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Genetics. Apr 2007; 175(4): 1813–1822.
PMCID: PMC1855142

Evolutionary Framework for Protein Sequence Evolution and Gene Pleiotropy


In this article, we develop an evolutionary model for protein sequence evolution. Gene pleiotropy is characterized by K distinct but correlated components (molecular phenotypes) that affect the organismal fitness. These K molecular phenotypes are under stabilizing selection with microadaptation (SM) due to random optima shifts, the SM model. Random coding mutations generate a correlated distribution of K molecular phenotypes. Under this SM model, we further develop a statistical method to estimate the “effective” number of molecular phenotypes (Ke) of the gene. Therefore, for the first time we can empirically evaluate gene pleiotropy from the protein sequence analysis. Case studies of vertebrate proteins indicate that Ke is typically ~6–9. We demonstrate that the newly developed SM model of protein evolution may provide a basis for exploring genomic evolution and correlations.

ALTHOUGH the wide availability of high-throughput data has greatly facilitated the study of evolutionary genomics, the foundation of molecular evolution has been relatively “static” (Kimura 1983). Basically, the evolutionary fate of a mutant is largely determined by the prespecified fitness value (the coefficient of selection) and the effective population size. Although this classical treatment has successfully removed the complexity of genotype–phenotype association, problems may appear when the association itself has been one of the research highlights. The recent debate about genomic correlations among evolutionary rate, interaction, expression, and dispensability has indicated the limitation of current evolutionary models (reviewed in Pal et al. 2006). Alternatively, researchers have to use the partial-correlation or principle-component methods to analyze major determinants governing protein evolution (e.g., Drummond et al. 2005; Wall et al. 2005; Wolf et al. 2006), because these approaches are not model dependent.

It is desirable to develop a novel framework that can provide an integrated view of protein evolution, but an unsolved key problem is how to deal with the sophisticated relationship between protein sequence and organismal fitness. We recognized that the substantial genotype–phenotype literature can be used to address this issue (e.g., Fisher 1930; Lande 1980; G. Wagner 1989; Waxman and Peck 1998; A. Wagner 2000; Poon and Otto 2000; Zhang and Hill 2003). As mutations are raw materials for evolution, the distribution of effects of mutations, f(s), has been a central issue in the genotype–phenotype study (Lynch et al. 1999; Bataillon 2000; Shaw et al. 2002). Under Fisher's geometric model (e.g., Fisher 1930; Hartl and Taubes 1998; Poon and Otto 2000) or multivariate model (e.g., Lande 1980; Welch and Waxman 2003; Martin and Lenormand 2006), f(s) can be predicted on the basis of selective and mutational assumptions. The dimensionality in each framework has been interpreted as the number of loci affecting a quantitative trait (reviewed in Kingsolver et al. 2001), or the number of affected phenotypic traits of a mutation, estimated from mutational-accumulation experiments (reviewed in Elena and Lenski 2003). On the other hand, an ad hoc gamma distribution for f(s) was introduced in the early days of the neutral theory of molecular evolution (Ohta 1973, 1977; Kimura 1983), which has been found useful to estimate the proportion of neutral or deleterious mutations (e.g., Nielsen and Yang 2003; Piganeau and Eyre-Walker 2003; Eyre-Walker et al. 2006).

Here we extend our recent work (Gu 2006). The principle behind our present work in modeling protein sequence evolution is as follows: First, multifunctionality of a gene (pleiotropy) is characterized by K distinct components in the fitness (called molecular phenotypes). Second, these K molecular phenotypes are under stabilizing selection (Fisher 1930; Waxman and Peck 1998) with microadaptations due to random shifts of the fitness optimum (Gillespie 1991). Third, random mutations in the coding region of the gene generate a mutational distribution for K molecular phenotypes. Correlated structures among K molecular phenotypes are taken into account at both selectional and mutational levels. More importantly, we develop a statistical method to estimate the “effective” number of molecular phenotypes (Ke) of the gene. Case studies of vertebrate proteins demonstrate that for the first time we can empirically evaluate gene pleiotropy only from phylogenetic sequence analysis.


