The nonsynonymous (amino acid–altering) to synonymous (silent) substitution rate ratio (ω = dN/dS) provides a measure of natural selection at the protein level, with ω = 1, >1, and <1, indicating neutral evolution, purifying selection, and positive selection, respectively. Previous studies that used this measure to detect positive selection have often taken an approach of pairwise comparison, estimating substitution rates by averaging over all sites in the protein. As most amino acids in a functional protein are under structural and functional constraints and adaptive evolution probably affects only a few sites at a few time points, this approach of averaging rates over sites and over time has little power. Previously, we developed codon-based substitution models that allow the ω ratio to vary either among lineages or among sites. In this paper we extend previous models to allow the ω ratio to vary both among sites and among lineages and implement the new models in the likelihood framework. These models may be useful for identifying positive selection along prespecified lineages that affects only a few sites in the protein. We apply those branch-site models as well as previous branch- and site-specific models to three data sets: the lysozyme genes from primates, the tumor suppressor BRCA1 genes from primates, and the phytochrome (PHY) gene family in angiosperms. Positive selection is detected in the lysozyme and BRCA genes by both the new and the old models. However, only the new models detected positive selection acting on lineages after gene duplication in the PHY gene family. Additional tests on several data sets suggest that the new models may be useful in detecting positive selection after gene duplication in gene family evolution.
The nonsynonymous/synonymous substitution rate ratio (ω = dN/dS) provides a sensitive measure of selective pressure at the amino acid level. An ω ratio greater than one means that nonsynonymous mutations offer fitness advantages and are fixed in the population at a higher rate than synonymous mutations. Positive selection can thus be detected by identifying cases where ω > 1. Previous studies have most often employed a pairwise approach, calculating dS and dN rates between two sequences by averaging over all codons (amino acids) in the gene and over the time period that separates the sequences. Because many amino acids in a functional protein might be largely invariable (with ω close to 0) because of strong structural and functional constraints, the average dN is rarely higher than the average dS. As a result, this approach has little power in detecting positive selection (e.g., Endo, Ikeo, and Gojobori 1996<$REFLINK> ; Sharp 1997<$REFLINK> ; Akashi 1999<$REFLINK> ; Crandall et al. 1999<$REFLINK> ).
The model of codon substitution of Goldman and Yang (1994)<$REFLINK> (see also Muse and Gaut 1994<$REFLINK> ) provides a framework for studying the mechanism of sequence evolution by comparing synonymous and nonsynonymous substitution rates. The original model assumes a single ω for all lineages and sites and has been extended to account for variation of ω either among lineages or among sites. The lineage-specific models (Yang 1998<$REFLINK> ; Yang and Nielsen 1998<$REFLINK> ; see also Muse and Gaut 1994<$REFLINK> ) allow for variable ωs among lineages and are thus suitable for detecting positive selection along lineages. They assume no variation in ω among sites; as a result, they detect positive selection for a lineage only if the average dN over all sites is higher than the average dS. The site-specific models (Nielsen and Yang 1998<$REFLINK> ; Yang et al. 2000<$REFLINK> ) allow the ω ratio to vary among sites but not among lineages. Positive selection is detected at individual sites only if the average dN over all lineages is higher than the average dS. If adaptive evolution occurs at a few time points and affects a few amino acids (Gillespie 1991, pp. 132–139<$REFLINK> ), both classes of models might lack power in detecting positive selection. It appears that averaging over sites is a more serious problem than averaging over lineages because the site-specific analysis has been successful in detecting positive selection in a variety of genes (Zanotto et al. 1999<$REFLINK> ; Bishop, Dean, and Mitchell-Olds 2000<$REFLINK> ; Yang et al. 2000<$REFLINK> ; Fares et al. 2001<$REFLINK> ; Haydon et al. 2001<$REFLINK> ; Swanson et al. 2001<$REFLINK> ). Computer simulations also confirmed the power of the site-specific analysis (Anisimova, Bielawski, and Yang 2001<$REFLINK> ; see Yang and Bielawski 2000<$REFLINK> for a review).
