Annotated Bibliography

A Single Determinant Dominates the Rate of Yeast Protein Evolution

D. Allan Drummond,* Alpan Raval,�� and Claus O. Wilke§ *Program in Computation and Neural Systems, California Institute of Technology, Pasadena; �Keck Graduate Institute, Claremont; �School of Mathematical Sciences, Claremont Graduate University; and §Section of Integrative Biology and Center for Computational Biology and Bioinformatics, University of Texas at Austin

A gene’s rate of sequence evolution is among the most fundamental evolutionary quantities in common use, but what determines evolutionary rates has remained unclear. Here, we carry out the frst combined analysis of seven predictors (gene expression level, dispensability, protein abundance, codon adaptation index, gene length, number of protein-protein interactions, and the gene’s centrality in the interaction network) previously reported to have independent infuences on protein evolutionary rates. Strikingly, our analysis reveals a single dominant variable linked to the number of translation events which explains 40-fold more variation in evolutionary rate than any other, suggesting that protein evolutionary rate has a single major determinant among the seven predictors. The dominant variable explains nearly half the variation in the rate of synonymous and protein evolution. We show that the two most commonly used methods to disentangle the determinants of evolutionary rate, partial correlation analysis and ordinary multivariate regression, produce misleading or spurious results when applied to noisy biological data. We overcome these diffculties by employing principal component regression, a multivariate regression of evolutionary rate against the principal components of the predictor variables. Our results support the hypothesis that translational selection governs the rate of synonymous and protein sequence evol- ution in yeast.

Introduction

A protein’s evolutionary rate, commonly measured by the number of nonsynonymous substitutions per site in its encoding gene, is routinely used to characterize functional importance, detect selection (Nei and Kumar 2000), create phylogenetic trees (Kurtzman and Robnett 2003), identify orthologous genes (Wall, Fraser, and Hirsh 2003), and infer the time of major evolutionary events. However, what determines a protein’s evolutionary rate has remained the subject of active speculation and ongoing research (Pál, Papp, and Hurst 2001; Akashi 2003; Rocha and Danchin 2004).

Recently, studies have found signifcant infuences on evolutionary rate from many disparate variables: proteins have been reported to evolve slower if their encoding genes have a higher expression level in mRNA molecules per cell (Pál, Papp, and Hurst 2001), if they have a higher codon adaptation index (CAI) (Rocha and Danchin 2004; Wall et al. 2005), more protein-protein interactions (higher ‘‘de- gree’’) (Fraser et al. 2002), shorter length (Marais and Duret 2001), a smaller ftness effect upon gene knockout (higher ‘‘dispensability’’) (Hirsh and Fraser 2001; Yang, Gu, and Li 2003; Zhang and He 2005), or a more central role in inter- action networks (‘‘betweenness centrality,’’ or simply ‘‘cen- trality’’) (Hahn and Kern 2005).

Here, we frst demonstrate that the analytical techni- ques widely used to establish independent roles for many effects, partial correlation and multiple regression, generate highly signifcant but entirely spurious effects given noisy data such as those available for evolutionary analyses. Then, using a technique which does not suffer from these problems, we carry out a comprehensive analysis designed to uncover the major independent correlates of evolutionary rate in the model eukaryote Saccharomyces cerevisiae. We

Key words: Saccharomyces cerevisiae, evolutionary rate, gene expression, protein-protein interactions, dispensability, translational selection.

E-mail: [email protected].

Mol. Biol. Evol. 23(2):327–337. 2006 doi:10.1093/molbev/msj038 Advance Access publication October 19, 2005

� The Author 2005. Published by Oxford University Press on behalf of the Society for Molecular Biology and Evolution. All rights reserved. For permissions, please e-mail: [email protected]

determine the number of such correlates, their strength, and their relationship to the biological variables used in previous studies. Finally, we ask what these correlates re- veal about the biological constraints on protein sequence evolution.

Materials and Methods Genomic Data

We obtained CAIs and evolutionary rates (nonsynon- ymous substitutions per site dN, synonymous substitutions per site dS, adjusted synonymous substitutions dS# [Hirsh, Fraser, and Wall 2005], and ratios dN/dS and dN/dS#) from four-way yeast species alignments for 3,036 S. cerevisiae genes (Wall et al. 2005, supporting information, Table 4). Deletion-strain growth rate data were downloaded from http://chemogenomics.stanford.edu/supplements/01yfh/fles/ orfgenedata.txt; the average growth rates of the homozy- gous deletion strains were used as dispensability measure- ments in our analysis. The fltered yeast interactome data set (Han et al. 2004) provided interaction network hub types for 199 genes and the number of interactions for 1,379 yeast genes. The latter data set was used to compute betweenness- centrality values, which quantify the frequency with which a network node lies on the shortest path between other nodes, as described by Hahn and Kern (2005). Genomic data for Saccharomyces paradoxus and Kluyveromyces waltii were obtained exactly as described by Drummond et al. (2005). Genome sequences for Escherichia coli K12 and Salmonella typhimurium LT2 were obtained from the Institute for Genomic Research (Peterson et al. 2001), with orthologs identifed and evolutionary rates computed exactly as described (Drummond et al. 2005). Gene ex- pression levels for E. coli measured in mRNAs per cell in Luria-Bertani (LB) and M9 media were obtained from Bernstein et al. (2002).

Statistical Analysis

We used R (Ihaka and Gentleman 1996) for statistical analyses and plotting. The package �pls� was used to perform

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328 Drummond et al.

principal component regression. We log transformed all variables except dispensability. We decided whether or not to log transform a variable based on whether log trans- formation led to a higher R2. For those variables that contained zeros, we added a small constant before the log transformation, as previously suggested (Wall et al. 2005). This constant was 0.001 for dN, dS#, and dN/dS# and 10

�7 for betweenness centrality. We scaled the predic-

tor variables to zero mean and unit variance before carrying out the principal component analysis. In all regression anal- yses (both against the original predictors and against the principal components), we determined the statistical signif- icance levels by starting with the full model and succes- sively dropping the least signifcant predictor until only signifcant predictors (P , 0.01) remained.