Molecular phenotypes and gene pleiotropy:

Pleiotropy of a gene (protein) function can be modeled by multiple (K) molecular phenotypes, denoted by (y1, …, yK). Each molecular phenotype yi represents a nontrivial component of genetic variation, contrasting with organismal fitness as a result of a specific (yet unknown) biological process (Figure 1). At one extreme, molecular phenotypes could correspond to subcomponents of protein function, regardless of the biological processes. At another extreme, molecular phenotypes could be determined mainly by distinct physiological processes. Since these underlying biological processes are usually intractable, the concept of molecular phenotypes based on fitness effects avoids this difficulty.

Figure 1.
Schematic presentation for the concept of molecular phenotypes that affect the fitness of an organism.

Following the widely used procedure (e.g., Lande 1980; Turelli 1985; Waxman and Peck 1998), we assume that the fitness function of molecular phenotypes y = (y1, …, yK)′ is Gaussian-like; i.e.,

equation M1

where μ is the optimum of the fitness function, and Σw is a (positive definite) symmetric matrix characterizing correlated stabilizing selection on K molecular phenotypes. The ith diagonal element equation M2 measures the strength of stabilizing selection on the ith molecular phenotype, while the ijth nondiagonal element σw,ij measures the correlated stabilizing selection on yi and yj.

Let y0 be the population mean of molecular phenotypes of a gene, which may not necessarily be at the optimum (μ). By definition (Fisher 1930), the coefficient of selection for the molecular phenotype y is given by

equation M3

Classical stabilizing selection (Turelli 1985; Waxman and Peck 1998) assumes that the population mean is always at the fitness optimum, i.e., y0 = μ, leading to purifying selection; that is, ρ(y | y0 = μ) ≤ 0 always holds.

Next, we consider the effects of mutations on molecular phenotypes. A random mutation in the coding region of a gene may affect molecular phenotypes in a correlated fashion (Figure 1). Such a pleiotropic mutational effect can be described by a multivariate normal distribution

equation M4

where ym is the mean mutational effect on molecular phenotypes, and the covariance matrix Σm measures the correlated mutational effects. The coefficient of selection ρ(y | y0, μ) and the distribution of mutational effects p(y) provide a connection between organismal fitness and protein sequence evolution through molecular phenotypes y (Figure 1). Below we discuss the classical stabilizing selection model and the newly developed stabilizing selection with microadaptation (SM) model. See Figure 2 for illustration.

Figure 2.
(A) Stabilizing selection model with microadaptation in the case of a single molecular phenotype. (B) Distribution of mutational effects on the molecular phenotype. In both cases, we set K = 1 for simplicity.

Stabilizing selection model:

Under the classical stabilizing model, the population mean of molecular phenotypes (y0) is always fixed at the optimum (μ). Without loss of generality, we assume that y0 = μ = 0. From Equation 2 one can show that equation M5, reflecting the stabilizing (purifying) selection against deleterious mutations, which necessarily results in a deviation from the optimum. A consequence of this model is that sequence evolution is dominated by the fixation of very slightly deleterious mutations (Ohta 1973; Kimura 1979, 1983).

For mathematical convenience, we assume that the mutational distribution, p(y) in Equation 3, is centered at the population mean, namely, ym = y0 = 0. In the following we consider mainly the selection intensity S(y) = 4Neρ0(y) = −2Ne(equation M6), where Ne is the effective population size. Given Equation 3 of p(y), the distribution of S, f(S), can be uniquely determined by the probability theory (see the appendix). Although the analytical form of f(S) is available only for a few special cases, the moments of S can be obtained readily. For instance, the mean of S can be computed by

equation M7

which is always negative.

