Assessing the Determinants of Evolutionary Rates in the Presence of Noise
Joshua B. Plotkin* and Hunter B. Fraser*Department of Biology, The University of Pennsylvania; and The Broad Institute of Harvard University and MIT
Although protein sequences are known to evolve at vastly different rates, little is known about what determines their rateof evolution. However, a recent study using principal component regression (PCR) has concluded that evolutionary rates
in yeast are primarily governed by a single determinant related to translation frequency. Here, we demonstrate that noisein biological data can confound PCRs, leading to spurious conclusions. When equalizing noise levels across 7 predictor
variables used in previous studies, we find no evidence that protein evolution is dominated by a single determinant. Our
results indicate that a variety of factors—including expression level, gene dispensability, and protein–protein
interactions—may independently affect evolutionary rates in yeast. More accurate measurements or more sophisticatedstatistical techniques will be required to determine which one, if any, of these factors dominates protein evolution.

Introduction
Proteins span more than 3 orders of magnitude in theirevolutionary rate, as quantified by the number of nonsynonymous substitutions per site. What determines a protein’srate of evolution has been actively debated over the past
several decades. Early hypotheses suggested that the evolutionary rate of a protein is governed by at least 2 factors:
the protein’s level of functional constraint (i.e., the densityof functionally active residues) and its overall importance(or ‘‘dispensability’’) to the organism (Wilson et al. 1977).
Arguments for the role of dispensability are mostly
based on theory (Ohta 1973), as very little empirical data
have been available until recently (Hurst and Smith
1999; Hirsh and Fraser 2001). A relationship between dispensability and evolutionary rate is hypothesized because
mutations in essential proteins are more likely to be deleterious. Such mutations are purged from the population,
thereby reducing the evolutionary rate of indispensable proteins (Ohta 1973).
In contrast to dispensability, the relationship between
functional constraint and evolutionary rate has been studied
empirically for over 40 years. Among the first to address therole of functional constraint was Ingram (1961), who observed that the polypeptide chains of hemoglobin should
be differentially constrained depending on the number ofother chains with which they physically interact (Ingram1961). Similarly, when studying cytochrome c, Dickerson(1971) observed that surface residues that interact withother proteins tend to be highly conserved. Although functional constraints are difficult to measure directly, or evendefine precisely, many studies have used as a proxy the
number of physical interactions in which a given protein
participates. Authors have recently confirmed early hypotheses of Ingram and Dickerson on a much larger scale using
curated sets of interacting protein crystal structures(Mintseris and Weng 2005).
The sudden plethora of sequenced genomes allows us
to compare orthologous coding sequences from related species, estimate evolutionary rates, and ask what features ofproteins covary with their evolutionary rates. The yeast
Saccharomyces cerevisiae has emerged as a model system
Key words: evolutionary rates, noise, pca, PCR, expression levels,dN, dS.
E-mail: jplotkin@sas.upenn.edu.
Mol. Biol. Evol. 24(5):1113–1121. 2007
doi:10.1093/molbev/msm044
Advance Access publication March 7, 2007
Ó The Author 2007. Published by Oxford University Press on behalf of
the Society for Molecular Biology and Evolution. All rights reserved.
For permissions, please e-mail: journals.permissions@oxfordjournals.org

for systematically studying the determinants of evolutionary rates. Saccharomyces cerevisiae was the first fully sequenced eukaryote, and its genome remains the mostcomprehensively annotated. Additionally, S. cerevisiae
has been the subject of thousands of functional genomic
experiments producing diverse information for evolutionary investigation.
Previous studies have reported a variety of functional,
biophysical, and fitness-related variables that correlate withthe evolutionary rates of proteins (Drummond et al. 2006):
proteins evolve more slowly if they have a higher number ofmRNA molecules per cell (expression) (Green et al. 1993;
Pal et al. 2001), if they have a higher number of protein
molecules per cell (abundance) (Drummond et al. 2006),
a higher codon adaptation index (CAI) (Pal et al. 2001;
Wall et al. 2005), more protein–protein interactions (degree) (Fraser et al. 2002), a larger fitness effect upon geneknockout (dispensability) (Hirsh and Fraser 2001), shorter
sequence length (Marais and Duret 2001), or a more central
role in the interaction network (centrality) (Hahn and Kern2005). But these predictor variables are themselves correlated with one another—raising the question of which variables are truly involved in determining evolutionary rates
and which variables happen to covary simply because they
are influenced by another, causal variable. It has been argued, for example, that the correlation between dispensability and evolutionary rate is simply a side effect of causal
relationships between expression level and evolutionaryrate and between expression level and dispensability (Palet al. 2003).
Drummond et al. (2006) undertook a comprehensive
analysis of the determinants of protein evolution in yeast.
Their work represents a significant advance towards identifying the major, independent correlates of evolutionary
rates (McInerney 2006; Pal et al. 2006; Rocha 2006). Prior
to the work of Drummond et al. (2006), many authors had
used the techniques of multiple regression and partial correlation to assess whether correlates of evolutionary rate are
independent of one another (Fraser et al. 2002; Bloom andAdami 2003; Rocha and Danchin 2004). Drummond et al.
(2006) demonstrated that colinearity of predictor variables
and measurement noise can cause partial correlations toyield spuriously significant results. In lieu of partial correlations, and in order to remove colinearity, Drummond et al.
(2006) used a principal component regression (PCR) to analyze evolutionary rates. Surprisingly, they found that a single component—comprised almost entirely of expression
level, abundance, and CAI—explained far more variation

