaches to study this question. First, we identified genes in the yeast network with closely related duplicates and used the ratio K a/K s of amino acid replacement to silent substitutions (Kimura 1977) to measure selective constraint. Specifically, we searched the genome of S. cerevisiae (Goffeau et al. 1996) for the nearest paralog (if any) of each gene in the yeast protein-interaction network (an intragenome search). The E. coli genome (Blattner et al. 1997) has too few duplicate genes for an intragenome analysis. Our second approach uses reference genomes to find orthologs of network genes in both the E. coli and the yeast networks. In S. cerevisiae we identified orthologous genes in the genomes of Schizosaccharomyces pombe (Wood et al. 2002) and Saccharomyces paradoxus (Kellis et al. 2003). For E. coli, we used Haemophilus influenzae (Fleischmann et al. 1995) as the reference species. We estimated the value of K a, the rate of amino acid replacement substitutions for each pair of orthologs, allowing us to determine the correlation between K a and protein degree. Because S. paradoxus and S. cerevisiae are very closely related (Kellis et al. 2003), we were also able to estimate the synonymous distance (K s) between orthologs for this comparison. To avoid mistaking gene duplicates for truly orthologous genes in all these between-genome comparisons, we retained only gene pairs where neither species contained any duplicates.

We used our previously described tool, GenomeHistory (Conant and Wagner 2002), to identify orthologs and paralogs. GenomeHistory uses gapped BLASTP (Altschul et al. 1997) to identify related genes which are globally aligned using ClustalW (Thompson et al. 1994). K a and K s are calculated by maximum likelihood estimation under the model of Goldman and Yang (1994). We accepted all BLAST hits with E-values less than 1 × 10−8 for the S. cerevisiae–S. pombe/S. paradoxus comparison and 1 × 10−7 for the E. coli–H. influenzae comparison. We further required 40% global amino acid identity between any two putative orthologs or paralogs. For the estimation of K a and K s, we removed gap residues, calculated base frequencies separately at each codon position, and estimated the transition/transversion ratio from the data.

Using the identified paralogs and orthologs, we calculated the correlation between evolutionary distance and the degree of protein connectivity (D). For the S. cerevisiae paralogs and orthologs in S. paradoxus, we calculated the correlation coefficients (both Pearson and Spearman) between K a/K s for the closest paralog and D using only unsaturated duplicate pairs with K s < 3. For the remaining two sets of orthologs (S. pombe and H. influenzae) we calculated the correlation coefficients between K a and D (using only pairs where K a < 1). We were unable to use K a/K s ratios for the latter two analyses, as most K s values showed saturation (see below). Excluding pairs with high divergence values (K s > 3, K a > 1) not only avoids the high variances associated with estimating such values, but also serves to exclude spurious gene pairs where homology is uncertain.

To determine if a relationship between selective constraint and degree of connectivity of yeast proteins depended upon protein function, we used a previously described high-level functional classification of the yeast genome (Conant and Wagner 2002) based on the Gene Ontology database (The Gene Ontology Consortium 2000). This classification system groups annotations into 10 high-level categories, which we have further reduced into five overall groups: (1) All Metabolism (includes Energy, Metabolism, and Protein Metabolism),(2) Cell Cycle/DNA Processing, (3) Cytoskeleton, (4) All Transcription (includes Transcription and Transcription Factors), and (5) Transport. Using the S. paradoxus orthologs, we calculated the statistical association within these five groups between D and K a/K s. In addition, we repeated the above intra- and intergenome analyses excluding ribosomal proteins and found that so doing did not change our results (not shown).

Significance tests for Pearson’s product–moment correlation require normally distributed data, an assumption violated by the broad-tailed distribution of the network degree, D (Wagner 2001; Wagner and Fell 2001). We thus adopted a permutation approach to test for significant correlations. Degree observations were randomly reordered with respect to evolutionary distances and the resulting Pearson and Spearman correlations calculated. The correlations from the original data were then compared to a distribution of correlations obtained from 10,000 permutations of those data.