Stabilizing selection with microadaptation model:

Consider the case when the fitness optimum (μ) of a gene is no longer fixed during the evolution. Rather, μ can be shifted either by environmental changes or by internal physiological perturbations. Each shift of μ results in a microadaptation toward a new fitness optimum (Gillespie 1991). Since the direction and strength of the μ-shift are unobservable, we treat μ as a (K-dimensional) random variable. Similar to the above, we assume that the population mean and the mutation mean satisfy y0 = ym = 0. Then, under the SM model, the coefficient of correlation defined by Equation 2 can be specified as follows:

equation M8

After assuming that μ follows a multivariate normal distribution [var phi](μ) with mean 0 and covariance matrix Σμ, one can integrate out the unobservable μ by equation M9. Therefore, one can show that ρ(y), the coefficient of selection of y averaged over μ-shifts during the evolution, is given by

equation M10

where the matrix equation M11 characterizes the correlated nature of molecular phenotypes in fitness after stabilizing selection and microadaptation. It follows that the selection intensity defined by S(y) = 4Neρ(y) is given by

equation M12

Apparently, the stabilizing selection model is a special case when Σμ = 0 so that U = Σw. Moreover, unlike the stabilizing selection model that S(y) is always negative, the selection intensity S(y) under the SM model can be positive when the μ-shift becomes huge. See analytical results for more details.

Evolutionary rate of protein sequence:

Suppose that a protein carrying one (nonsynonymous) mutation has the molecular phenotype vector y. The selection intensity for this mutation, S(y), is given by Equation 7. Hence, the well-known formula for the evolutionary rate (Kimura 1983) depends on the molecular phenotypes y; that is,

equation M13

where v is the mutation rate. At the gene level, molecular phenotypes y are random variables, resulting from random mutations at nucleotide sites (or amino acid residues). Therefore, the evolutionary rate is a random variable that varies among sites. Under the SM model, the mean evolutionary rate of a protein sequence is given by

equation M14

given the distribution of mutational effects on y, p(y).


Here we present analytical results under the SM model, which are useful to study the underlying mechanisms governing protein sequence evolution and to develop a statistical method for estimating important parameters such as K, the number of molecular phenotypes of a gene.

Single molecular phenotype (K = 1):

When K = 1, the fitness function in Equation 1 can be reduced as equation M15 and the selection coefficient in Equation 6 as equation M16, where equation M17 measures the strength of single stabilizing selection. Similarly, the shift of μ follows a single normal distribution [var phi](μ) with mean 0 and equation M18. Together, Equation 8 can be simplified as

equation M19

Given the distribution of mutational effects equation M20, one can show the mean of selection intensity, equation M21, as follows:

equation M22

Several interesting results need to be mentioned. First, equation M23 is determined by the mutational variance (equation M24) and the variance of microadaptation (equation M25) relative to the coefficient of stabilizing selection (equation M26). Hence, the scale of molecular phenotype is not important. Second, equation M27 if equation M28 < equation M29, and vice versa. In particular, the microadaptation model is reduced to the stabilizing selection model when equation M30 = 0 (Gu 2007). Third, microadaptation reduces the sequence conservation by decreasing the absolute values of equation M31, although equation M32 may remain negative. In other words, microadaptation may increase the evolutionary rate of protein sequence. And fourth, from Equation 10, one can show that S follows a negative gamma distribution with the shape parameter ½. For instance, when equation M33 < equation M34, S follows a negative gamma distribution

equation M35

K molecular phenotypes:

For K ≥ 2, the distribution of selection intensity S, f(S), is generally not analytically tractable. Nevertheless, one can derive the analytical forms of moments on the basis of the theory of multiple normal distribution. These results are crucial for estimating K from the sequence data.

Mean and variance of S—general results:

Denote matrix A = U−1Σm = ZΩ, where equation M36 and equation M37. Briefly, under stabilizing selection, matrix Z describes correlated mutational effects, while Ω describes correlated microadaptations. Together, matrix A characterizes the net effects of correlated mutations on fitness under the SM model. In the appendix (also see Martin and Lenormand 2006 for phenotypic data), we show that the joint effects of all selective, microadaptive, and mutational covariances can be reduced to K eigenvalues of matrix A, α1, …, αK. Consequently, the mean and variance of S are given by

equation M38


equation M39

respectively. In short, each αi corresponds to an independent molecular phenotypic direction, on which mutation, microadaptation, and stabilizing selection act with an average effect −2Neαi on the mean selection intensity (equation M40) of the gene.