1114 Plotkin and Fraser

FIG. 1.—PCRs of evolutionary rate against 7 explanatory variables. Components are ordered according to the amount of variation they explain. a) A
regression on 568 genes previously reported by Drummond et al. (2006) shows a dominant component—consisting primarily of expression, abundance,
and CAI—that explains 42.8% of the variation in dN. (b) A regression over 242 genes using a different data set of protein–protein interactions (Ho et al.
2002) instead of Han et al. (2004) also shows a dominant explanatory component. (c) A regression on the same data as in (b) but with noise levels equalized
across predictor variables. Each panel also shows the loadings of predictor variables on the first principal component (±one standard deviation in panel c).
When noise levels are equalized, there is no evidence of a dominant explanatory component, and all predictor variables explain similar total amounts ofvariation in dN. Furthermore, when noise levels are equalized, the first component contains roughly equal contributions from each predictor variable.

in evolutionary rates than any other component. On thebasis of these results, Drummond et al. proposed that asingle determinant, namely, selection against translationerror–induced protein misfolding, dominates protein evolution in yeast (Drummond et al. 2005, 2006). This recent
finding has already changed how researchers think aboutprotein evolution (Koonin and Wolf 2006; McInerney2006; Pal et al. 2006; Rocha 2006).
Here, we reinspect the analyses of evolutionary rates
presented by Drummond et al. (2006). We show that the
PCR utilized by Drummond et al. can be confounded when
predictor variables have been measured with different
amounts of noise. We assess the amount of noise associatedwith each of the 7 predictor variables used in previous studies of yeast protein evolution. We show that after equalizing
noise levels across the predictor variables, each predictor
has a roughly equal contribution to the evolutionary rateof proteins, and there is no evidence for a single, dominantfactor driving protein evolution. Finally, we present a simple mathematical model of evolutionary rates, with parameter values determined by empirical yeast data. Our model
demonstrates that the apparent predominance of translational selection as the determinant of yeast protein evolution may be a spurious artifact arising from the variableaccuracy of functional genomic measurements.

however, we have only noisy estimates of these variables,
denoted E*, and D*. Noise in expression data, for example,
arises both from variability in expression levels between
cells and from inaccuracies in measurement. We refer toboth sources of variability as noise.
Drummond et al. (2006) have shown that the methodof partial correlations can yield spuriously significant results when applied to noisy variables—that is, r(D*, R |E*) may show a significant departure from zero even whenr(D, R | E) equals zero. In other words, as a result of noise,
the relationship between protein dispensability and evolutionary rate may appear to be independent of expression
levels, even when the underlying, noiseless variables areuncorrelated when controlling for expression. In order toavoid the pitfalls of partial correlations, which are caused
by noise and colinearity, Drummond et al. regressed evolutionary rates against the principal components of the 7predictor variables. The principal component regression
corrects for colinearity among predictor variables, but
the PCR implicitly assumes that all predictors have beenmeasured with the same amount of noise. In the following
sections, we show that the assumption of equal noise is
not valid for the data sets analyzed by Drummond et al.,and we demonstrate that the apparent dominant determinant of protein evolution may be an artifact of this invalid
assumption.

Results

Quantifying Noise in Functional Genomic Data

We have reanalyzed the data sets studied by Drummondet al. (2006), consisting of evolutionary rates (dN) for 3,036
yeast protein sequences (Hirsh et al. 2005) and 7 variables that
correlate with evolutionary rates: expression, abundance,
CAI, length, dispensability, degree, and centrality.
Given the large number of variables correlated withevolutionary rate, we wish to know which correlations
are independent of one another. For example, let R, E,
and D denote evolutionary rate, mRNA expression level,and degree of protein–protein interactions, respectively.
If these variables are noise free, then the partial correlation
coefficient r(D, R | E) describes the relationship between
degree and rate, controlling for expression. In practice,Some measurable features of genes are virtually free ofnoise, such as gene length and CAI. But other variables
contain significant noise, such as mRNA expression levels,
protein abundance, the fitness effects of knockouts, and thenumber of protein–protein interactions. How can we quantify and estimate the amount of noise associated with eachof these genome-wide measurements?
The most straightforward way to quantify noise is tocalculate the correlation coefficient between 2 (or more)
independent measurements of the same quantity. For example, mRNA expression levels in yeast are quite reproducible: the correlation between 2 independent measurementsof mRNA expression levels, using the same oligonucleotide