Results Correlation and Partial Correlation Analysis

We used the yeast S. cerevisiae to examine the deter- minants of evolutionary rate because it has been the subject of many previous analyses (e.g., Pál, Papp, and Hurst 2001; Fraser 2005) and has an enormous amount of available ge- nomic, proteomic, and functional data. We frst examined the raw correlation of six previously assessed biological variables (expression, CAI, length, dispensability, degree, and centrality) with protein evolutionary rate, as measured by the number of nonsynonymous substitutions per site in the underlying gene. A seventh variable, the number of pro- tein molecules per cell (‘‘abundance’’), was also considered. Table 1 shows that all variables except centrality correlated signifcantly with evolutionary rate, as previously reported.

Expression level strongly correlates with evolutionary rate, and higher expressed genes have higher CAIs (Akashi 2001), are less dispensable (Gu et al. 2003), more abundant (Ghaemmaghami et al. 2003), and more likely to be found in protein-protein interaction experiments (Bloom and Adami 2003) than lower expressed genes. No inverse rela- tionships have been posited by which these variables alter the expression level. Thus, it is imperative to establish whether these variables play a role independent of expres- sion level. Following previous analyses (Pál, Papp, and Hurst 2003; Lemos et al. 2005; Wall et al. 2005), we com- puted the partial correlation of our seven variables with

Table 1 Partial Correlation Analysis of Seven Putative Determinants of Evolutionary Rate

Correlation Partial Correlation Variable X rX,dN rX,dNjgene expression VIF

Gene expression �0.537*** 0 2.72 CAI �0.565*** �0.338*** 2.46 Protein abundance �0.478*** �0.232*** 2.05 Gene length 0.136*** 0.010 1.25 Gene dispensability 0.265*** 0.183*** 1.08 Degree (number of �0.246*** �0.127* 1.70

protein-protein interactions) Protein centrality �0.098# �0.082 1.64

(frequency on node-node shortest paths)

# NOTE.— P , 0.01; * P , 10�3; *** P , 10�9 .

evolutionary rate, controlling for expression level. Table 1 shows that CAI, dispensability, and degree all showed re- duced but highly signifcant partial correlations, consistent with previous studies (Hirsh and Fraser 2003; Wall et al. 2005), as did abundance.

Partial Correlations and Noisy Data

What can we conclude from highly signifcant partial correlations? Yeast expression-level measurements from multiple groups, even two using the same commercial ol- igonucleotide array, correlated with coeffcients of only 0.39–0.68 (Coghlan and Wolfe 2000), demonstrating that expression-level measurements are inaccurate and/or simply refect the variability of gene expression across growth conditions and strains. We refer to all such variabil- ity as noise, regardless of its source. Noisy data are the rule in genome-wide molecular studies, leading us to explore what effect noise has on partial correlation analyses. As a concrete example, CAI is so tightly bound to expression level that a recent analysis used CAI as its preferred expres- sion-level measurement (Wall et al. 2005). Might CAI’s signifcant partial correlation only refect our inability to control for the true (i.e., evolutionarily relevant) underlying expression level? More generally, we can ask: what is the expected partial correlation of two variables, controlling for a third, when (1) the two variables relate only through de- pendence on the third ‘‘master’’ variable and (2) all meas- urements contain noise?

Given these conditions, we derive explicit formulas for the expected partial correlation, its statistical signif- cance, and its behavior under various limiting cases in the Appendix. The expected partial correlation is, in gen- eral, larger than zero because the full correlation refects the true underlying master variable’s infuence, while par- tial correlations can only remove the portion of this infu- ence that is visible through a noisy measurement (box 1). We show that, surprisingly, if measurements of an underly- ing causal variable (e.g., expression level) are noisy, highly signifcant partial correlations of virtually any strength be- tween the dependent predictors can be obtained.

As a case in point, dispensability’s role has been vig- orously debated (Hirsh and Fraser 2003; Pál, Papp, and Hurst 2003; Wall et al. 2005) with correlation and partial correlations acting as key analytical tools. Given a model in which expression level X and noise completely determine dispensability D and evolutionary rate K (see box 1), what is the observed partial correlation rDKjX# if we ft variables to approximately match the observed correlations between X#, D, and K? As a concrete example, previous reports show that, using parametric Pearson’s correlations, rX#K ’ �0.6 (Pál, Papp, and Hurst 2001; Wall et al. 2005), rDK ’ 0.25 (Wall et al. 2005), rDX# ’ 0.2 (Pál, Papp, and Hurst 2003), and rDKjX# ’ 0.24 (Wall et al. 2005). We can obtain roughly the reported full correlations and rDKjX# ’

10 �9

0.23 6 0.02, P with 3,000 observations if the true expression level X is normally distributed with mean 0.5 and standard deviation (SD) 0.25, and the observable predictors X#, D, and K are equal to X plus zero mean nor- mally distributed noise with SDs of 0.3, 0.7 and 0.1, respec- tively. This highly signifcant partial correlation is entirely

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A Single Determinant Dominates the Rate of Yeast Protein Evolution 329

Table 2 Results of Principal Component Regression Analysis on Seven Predictors and Five Measures of Evolutionary Rate for 568 Saccharomyces cerevisiae genes