Canonical representation of S:

Although Equation 13 is mathematically elegant, we wish to find an equivalent representation of equation M41 that is biologically more interpretable. Due to the arbitrary nature of the original K molecular phenotypes, without loss of generality one may choose a convenient coordinate system. Here we adopt a canonical form of K molecular phenotypes, to define independent phenotypes experiencing stabilizing selection. This leads to a diagonal matrix Σw; each diagonal element equation M42 measures the independent stabilizing selection on the ith (canonical) molecular phenotype. Let equation M43 be the ith diagonal element of matrix Σw or the mutational variance for the ith (canonical) molecular phenotype. We have shown that the canonical presentation for the mean of selection intensity is given by

equation M44

where the parameter γi measures the net effect of microadaptation on the ith (canonical) molecular phenotype. In particular, when the mutational effect or microadaptation is independent among the canonical molecular phenotypes, we have equation M45. It should be noted that the pleiotropic nature of mutations affects the variance of S, Var(S) (see the appendix), while equation M46 is not affected, as shown in Equation 15.

Model classification:

As the stabilizing selection acts against nucleotide substitutions (negative selection), microadaptation may provide an opposite force (positive selection) to increase the evolutionary rate. The mean selection intensity given by Equations 13 and 15 can be used to classify the pattern of sequence conservation under the SM model:

  1. Strong stabilizing selection with weak microadaptation (SMw): In this case, the magnitude of μ-shift of molecular phenotypes during the evolution is small, relative to the strength of stabilizing selection. It indicates a dominant purifying selection in the protein sequence evolution. The SMw model can be defined as having matrix A positive definite, such that each eigenvalue αi > 0 and equation M47. In the canonical form of Equation 15, it means that γi < 1 for any i = 1, … , K. The pure stabilizing selection is a special case of γi = 0 (no microadaptation). Moreover, equation M48 implies that the ratio of nonsynonymous to synonymous substitutions (dN/dS) is <1. Therefore, the SMw model may describe the general pattern in protein sequence evolution, evidenced by genomewide analyses that have shown dN/dS < 1 for most genes.
  2. Episodic microadaptation under strong stabilizing selection (SME): Some genes may have experienced episodic adaptive processes in a few molecular phenotypes, resulting in negative eigenvalues of matrix A (i.e., αi < 0 for some i's, but the mean selection intensity remains negative, equation M49). That is, matrix A is no longer positive definite but a positive trace equation M50 remains. In the canonical form, some of γi are >1, although equation M51. Apparently, the dN/dS ratio under the SME model may be increased by episodic microadaptations, but dN/dS < 1 remains. A special case is the neutral-like behavior of protein sequence evolution when equation M52, resulting in equation M53. That is, the observed neutral-like evolution may reflect the zero net effect on fitness, canceling each other between episodic microadaptation and stabilizing selection.
  3. Strong microadaptation under stabilizing selection (SMs): In a few genes, positive selection and adaptation may dominate the evolution in many molecular phenotypes. In these cases, equation M54 results in equation M55 and the ratio dN/dS > 1. An extreme case occurs if αi < 0 or γi > 1 (the canonical form) for all i = 1, …, K. In this case, matrix A is negative definite, indicating the overwhelming adaptation forces for virtually all of the substitutions.

The W-model for S-gamma distribution:

Under some strong assumptions, Gu (2007) showed that S is distributed as a (negative) gamma. A more general condition for S-gamma distribution exists when stabilizing selection, mutations, and microadaptation have very similar correlation structures among molecular phenotypes. Indeed, when the three corresponding matrices can be written as equation M56, equation M57, and equation M58, respectively, where W is a (positive-definite) symmetric matrix, the eigenvalues of matrix A become identical; i.e., α1, …, αK = α, as given by

equation M59

Then, one can easily show that when α > 0 (the SMw model), f(S) follows a negative gamma distribution

equation M60

where the parameter h = 1/(4Neα). Although it is rare, when α < 0 (the SMs model), f(S) follows a (positive) gamma distribution. In both situations, the gamma shape parameter of S is the (half) number of molecular phenotypes (K/2) (Gu 2007).