It appears worthwhile to develop models that allow the ω ratio to vary both among sites and among lineages. In this paper, we implement two such models. Our main objective is to improve the power of the likelihood ratio test (LRT) to detect positive selection along prespecified lineages. A major use of these new models might be to analyze the evolution of gene families, where functional divergence after gene duplication might have caused adaptive evolution (Ohta 1993<$REFLINK> ). We implement the new models in the likelihood framework and apply them to analyze three data sets: the lysozyme genes from primates (Messier and Stewart 1997<$REFLINK> ; Yang 1998<$REFLINK> ), the cancer suppressor BRCA1 genes from primates (Huttley et al. 2000<$REFLINK> ), and the phytochrome (PHY) gene family in angiosperms (Alba et al. 2001<$REFLINK> ).
We assume that the phylogeny is known or independently estimated, and the branches that might be expected to be under positive selection are specified a priori. For example, in the analysis of a gene family, we are interested in testing whether positive selection has occurred along the lineages right after gene duplication. For convenience, we refer to branches for which we test positive selection as the “foreground” branches and all others the “background” branches.<$REFLINK> ). The scale factor μ is defined by the requirement that the average substitution rate is one: <$REFLINK> ).
We assume that the ω ratio varies among codon (amino acid) sites, and there are four site classes in the sequence. The first class of sites are highly conserved in all lineages with a small ω ratio, ω0. The second class includes neutral or weakly constrained sites at which ω = ω1, where ω1 is near or smaller than 1. In the third and fourth classes, the background lineages have ω0 or ω1, but the foreground branches have ω2, which may be greater than 1. In other words, there are two site classes with the ratios ω0 or ω1 along the background branches, but along the lineages of interest, a certain event caused some sites to come under positive selection with the ratio ω2 > 1 (table 1 ). We assume that when positive selection occurs along the foreground lineages, it is equally likely to involve a site from site class 0 as a site from class 1; the proportions of sites from classes 2 and 3 are the same as those from classes 0 and 1 (table 1 ). This assumption can be relaxed by introducing an additional proportion parameter, but this is not pursued here.
We implement two versions of the model and refer to them later as models A and B. In model A, we fix ω0 = 0 and ω1 = 1. This model is an extension of the site-specific “neutral” model (M1) of Nielsen and Yang (1998)<$REFLINK> , which assumes two site classes with ω0 = 0 and ω1 = 1 in all lineages. In model B, we estimate ω0 and ω1 from the data as free parameters. Although we envisage ω0 and ω1 to be smaller than one, we do not place this constraint in the implementation. Model B is an extension to the site-specific “discrete” model (M3) of Yang et al. (2000)<$REFLINK> with K = 2 site classes. In both models A and B, the proportions p0 and p1 as well as the ratio ω2 are estimated from the data by maximum likelihood (ML).
Let the number of sites (codons) in the sequence be n and the observed data at site h be xh (h = 1, 2, …, n); xh is a vector of codons at site h across all sequences in the alignment. Let yh (=0, 1, 2, or 3) be the site class that site h belongs to. We assume that there are different classes of sites in the gene, but we do not know which class each site is from. Given the site class yh, the conditional probability of observing data xh at the site, f(xh|yh), can be calculated using previous algorithms. If the site is from classes 0 or 1 (if yh = 0 or 1), all branches on the phylogeny have the same ω ratio, and f(xh|yh) can be calculated according to Goldman and Yang (1994)<$REFLINK> . If the site is from classes 2 or 3 (if yh = 2 or 3), the ω ratios are different for the background and foreground branches, and f(xh|yh) can be calculated according to Yang (1998)<$REFLINK> . The unconditional probability is an average over the site classes:<$REFLINK> ).
Models implemented here assume that the synonymous rate is constant across all sites, and only the nonsynonymous rate varies among site classes. The branch length (t), measured by the expected number of nucleotide substitutions per codon, is defined as an average across the site classes (Nielsen and Yang 1998<$REFLINK> ). The scale factor μ in equation (1) differs between the foreground and background branches.
After ML estimates of parameters are obtained, an empirical Bayes approach can be used to infer which class a site is most likely from (Nielsen and Yang 1998<$REFLINK> ). The posterior probability that site h with data xh is from site class k isequation (5) .