Identifying Gene Pairs with Saturated Synonymous Substitutions

Saturation is a problem when estimating the fraction of synonymous substitutions, K s, that occurred between two genes. Saturation occurs when every synonymous site has undergone at least one substitution, and it makes accurate estimates of K s impossible. However, because K s gives only the average number of synonymous substitutions per synonymous site, it is misleading to speak of saturation occurring at a certain value of K s (such as K s = 1). Instead, one must also account for the degree of variation among sites and the sequence length. Analytical calculation of the probability of saturation is possible for simple evolutionary models such as the Jukes–Cantor model (Li 1997) but cannot be done easily for the codon-based models needed to calculate K s. Instead, we use a heuristic saturation test. It relies on the fact that we have obtained maximum likelihood estimates of (MLEs) K s, such that (for unsaturated sequences) no other value of K s will give a higher likelihood than the MLE. Saturated sequences may have undergone an arbitrary number of substitutions and thus have no unique MLE value of K s. Put differently, under saturation there should be no decrease in likelihood if the divergence (K s) of two sequences increases beyond the MLE estimate. This concept is illustrated in Fig. 1 In practice, two distinct cases of saturation must be distinguished. First, if two sequences are “exactly” saturated (such as genes whose sequence identity equals exactly 25%), then the likelihood, L, no longer changes past some critical value K s. Second, if the sequences are “oversaturated,” such as when a (chance) nucleotide identity of less than 25% is observed, then L will asymptotically increase as K s approaches infinity (see Fig. 1B). We take advantage of these facts by calculating, for each sequence pair, the MLE K s value (determined by numerical optimization and therefore always finite), as well as the likelihood of a K s value 10 times larger. To use our numerical analysis method, we make the implicit assumption that the likelihood function is free of local maxima in the neighborhood of the MLE. If the likelihood of (10K s) is greater than or equal to the likelihood of K s, then we call a sequence pair saturated and exclude it from further analysis.

Figure 1
figure 1

Diagrammatic view of the principle upon which our heuristic test for saturation in K s is based. A In cases where K s is not saturated, there exists a distinct likelihood maximum which can be identified numerically. B When synonymous substitutions have saturated, the likelihood will remain constant or actually increase (the illustrated case) as K s approaches infinity.

Full size image

Results and Discussion

The Yeast Protein-Interaction Network

In the S. cerevisiae protein-interaction network, two proteins are neighbors if they physically interact in vivo. Our primary dataset consists of data (Pooled Data I; see also Materials and Methods) on such pairwise interactions from large-scale two-hybrid experiments (Ito et al. 2001; Uetz et al. 2000) and from experiments not using the two-hybrid approach (Mewes et al. 1999). We first calculated K a/K s (the ratio of amino acid replacement to silent substitutions per site) between each network protein and its closest paralog in the yeast genome. K a/K s is the most reliable indicator of selective constraint on protein evolution because it accounts for mutation rate variation among genes. For the paralog comparison, we used a total of 65 gene pairs, with 1 ≤ D ≤ 13 and 0.006 ≤ K a/K s ≤ 0.74. Figure 2A shows that there is a weakly significant Pearson’s correlation between protein degree D and K a/K s but no significant Spearman’s correlation (Pearson’s r, −0.187, p = 0.047; Spearman’s s, −0.151, p = 0.12). Using paralogs for this analysis has shortcomings, including the fact that paralogs may diverge at different rates and that recent duplicates (rare in our data) diverge at rates different than old duplicates (Lynch and Conery 2000). In addition, the range of D values in this dataset is quite small.

Figure 2
figure 2

Connectivity (D) in the yeast protein-interaction network (Pooled Data I) vs. selective constraint (K a/K s) of a gene and (A) its nearest paralog or (B) its ortholog in the S. paradoxus genome.