The newly developed SM model is useful in practice only if the underlying parameters can be estimated. But this is difficult to accomplish using conventional statistical approaches, since the number of unknown parameters may be huge. Instead, we attempt to focus on two important parameters, K, critical for understanding gene pleiotropy, and equation M61, the measure for overall sequence conservation. Our purpose here is to estimate K and equation M62, without specifying other more detailed parameters.

Moments of evolutionary rate:

Under the SM model, the mean evolutionary rate of a protein sequence (λ) is given by Equation 9. Therefore, any kth moment of the evolutionary rate is given by

equation M63

To apply Equation 18 to estimate unknown parameters, we use the approximation S/(1 − eS) ≈ e−|S|(1 + c|S|), where c ≈ 0.5772 is Euler's constant. As shown by Figure 3, this approximation is satisfactory for S ≤ 0. Thus, under the SMw model (stabilizing selection with weak microadaptation), which assures S(y) = −2Ney′U−1y < 0, the kth moment of the evolutionary rate λ can be approximated as follows:

equation M64
Figure 3.
(A) The λ/vS (the rate ratio of evolution to mutation vs. the selection intensity) relationship, for the accurate formula and the approximation. (B) The gK ~ K plotting for estimating the effective number of molecular phenotypes. ...

As shown in the appendix, the mean evolutionary rate (k = 1) is given by

equation M65

and the second moment of λ (k = 2) by

equation M66

Effective number (Ke) of molecular phenotypes:

In general, the mean evolutionary rate depends on K distinct eigenvalue parameters (Neαi's), while the second moment depends on Neαi's and equation M67, and so forth. To develop a simple method for estimating K on the basis of Equations 20 and 21, we invoke the assumption of Neαi [dbl greater-than sign] 1, leading to

equation M68


equation M69

respectively. Notably, the first and second moments of evolutionary rate depend only on two parameters, K and the product equation M70. Further, we found that the ratio of the second moment equation M71 to the first moment (equation M72), denoted by equation M73, is given by

equation M74

which is only K dependent. As shown in Figure 3, gK decreases when K increases; gK = 1 when K = 0, and equation M75 when equation M76.

Equations 2224 provide the theoretical foundation for estimating K when Neαi [dbl greater-than sign] 1. In this case, the parameter K is interpreted as the number of molecular phenotypes that have experienced strong stabilizing selection. Thereafter we refer to Ke as the effective number of molecular phenotypes, which is less than the true number of molecular phenotypes. If the ratio gK is known, Ke can be estimated according to Equation 24.

A widely used measure for equation M77 is dN/dS, the ratio of nonsynonymous to synonymous substitutions, under the assumption that synonymous substitutions are almost neutral. We have realized that the second moment of rate equation M78 is related to the H-measure (Gu et al. 1995) for the rate variation among sites (Uzzel and Corbin 1971), defined by equation M79. Ranging from 0 to 1, a higher value of H means a greater rate variation among sites and vice versa. One can easily verify

equation M80

Therefore, the effective number of molecular phenotypes (Ke) is the solution of the following equation:

equation M81

Effective selection intensity (Se):

Equation 13 indicates that the mean selection intensity equation M82 can be written as equation M83, where equation M84 is the (arithmetic) average of selection intensity over molecular phenotypes. However, without knowing each Neαi, it is difficult to estimate equation M85. After realizing that the product equation M86 is actually estimable, as implied by Equation 22, we define the effective selection intensity as equation M87, where equation M88 is the geometric average of selection intensity per molecular phenotype; that is,

equation M89

In general, equation M90, and equation M91 when all αi's are roughly the same. At any rate, both geometric and arithmetric means of selection intensity have virtually the same effect on the rate of protein evolution. Replacing equation M92 by dN/dS in Equation 22 and K by Ke, the effective selection intensity can be estimated by

equation M93

Estimation procedure:

We propose an analytical pipeline to estimate the effective number of molecular phenotypes (Ke) and the effective selection intensity (Se). Although it needs to be refined both statistically and computationally, our procedure consists of the following steps:

  1. Infer the phylogenetic tree from a multiple alignment of homologous protein sequences. Although there is no methodological preference, we require that the inferred tree topology should be roughly the same over methods.
  2. Estimate the nonsynonymous to synonymous ratio (dN/dS) from closely related coding sequences that satisfy dN/dS < 1.
  3. Estimate the H-measure for rate variation among sites: Use the method of Gu and Zhang (1997) to infer the (corrected) number of changes at each site, under the inferred phylogeny.