Assessing the Determinants of Evolutionary Rates 1115

arrays, has been reported as rexpr 5 0.72 (n 5 5, 555)
(Drummond et al. 2005). We compared expression levels
measured by different investigators 3 years apart (Holstege
et al. 1998; Causton et al. 2001) and found an even stronger
correlation: r 5 0.90 (n 5 5, 460). Nevertheless, we will
use the value rexpr 5 0.72 so that our analysis is conservative and our estimate of expression noise agrees with that ofDrummond et al. (2005).
Protein abundance data are apparently less noisy thanmRNA abundance data. In the only systematic study reporting abundances for a large number of yeast proteins,
Ghaemmaghami et al. (2003) measured 206 proteins in triplicate. Assuming the noise to be normally distributed, theaverage correlation in abundance between any 2 sets of
206 replicates is r 5 0.98. This estimate of noise for protein
abundances is not as conservative as for mRNA abundances
above—because these replicates were performed using thesame method by the same investigators. But this estimate
nevertheless suggests that protein abundance measurements
in yeast contain relatively little noise. In order to be conservative (see Discussion), we will assume that protein abundances contain approximately the same amount of noise as
mRNA levels, rabund 5 0.72, even though this assumption
likely overestimates the noise in protein abundances.
Gene dispensability data in yeast are more noisy thanprotein abundances or expression levels. The correlation coefficient between 2 independent measurements of the fitness
effects of knockouts (Warringer et al. 2003; Deutschbauer
et al. 2005) made using the same set of viable single-genedeletion strains is rdisp 5 0.56 (n 5 4, 156). In order to facilitate comparison with Drummond et al. (2006), we analyze the same data set of gene dispensabilities (Deutschbauer
et al. 2005), despite the fact that the other data set correlates
more strongly with evolutionary rates (Wall et al. 2005).
The number, or degree, of protein–protein interactions
is by far the most noisy variable analyzed in this study andprevious related studies. Bloom and Adami (2003) recently
summarized the results from 9 protein interaction data sets.
Performing all 36 possible pairwise comparisons of thesedata sets, we find that as many pairs exhibit negative correlations as exhibit positive correlations. Despite the discouraging discordance among the 9 protein interaction
data sets, it is well established that some data sets containless noise than others (Kemmeren et al. 2002; von Mering
et al. 2002). Protein interaction data sets assembled from
low-throughput studies are difficult to use in this context
because of the unknown but certainly extreme bias in which
proteins have been studied by individual investigators(Reguly et al. 2006). For this reason, we chose to compare
2 of the highest quality high-throughput data sets that weregenerated by a single method: mass spectrometry (Gavinet al. 2002; Ho et al. 2002). Any systematic biases inherent
in the mass spectrometry method will artificially inflate thecorrelation between these 2 measurements, leading to a conservative underestimate of noise. The observed agreement
between the 2 measurements of protein interaction degree isrdegree 5 0.11 (n 5 524). Whereas this correlation is muchgreater than the median correlation among all 36 pairs ofinteraction data sets (r 5 0.002), the reproducibility ofthe protein interaction data is much lower than for all othertypes of data in this study.