Principal Components

1 2 3 4 5 6 7 All

Percent variance explained in dN 42.76*** 0.05 0.50 0.19 0.14 0.47 0.48 44.60*** dS 50.77*** 2.13** 0.88* 0.08 6.55*** 0.37 1.14* 61.92*** dN/dS 24.82*** 0.05 0.67 0.82 0.00 0.00 0.05 26.42*** dS# 6.70*** 0.19 7.31*** 0.26 0.06 0.14 1.25# 15.92*** dN/dS# 42.34*** 0.09 0.13 0.28 0.13 0.40 0.70# 44.07***

Percent contributions Expression 32.8 1.2 1.5 0.1 1.2 11.2 52.1 CAI 28.3 3.1 8.4 0.9 2.7 17.6 39.0 Abundance 29.2 2.0 1.6 0.3 15.4 51.4 0.1 Length 2.0 1.1 86.4 0.0 2.1 0.3 8.2 Dispensability 1.8 13.0 0.0 84.0 0.0 0.9 0.3 Degree 5.0 36.7 1.9 6.2 38.9 10.9 0.4 Centrality 0.9 42.9 0.3 8.5 39.6 7.7 0.0

# ** *** NOTE.— P , 0.01; * P , 10�3; P , 10�6; P , 10�9. Bold indicates that the indicated predictor contributes at least 20% to

the indicated component.

spurious: in this model, expression level and random noise completely determine dispensability. Thus, the observed statistical relationship between dispensability and evolu- tionary rate, established by correlation and partial correla- tion, would arise even if no actual relationship existed except mutual dependence on noisily measured expres- sion level.

Multivariate Regression Analysis

Because partial correlation analysis is not applicable to the problem at hand, what other methods can we use to de- termine the relative infuence of different predictors on the rate of evolution? One obvious choice is multivariate re- gression analysis, a method with the added beneft that we can look at the infuence of all potential predictor var- iables at the same time and can eliminate step by step those predictors that contribute the least to the regression model. Indeed, several authors have followed this route (Rocha and Danchin 2004; Agrafoti et al. 2005). Regressing dN simul- taneously against the seven predictors we consider here, we fnd that all but centrality make a signifcant contribution to the regression and that the overall R2 5 0.45.

Unfortunately, ordinary multivariate regression is not appropriate to analyze the infuence of the various predic- tors on the evolutionary rate either (box 1). The problem is that the predictors intercorrelate, while multivariate regres- sion implicitly assumes that the predictors are statistically independent. This problem is widely discussed in the sta- tistical literature, mostly in the context of ‘‘collinear’’ or ‘‘nearly collinear’’ predictors (Gunst and Mason 1977a, 1977b; Mandel 1982; Næs and Martens 1988). The vari- ance infation factor (VIF) may be used to quantify the de- gree of predictor collinearity, and table 1 reports VIFs for our data. These VIFs indicate some collinearity but are not high enough to raise signifcant concerns. However, for our toy model (box 1) in which the two predictors refect the same underlying variable plus noise, the VIFs are only 1.21 in both cases, yet the analysis demonstrates that mul- tivariate regression and partial correlation break down any-

way. Collinearity and noise work together to undermine these techniques.

Principal Component Regression Analysis

An alternative approach is to frst identify the indepen- dent sources of variation in the data, and then determine the contribution of each biological predictor to each source. The technique of principal component regression offers a standard way to carry out such an analysis.

In principal component regression (Mandel 1982), multiple linear predictors (e.g., expression level and dis- pensability) are scaled to zero mean and unit variance, inserted in a matrix, and rotated such that the new coordi- nate axes point in the directions of greatest predictor var- iation. The new axes defne variables, called principal components, which are linear combinations of the original predictors. Subsequent linear regression of the response (e.g., dN) on the rotated predictor data yields several pieces of information per principal component: the proportion of the response’s variance, R2, explained by the component, the signifcance of this R2, and the fractional contribution of each original predictor to the component. Because all principal components are orthogonal and independent, the total proportion of response variance explained by the data is the sum of the component R2 values. Principal component regression thus circumvents the debilitating problems of partial correlation and multivariate regression analyses (box 1) while yielding results which are, in some ways, easier to interpret.

We carried out principal component regression on the seven predictors analyzed above. Because the determina- tion of principal components involves only the predictors and not the response (i.e., dN or dS), there is only one set of components and contributions from biological predic- tors. The regression analysis generates response-specifc results, in particular, the proportions of variance in dN, dS, and so on, which each component explains. Table 2 shows numerical data from the analysis of dN and dS using the seven predictors of expression, CAI, abundance, length,

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330 Drummond et al.

FIG. 1.—Principal components regression on the rate of protein evo- lution (dN) in 568 yeast genes reveals a single dominant underlying com- ponent. (a) Of the seven principal components only one (starred) explained a statistically signifcant proportion of the variation in dN. This component explained 43% of the variance, while no other component explained more than 1%. Expression level, CAI, and protein abundance determined most of this dominant component (labeled), while the remaining predictors (in order from top to bottom: length, dispensability, degree, and centrality) determined ,10% of the component’s R2. See table 2 for numerical data. (b) A larger data set (1,939 genes) excluding protein-protein interaction predictors showed the same patterns as in (a).

dispensability, degree, and centrality; fgures 1a and 2a show these data graphically.

Strikingly, for the rate of protein evolution, dN, one principal component explained 43% of the variance with high signifcance, while all other components explained less than 1% (fg. 1). The single dominant component was almost entirely (.90%) determined by roughly equal contributions from three predictors: expression level, abun- dance, and CAI.