Real Data Analysis
Lysozyme Genes from Primates
The lysozyme c gene sequences of 24 primate species analyzed by Messier and Stewart (1997)<$REFLINK> are used. The phylogenetic tree of the species is shown in figure 1 and used in later analysis. Only the 19 distinct sequences are used, each with 130 codons. In many mammals, such as humans and rats, the lysozyme performs the function of fighting invading bacteria and exists mainly in secretions like tears and saliva as well as in white blood cells and tissue macrophages. Colobine monkeys (such as the langur) have fermentative foreguts, where high levels of lysozyme are present and where its function is to digest bacteria that pass from the foreguts into the true stomach (Stewart, Schilling, and Wilson 1987<$REFLINK> ). Messier and Stewart (1997)<$REFLINK> suggested that positive selection occurred along the lineage ancestral to colobine monkeys (branch c in fig. 1 ). We apply the new models developed here to these data and treat branch c as the foreground branch and all other branches in the phylogeny as background branches (fig. 1 ).
Yang (1998)<$REFLINK> has performed a branch-specific likelihood analysis of the data, assuming that all sites in the sequence have the same ω ratio. The two-ratios model assigns the ratio ωc for branch c and the ratio ω0 for all other branches (table 2 ). This model fits the data significantly better than the one-ratio model of Goldman and Yang (1994)<$REFLINK> . The LRT statistic for this comparison is 2Δℓ = 2 × 2.13 = 4.26, with P = 0.039 and df = 1 (table 2 ). So the ω ratio for branch c is significantly different from that for all other branches. To test whether ωc is significantly higher than 1, the log likelihood value was calculated under the two-ratios model but with ωc = 1 fixed, giving the log likelihood value −1,042.50. The two-ratios model that does not place the constraint on ωc (table 2 ) is not significantly better; the test statistic is 2Δℓ = 2 × 1.33 = 2.66, and P = 0.10 with df = 1. So ωc is not significantly greater than 1 at the 5% significance level (see Yang 1998<$REFLINK> ).
We also applied the site-specific likelihood models (Nielsen and Yang 1998<$REFLINK> ; Yang et al. 2000<$REFLINK> ) to the lysozyme data (table 2 ), which assume variable selective pressures among sites but no variation among branches in the phylogeny. We use three pairs of models, forming three LRTs: M1 (neutral) and M2 (selection), M0 (one-ratio) and M3 (discrete), and M7 (beta) and M8 (beta&ω) (Nielsen and Yang 1998<$REFLINK> ; Yang et al. 2000<$REFLINK> ). Model M1 (neutral) assumes two site classes with ω0 = 0 and ω1 = 1 fixed and with the proportions p0 and p1 estimated. Model M2 (selection) adds a third site class with the ratio ω2 estimated. This model suggests that about 7% of sites are under positive selection with omegâ2 = 3.7. Because model M2 (selection) is an extension of M1 (neutral), the two models can be compared using an LRT. The test statistic is 2Δℓ = 2 × ((−1,035.83) − (−1,037.21)) = 2 × 1.38 = 2.76, with P = 0.25 and df = 2. So model M2 is not significantly better than M1. The discrete model (M3) with K = 2 site classes suggested that 18% of sites are under positive selection with omegâ1 = 2.6 and identified six amino acid sites under positive selection at the 95% cutoff. Using K = 3 site classes produced the same estimates. M3 (K = 2) was significantly better than the one-ratio model; the test statistic is 2Δℓ = 17.20 and P < 0.001 with df = 2. Model M7 (beta) assumes a beta distribution for ω over sites. The beta distribution is limited to the interval (0, 1) and provides a flexible null hypothesis for testing positive selection. The estimates suggest that the distribution reduced to the neutral model (M1). Model M8 (beta&ω) adds another site class to M7 (beta), with the ω ratio estimated from the data. The model suggested 16% of sites to be under positive selection with omegâ = 2.5 and identified seven sites under positive selection (the same six sites as under M3 plus site 17M). However, the difference between M7 and M8 is not statistically significant as 2Δℓ = 3.30, with P = 0.19 and df = 2. We note that the M0-M3 comparison is more a test of variability in the ω ratio among sites and does not constitute a rigorous test of positive selection. The lysozyme gene has only 130 codons, and the nonsignificant results of the M1-M2 and M7-M8 comparisons might well be caused by the short sequence and lack of power of the LRTs. It is worth noting that parameter estimates under all the models, M2 (selection), M3 (discrete), and M8 (beta&ω), suggest presence of sites under positive selection.