Full size image

To alleviate these shortcomings, we carried out complementary analyses using not paralogous but orthologous genes from various species. We first used the recently published genome sequence of the yeast species S. paradoxus, which is closely related to S. cerevisiae (Kellis et al. 2003). Use of this species allows us to compare protein connectivity, D, to the ratio K a/K s for many more orthologous genes than the 65 paralogs above. Specifically, we identified 1393 orthologous gene pairs with protein-interaction information from Pooled Data I. In these data, D ranges between 1 and 76, while K a/K s ranges between 0.003 and 0.64. There was a weak but highly significant correlation between D and the selective constraint (K a/K s) experienced by a gene (r, −0.10; s, −0.11; p < 0.0001 for both) (Fig. 2B and Table 1). (Comparing just K a to D in S. paradoxus using Pooled Data I produces a similar Spearman correlation: s, −0.10.)

Table 1 Correlations between connectivity and selective constraint in yeast
Full size table

We also used a reference taxon other than S. paradoxus to compare our results with data recently presented by other researchers. (Fraser and coauthors 2002, 2003), using either C. elegans or S. pombe as a reference taxon, found a weak but significant negative correlation between amino acid distance and D.These results are in contrast to those of Jordan and coauthors 2003a, b), who found no such correlation. We examined the relationship between D and amino acid distance (measured by K a) in different protein-interaction datasets with S. pombe as a reference. Using our Pooled Data I, which includes no data from purified protein complexes, we found no relationship between D and K a (r, −0.02 [p = 0.27]; s, 0.04 [p = 0.17]; n = 605), a result in accord with Jordan and coauthors, who used a similar dataset. In contrast to these authors, Fraser and coauthors (2003) used data from multiple experimental methods including mass spectrometry of purified protein complexes in their analyses. We analyzed a similar dataset from von Mering and collaborators (2002; Pooled Data II), which contains both data on pairwise protein-interactions and data from protein complexes. Analyzing 1175 gene pairs from S. cerevisiae and S. pombe with this dataset, we indeed found a correlation similar in magnitude to that obtained by Fraser and coauthors. (r, −0.21; s, −0.22; p < 0.0001 for both).

We are thus able to explain the discrepancy in results between (Jordan and coauthors 2003a, b) and Fraser and coauthors 2002, 2003) by using slightly different protein-interaction datasets. The reference taxon used by both of these groups, S. pombe, is less than ideal because it is only a distant relative of S. cerevisiae, with a most recent common ancestor 0.3–1.3 billion years ago (Wood et al. 2002). We thus repeated the analysis with the much more closely related S. paradoxus as the outgroup. In addition to the identification of many more orthologs, use of S. paradoxus allows us to control for mutation rate biases among genes by using the ratio K a/K s instead of simply K a. Pooled Data II (cf. Fraser et al. 2003) again yields significant correlations (r, −0.14; s, −0.24; p < 0.0001 for both) (Fig. 3A and Table 1) that are stronger than that using our Pooled Data I (r, −0.10; s, −0.11; p < 0.0001 for both; see also Table 1). The major difference between Pooled Data I and Pooled Data II is the inclusion of protein interactions from purified protein complexes in the latter. We therefore estimated the statistical association of K a/K s and connectivity, D, when D was computed strictly from the protein complex data of Gavin and coauthors (2002) and Ho and coauthors (2002) included in Pooled Data II. These data suggest a stronger (Spearman’s) association than does our Pooled Data I (r, −0.04 [p = 0.18]; s, −0.14 [p < 0.0001] (Table 1). Figure 3B illustrates the effect of using either Pooled Data II or just the protein complex data (compare to Fig. 2B). We conclude that the strong correlations seen by (Fraser and coauthors 2002, 2003) and our own analyses of Pooled Data II data are due at least in part to the inclusion of data from purified protein complexes.