While the causes of dNs variation have remained un- clear, dS is constrained by translational selection: selection for preferred codons, which correspond to abundant tRNAs and are translated faster and more accurately (Akashi 1994, 2001), makes many synonymous changes unfavorable and thus reduces dS (Hirsh, Fraser, and Wall 2005). Figure 2 shows that the dS results mirror those using dN: the frst component, which is determined almost entirely by expres- sion, abundance and CAI, is overwhelmingly dominant (50.8% of dS variation). A second highly signifcant com- ponent of modest size (6.6% of dS variation) appears but is 88.4% determined by abundance and CAI. Astonishingly, the seven biological predictors explain a cumulative 61.9% of the total variance in dS, with three predictors (expression, abundance, and CAI) contributing roughly equal amounts and accounting for 87% of the total variance explained. Be- cause synonymous sites are thought to be under relatively weak selection, we would expect random fuctuations (noise) to contribute a large proportion of variation in dS, yet our analysis suggests that selective pressures, even those revealed using noisy data, account for almost two- thirds of the dS variation among these genes.

The size of the seven-component data set (568 genes) was severely limited by the requirement for genes having measures for all seven predictors. In particular, we used high-quality interactions measurements (Han et al. 2004) for degree and betweenness centrality; eliminating these

FIG. 2.—Principal components regression on the rate of synonymous site evolution (dS) in 568 yeast genes reveals a single dominant underlying component. (a) Seven-predictor variables (see text) yielded seven principal components, of which six (starred) explained a statistically signifcant pro- portion of the variation in dS. The dominant component explained 51% of the variance, while no other component explained more than 7%. See fgure 1 caption for the breakdown of predictor contributions. (b) A larger data set (1,939 genes) excluding protein-protein interaction predictors showed the same patterns as in (a).

measurements, which apparently contribute negligible amounts to evolutionary rate, more than triples the data set size to 1,939 genes. We performed the same analysis on this expanded set and obtained similar results (table 3, and fgs. 1b and 2b).

To examine the possible effects of assuming a linear model in the regression, we repeated our analyses using only data ranks for the predictors and each response. The results of this nonparametric analysis were virtually un- changed from the parametric case (data not shown), indi- cating that little information is contained in the relative magnitudes of the variables.

It is common practice to interpret dS as the rate of se- lectively neutral divergence and the ratio dN/dS as the de- viation of protein evolutionary rate from neutral, putatively allowing detection of purifying selection or adaptive evo- lution. We analyzed dN/dS and found trends that were sim- ilar to those observed in dN and dS alone (tables 2 and 3). The dominant principal component explained only half the variation in dN/dS compared to dN or dS, but the reason seems obvious in light of our results: dN and dS appear to refect the same underlying selective force, so dividing one by the other removes much of the shared infuence. In yeast, as in many other organisms, dS does not refect neutral divergence but rather divergence constrained by translational selection for preferred codons, as previous authors have noted (Hirsh, Fraser, and Wall 2005). These authors proposed an adjusted measure of dS, denoted dS#, from which the infuence of codon preference has been extracted (Hirsh, Fraser, and Wall 2005). We thus analyzed dS# and dN/dS# (tables 2 and 3), and found that for dS# the dominance of the frst principal component was obliterated. While two components (component 1, mostly CAI, expres- sion and abundance; and component 2, mostly dispensabil- ity) appeared to make small but possibly meaningful contributions (R2 . 6%) in the smaller seven-predictor data set, these contributions were effectively eliminated in the

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A Single Determinant Dominates the Rate of Yeast Protein Evolution 331

Table 3 Results of Principal Component Regression Analysis on Five Predictors and Five Measures of Evolutionary Rate for 1,939 Saccharomyces cerevisiae genes

Principal Components

1 2 3 4 5 All

Percent variance explained in dN 36.94*** 0.05 0.03 0.22 0.60*** 37.85*** dS 39.33*** 0.73** 0.09 1.93*** 1.92*** 44.01*** dN/dS 22.39*** 0.28 0.21 0.00 0.21 23.10*** dS# 1.26** 2.52*** 2.58*** 0.00 1.54** 7.91*** dN/dS# 37.61*** 0.28 0.00 0.14 1.16** 39.20***

Percent contributions Expression 33.2 1.7 0.1 24.2 40.8 CAI 31.4 1.0 9.4 9.0 49.2 Abundance 31.3 0.6 0.4 65.8 1.9 Length 2.0 61.0 29.6 0.4 7.0 Dispensability 2.1 35.7 60.5 0.6 1.1

# ** *** NOTE.— P , 0.01; P , 10�6; P , 10�9. Bold indicates that the indicated predictor contributes at least 20% to the

indicated component.

larger fve-predictor data set (R2 , 3%), even though the major contributing predictors were still present. This sam- ple size dependence suggests that the contributions of com- ponents 1 and 2 are artifacts. Overall, our results are consistent with the previous claim that dS# has been purged of the infuence of selection on synonymous sites (Hirsh, Fraser, and Wall 2005). As additional support, the dN/ dS# regression was nearly instinguishable from that of dN (tables 2 and 3).

To assess the importance of phylogenetic distance on our results, we carried out principal component regression on dN and dS values calculated using two relatives of S. cerevisiae, S. paradoxus and K. waltii, which diverged roughly 5 and 100 MYA, respectively (Drummond et al. 2005).

For S. paradoxus, we obtained almost identical results for dN as for the data of Hirsh, Fraser, and Wall (2005). However, dS showed a much weaker, though still dominant, frst component that explained 15% of the dS variance in- cluding interaction data and 6% without these data, fvefold more than any other variable. We traced the weaker dS signal to differences in gene fltering (the smaller data set of Hirsh, Fraser, and Wall (2005) omits sequences whose gene-level phylogeny did not match the species-level pattern and se- quences containing introns and potential frameshifts) and in codon frequency estimates. Controlling for gene fltering, the nine-free-parameter codon frequency model used by Hirsh, Fraser, and Wall (2005) produced a larger signal than the sixty-free-parameter model used by Drummond et al. (2005), indicating that analyses of dS may be sensitive to estimation methodologies (data not shown).