The new branch-site models implemented in this paper are applied to the lysozyme data, with branch c of figure 1 considered as the foreground branch and all other branches in the tree as background branches. Model A does not allow for sites under positive selection across all lineages and suggests that a large proportion of sites (40%) are under positive selection along branch c with omegâ2 = 4.8. This model can be compared with the site-specific model M1 (neutral); the LRT statistic is 2Δℓ = 2 × 1.68 = 3.36, with P = 0.19 and df = 2. So model A does not fit the data significantly better than model M1. Estimates under model B suggest sites under positive selection across all lineages (omegâ1 = 2.3) as well as sites under selection along branch c only (omegâ2 = 4.3). The comparison between model B and the site-specific model M3 (discrete with K = 2) gave 2Δℓ = 2 × 0.96 = 1.92, with P = 0.38 and df = 2. So model B does not fit the data significantly better than the site-specific model M3. However, this comparison is not interesting biologically because the null model allows for positive selection as well.
In sum, the selective pressure in the lysozyme is highly variable among sites. Some sites are under strong selective constraints throughout the primate phylogeny, whereas others appear to be under positive selection, with an excess of nonsynonymous substitutions relative to synonymous substitutions. The branch ancestral to the colobine monkeys (branch c in fig. 1 ) is under much stronger positive selection pressure. We note, however, that some of the LRTs fail to provide significant support for positive selection, possibly because of the short sequence and low divergence in the lysozyme data, resulting in low power in the tests.
The posterior probabilities that each site is from the four site classes of models A and B are calculated at the ML estimates of parameters (eq. 5 ). The probabilities for classes 2 and 3 are combined (see table 1 ). The sites identified in this way are listed in table 2 . We also performed ML reconstruction of ancestral sequences (Yang, Kumar, and Nei 1995<$REFLINK> ) using the codon model of Goldman and Yang (1994)<$REFLINK> . The reconstruction suggested nine amino acid replacement changes along branch c in the tree: 14RK, 21RK, 23IV, 37DG, 41QE, 50QE, 62HR, 87DN, 126QK (see also Messier and Stewart 1997<$REFLINK> ). These sites are essentially the same as those predicted to be under positive selection by the branch-site models (table 2 ).
Tumor Suppressor BRCA1 Gene from Primates
The tumor suppressor gene BRCA1 plays a role in the maintenance of genomic integrity, including recombinational and transcription-coupled DNA repair and in transcription regulation. Mutations in BRCA1 confer an increased risk of female breast cancer. The BRCA1 locus has a complex structure of 24 exons spanning more than 80 kb, with the majority (∼60%) of the protein encoded by exon 11 (Huttley et al. 2000<$REFLINK> ). Huttley et al. (2000)<$REFLINK> performed a lineage-specific ML analysis of the nucleotide sequences from exon 11 of human and non-human primates and suggested that the human and chimpanzee lineages are under positive selection (fig. 2 ). The authors hypothesized that the BRCA1 has a modified function in humans and chimpanzees relative to its homologues in other primates.
The alignment of Huttley et al. (2000)<$REFLINK> was modified slightly to accommodate the coding structure of the genes. The rat and mouse sequences used by the authors appear too divergent from the primate sequences, so that the alignment does not seem reliable in certain regions. Only the primate sequences are used in this paper. The alignment had 1,160 codons, but some regions had gaps, which are treated as ambiguity characters in the likelihood calculation (Yang 1997<$REFLINK> ). The phylogenetic tree for the sequences is shown in figure 2 .
The one-ratio model (M0) gives a log likelihood of −9,565.22, with the estimate omegâ = 0.624. The two-ratios model assigns two different ω ratios for the human and chimpanzee branches (ω1) and for all other branches (ω0). The log likelihood under this model is −9,561.06, with parameter estimates omegâ0 = 0.604 for the background branches and omegâ1 = 2.676 for the foreground branches. This model is significantly better than the one-ratio model (2Δℓ = 2 × 4.16 = 8.32, P = 0.0039 and df = 1). To test whether ω1 is significantly greater than 1, the two-ratios model is fitted to the data with ω1 = 1 fixed, giving a log-likelihood value of −9,562.72. This model is not significantly worse than the two-ratios model of table 2 without constraining ω1 = 1; the test statistic is 2Δℓ = 3.32, with P = 0.068 with df = 1. Huttley et al. (2000)<$REFLINK> obtained a slightly larger test statistic, 2Δℓ = 4.3, and their test was marginally significant (P = 0.04). This minor discrepancy seems to be caused by the different alignments.