Protein interactions measured by mass spectrometry(Ho et al. 2002) are very similar to the composite data setof interactions analyzed by Drummond et al. (2006). Using
the data from Ho et al. in place of the interaction data analyzed
by Drummond et al. does not significantly alter the PCR of dN(compare fig. 1a and b, below). Therefore, because we canestimate the noise in the mass spectrometry data, but notin the composite data set used by Drummond et al., we will
use the spectrometry data for all subsequent analyses. (Results are unchanged if we use the composite data set of interactions in our analysesinstead ofthe mass spectrometry data.)
Lastly, we assume that a protein’s centrality in the interaction network—a quantity based entirely on protein–protein interaction data—has the same noise level as theunderlying interaction data. Relaxing this assumption byraising or lowering the amount of noise in the centrality data
does not significantly affect our results.
Equalizing Noise across Predictor Variables
As we have shown above, the 7 correlates of evolutionary rate analyzed by Drummond et al. (2006) containwidely different amounts of noise. What conclusions, then,can we draw from PCRs, given that the PCR method assumes equal noise across all predictors? One way to answer
this question is by artificially equalizing the noise levels
across the predictor variables and repeating the PCR analysis. (A second way to answer the question is presented ina subsequent section.) We focus on the PCR because this is
the technique employed by Drummond et al. (2006) toreach their conclusions about protein evolution in yeast.
The degree of protein interactions is by far the most noisyof the 7 predictor variables in our study. In order to match thelevel of noise in interaction degree, we can add an appropriate
amount of extra noise to each of the other 6 predictors. As described in Appendix, we can solve analytically for the appropriate amount of Gaussian noise to be added to each predictor
so that the resulting variables have the same amount of noise asdegree, namely, rdegree 5 0.11.
Figure 1 shows PCR analyses of the original predictor
variables alongside analyses of modified predictors whose
noise levels have been equalized. There are 3 features ofeach regression that are important to note: 1) the amountof variation in evolutionary rate explained by the dominantcomponent; 2) the amount of variation explained by the secondmost dominant component; and 3) the loadings of predictor variables on components.
As figure 1 shows, variable noise among predictors
dramatically affects the PCR analysis of evolutionary rates.
Without correcting for variable noise, the PCR identifies
a single component that explains at least 20-fold morevariance in evolutionary rate than any other component(Drummond et al. 2006); whereas after equalizing noiselevels, the dominant component does not explain significantly more variance than the subdominant component(i.e., not more than 2 standard deviations). In other words,
after correcting for noise levels, there is no evidence of a single, dominating determinant of evolutionary rates in yeast.
Variable noise levels affect the PCR analysis in several
other, important ways. Without correcting for noise, the dominant explanatory component consists almost exclusively of1116 Plotkin and Frasertranslation-related variables: mRNA expression, protein
abundance, and CAI. The translation-related variables eachexplain more than 4 times the total variation in evolutionaryrate than any of the other predictor variables. These results
have been interpreted as conclusive evidence that translational selection governs the rate of protein evolution (Drummond et al. 2006). By contrast, after equalizing noise levels,
the dominant explanatory component contains roughly equalloadings from all 7 predictor variables (fig. 1c), and each of
the predictor variables explains roughly the same amount of
total variation in evolutionary rate (within one standard deviation). In other words, after correcting for noise levels, there
is no evidence of a dominant, translation-related determinantof evolutionary rates.
A Simple Model of Evolutionary Rates
Aswithothertechniquesformultipleregression,thePCR
method assumes equal noise levels across predictors, and it issensitive to violationsof this assumption. As seenabove, when
we equalize noise levels among predictor variables, the resulting PCR paints a very different picture of yeast protein evolution than has been previously reported (Drummond et al.
2006). These results still beg the following question: given
the known amount of noise associated with each predictor
variable, how much variation in evolutionary rate would be
explained by the underlying, noiseless predictors?
In this section, we provide one possible answer to thisquestion by using a simple mathematical model. We presenta phenomenological model of evolutionary rates, and wechoose parameters consistent with most important featuresof the observed yeast data. The purpose of this model is not
to recapitulate every detail of the empirical data but rather toexplore what underlying patterns are consistent with the salient features of the observed, noisy data.
For the sake of simplicity, we focus on the 4 variables
that explain the most rate variation: expression (E), abundance (A), CAI (C), and protein–protein interaction degree(D). We demonstrate that the observed, noisy data are consistent with a model in which there are multiple independent determinates of evolutionary rates and in which theunderlying (noiseless) protein interactions explain morevariance in evolutionary rate than expression level, abundance, or CAI.
We specify our model so as to reflect several importantbiological features of protein evolution (Drummond et al.
2006): 1) mRNA expression, protein abundance, and
CAI all covary because they all reflect, in part, the amountof translation events experienced by a gene; 2) expression,abundance, and CAI also covary with the degree of protein–protein interactions, for reasons unrelated to translation; 3)
the amount of translation and the degree of protein interactions both influence the evolutionary rate. These features
lead to the following model equations:
E5aZ1 þ bZ2 þ Z3;
A5aZ1 þ bZ2 þ Z4;
C5aZ1 þ bZ2 þ Z5 ;D5aZ1 þ Z6 ;R5Z2 þ Z6 ;where each Zi is an independent, normally distributed random variable with mean zero and variance one. The term
Z1 represents a source of covariation shared by expression,abundance, CAI, and degree. The term Z2 represents theamount of translation, which contributes to covariation in
E, A, and C. The terms Z3 through Z6 represent sources ofvariation present in each predictor variable but not shared
between them. The parameters a and b quantify the
relative importance of translation versus other sources ofvariation in the predictor variables. In our model, theevolutionary rate (R) is determined by the amount oftranslation (Z2) and by the variation in protein interactions
unrelated to other variables (Z6). (Similar results are
obtained under related models, such as R 5 aZ1 þ bZ2;
see also Supplementary Materials online.)
In addition to the underlying model, we also specify
equations that describe noisy versions of the predictor
variables representing measurements of expression (E*),
measurements of abundance (A*), measurements of CAI(C*), and measurements of interaction degree (D*):
E 5E þ nE W1 ;A 5A þ nE W2 ;C 5C;
D 5D þ nD W5 ;where each Wi is an independent, normally distributed
random variable with mean zero and variance one. The
parameter nE determines the amount of noise in themeasured expression data, E*. We conservatively assume
that the amount of noise in protein abundance measurements equals the amount of noise in mRNA abundance
measurements (even though our calculations above
suggest that abundance has less noise). The parameternD determines the amount of noise in measured interaction
degree data. Note that CAI is measured without noise (i.e.,
C* 5 C).
We choose the 4 parameters of our model so as tomatch the most important empirical features of the yeast
data: 1) the noise in mRNA expression data, rexpr 50.72; 2) the noise in protein interaction data, rdeg 5
0.11; 3) the correlation between measured expression levels
and evolutionary rate, r(E*, R) 5 0.56 (n 5 2, 840); and 4)
the correlation between measured interaction degree andevolutionary rate, r(D*, R) 5 0.23 (n 5 692). Usinga straightforward parameterization technique (see Appendix), we find parameters so as to match all 4 of these observed features in the yeast data.
It is instructive to compare a PCR analysis of the underlying variables in our model against a PCR analysis of thenoisy variables, which represent measurable quantities.
When applied to the noisy variables (fig. 2a), the PCR indicates a dominant explanatory component consisting almost
entirely of the translation-related variables—expression(E*), abundance (A*), and CAI (C*). Each of these variables
explains significantly more variation in evolutionary rate
than protein interactions, which appear in a secondary minor
component; this situation is analogous to that seen in the real
data (fig. 1a and b). By contrast, when applied to the underlying noiseless variables, the PCR reveals a dramaticallyAssessing the Determinants of Evolutionary Rates 1117

FIG. 2.—A PCR analysis of the noisy (a) and underlying (b) variables in a simple model of evolutionary rates. When applied to the noisy variables,
which represent measurable quantities, the regression indicates a single, dominant determinant of evolutionary rates, and the degree of protein–protein
interactions appears to explain significantly less variation in rate than CAI, abundance, or expression. But when applied to the underlying noiselessvariables, the PCR analysis reveals the opposite conclusion: no single component dominates evolutionary rates, and protein interactions explain
significantly more variation than CAI, abundance, or expression.