Let equation M94 and V(x) be the mean and variance of number of changes over sites, respectively. Assuming a Poisson process at each site, we obtain the mean evolutionary equation M95, where T is the total evolutionary time along the tree. Similarly, the variance of evolutionary rate among sites is given by equation M96. Then, H can be estimated by

equation M97

And finally, one can estimate Ke and Se according to Equations 26 and 28, respectively.


For case studies, we compiled 20 vertebrate protein families. Each family includes eight complete sequences from human, mouse, dog, cow, chicken, Xenopus, fugu, and zebrafish, respectively, which were downloaded from Ensembl EnsMar (http://www.ensembl.org/Multi/martview) (Kasprzyk et al. 2004). The synonymous distance (dS) and the nonsynonymous distance (dN) between mammalian orthologs were estimated by codeml of the PAML package (Yang 1997). We used the same phylogeny (Figure 4) to estimate the H-index of rate variation among sites; using an alternative tree gave a very similar result (not shown).

Figure 4.
Phylogenetic tree of vertebrates used in our data analysis.

Table 1 shows the results. The dN/dS ratio varies substantially among genes (from 0.001 to 0.274), and the H-index ranges from 0.286 to 0.886. Consequently, the estimated effective number of molecular phenotypes Ke ranges from 2.3 to 20.4, with an average of 8.77 (the median is 8.23). Our analysis suggests that most genes are highly pleiotropic. That is, there are on average eight distinct components in the fitness that may be related to a gene. Roughly, ~73% of the among-gene variation in the dN/dS ratio can be explained by the variation in Ke among genes. In other words, the level of gene pleiotropy associated with a gene may be the dominant factor for predicting sequence conservation, as predicted by Fisher (1930). The effective mean selection intensity (Se) ranges from 5.39 to 23.63, with a median of 12.6. Note that dN/dS ratios in Table 1 were estimated from the human and mouse orthologous genes; using other mammals or taking an average would lead to ~5% difference in the Ke estimation.

Estimation of Ke and Se for 20 vertebrate proteins


Estimation of gene pleiotropy:

Although the theory of gene pleiotropy, the capacity of a gene to affect multiple phenotypic characters, has been used to explain many biological phenomena (Fisher 1930; Wright 1968; Barton 1990; Waxman and Peck 1998; Otto 2004; MacLean et al. 2004; Dudley et al. 2005), the characteristic level of pleiotropy remains largely unknown. Many phenotype–genotype models (e.g., Fisher 1930; Wright 1968; Hartl and Taubes 1996, 1998; Waxman and Peck 1998; Poon and Otto 2000) have linked gene pleiotropy to the dimensionality of the fitness function of the organism. The major contribution of this article is to formulate a statistical framework for estimating the capacity of a gene to significantly affect distinct components in the fitness, the effective number of molecular phenotypes (Ke). Thus, for the first time we have an empirical measure of gene pleiotropy. Examples in Table 1 show that Ke is typically ~6–9, the number of distinct fitness components a gene may affect, which has been further confirmed by a large-scale data analysis (not shown).

Our SM model provides an approach to understanding the role of gene pleiotropy in a gene network. Although a detailed analysis will be published elsewhere, we test the hypothesis that gene pleiotropy underlies genomic correlations related to the evolutionary rate of protein sequence (He and Zhang 2006; Pal et al. 2006; Wolf et al. 2006). That is, (i) evolutionary rate is inversely related to gene pleiotropy (K), (ii) gene pleiotropy determines gene dispensability, and (iii) gene pleiotropy underlies “hub” genes. In short, estimation of gene pleiotropy (Ke) provides a model-based data-analysis approach that avoids the interpretative difficulty in principle-component or partial-correlation analysis.