We also applied the site-specific models (Nielsen and Yang 1998<$REFLINK> ; Yang et al. 2000<$REFLINK> ) to these data (table 3 ). The selective pressure on the protein varies greatly among amino acid sites. For example, using K = 2 site classes in the discrete model (M3) fits the data significantly better than the one-ratio model (M0); the test statistic is 2Δℓ = 458.64, and P = 0.000 with df = 2. Model M3 suggests 17% of sites to be under positive selection with omegâ1 = 2.24 and identifies seven amino acid sites under positive selection at the 95% cutoff (table 3 ). Model M8 (beta&ω) also suggests about 16% of sites under positive selection with omegâ = 2.25 and identifies the same seven sites under positive selection as model M3. Furthermore, M8 provides significantly better fit to the data than M7 (2Δℓ = 15.24, P = 0.00049, and df = 2). These tests provide significant evidence for presence of sites under positive selection. Unlike M3 (discrete) and M8 (beta&ω), model M2 (selection) does not suggest positive selection. As discussed by Yang et al. (2000)<$REFLINK> , this pattern is because M1 (neutral) does not allow for sites with 0 < ω < 1; as a result, the extra site class in M2 (selection) is forced to account for such sites.
The branch-site models of this paper suggest sites under positive selection along the lineage of interest (table 3 ). Parameter estimates under model A suggest that 11% of sites are highly conserved across all lineages with omegâ0 = 0, and 24% of sites are nearly neutral with ω1 = 1, whereas as high as 65% of sites are under strong positive selection along the human and chimpanzee branches with omegâ2 = 3.7. Model A can be compared with the neutral model (M1) by an LRT. The statistic is 2Δℓ = 2 × 4.30 = 8.60, with P = 0.014 and df = 2. This improvement is statistically significant. Parameter estimates under model B (table 3 ) suggest that 15% of sites are under positive selection in all lineages with omegâ1 = 2.1, whereas 22% of sites are under even stronger positive selection in the human and chimpanzee branches with omegâ2 = 6.4. The LRT comparing the branch-site model B and the site-specific model M3 (K = 2) gave 2Δℓ = 2 × 2.77 = 5.54, and P = 0.063 with df = 2. This comparison is close to being significant. Because the null model in this comparison suggests positive selection at some sites along all lineages with ω1 > 1, it is safe to suggest positive selection along the human and chimpanzee lineages.
We examined the posterior probabilities for site classes under model A to infer which sites are likely to be under positive selection along the human and chimpanzee branches. Probabilities for site classes 2 and 3 were combined (see table 1 ). At the P > 85% level, the following sites are identified: 30A, 179E, 233I, 244K, 285P, 317T, 449G, 456A, 489K, 509K, 604N, 653P, 660A, 671S, 672G, 686C, 691E, 716G, 725N, 763R, 853I, 892G, 919Y, 975K, 1027N. We also applied ML reconstruction of ancestral sequences using the codon model of Goldman and Yang (1994)<$REFLINK> and Yang, Kumar, and Nei (1995)<$REFLINK> . The reconstruction suggested the following amino acid changes along the human branch: 30TA, 179GE, 244RK, 285AP, 317MT, 449DG, 489NK, 604TN, 653LP, 660PA, 672RG, 716CG, 725YN, 763HR, 853VI, 919CY, 975RK, 1027SN and the following changes along the chimpanzee branch: 233IT, 456AV, 509KE, 671SC, 686CR, 691EQ, 892GE. The list identified by the branch-site model simply consists of the amino acids that probably changed along the foreground branches as suggested by the ancestral reconstruction.