Figure 3
figure 3

Effects of dataset selection on the magnitude of statistical association between D and K a/K s (measured with S. paradoxus orthologs). A Protein interaction data taken from the medium- and high-confidence interactions of von Mering and coauthors (2002; Pooled Data II). B Protein interaction data taken from mass spectrometry experiments on purified protein complexes (Gavin et al. 2002; Ho et al. 2002) included in Pooled Data II.

Full size image

Mass spectrometry assays such as those of Gavin and coauthors (2002) and Ho and coauthors (2002) identify all members of a protein complex, rather than direct interactions between two proteins. Such data are less appropriate for our analysis because many member proteins of a large complex need not interact directly and because any one protein need not interact with a large number of other proteins to be part of a large complex. We also note that distributions of protein-interaction connectedness have significantly different statistical properties if data from purified complexes are included. We calculated maximum likelihood estimates (MLEs) for the power-law exponent b for both Pooled Data I and Pooled data II, as well as for data stemming only from purified complexes (see Materials and Methods and Eq. [1]). The purified complex data and the Pooled Data II have b = 1.64 and b = 1.71, respectively, values significantly different from the MLE of our Pooled Data I (b = 2.74, p < 0.0001) (Table 1). Qualitatively, this implies that there are many more genes with a large number of interactions in the purified complex data, exactly as one would expect from such data. We suspect that analyses using protein-complex data may therefore overstate the magnitude of the association between evolutionary constraint and network connectivity.

The E. coli Metabolic Network

The E. coli core metabolic network encompasses the catabolic and biosynthetic metabolism central to the cell’s function. Wagner and Fell (2001) constructed a graph representation of this network in which genes are connected by an edge if the chemical reactions their products catalyze share at least one substrate. We measured K a by comparing genes in the network to their orthologs in the closely related Haemophilus influenzae genome (see Materials and Methods). Of the 133 genes in the E. coli metabolic network with related genes in H. influenzae, we used only gene pairs that have no paralogs in either species. Figure 4 shows the relationship between connectedness and K a for the E. coli genes and their orthologs (Pearson’s r, −0.056; Spearman’s s, 0.146; n = 108; 1 ≤ D ≤ 41; 0.111 ≤ K a ≤ 0.731). Permutation analyses indicate that neither correlation is significant (r, p = 0.29; s, p = 0.067).

This dataset includes many fewer genes than the protein-interaction comparisons and may, therefore, suffer from reduced statistical power. However, the lack of a correlation between selective constraint and degree of connectedness in the E. coli metabolic network is consistent with the results of (Fraser and coauthors 2002, 2003) for the protein-interaction network. These researchers found that the weak negative correlation between evolutionary rate and protein connectivity was not due to the fact that highly connected genes are more important to the organism but, instead, due to the greater proportion of a highly connected protein’s surface that is in contact with other proteins. In a metabolic network, where genes are connected not through physical contact but by shared metabolites, there should be no such effect. This is exactly what we observe.

Figure 4
figure 4

Amino acid sequence divergence (K a) from H. influenzae orthologs vs. connectivity (D) in the core intermediary metabolic network of E. coli (Wagner and Fell 2001).

Full size image

How Strong Is the Effect of Protein Connectivity on Selective Constraint?

Many factors may contribute to differences in evolutionary rates among genes. Rates may differ simply because of differences in structure or function; other candidate factors determining rates of evolution include physical position in the genome (Williams and Hurst 2000), expression level (Akashi 2001), and “essentialness” (Hurst and Smith 1999), as well as the estimation errors introduced by gene length and GC-content (Kumar and Subramanian 2002). This makes it difficult to measure the effects of a single factor on selective constraint. In order to determine the influence of several factors other than connectivity, we carried out two further analyses. First, we used an analysis of variance (ANOVA) for our primary yeast dataset, Pooled Data I, and for the E. coli data. Each ANOVA included degree of connectedness (D), codon adaptation index (CAI; a correlate of expression level [Akashi 2001]), gene length, and GC-content as separate effects in the model (all analyses carried out in JMP, SAS Institute, Inc.). Second, we broke the yeast Pooled Data I protein-interaction dataset into five major functional categories to see whether there were function-specific correlations (see Materials and Methods for details).