For the distant relative K. waltii, we again obtained nearly identical results for dN. For the 2,412 genes without (and 752 genes with) interaction data, one principal com- ponent determined by CAI, abundance, and expression ex- plained 41% of the variance in dN, while all other components explained ,2%. For dS, no dominant compo- nent emerged, and the best component (mostly expression and CAI) explained 1.7% of the variance. The lack of any predictive signal for dS is not surprising because the dS val- ues relative to K. waltii average more than 14 substitutions

per synonymous site, far beyond the range of reliable esti- mation. These high dS values may result from a combina- tion of the large amount of time separating the species, changes in synonymous pressures, and diffculties in ortho- log identifcation and alignment. The robust dN results lend weight to the frst two explanations. We expect that as even more distant relatives are analyzed, the dN results will be attenuated by noise, alignment degradation, and phenotypic changes that must, in some cases, be linked to changes in relative gene expression levels.

To assess whether the trends we identifed for yeast extend to other species, we examined evolutionary rates in 2,605 E. coli genes relative to S. typhimurium. Lacking global protein abundance, interaction, and dispensability data for E. coli, we used length, two measures of expression level, refecting growth in minimal M9 and rich LB media, and two measures of codon optimization, CAI and the fre- quency of optimal codons Fop (Ikemura 1985), as predic- tors. Again, a dominant component emerged which explained 36% of the dN variance (16-fold more than any other) and 25% of the dS variance (38-fold more than any other). Because most of the included predictors are translation oriented in some way, our results offer no con- clusion as to the possible infuence of other predictors in E. coli. However, the remarkable similarity to the yeast re- sults, including the large portion of variance explained, sug- gests that similar selective forces have shaped evolutionary rates in this prokaryotic organism.

Analysis of Binary Variables Using Analysis of Covariance

In all the above analyses, we found that protein-protein interactions and gene dispensability showed little or no ap- parent infuence on the rate of protein evolution (dN) and synonymous site evolution (dS), contrary to previous reports (Hirsh and Fraser 2001; Fraser et al. 2002; Fraser and Hirsh 2004; Wall et al. 2005). Perhaps these measures, as continuous predictors describing complex and poorly un- derstood phenomena, display false precision, i.e., they re- fect real underlying effects but quantify them in overly precise ways that introduce noise. We reasoned that simpler

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332 Drummond et al.

FIG. 3.—Binary analyses of the infuence of hub type and essentiality on dN reveal subtle relationships masked by continuous analyses. (a) Party hubs (dark points, solid line), which interact with many partners at once, evolve at 60% of the average rate for all genes with measured interactions (light points, dashed line). (b) Date hubs (dark points, solid line), which interact with many partners sequentially, evolve at 92% of the rate (not signifcant) for genes with measured interactions (light points, dashed line). (c) Essential genes (dark points) evolve at 84% of the genome average rate (light points, dashed line).

measures, such as essentiality (the limiting case of dispens- ability, where essential genes are indispensable and nones- sential genes include all those with dispensability .0) and type of network interaction hub (‘‘date’’ hubs interact with many partners individually, while ‘‘party’’ hubs do so si- multaneously) (Han et al. 2004; Fraser 2005), subjected to a category-based analysis, might reveal relationships ob- scured by their noisy continuous counterparts.

Because expression level, CAI, and abundance have such an important effect on evolutionary rate, we have to carry out the category-based analysis controlling for the effect of these three predictors. We chose to perform an analysis of covariance (‘‘ANCOVA’’). As the continu- ous variable we used the principal component of the three quantities expression level, CAI, and abundance, while we encoded the category (gene is or is not a party hub, is or is not a date hub, is or is not essential) as a binary variable. We found that party hubs evolve on average at 60% of the rate of genes with known interactions (P , 10�5), date hubs at 92% of the rate of genes with known interactions (not sig- nifcant), and essential genes at 84% of the speed of all genes (P , 10�6) (fg. 3). The effect of gene essentiality is signifcant but small in magnitude, and date hubs show no signifcant rate constraint. As previously reported (Fraser 2005), party hubs do indeed experience a notable 40% reduction in evolutionary rate. However, evolutionary rates in yeast span three orders of magnitude; interactions play at best a minor role in constraining rates.

Discussion

We have carried out the most comprehensive compar- ative analysis to date of potential determinants of non- sysnonymous (dN) and synonymous (dS) yeast gene evolutionary rates. We used a published data set of evolu- tionary rates, previously used to establish an independent role for dispensability (Wall et al. 2005) and to correct dS for translational selection (Hirsh, Fraser, and Wall

2005), to highlight the methodological improvements intro- duced here. We fnd that a single underlying component explains roughly half the variation in both dN and dS, and that this dominant component is almost entirely deter- mined by the gene expression level, protein abundance, and codon bias as measured by the CAI. Our results generalize to E. coli despite use of a reduced set of predictors.

The predictors we included in our analysis appear to explain roughly half the variation in dN and dS. Some other predictors could explain the remaining half, but this seems quite unlikely for a variety of reasons. First, a signifcant portion of evolutionary rate variations are probably random because the evolutionary process is inherently stochastic. Second, our R2 estimates constitute a lower bound because the R2 values we fnd are attenuated by measurement noise, for example, on microarray readings of gene expression (Coghlan and Wolfe 2000), by systematic error, e.g., in some protein-protein interactions data (Bloom and Adami 2003), and by time variation, for example, in expression over the cell cycle (Cho et al. 1998). Finally, the true rela- tionship between any of the predictors we examine and dN or dS is unlikely to be perfectly linear, and deviations from linearity reduce parametric R2.