In the future, we shall refine the analytical pipeline for estimating Ke. We shall examine how statistical properties of dN/dS and H would affect the estimation efficiency of Ke. In addition to the delta method, another method to estimate the sampling variance of the estimated Ke is nonparametric bootstrapping. Under the W-model when the selection intensity S follows a (negative) gamma distribution, a likelihood approach can be developed. Since the method of Nielsen and Yang (2003) is applicable only for closely related sequences, we shall develop a protein sequence-based likelihood approach.

Gene pleiotropy and allelic pleiotropy:

It should be noted that the effective number of molecular phenotypes (Ke) measures the functional pleiotropy of a gene, rather than a single mutation. For instance, Ke = 8.23 for trans-aldolase is the estimated number of distinct fitness dimensions implied by all random mutations in the coding sequence. Although the estimated Ke indicates that most genes may be highly pleiotropic, what it exactly means is very much gene specific.

The number of fitness dimensions affected by a single mutation (Ω) could be much less than Ke, say, ~2–4 (Dudley et al. 2005). Waxman and Peck (1998) argued that a single optimal gene sequence may become common when three or more characters (as molecular phenotypes here) are affected by each mutation, (i.e., Ω ≥ 3). The problem of estimating Ω effectively from sequence data remains a question that requires further study.

Several studies (Nielsen and Yang 2003; Piganeau and Eyre-Walker 2003; Eyre-Walker et al. 2006) estimated the shape parameter of gamma distribution for S. Nielsen and Yang (2003) obtained the shape parameter of 3.22 from the primate mitochondrial genome sequences. Under the SM framework, it can be interpreted as Ke/2; see the above section for the W-model. That is, Ke = 2 × 3.22 = 6.44 for primate mitochondria proteins, consistent with our results (Table 1). In contrast, Eyre-Walker et al. (2006) obtained a much smaller value (0.23) of the shape parameter from the human SNP data. We speculate that the estimate from population genetics data perhaps reflects the allelic pleiotropy (Ω). Hence, Eyre-Walker et al. (2006) may provide a rough estimate for Ω = 2 × 0.23 = 0.46.

Assumptions and alternative models:

The SM model involves several simplifying assumptions, since an exhaustive consideration of all possible mutational effects on molecular phenotypes is intractable. Many are shown with previous phenotype–genotype models. One may refer to Martin and Lenormand (2006) for a recent summary about rationales and criticisms. The first assumption is the single fitness optimum. Under this condition we assume that disruptive selection with alternative optima following an adaptive process (reviewed in Kingsolver et al. 2001 and Elena and Lenski 2003) occurs rarely at the molecular level. Moreover, we used the Gaussian fitness function (see Equation 1) to measure the fitness consequences of a deviation from the optimum. Lande (1980) showed that when the population mean is close to the optimum, a Gaussian-like function can be a local kernel approximation for many arbitrary fitness functions.

Second, the effect of mutations on molecular phenotypes is assumed to be continuous and symmetric. The continuous property is consistent with some empirical evidence from random mutations of genes (Keightley 1994; Keightley and Caballero 1997; Imhof and Schlotterer 2001). The symmetric assumption implies that even if mutations bias toward higher or lower values of molecular phenotypes, the trend should be small compared to the mutational variance. In practice we used a multiple Gaussian distribution for mutational effects. In the non-Gaussian case, it is possible to reduce kurtosis after an appropriate transformation such that mutational effects on a “transformed” molecular phenotype are roughly Gaussian-like. In short, we conclude that in the absence of practically useful alternatives, single optimum, Gaussian fitness function, and continuous symmetric effect of mutations remain useful working assumptions. Further work will examine the effects of these assumptions using computer simulations.