As in the above analysis, only the yeast ortholog comparison showed a significant effect of protein connectedness on selective constraint in the ANOVA for Pooled Data I (S. cerevisiae–S. paradoxus [F = 16.2, p < 0.0001], S. cerevisiae paralogs [F = 1.7, p = 0.19], E. coli–H. influenzae orthologs [F = 0.05, p = 0.82]). CAI and GC-content also had significant effects in the Pooled Data I ortholog comparison (CAI: F = 59.5, p < 0.0001; GC-content: F = 7.1, p = 0.008), but only CAI was a significant effect in the other two comparisons (yeast paralogs: F = 4.56, p = 0.037; E. coli–H. influenzae orthologs: F = 18.1, p < 0.001). The fact that CAI is associated with evolutionary constraint is not surprising, since it is known from a number of organisms that expression level and evolutionary rate are negatively correlated (Akashi 2001). Degree of connectedness and CAI were significantly correlated with each other in the E. coli metabolic network (r = 0.168, p = 0.08; s = 0.265, p = 0.006) (highly connected genes were expressed at higher levels), but they were not significantly correlated in the yeast protein-interaction network. The full ANOVAs have controlled for many effects in attempting to detect the correlation between connectivity and selective constraint; nonetheless, they explain very little of the variation in K a/K s. The R 2 for the full Pooled Data I S. cerevisiae—S. paradoxus ANOVA is 0.061; using only connectivity as an effect in the model gives an R 2 = 0.01.

Using S. cerevisiae–S. paradoxus orthologs taken from Pooled Data I, we separated genes by function and calculated individual correlations for five major categories: cytoskeleton, metabolism, transport, cell cycle/DNA processing, and transcription. Interestingly, the relationship between network degree and evolutionary constraint is highly dependent on a gene’s function. Genes involved in metabolism, transport, and the cytoskeleton show no significant relationship between D and K a/K s (p always > 0.05) (see Table 2). However, genes involved in the cell cycle and transcriptional processes show a significant, though weak, effect (s: −0.20 and −0.25) (Table 2). The overall correlation that we observe for these data, therefore, is an average of these two cases, as the cell cycle and transcription correlations are higher than the overall correlation (s: −0.11) (Table 1).

Table 2 Function-specific correlations between connectivity and selective constraint in the yeast protein-interaction network
Full size table

Is there any reason to expect certain functional categories to show stronger correlation than others? The connectivity distributions do not consistently differ between the categories (Table 2), so there do not appear to be systematic differences in the numbers of interactions the genes in each category take part in. One possibility is that the divergent roles proteins take on in the different categories are responsible for this disparity. For instance, the functions of metabolic enzymes are defined by their specific interactions with small molecules, whereas the function of transcription factors and cell-cycle regulators often involves mediation of protein interactions in larger protein complexes. In this context, it is surprising that cytoskeletal proteins do not show a strong effect, but we also note that these proteins constitute our smallest sample, rendering statistical inference more difficult.

Are Highly Connected Proteins Especially Important to the Cell?

Biological networks are extremely robust to perturbation (Albert et al. 2000; Edwards and Palsson 2000; Hartwell et al. 1999; Jeong et al. 2001; Wagner 2000; Watts 1999). Both drastic environmental changes and loss-of-function mutations in a variety of genes often have no detectable effect on the phenotype of an organism, even if the lost gene is considered a priori to be important for cellular activity (Ross-Macdonald et al. 1999; Smith et al. 1996; Winzeler et al. 1999). This robustness may be a function of a network’s architecture: broad-tailed distributions (such as power laws) of the number of interactors observed in genetic networks have been hypothesized to confer robustness against mutations (Albert et al. 2000). However, these networks are vulnerable to loss of highly connected nodes. Here we have tested the robustness hypothesis through one of its key predictions: highly connected proteins should be subject to greater selective constraint. The S. cerevisiae protein-interaction network shows evidence of a significant association, but its magnitude is small. The E. coli metabolic network shows no significant association.