Phytochrome Gene Family from Angiosperms
Light is of critical importance to plant metabolism and development, and plants employ a number of mechanisms to detect light. Phytochromes are the best-characterized plant photosensors. They are chromoproteins that regulate the expression of a large number of light-responsive genes and many photomorphogenic events, including seed germination, flowering, fruit ripening, stem elongation, and chloroplast development (Alba et al. 2001<$REFLINK> ). All angiosperms characterized to date contain a small number of different PHY apoproteins encoded by a small PHY gene family. Alba et al. (2001)<$REFLINK> characterized the five loci in the PHY family in the tomato and performed phylogenetic analysis of the PHY family in angiosperms. The study characterized conserved and fast-evolving regions of the gene and demonstrated elevated evolutionary rates in between-subfamily comparisons. The authors pointed out that the rapid evolution is compatible with both positive selective pressure and relaxed evolutionary constraint. Here we apply the models of this paper to these data to detect possible positive selection driving the evolution of new functions after gene duplication. We use the alignment and phylogenetic tree published by Alba et al. (2001<$REFLINK> ; fig. 4). To reduce the level of sequence divergence, we include the 16 sequences from the A and C-F subfamilies in one data set and the 11 sequences from the B-D and E subfamilies in another data set (see fig. 3A and B ). Some sites in each data set had alignment gaps in all sequences, and we kept those sites to maintain the same site numbering (Alba et al. 2001<$REFLINK> ). See Alba et al. (2001)<$REFLINK> for detailed sequence information and accession numbers. We test whether the gene was under positive selection along the branch separating the A and the C-F subfamilies (fig. 3A ) and along the branch separating the B-D and E subfamilies (fig. 3B ). Those branches are the foreground branches in our analysis. The results are reported in tables 4 and 5 for the ACF and BDE data sets, respectively.
The A and CF Subfamilies
The one-ratio model (M0) gives an estimate omegâ = 0.089 (table 4 ). This low average ratio suggests the dominating role of purifying selection in the evolution of the phytochrome gene family. The branch-specific two-ratios model assigns two ω ratios for the foreground (ω1) and background branches (ω0). Although the estimates omegâ1 > omegâ0 (table 4 ), the two-ratios model does not fit the data significantly better than the one-ratio model.
The site-specific models (Nielsen and Yang 1998<$REFLINK> ; Yang et al. 2000<$REFLINK> ) revealed a huge amount of variation in selective pressure among sites. For example, two site classes (M3, K = 2) fitted the data better than one site class (M0) by 570.42 log likelihood units. However, none of the site-specific models suggest the presence of sites under positive selection with ω > 1, and all of them suggest that most sites in the sequence are under very strong selective constraints (table 4 ).
Branch-site model A adds an additional site class to the neutral model, where the selective pressure is different among the foreground and background branches. Although model A fits the data better than the neutral model, both models are highly unrealistic and even much worse than the one-ratio model (M0). There appears to be a large proportion of sites with ω small but positive, and neither model accounts for them. Branch-site model B is an extension to the discrete model with K = 2 site classes. The difference between those two models is significant, with 2Δℓ = 2 × 13.86 = 27.72, and P = 10−6. Furthermore, parameter estimates under model B suggest a small proportion (6%) of sites under positive selection along the foreground branch with omegâ2 = 7.0 (table 4 ). Therefore, model B detected positive selection along the foreground branch when the old models failed.
To examine how sensitive this result is to assumptions made in the model, we performed the LRT comparing M3 and model B under two assumptions about codon usage: the Fequal model, which assumes equal codon frequencies and thus ignores codon usage bias and the Fcodon model, which uses all the 61 codon frequencies as parameters. The results are shown in table 6 . Previous studies suggested that codon usage bias can have a drastic effect on estimation of synonymous and nonsynonymous rates, much more important than the transition-transversion rate bias (see e.g., Yang and Bielawski 2000<$REFLINK> ). It is thus interesting to examine the effect of codon usage bias on the LRT of positive selection. Results of table 6 suggest that although the LRT statistic varies according to the model assumption about codon usage, all of them are highly significant, supporting the presence of sites under positive selection among the foreground branch. The parameter estimates are quite similar and so are the calculated posterior probabilities (not shown). We also note that the two simpler models, Fequal and F3×4, are both rejected compared with Fcodon, indicating that codon usage is highly uneven.