A more important issue is whether the estimation procedure for the effective number of molecular phenotypes (Ke) is robust under various alternative models. The first alternative model is Fisher's geometric model, in which adaptive change is represented by stepwise movement of a point toward the center of the hypersphere. In this case, the number of molecular phenotypes (K) is Fisher's geometric dimension. Several studies (Hartl and Taubes 1996, 1998; Sella and Hirsh 2005) showed that at equilibrium, Fisher's geometric model can be approximated by allowing the current population to vary randomly around the fixed optimum. A second alternative model is to define the fitness to be a function of flux through a metabolic pathway (Kacser and Burns 1979; Hartl and Taubes 1996), rather than a Gaussian-like function. Preliminary results have indicated that in both cases, estimation of Ke varies only slightly. It seems that Ke is a robust measure of gene pleiotropy, which may not be very sensitive to the specific evolutionary model. We shall address this issue more extensively in future studies.


The author is grateful to Adam Eyre-Walker and the reviewing editor Antony Long for constructive comments on the manuscript. Assistance in data analysis from Dongping Xu and Zhixi Su is appreciated.


Mean and variances of S—derivation of Equations 13 and 14:

For the distribution of y that follows a multi(K)-variate normal distribution,

equation M98

we note that for any quadratic function y′U−1y, the first and second moments can be expressed as

equation M99

respectively, where the operator tr[.] means the trace of a square matrix. From Equation 7 it is straightforward that equation M100 and equation M101. Denote the matrix A = U−1Σm. After noting that the trace of the matrix is the sum of the eigenvalues, i.e., equation M102 and equation M103, we obtain Equations 13 and 14.

Canonical representation of S—derivation of Equation 15:

The canonical representation will choose molecular phenotypes such that Σw = diag(equation M104,1, …, equation M105). Recall that A = U−1Σm = ZΩ, where equation M106 and equation M107. Then, one can verify equation M108, since the trace of a matrix is the sum of its diagonal elements. Next, matrix Ω can be written as equation M109. It follows that equation M110, where equation M111 is the ith eigenvalue of matrix ΣμΣm. From the relation tr[A] = tr[Z] − tr[Ω], together we have

equation M112

leading to Equation 15, where equation M113/[equation M114].

The canonical form for the variance of S can be obtained similarly, but the algebra is somewhat tedious, largely due to the detail of A2 = Z2 + Ω2 − 2. Let equation M115. One thus can show the trace equation M116 and equation M117. Noting that = diag(equation M118,1, …, equation M119,K)ΣmΣμΣm, we obtain equation M120, where equation M121 is the ith eigenvalue of matrix ΣmΣμΣm. Therefore, from the fact tr[A2] = tr[Z2] + tr[Ω2] − 2 tr[], we have

equation M122

Condition of S-gamma distribution:

Given p(y) characterized by the covariance Σw, the distribution of selection intensity, f(S), can be uniquely determined by the probability theory that for any S* and y* that satisfy S* = −2Ne(y*′equation M123y*) in Equation 7, the relationship equation M124 holds. However, the analytical form of f(S) is not available, except for the case when the matrix U in the quadratic form and the covariance in p(y), Σm, satisfies A = ΣmU−1 = αI. In other words, the matrix A is diagonal, having the same eigenvalues α.

Moments of evolutionary rate—derivation of Equations 20 and 21:

Here we show several analytical integral results. First we consider equation M125, which is given by

equation M126

where equation M127. It follows that

equation M128

In the same manner, we have equation M129 given by

equation M130

Next we consider equation M131. From the derivation of I1, one can verify

equation M132

From Equation A2, we obtain

equation M133

In the same manner, we show that equation M134 as follows:

equation M135

Finally, we consider the integral equation M136. Similar to the derivations of I1I5, one can write

equation M137

where matrix B* = (equation M138 + 8NeU−1)−1. Since U−1B* = (equation M139U + 8NeI)−1 = (A−1 + 8NeI)−1, and |B*/Σm| = |I + 8NeA|, I5 can be written as follows:

equation M140

From Equation 19, one can show that equation M141 and equation M142, both of which are determined a single matrix A. Hence, we have derived Equations 20 and 21 on the basis of the matrix theory that, for a square matrix A, the trace is the sum of the eigenvalues, the determinant is the production of the eigenvalues, and any eigenvalue of the matrix that is a function of A is the function of the corresponding eigenvalue of A.


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