different picture (fig. 2b): no single component dominates
evolutionary rates. Moreover, the true degree of protein interactions explains significantly more variation in rate thanexpression, abundance, or CAI.
The simple model developed here is consistent withthe observed noise levels in yeast genomic data and withthe observed correlations with evolutionary rates. Under
this model, protein interaction degree explains significantly
more variation in evolutionary rates than expression level,
abundance, or CAI. Nevertheless, if one were to analyze thenoisy versions of predictor variables (disregarding the factthat such a PCR analysis violates the assumption of equal
noise levels), one would reach the opposite conclusion.
Thus, our model highlights the danger of using a principle
component regression to analyze noisy biological data
without accounting for known variation in noise levels
across predictor variables.

Other Evidence on the Determinants of Evolutionary Rates
The preceding sections demonstrate that the PCR is
not robust to violating the assumption of equal noise levels
across predictor variables. This weakness is not unique tothe PCR, but it is likely shared by most other multiple regression techniques. As a result of this difficulty, however,given the known variation in noise levels across predictors,
there is no evidence at present for a single determinant of
evolutionary rates in yeast.
In this section, we demonstrate another related line ofevidence against our ability to deduce a single determinantof evolutionary rates: namely, the PCR depends strongly on
which predictors are included in the regression.
A gene’s expression, abundance, and CAI are all related to the total amount of translation events it experiences(Drummond et al. 2006). Therefore, Drummond et al. interpret the dominant explanatory component in their regression—comprised equal parts expression, abundance, andCAI—as the amount of translation, and they conclude that
selection against translation-error–induced protein misfold-ing is the predominant determinant of protein evolution inyeast (Drummond et al. 2005, 2006). If these conclusions
were robust, a PCR analysis of the same data excluding
CAI, for example, should yield a very similar result—
except that the resulting dominant component would becomprised abundance and expression.
As seen in figure 3, a regression excluding CAI paints
a very different picture of protein evolution than expectedunder the translational-selection hypothesis. According tothis regression, there is no evidence that translational processes dominate protein evolution. Instead, multiple independent components explain significant variation in
evolutionary rates. Moreover, the degree of protein interactions explains more variance in evolutionary rate than protein abundances, and it is more strongly represented in thefirst component. Regressions excluding mRNA expression,
protein abundance, or combinations of these variables yieldvery similar results. None of these results would be observed if the PCR method were robust and if translational
processes dominated protein evolution.
Finally, we note that other techniques for analyzing
collinear predictors, such as the sliced inverse regression(Duan and Li 1991), also fail to implicate a single, predominant determinant of evolutionary rates (not shown).

Discussion
What independent factors influence the rate of protein
evolution remains an outstanding question. The work ofDrummond et al. (2006) is critically important because it
demonstrates that partial correlations can yield spurious results, due to noise in a predictor variable. By the same token, we have demonstrated that multiple regressions canyield spurious results due to different levels of noise in predictor variables. At present, a conservative application of
the PCR method requires that we equalize noise levels
across predictors—in which case there is no evidence thatthe rate of protein evolution in yeast is dominated by a single
determinant.

1118 Plotkin and Fraser

FIG. 3.—PCRs of evolutionary rates. (a) A regression of dN against 7 predictor variables, previously reported by Drummond et al. (2006). (b) A
regression of the same data as in (a), omitting CAI as a predictor variable. When CAI is removed, there is no evidence of a dominant explanatorycomponent related to translational selection. In fact, when CAI is omitted, the degree of protein–protein interactions is more strongly weighted on thefirst component, and it explains more total variance in evolutionary rate than protein abundance. (Black corresponds to centrality; white todispensability; and grey to length.)

We emphasize that the limitations of the PCR in theface of predictor variables with different degrees of noise
are not unique to the PCR method. The same limitations
apply to virtually all methods of multiple regression, which
typically assume equal noise levels across predictors. We
have chosen to focus on the PCR only because this was
the method used by Drummond et al. (2006) to reach theirconclusions about protein evolution in yeast.
Our analysis has formally demonstrated a concept thatis intuitively clear: standard regression techniques cannot
meaningfully compare the explanatory power of predictors
when the predictors contain different amounts of measurement noise. It is less clear how best to deal with this difficulty, which comes hand in hand with diverse genomic
data. Ideally, information about the known noise levelsof predictors should be incorporated into statistical methods
when partitioning phenotypic variance into independent
contributions. Unfortunately, we know of no method that
performs such a partitioning while accounting for variablenoise levels. The simplistic approach used here for equalizing noise levels may seem Draconian because it throws
out some signal contained in the less noisy predictors. Nevertheless, this conservative approach is necessary until
more sophisticated statistical methods, which account forvariable noise levels, are developed. Indeed, the inadequacyof the PCR in this context is further demonstrated by thefact that its results are not robust to removing predictor variables from the regression.
Our procedure for equalizing noise levels is conservative with respect to our conclusions because we have used
one of the lowest estimates of noise in protein interaction
data and some of the highest estimates of noise in expression and abundance data. In particular, we have estimatedexpression noise using the same correlation coefficient reported by Drummond et al. (2005), and our estimate ofnoise in protein interactions is significantly smaller thanthe median estimate across 36 pairwise comparisons.
Aside from adding noise to empirical data, we have alsopresented a simple model of evolutionary rates that is consistent with the observed yeast data. The purpose of this model is