ML reconstruction of ancestral sequences (Yang, Kumar, and Nei 1995<$REFLINK> ) using the codon model of Goldman and Yang (1994)<$REFLINK> suggested as many as 277 amino acid substitutions along the foreground branch, which separates the A subfamily and the C-F subfamily. At the P > 50% level (eq. 5 ), the branch-site model B identifies the following 26 sites as potentially under positive selection along the branch: 52Q, 55R, 102T, 105S, 117P, 130T, 171T, 216E, 221T, 227F, 252I, 304I, 305D, 338V, 440L, 621N, 650T, 655S, 700A, 736K, 751G, 787N, 802V, 940T, 1069E, 1087D. Those are a subset of the sites predicted to have changed along the branch in ancestral reconstruction.
The BD and E Subfamilies
Results obtained from the B-D and E subfamilies are presented in table 5 . The results are rather similar to those for the A and C-F subfamilies (table 4 ). For example, the overall ω ratio under model M0 (one-ratio) is 0.087, almost identical to the estimate from the ACF data set (0.089). The two-ratios model gave about the same fit to the data as the one-ratio model. The site-specific models indicated considerable variation in selective pressure among sites. For example, two site classes (M3, K = 2) fit the data better than one class (M0) by 638.25 log-likelihood units. However, none of the site-specific models suggested presence of sites under positive selection. Most sites appear to be under strong purifying selection. Branch-site model A is even much worse than the one-ratio model. Model B is much more realistic and fits the data better than M3 (discrete, K = 2), with 2Δℓ = 2 × 54.90 = 109.8, and P < 0.000, df = 2. Furthermore, parameter estimates under Model B suggested a small proportion of sites (3%) to be under positive selection on the foreground branch with omegâ2 = 3.9. Hence, the new model B detected positive selection, whereas all old models failed. We also conducted this same test under two additional models of codon usage, Fequal and Fcodon, with the results shown in table 6 . Although codon usage is highly biased, the parameter estimates are quite stable among those models, and the results of the LRT are robust to assumptions about codon usage bias.
We also applied the empirical Bayes procedure to calculate the posterior probabilities that each site comes from the four site classes, with the probabilities for site classes 2 and 3 combined (eq. 5 , table 1 ). At the 50% level, 10 codons are identified (table 5 ). For comparison, ML reconstruction of ancestral sequences under the codon model of Goldman and Yang (1994)<$REFLINK> suggested about 226 amino acid substitutions along the branch separating the B-D and the E subfamilies of the PHY gene. The 10 sites identified by model B are a subset of the sites predicted to have changed in ancestral reconstruction.
In sum, LRTs based on the branch-site model B confirmed the hypothesis that position selection was operating to drive functional divergence after gene duplication in both the ACF and the BDE data sets (fig. 3A and B and tables 4 and 5 ). This conclusion is also robust to assumptions about codon usage bias. However, we expect the prediction of sites under positive selection to be much less reliable (see Discussion). Sequencing more species to break the long branches in the phylogenies might provide information about which of the many amino acid changes were responsible for functional divergences.
In the analysis of the lysozyme and the BRCA1 data sets, the previous branch and site models already detected positive selection, with different levels of statistical support. Although some of the LRTs were not significant in the lysozyme data set, we suspect that the nonsignificance may be because of the short sequence and lack of power of the tests. The phytochrome gene family is clearly under strong purifying selection, and previous models averaging rates either over branches or over sites failed to detect positive selection. However, the new branch-site model B detected positive selection causing adaptive changes at certain amino acid sites along branches right after the gene duplications that separated the A from the C-F subfamilies (fig. 3A ) and that separated the B-D and the E subfamilies (fig. 3B ).
The branch-site models have also been applied to several other gene families. In an analysis of the visual pigment gene family in vertebrates, the branch-site models detected positive selection along the lineage separating the rod and cone opsins, demonstrating the selective pressure exerted by the requirement of the new function of the ancestral rod opsin (B.S.W. Chang, personal communication). Bielawski and Yang (unpublished data) used the branch-site models to detect positive selection after gene duplication in the vertebrate Troponin C gene family, which created two distinct muscle isoforms for Troponin C: the fast skeletal muscle isoform and the cardiac and slow skeletal muscle isoform. A duplication of the caffeic acid O-methyl transferase gene apparently gave rise to the gene for isoeugenol O-methyl transferase, an enzyme that makes a floral scent compound important for attracting pollinators to flowers of the plant Clarkia breweri in the evening primrose family (Onagraceae). Todd Barkman (personal communication) applied the branch-site models to detect positive selection after that gene duplication event. In none of those data sets did previous branch and site models detect positive selection. Those case studies suggest that the new models have improved power in at least some data sets. Interestingly, all those examples involve gene family evolution.