Our results point to a single dominant cause for most of the 1,000-fold variation in evolutionary rates among yeast genes, and the dominant component’s three biological contributors suggest that the cause is translational selection. We hypothesize that the number of translation events a gene experiences determines its evolutionary rate and that ex- pression, abundance, and CAI are all roughly equally good predictors of the number of translation events. Translation is remarkably error prone, with roughly 19% of average length yeast proteins carrying a missense error (Drummond et al. 2005), and these errors can cause protein misfolding that imposes a well-known burden on the cell (Goldberg 2003) which scales with the number of translation events. Selection to reduce the number of error-induced misfolded proteins could constrain both synonymous site evolution

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A Single Determinant Dominates the Rate of Yeast Protein Evolution 333

(dS), e.g., through pressure for preferred codons which re- duce mistranslation of proteins (increased translational ac- curacy) (Akashi 1994), and protein sequence evolution (dN), e.g., through pressure for protein sequences that fold properly despite mistranslation (increased translational robustness) (Drummond et al. 2005). In this way, a single underlying cost can govern both synonymous and nonsy- nonymous evolutionary rates, consistent with our fndings.

We used principal component regression for our anal- ysis because, as we demonstrate, the more commonly employed techniques of partial correlation analysis and multivariate regression are inapplicable by assumption (in the latter case) and prone to produce spurious effects in the presence of noisy correlated data (in both cases). By contrast, under principal component regression, the transformed predictors are orthogonal and uncorrelated, so that their relative contributions to the overall regression model can be evaluated independently and reliably. More- over, we can extend this method to assess the infuence of additional binary predictors, by carrying out an ANCOVA in which the covariable is given by the principal component that explains the majority of the response-variable variance.

Wall et al. (2005) use a structural equation model to examine the infuence of measurement inaccuracy on their partial correlation analysis of the effects of expression level and dispensability on dN. Given their analysis, they admit an inability to determine the relative importance of these two predictors but conclude that dispensability has an in- dependent effect on dN. We claim to be able to determine relative importance and come to an opposite conclusion for two reasons. First, a general advantage of principal compo- nent regression over partial correlation is the ability to fnd predictors not originally included in the analysis. We were fortunate in this case that the dominant predictor is not ex- pression level, CAI, or abundance, but rather a variable (likely the frequency of translation) that these three predic- tors measure with roughly equal accuracy. Partial correla- tion can never fnd such underlying variables. Second, the structural equation model of Wall et al. (2005) attempts to quantify how much the predictors could explain given the hypothetical levels of measurement inaccuracy, but with principal component regression, we are asking how much the given predictors can explain, whatever their accuracy. Here we were doubly fortunate. Three of our predictors (CAI, abundance, and expression) triangulate on the same underlying variable, increasing accuracy essentially by measuring it in triplicate; this variable happens to explain a large portion, perhaps most, of dN’s explainable variance.

How much dispensability and degree infuence evolu- tionary rate has been a contentious issue. Regarding the for- mer, the literature refects disagreement over whether dispensability has any effect whatsoever on the rate of evo- lution, with partial correlation analyses playing a prominent evidentiary role (Hirsh and Fraser 2003; Pál, Papp, and Hurst 2003; Wall et al. 2005). Our analysis, which avoids problematic partial correlations but uses the same data as in previous analyses that appeared to confrm a signifcant role for dispensability (Wall et al. 2005), is quite clear: dispens- ability neither constitutes an independent source of varia- tion in dN nor contributes meaningfully to the dominant component that does infuence dN. In the case of degree,

the disagreement has pivoted on whether experimental sur- veys are biased toward detecting interactions more often in highly expressed proteins (Bloom and Adami 2003, 2004; Fraser and Hirsh 2004), leading to a true but biologically irrelevant degree-dN relationship. Our analysis shows that degree does not contribute independently, but makes a small, signifcant contribution to the variable dominated by expression, abundance, and CAI, as expected under the expression-bias hypothesis and inconsistent with a true con- straint from the number of interactions. In short, our results suggest neither degree nor dispensability make much differ- ence in dN and point out precisely why previous authors have been led to the opposite conclusion.

By contrast, our ANCOVA offers support for the ob- servation that proteins that interact with multiple partners simultaneously, so-called party hubs, evolve slower (Fraser 2005), even after accounting for the translational effect we identify here. Party hubs epitomize the intuition behind the interactions hypothesis: while interactions presumably con- strain residues, it appears that in order to slow a protein’s overall rate of evolution, interactions must involve a signif- icant proportion of the protein’s residues, as expected for party hubs. Date hubs, which would include proteins that interact with many partners serially at the same site, appeared signifcantly slower evolving in the previous anal- ysis (Fraser 2005), but our results suggest this fnding refected a failure to properly control for expression-linked effects (again, partial correlation was used). (Note that ANCOVA enjoys a crucial and useful advantage over par- tial correlation aside from its greater reliability, namely the ability to control simultaneously for multiple intercorre- lated variables such as expression and CAI.)

The rates dN and dS are routinely used to carry out analyses on selection, often under the assumption that dN/dS . 1 indicates adaptive protein evolution and dN/ dS , 1 indicates purifying selection, and generally with the intent of quantifying functional pressures. Our results suggest that both evolutionary rates are determined by translational selection and are therefore likely poor predic- tors of functional selection because translational selection by defnition operates before a protein becomes functional. In yeast, dS does not measure neutral divergence, and thus, in the absence of a quantitative description of the relative strengths of selection on nonsynonymous and synonymous sites, the measure dN/dS is meaningless. Recently, a method for correcting dS for synonymous site selection was pro- posed (Hirsh, Fraser, and Wall 2005), and we found that the adjusted measure of neutral divergence, dS#, indeed appears free of the infuence of the dominant variable we link to translational selection. Our dominant variable shows virtually identical predictive power for dN and dN/dS#, in- dicating that dividing out the neutral divergence, which for example might be due to variable mutation rates, makes lit- tle difference when analyzing the rate of protein evolution. Our results suggest that using overall gene evolutionary rates to characterize functional selection is unwise; exam- ining dN and dS at particular sites remains a powerful and important tool in evolutionary analyses of genes.