not to recapitulate all details of the empirical data but rather toexplore what underlying patterns are consistent with the salient features of the observed, noisy data. According to thismodel, the underlying, noiseless variable for protein interaction degree explains more variance in evolutionary rate than
all other variables—despite the fact that a PCR analysis on the
noisy, measured variables would yield the opposite conclusion. We emphasize that this model does not establish that
protein–protein interactions actually have a stronger influence on yeast evolutionary rates than expression, abundance,or CAI. Rather, the model simply demonstrates that we cannot yet rule out the possibility of an important, independentrole for protein–protein interactions (or other noisy variables)in determining evolutionary rates.
We emphasize that our analysis does not rule out thepossibility of a single determinant for evolutionary rates.
Indeed, in the future, we may conclude definitively thattranslational processes explain more variance in evolutionary rates than any other feature of yeast proteins. At present,
however, given the variable amounts of noise associatedwith existing genome-wide measurements, PCRs do notprovide sufficient evidence to reach this conclusion. It appears that more accurate measurements, or more sophisticated statistical techniques, will be required to tease apartthe underlying determinants of protein evolution.

Appendix
Data Sets
All data sets were taken directly from Drummond et al.
(2006), with the exception of the protein interaction data.
The protein interaction data were measured by mass spectrometry (Ho et al. 2002), and they were compared withanother mass spectrometry data set (Gavin et al. 2002) inorder to estimate the amount of noise. Our results remainunchanged if we use the interaction data from Drummondet al. instead of the mass spectrometry data. An additional
data set (Warringer et al. 2003) was used to estimate thenoise in gene dispensability.

Assessing the Determinants of Evolutionary Rates 1119

Equalizing Noise Levels across Data Sets
Before applying regressions, all variables were logtransformed (except dispensability), centered, and variancenormalized, as in Drummond et al. (2006). Our results remain essentially unchanged using rank regressions insteadof parametric regressions.
Of the data sets used in this study, the protein–protein interaction data are the least precise. In order toequalize noise across predictor variables, we add an appropriate amount of noise to each transformed variable
and then rescale each variable by its variance. In order tomatch the noise level in protein interaction data, rdegree 5
0.11, we must add enough extra noise to each other variable sothat, if we were to add the noise twice independently (so as toimitate 2 independent measurements, each with their own
source of noise), the resulting correlation would equal rdegree.
To be more explicit, consider a predictor variable Ewith original noise level, rE , rdegree, given by the correlation between 2 independent measurements: rE 5rðE1 ; E2 Þ.
We must find the value of k such that
r scaleðE1 Þ þ kZ1 ; scaleðE2 Þ þ kZ2 5rdegree ;where each Zi is an independent Gaussian random variablewith mean zero and variance one and scale(E) 5 (E  lE)/rE.
Letting A5scaleðE1 Þ þ kZ1 and B5scaleðE2 Þ þ kZ2 andsubstituting into the expression for the correlation coefficient, we must solveEðABÞ  EðAÞEðBÞ
5rdegree :rA rB
Because A and B both have expected value zero,and each Zi is uncorrelated with E1 and E2 , our equation
reduces to
E½scaleðE1 Þ scaleðE2 Þ
5rdegree :1 þ k2
Because the numerator in the equation above equals
rE, we may writerﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
rE
 1:
k5
rdegreeThis equation gives a simple expression for theamount of noise, k, that we must add to a variable E so
as to equalize its noise level with that of protein interaction
degree.
For our variables of interest, we have rexpr 5 rabund 5
0.72, rdegree 5 rcentrality 5 0.112, rdisp 5 0.561, andrCAI 5 rlength 5 1. As a result, kexpr 5 kabund 5 2.329,
kdegree 5 kcentrality 5 0, kdisp 5 2.003, and kCAI 5
klength 5 2.815. Figure 1c shows the mean results from
PCR analyses of the predictor variables after adding noise.
Standard deviations were calculated from .2000 independent random draws.
Principal Components Regressions
All regressions were performed in R (www.r-project.
com). In all cases, we have retained all the components in