In theory, the branch-site models implemented in this paper can be used to address two questions. The first is whether there are some sites in the gene that are under positive selection along the branches of interest; this is addressed by the LRT. The second is to identify positively selected sites when they exist. This is achieved using the Bayes prediction (eq. 5 ).
Real data analyses discussed earlier in the article suggest that the LRT based on the branch-site models has improved power in detecting positive selection than the branch models. Nevertheless, the new models make very specific assumptions, and it is advisable to examine the sensitivity of the LRTs to model assumptions and to sampling of sequences in the data. The models allow for only two types of lineages in the phylogeny. They will be difficult to apply if, for example, two branches are under positive selection but the selective pressure affects different amino acid sites along the two lineages. Fixing ω0 = 0 and ω1 = 1 in model A seems to lead to very poor fit in some data sets, especially when the gene is under strong purifying selection. Model B appears more widely applicable. We suggest that the null model M3 with only K = 2 site classes is not so restrictive as it might look. Although using two site classes with different ω ratios appears always to fit the data much better than the one-ratio model, it is seldom possible to fit more than three classes to real data; the parameter estimates simply collapse and the model reduces to one of fewer classes. Indeed use of three classes in M3 appears to fit any data set just as well as any parameter-rich continuous distributions (see Yang et al. 2000<$REFLINK> for examples). Even use of three classes sometimes leads to collapse of the model into two classes, as in the lysozyme and BRCA1 data sets (tables 2 and 3 ). We also emphasize that the branches to be tested for positive selection should represent a priori biological hypotheses (Yang 1998<$REFLINK> ). The significance values will not be correct if the LRT is applied to all branches in the phylogeny or if the same data are used to identify branches for test by the branch-site models. In particular, it is improper to use the “free-ratios” model of Yang (1998)<$REFLINK> to identify branches with high ω ratios and then use the branch-site models to test whether those branches are under positive selection.
Identifying amino acid residues under positive selection along the lineages of interest is clearly much more difficult than testing for the presence of such sites. We suspect that typical data sets might not contain enough information for the Bayes prediction or any other method to be reliable. Intuitively, the methods accumulate information about whether each site is under selection by comparing the numbers of synonymous and nonsynonymous substitutions at that site. In the site-specific analysis, many changes might have accumulated along branches of the phylogeny when many sequences are included in the alignment. The branch-site models, however, focus on only one or a few lineages of interest. If there is not enough opportunity for multiple changes at each site along those few lineages, the data will not allow reliable estimation of parameters in the model, and calculation of the posterior probabilities using equation (5) will be affected. Indeed, Bayes prediction of sites does not seem to offer much advantage over simple ancestral sequence reconstruction. The two approaches returned identical or highly similar results in the three data sets analyzed in this paper.
The branch-site models are implemented in the codeml program in the PAML package (Yang 1997<$REFLINK> ; http://abacus.gene.ucl.ac.uk/software/paml.html). ML estimation by numerical optimization is noted to be problematic for some of the data sets analyzed in this paper, and it is advisable to run the program multiple times to ensure convergence to the global optimum.
Keith Crandall, Reviewing Editor
Keywords: gene duplication maximum likelihood molecular adaptation nonsynonymous substitution positive selection synonymous substitution
Address for correspondence and reprints: Ziheng Yang, Department of Biology, 4 Stephenson Way, London NW1 2HE, UK. firstname.lastname@example.org
We thank Lee J. Pratt for kindly sending us the phytochrome gene alignment. Keith Crandall and two anonymous reviewers made many constructive comments. This study has been supported by UK Biotechnology and Biological Sciences Research Council grant 31/G14969 to Z.Y., US National Science Foundation Grant DEB-0089487 to R.N., and EU Human Frontier Science Program Grant RGY0055/2001-M to R.N. and Z.Y.