We have found that yeast coding sequences accu- mulate substitutions according to a surprisingly simple formula: more predicted translation events means slower

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334 Drummond et al.

evolution. In recent years, evidence has accumulated that translation-linked variables, in particular expression levels, govern the evolutionary rate of proteins across all life, from bacteria (Rocha and Danchin 2004) to fungi (Pál, Papp, and Hurst 2001), plants (Wright et al. 2004) and animals (Duret and Mouchiroud 2000) including humans (Subramanian and Kumar 2004), but translational selection has only re- cently been proposed as an explanation for this puzzling trend (Akashi 2003; Drummond et al. 2005). Our results suggest that translational selection dominates the rate of protein evolution, and by extension suggest that transla- tional selection operates across the tree of life, from prokar- yotes to humans. Future work must illuminate the precise biophysical effects that constrain molecular evolution, but we have shown that, at least in yeast, the answers may be found in translation.

Box 1 Comparing Partial Correlation, Multivariate Regression, and Principal Component Regression

How do the three analytical techniques considered here fare given a case where only one variable determines evolutionary rate? For each technique, what would we con- clude about the number and strength of the rate determi- nants? Consider a simple model in which a variable X (e.g., expression level) determines two other variables, a putative determinant D (e.g., dispensability) and a response K (evolutionary rate), so that D 5 X 1 eD and K 5 X 1 eK, where eD and eK are noise terms with mean 0 and variances

2 2r and rK: Further, assume that we cannot measure X butD only a noisy correlate, X# 5 X 1 eX#. In this model, X is responsible for all the correlation between D and K. We let X be normally distributed with mean 0.5 and SD 0.25 (so that X values span the unit interval) with the observable predictors X#, D, and K equal to X plus zero mean normally distributed noise with SD of 0.3. We ran each analysis 100 times with 3,000 measurements each.

Partial correlation analysis suggests that both D and X# contribute to the rate K independently and with equal strength:

Partial Correlation with K P Value

rDKjX#50:296 6 0:03 10 �9

rX#KjD50:291 6 0:02 10 �9

Multivariate regression similarly suggests that both D and X# independently infuence the rate K:

Predictor Percent Variance in K Explained (R2) P Value

X# 16.9 6 2 10�9

D 17.3 6 2 10�9

Principal component regression, however, properly identifes only one component which contributes signif- cantly to the rate K. The two components identifed are X# 1 D, which measures mostly X, and X# � D, which measures mostly noise. Component 1 alone carries predic- tive value for K.

Component Percent Variance in K Explained (R2) P Value

1 (X# 1 D) 21.3 10�9

2 (X# � D) 0 0.7

We may proceed with the confdence that we have properly identifed the number and strength of the underly- ing determinants of K.

In general, the underlying variable represented by the dominant component is not known a priori and its identi- fcation requires additional insight. In this case, we know it is X, which is accurately captured by the principal compo- nent regression method, but not by the other methods. Other methods are therefore likely to lead to erroneous results when faced with the problem of trying to fnd true predic- tors within noisy data. Principal component regression, as shown here, is unlikely to do so.

Our toy model underscores a key observation: in the presence of noisy and correlated data, nonzero partial cor- relations and R2 values from multivariate regression, even those with very high statistical signifcance, must not be taken as evidence for independent effects, contrary to pre- vious studies (Lemos et al. 2005; Wall et al. 2005).

Supplementary Material

All data and scripts used to perform our analyses are available as supplementary material at Molecular Biol- ogy and Evolution online (http://www.mbe.oxfordjournals. org/).

Acknowledgments

We are grateful to Frances H. Arnold for insightful comments on the manuscript. C.O.W. was supported by National Institutes of Health (NIH) grant AI 065960 and D.A.D. was supported by NIH National Research Service Award 5 T32 MH19138.

Appendix Spurious Partial Correlations from Noisy Data

Consider a model in which a variable X determines two other variables D and K in a linear fashion. Then we can write, without loss of generality,

D 5 X 1 eD; K 5 X 1 eK ; ð1Þ where eD and eK are noise terms with mean 0 and variances r2 and r2 ; respectively. We assume that these noise terms D K are each independent of X and of each other. In the follow- ing analysis, we also assume for convenience that X has variance 1; results for the case when X has an arbitrary var- iance r2 follow by dividing all other variances in the prob- lem by r2 in the equations below.

The partial correlation between D and K given X is defned as

rDK � rDXrKX rDKjX 5 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi; ð2Þ

ð1 � r 2 Þð1 � r 2 Þ DX KX

where rij is the standard Pearson correlation between var- iables i and j. Given the model (1), it is intuitively obvious

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A Single Determinant Dominates the Rate of Yeast Protein Evolution 335

that rDKjX should be 0 because eD and eK are independent noise sources. Indeed, we fnd in this case

EðDKÞ � EðDÞEðKÞ 2 2 �1=2 rDK [ 5 ½ð1 1 rDÞð1 1 rK Þ� ; rDrK

2 �1=2 2 �1=2 rDX 5 ð1 1 rDÞ ; rKX 5 ð1 1 rK Þ ; ð3Þ

giving rDKjX50: In the above, E( ) denotes expected value. We now consider the question of whether it is possible

that we observe a spurious partial correlation between D and K if we are given a somewhat noisy version of X instead of X itself. Thus, we introduce the new variable X#5X1eX#; where the noise eX# is assumed to have mean 0 and variance

2rX#; and now compute the partial correlation rDKjX# because this is the actual partial correlation we would observe if we were given noisy samples of X. We fnd that rDK remains the same as in equation (3), while rDX# and rKX# are now given by