the PCRs. Although there are techniques designed to assess
the appropriate number of ‘‘nondegenerate’’ components,
such techniques are inherently subjective (Jackson 1993),
and so their utility in this context is unclear.
Parameterizing the Model
Our phenomenological model of expression level (E),abundance (A), CAI (C), protein interaction degree (D), andevolutionary rate (R) depends on 4 parameters according to
the equations
E5aZ1 þ bZ2 þ Z3;
A5aZ1 þ bZ2 þ Z4;
C5aZ1 þ bZ2 þ Z5 ;D5aZ1 þ Z6 ;R5Z2 þ Z6 ;where each Zi is an independent, normally distributed randomvariable with mean zero and variance one. In addition to theunderlying model, we also specify equations that describe
noisy versions of the predictor variables, representing 2
independent measurements of expression ðE1 ; E2 Þ; a measurement of abundance (A*), a measurement of CAI (C*), and2 independent measurements of interaction degree ðD1 ; D2 Þ:
E1 5E þ nE W1 ;E2 5E þ nE W2 ;A 5A þ nE W3 ;C 5C;
D1 5D þ nD W4 ;D2 5D þ nD W5 :In these equations, each Wi is an independent Gaussianvariable, and we have conservatively assumed that abundance data are as noisy as expression data (nA 5 nE). We wish
to choose the 4 parameters a, b, nE, and nD so as to match
important features of the yeast data: 1) the noise in mRNA
expression data, rexpr 5 0.72; 2) the noise in protein interaction data, rdeg 5 0.11; 3) the correlation between measured
expression levels and evolutionary rate, rðE1 ; RÞ50:56; and4) the correlation between measured interaction degree
and evolutionary rate, rðD1 ; RÞ50:23. In other words, ourparameters should satisfy the following 4 conditions as
precisely as possible:rðE1 ; E2 Þ50:72;
rðD1 ; D2 Þ50:11;
rðE1 ; RÞ50:56;
rðD1 ; RÞ50:23:
Under the assumptions of our model, we can write
analytic expressions for left hand sides of these equations,in terms of our 4 parameters:rðE1 ; E2 Þ5ða2 þ b2 þ 1Þ ðrE Þ2 ;22rðD1 ; D2 Þ5ða Þ ðrD Þ ;rðE1 ; RÞ5b ðrE  rR Þ;rðD1 ; RÞ51 ðrD  rR Þ;1120 Plotkin and Fraser

where
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
a2 þ b 2 þ 1 þ n2 ;pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ E
rD 5pﬃﬃﬃ a2 þ 1 þ n2D ;rR 5 2:rE  5Given these analytic expressions, we numerically minimize the square Euclidean distance½rðE1 ; RÞ  0:562 þ ½rðD1 ; RÞ  0:232
þ ½rðE1 ; E2 Þ  0:722 þ ½rðD1 ; D2 Þ  0:112
and thereby find parameters that satisfy our desiredconditions to 2 significant digits:
a50:1992;
b52:651;
nE 51:771;
nD 52:900:
Supplementary Material
Supplementary materials are available at Molecular Biology and Evolution online (http://www.mbe.oxfordjournals.
org/).
Acknowledgments
We thank D. A. Drummond, C. O. Wilke, and A. Ravalfor productive conversations and contributions to the manuscript. J.B.P. acknowledges support from the Burroughs
Wellcome Fund. Note added in proof: recently Kim andYi (Genetica 2007, DOI 10.1007/s10709-006-9125-2) have
independently demonstrated that principle component regression can yield spurious results when predictor variables
contain different amounts of noise.

Literature Cited
Bloom JD, Adami C. 2003. Apparent dependence of protein
evolutionary rate on number of interactions is linked to biases
in protein-protein interactions data sets. BMC Evol Biol. 3.
Causton HC, Ren B, Koh SS, Harbison CT, Kanin E,
Jennings EG, Lee TI, True HL, Lander ES, Young RA.
2001. Remodeling of yeast genome expression in response toenvironmental changes. Mol Biol Cell. 12:323–337.
Deutschbauer AM, Jaramillo DF, Proctor M, Kumm J,
Hillenmeyer ME, Davis RW, Nislow C, Giaever G. 2005.
Mechanisms of haploinsufficiency revealed by genome-wide
profiling in yeast. Genetics. 169:1915–1925.
Dickerson RE. 1971. The structures of cytochrome c and the rates
of molecular evolution. J Mol Evol. 1:26–45.
Drummond D, Bloom JD, Adami C, Wilke CO, Arnold FH.
2005. Why highly expressed proteins evolve slowly. Proc
Natl Acad Sci USA. 102:14338–14343.
Drummond DA, Raval A, Wilke CO. 2006. A single determinant
dominates the rate of yeast protein evolution. Mol Biol Evol.
23:327–337.
Duan N, Li KC. 1991. Slicing regression: a link-free regression
method. Ann Stat. 19:505–530.