We cannot completely exclude the possibility that genes in these two networks have acquired their functions (and numbers of network interactors) very recently, after the divergence of E. coli from H. influenzae or S. cerevisiae from S. paradoxus. In that case, estimates of selective constraint might reflect past rather than present function. However, the H. influenzae core metabolic network is likely very similar to that of E. coli, not only because of their close evolutionary relationship, but also because core metabolism is nearly universal among free-living nonextremophiles (Edwards and Palsson 1999; Morowitz 1992; Tatusov et al. 1996). In addition, 56% (133) of the genes in the E. coli metabolic network have a similar gene in H. influenzae. This percentage is much larger than the approximately 26% of genes shared overall between the two species (Blattner et al. 1997). S. cerevisiae is closely related to S. paradoxus and shares an even larger set of genes and interactions (Kellis et al. 2003).

We have used K a and K a/K s to test the prediction that highly connected genes are more important to an organism. That is, we have used these measures of evolutionary constraint (the number of mutations tolerable in a coding region) as indicators of the importance of a gene. Of course, alternative indicators of importance are conceivable. For example, removing highly connected genes from a network may be more detrimental than removing less-connected genes, which could indicate the importance of highly connected genes. Evidence for this was obtained from gene knockout data and the protein-interaction network from yeast, where Jeong and coauthors (2001) found that highly connected proteins are much more likely to be essential for survival than less-connected proteins. Using gene-knockout data to assess the importance of highly connected genes or proteins, however, has disadvantages. First, even apparently neutral knockouts may have subtle but undetectable fitness effects. In the huge populations characteristic of microbes like yeast, growth rate differences of 10−6 between mutant and wild-type may be evolutionarily important, but chemostat experiments have difficulty resolving differences smaller than 10−3 (Dykhuizen and Hartl 1983). Second, laboratory experiments cannot assess fitness differences over the entire environmental spectrum important for life in the wild. An apparently neutral knockout mutation in one environment may have severe fitness effects in environments not easily re-created in the laboratory. For these reasons, we think that our results are a more accurate reflection of the importance of genes in a genetic network. Using a protein’s evolutionary rate takes advantage of an evolutionary record of mutations experienced over millions of years, mutations whose effects manifested themselves in the environments the organism experienced during its evolution.

Our results, as well those of others (Fraser et al. 2002, Wagner 2003), show that a gene’s position in a network has only a very limited impact on its importance as defined by evolutionary constraints. There is some controversy as to the exact level of association, with Fraser and coauthors arguing for a stronger relationship in yeast than Jordan et al (2002a, b). We have shown here that the choice of protein-interaction datasets matters when looking for correlations with distant reference taxa but that the choice of datasets has less of an effect when using more closely related organisms. Our analyses of multiple types of protein-interaction data, using a more closely related reference taxon unavailable to earlier authors, come down somewhere in the middle of previous studies, with a very weak but statistically significant association (r or s ≈ –0.1). Our results also show that the correlation between connectivity and evolutionary rate is function-specific: only certain functional classes of proteins show a significant correlation. And they fail to support the claim that power-law connectivity in cellular networks reflects selection for robust network diameters: analysis of a metabolic network, where genes do not physically interact, failed to show any correlation. What, then, can we learn from a purely qualitative, topological analysis of genetic networks? The work of Rausher and coauthors (1999) on anthocyanin biosynthesis genes raises the possibility that a gene’s role in controlling flux through a metabolic pathway may determine its rate of evolution. This suggests that a gene’s position in a network, although uninformative on its own, may become informative when supplemented by additional biological information. Future genetic network analyses that include details on enzyme kinetics and rates of flux across nodes may provide the information needed to better understand how the position, function, and interactions of proteins are likely to influence which genes are targeted by natural selection.