2 2 �1=2 rDX# 5 ½ð1 1 rDÞð1 1 rX#Þ� ; ð4Þ

2 2 �1=2 rKX# 5 ½ð1 1 rK Þð1 1 rX#Þ� ;

and the new partial correlation is

2 r

X#qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi: ð5ÞrDKjX# 5 2 2 2 2 2 2 2 2ðr 1 r 1 r r Þðr 1 r 1 r r Þ D X# D X# K X# K X#

Thus, the presence of noise in the samples of X, char- 2acterized by the variance rX#; leads to a nonzero spurious

partial correlation between D and K. Because equation (5) is 2quadratic in rX#; we can also write it in terms of the amount

of noise on X that would result in a given spurious partial correlation:

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 2 2 2 �2 2 2

r ð1 1 r Þ 1 r ð1 1 r Þ 1 ðr � r Þ 1 4r r K D D K K D DKjX#rK D

r 2 5 :

X# �2 2 2 2ðr � ð1 1 r Þð1 1 r ÞÞ

DKjX# D K

ð6Þ

We may alternatively specify a signifcance level as a P value P, for the desired spurious correlation and the number of data points n, and ask what noise level in X is required to achieve the given signifcance. Based on as- ymptotic (large n) results, the P value for testing signif- cance of the partial correlation is given (Wall et al. 2005) as rffiffiffiffiffiffiffiffiffiffiffiffi

n � 3 P 5 2½1 � UðjtjÞ� ; t 5 r ; ð7Þ

1 � r 2

where U(x) denotes the standard normal cumulative distri- bution function, and r is a partial correlation. Solving for the partial correlation in terms of P and n, we obtain

�2 n � 3 r 5 1 1 2 ; ð8Þ

z 1�P=2

where zc is the 100 3 cth percentile of the standard normal distribution. Given a P value P and the number of data points n, we may therefore use equation (8) to fnd the cor- responding partial correlation r and then substitute this partial correlation in equation (6) to fnd the noise level

in X that would produce that partial correlation to the de- sired signifcance.

There are a number of important consequences and special cases of equations (5) and (6):

(i) For fxed rD and rK; rDKjX# is maximized when 2rX#/N: This maximum achievable partial correla-

tion is given by

2 2 �1=2 rDKjX#/½ð1 1 r Þð1 1 r Þ� : ð9ÞD K

(ii) As expected, rDKjX# increases as the noise level on X increases. Perhaps, less intuitive is the fact that the par- tial correlation also increases as the noise levels on D and K become smaller. In the limit that D and K are perfectly clean ðrD;K /0Þ; we fnd a perfect but spu- rious partial correlation of 1 with an arbitrary, nonzero rX#: Thus, if we are given noisy samples of X, we would be falsely led to conclude that D and K are well-correlated even if we control for X.

(iii) When rK and rD are comparable, i.e., rK ffi rD5rc; equation (6) simplifes to yield

2 r

cr 2 5 : ð10Þ X# �1 2

r � ð1 1 r Þ DKjX# c

With rc51 (almost a worst-case scenario: in most realistic situations, rc will be much smaller than the variation in X, and we would then need a much smaller noise level in X in order to obtain a signifcant rDKjX#) we fnd, using equation (8) with P 5 0.05 (95%

2signifcance) and equation (10), that rX#50:11 for n 5 2500 and rX#50:07 for n 5 1,000. Thus, for 1,000 data

points, and assuming a large noise on D and K, we only need a modest (;7% of the variance in X) amount of noise to achieve a 95% signifcant partial correlation. This noise level would, of course, be much lower if D and K were less noisy.

(iv) When n is large, equation (8) implies that a small par- tial correlation is needed to achieve signifcance. We may therefore assume that the r�2 terms dominate in equation (6), which may then be combined with equa- tion (8) to yield, for large n,

2 rK rDz1�P=2 r ffi pffiffiffi ; ð11Þ

X# n

which directly expresses the noise level rX# required to achieve signifcant partial correlation in terms of n and P. This shows that for fxed rD; rK ; and P the amount of noise on X that is required to achieve signifcancepffiffiffi decreases as 1= n for large n.

Partial Correlations in Terms of Measurable Quantities

Because all variances in equations (5) and (6) are re- ally ratios with respect to the true variance of X, it may ap- pear that we need to know the true variance of X, an unmeasurable quantity, in order to fnd rDKjX#: This is, how- ever, not the case. Suppose we make two measurements of a variable Y5X1eY (here, Y could stand for any of X#, D, or

2K), where eY is a noise source with mean 0 and variance rY :

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336 Drummond et al.

These two measurements, say, Y1 and Y2; can be expressed as Y15X1e1; Y25X1e2; where e1 and e2 are independent, identically distributed noise sources with the same distribu- tion as eY : The Pearson correlation between the two meas- urements Y1 and Y2 is then given by

EðY1Y2Þ � EðY1ÞEðY2Þ rY [ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 ð1 1 r 2 Y Þ�1 ; ð12Þ

VarðY1ÞVarðY2Þ where we have assumed, as before, that the variance of X itself is set to 1. We can therefore use equation (12) to ex- press the variances of X#, D, and K in equations (5) and (6) in terms of the two-measurement correlations rX#; rD; and rK; respectively, as

r 2 5 r �1 � 1: ð13Þ X#;D;K X#;D;K

Such a substitution ensures that the true variance of X does not appear in equations (5) and (9) and that these equa- tions are expressed directly in terms of measurable quanti- ties. In particular, equation (9) takes the particularly simple pffiffiffiffiffiffiffiffiffi 2form rDKjX#/ rDrK as rX#/N (or rX#/0), and equation (10) becomes, with rc5rD or rc5rK :

�1 1 � rDK j X#r

rX# 5 c : ð14Þ

1 � rDK j X#

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Accepted October 10, 2005

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