Fraser HB, Hirsh AE, Steinmetz LM, Scharfe C, Feldman MW.
2002. Evolutionary rate in the protein interaction network.
Science. 296:750–752.
Gavin AC, Bosche M, Krause R, et al. (38 co-authors). 2002.
Functional organization of the yeast proteome bysystematic analysis of protein complexes. Nature. 415:
141–147.
Ghaemmaghami S, Huh WK, Bower K, Howson RW, Belle A,
Dephoure N, O’Shea EK, Weissman JS. 2003. Global
analysis of protein expression in yeast. Nature. 425:737–
741.
Green P, Lipman D, Hillier L, Waterston R, States D,
Claverie JM. 1993. Ancient conserved regions in new genesequences and the protein databases. Science. 259:
1711–1716.
Hahn MW, Kern AD. 2005. Comparative genomics of centrality
and essentiality in three eukaryotic protein-interaction networks. Mol Biol Evol. 22:803–806.
Han JD, Bertin N, Hao T, et al. (11 co-authors). 2004. Evidence
for dynamically organized modularity in the yeast proteinprotein interaction network. Nature. 430:88–93.
Hirsh AE, Fraser HB. 2001. Protein dispensability and the rate ofevolution. Nature. 411:1046–1049.
Hirsh AE, Fraser HB, Wall DP. 2005. Adjusting for selection onsynonymous sites in estimates of evolutionary distance. Mol
Biol Evol. 22:174–177.
Ho Y, Gruhler A, Heilbut A, et al. (46 co-authors). 2002.
Systematic identification of protein complexes in Saccharomyces cerevisiae by mass spectrometry. Nature. 415:
180–183.
Holstege FC, Jennings EG, Wyrick JJ, Lee TI, Hengartner CJ,
Green MR, Golub TR, Lander ES, Young RA. 1998.
Dissecting the regulatory circuitry of a eukaryotic genome.
Cell. 95:717–728.
Hurst LD, Smith NG. 1999. Do essential genes evolve slowly?
Curr Biol. 9:747–750.
Ingram VM. 1961. Gene evolution and the haemoglobins.
Nature. 189:704–708.
Jackson DA. 1993. Stopping rules in principal components
analysis: a comparison of heuristical and statistical approaches. Ecology. 74:2204–2214.
Kemmeren P, van Berkum NL, Vilo J, Bijma T, Donders R,
Brazma A, Holstege FC. 2002. Protein interaction verificationand functional annotation by integrated analysis of genomescale data. Mol Cell. 9:1133–1143.
Koonin EV, Wolf YI. 2006. Evolutionary systems biology: links
between gene evolution and function. Curr Opin Biotechnol.
17:481–487.
Marais G, Duret L. 2001. Synonymous codon usage, accuracy oftranslation, and gene length in caenorhabditis elegans. J Mol
Evol. 52:275–280.
McInerney JO. 2006. The causes of protein evolutionary ratevariation. Trends Ecol Evol. 21:230–232.
Mintseris J, Weng Z. 2005. Structure, function, and evolution oftransient and obligate protein-protein interactions. Proc Natl
Acad Sci USA. 102:10930–10935.
Ohta T. 1973. Slightly deleterious mutant substitutions inevolution. Nature. 246:96–98.
Pal C, Papp B, Hurst LD. 2001. Highly expressed genes in yeast
evolve slowly. Genetics. 158:927–931.
Pal C, Papp B, Hurst LD. 2003. Genomic function: rate ofevolution and gene dispensability. Nature. 421:496–497.
Pal C, Papp B, Lercher MJ. 2006. An integrated view of protein
evolution. Nat Rev Genet. 7:337–348.
Reguly T, Breitkreutz A, Boucher L, et al. (20 co-authors). 2006.
Comprehensive curation and analysis of global interaction
networks in saccharomyces cerevisiae. J Biol. 5:11.

Assessing the Determinants of Evolutionary Rates 1121

Rocha EP, Danchin A. 2004. An analysis of determinants ofamino acids substitution rates in bacterial proteins. Mol Biol
Evol. 21:108–116.
Rocha EPC. Forthcoming 2006. The quest for the universals ofprotein evolution. Trends Genet. 22:412–416.
von Mering C, Krause R, Snel B, Cornell M, Oliver SG, Fields S,
Bork P. 2002. Comparative assessment of large-scale data setsof protein-protein interactions. Nature. 417:399–403.
Wall DP, Hirsh AE, Fraser HB, Kumm J, Giaever G, Eisen MB,
Feldman MW. 2005. Functional genomic analysis of the rates
of protein evolution. Proc Natl Acad Sci USA. 12:5483–5488.

Warringer J, Ericson E, Fernandez L, Nerman O, Blomberg A.
2003. High-resolution yeast phenomics resolves different
physiological features in the saline response. Proc Natl Acad
Sci USA. 100:15724–15729.
Wilson A, Carlson SS, White TJ. 1977. Biochemical evolution.
Annu Rev Biochem. 46:573–639.

Michele Vendruscolo, Associate Editor
Accepted February 8, 2007

Assessing the determinants of evolutionary rates in the presence of noise
Joshua B. Plotkin and Hunter B. Fraser
Supplementary Information
In this supplement we present the results of an alternative model of evolutionary rates
that is slightly more detailed that the model presented in the paper:

100

100

80

80

b (clean)

60

C

40

60

64.1

E

C*

E*

27.8

A

D

Comp 1

Comp 2

D*

0

1.5

0

A*

20

40

46.1

20

Explained variance in rate, R(%)

a (noisy)Comp 1

Comp 2Comp 3Comp 4Comp 3Comp 4

Supplementary Figure A. PCR analysis of the noisy (a) and underlying (b)variables in a simple model of evolutionary rates. When applied to the noisy
variables, which represent measurable quantities, the regression indicates asingle, dominant determinant of evolutionary rates; and the degree ofprotein-protein interactions appears to explain significantly less variation inrate than CAI, abundance, or expression. But when applied to the underlying
noise-less variables, the PCR analysis reveals the opposite conclusion: no
single component dominates evolutionary rates; and the degree ofinteractions explains significantly more variation than CAI, abundance, or
expression.
The model shown above is identical to the one presented in the paper, except for an
additional term in the degree equation: D = α Z1 + β Z2 + γ Z6. As a result of this minor
modification, the first component of the noisy regression contains a small loading bydegree, similar to that observed in the empirical data (Fig. 1a). This modified model
demonstrates the same conclusion as the model presented in the paper: the noisy data
implicate a single determinant, whereas the underlying variables reveal multiple
determinants. Parameters are given by α=0.199, β=2.651, γ=0.5, nE=1.77, nD =2.0.

Color version of Figure 1

Color version of Figure 2

Color version of Figure 3