Identifying site-specific substitution rates

A maximum likelihood framework for estimating site-specific substitution rates is presented that does not require any prior assumptions about the rate distribution. We show that, when the branching pattern of the underlying tree is known, the analysis of pairs of positions is sufficient to estimate site-specific rates. In the abscense of a known topology, we introduce an iterative procedure to estimate simultaneously the branching pattern, the branch lengths, and site-specific substitution rates. Simulations show that the evolutionary rate of fast-evolving sites can be reliably inferred and that the accuracy of rate estimates depends mainly on the number of sequences in the data set. Thus, large sets of aligned sequences are necessary for reliable site-specific rate estimates. The method is applied to the complete mitochondrial DNA sequence of 53 humans, providing a complete picture of the site-specific substitution rates in human mitochondrial DNA.     

Evaluation of several methods for estimating phylogenetic trees when substitution rates differ over nucleotide sites

Several maximum likelihood and distance matrix methods for estimating phylogenetic trees from homologous DNA sequences were compared when substitution rates at sites were assumed to follow a gamma distribution. Computer simulations were performed to estimate the probabilities that various tree estimation methods recover the true tree topology. The case of four species was considered, and a few combinations of parameters were examined. Attention was applied to discriminating among different sources of error in tree reconstruction, i.e., the inconsistency of the tree estimation method, the sampling error in the estimated tree due to limited sequence length, and the sampling error in the estimated probability due to the number of simulations being limited. Compared to the least squares method based on pairwise distance estimates, the joint likelihood analysis is found to be more robust when rate variation over sites is present but ignored and an assumption is thus violated. With limited data, the likelihood method has a much higher probability of recovering the true tree and is therefore more efficient than the least squares method. The concept of statistical consistency of a tree estimation method and its implications were explored, and it is suggested that, while the efficiency (or sampling error) of a tree estimation method is a very important property, statistical consistency of the method over a wide range of, if not all, parameter values is prerequisite.     

Estimating synonymous and nonsynonymous substitution rates under realistic evolutionary models

Approximate methods for estimating the numbers of synonymous and nonsynonymous substitutions between two DNA sequences involve three steps: counting of synonymous and nonsynonymous sites in the two sequences, counting of synonymous and nonsynonymous differences between the two sequences, and correcting for multiple substitutions at the same site. We examine complexities involved in those steps and propose a new approximate method that takes into account two major features of DNA sequence evolution: transition/transversion rate bias and base/codon frequency bias. We compare the new method with maximum likelihood, as well as several other approximate methods, by examining infinitely long sequences, performing computer simulations, and analyzing a real data set. The results suggest that when there are transition/transversion rate biases and base/codon frequency biases, previously described approximate methods for estimating the nonsynonymous/synonymous rate ratio may involve serious biases, and the bias can be both positive and negative. The new method is, in general, superior to earlier approximate methods and may be useful for analyzing large data sets, although maximum likelihood appears to always be the method of choice.     

Efficient biased estimation of evolutionary distances when substitution rates vary across sites

This paper deals with phylogenetic inference when the variability of substitution rates across sites (VRAS) is modeled by a gamma distribution. We show that underestimating VRAS, which results in underestimates for the evolutionary distances between sequences, usually improves the topological accuracy of phylogenetic tree inference by distance-based methods, especially when the molecular clock holds. We propose a method to estimate the gamma shape parameter value which is most suited for tree topology inference, given the sequences at hand. This method is based on the pairwise evolutionary distances between sequences and allows one to reconstruct the phylogeny of a high number of taxa (>1,000). Simulation results show that the topological accuracy is highly improved when using the gamma shape parameter value given by our method, compared with the true (unknown) value which was used to generate the data. Furthermore, when VRAS is high, the topological accuracy of our distance-based method is better than that of a maximum likelihood approach. Finally, a data set of Maoricicada species sequences is analyzed, which confirms the advantage of our method.     

A simulation comparison of phylogeny algorithms under equal and unequal evolutionary rates [published erratum appears in Mol Biol Evol 1995 May;12(3):525]

Using simulated data, we compared five methods of phylogenetic tree estimation: parsimony, compatibility, maximum likelihood, Fitch- Margoliash, and neighbor joining. For each combination of substitution rates and sequence length, 100 data sets were generated for each of 50 trees, for a total of 5,000 replications per condition. Accuracy was measured by two measures of the distance between the true tree and the estimate of the tree, one measure sensitive to accuracy of branch lengths and the other not. The distance-matrix methods (Fitch- Margoliash and neighbor joining) performed best when they were constrained from estimating negative branch lengths; all comparisons with other methods used this constraint. Parsimony and compatibility had similar results, with compatibility generally inferior; Fitch- Margoliash and neighbor joining had similar results, with neighbor joining generally slightly inferior. Maximum likelihood was the most successful method overall, although for short sequences Fitch- Margoliash and neighbor joining were sometimes better. Bias of the estimates was inferred by measuring whether the independent estimates of a tree for different data sets were closer to the true tree than to each other. Parsimony and compatibility had particular difficulty with inaccuracy and bias when substitution rates varied among different branches. When rates of evolution varied among different sites, all methods showed signs of inaccuracy and bias.      

Factors affecting the errors in the estimation of evolutionary distances between sequences

Phylogenetic methods that use matrices of pairwise distances between sequences (e.g., neighbor joining) will only give accurate results when the initial estimates of the pairwise distances are accurate. For many different models of sequence evolution, analytical formulae are known that give estimates of the distance between two sequences as a function of the observed numbers of substitutions of various classes. These are often of a form that we call "log transform formulae". Errors in these distance estimates become larger as the time t since divergence of the two sequences increases. For long times, the log transform formulae can sometimes give divergent distance estimates when applied to finite sequences. We show that these errors become significant when t approximately 1/2 |lambda(max)|(-1) logN, where lambda(max) is the eigenvalue of the substitution rate matrix with the largest absolute value and N is the sequence length. Various likelihood-based methods have been proposed to estimate the values of parameters in rate matrices. If rate matrix parameters are known with reasonable accuracy, it is possible to use the maximum likelihood method to estimate evolutionary distances while keeping the rate parameters fixed. We show that errors in distances estimated in this way only become significant when t approximately 1/2 |lambda(1)|(-1) logN, where lambda(1) is the eigenvalue of the substitution rate matrix with the smallest nonzero absolute value. The accuracy of likelihood-based distance estimates is therefore much higher than those based on log transform formulae, particularly in cases where there is a large range of timescales involved in the rate matrix (e.g., when the ratio of transition to transversion rates is large). We discuss several practical ways of estimating the rate matrix parameters before distance calculation and hence of increasing the accuracy of distance estimates.     

Site-by-site estimation of the rate of substitution and the correlation of rates in mitochondrial DNA

A maximum likelihood method for independently estimating the relative rate of substitution at different nucleotide sites is presented. With this method, the evolution of DNA sequences can be analyzed without assuming a specific distribution of rates among sites. To investigate the pattern of correlation of rates among sites, the method was applied to a data set consisting of the protein-coding regions of the mitochondrial genome from 10 vertebrate species. Rates appear to be strongly correlated at distances up to 40 codons apart. Furthermore, there appears to be some higher order correlation of sites approximately 75 codons apart. The method of site-by-site estimation of the rate of substitution may also be applied to examine other aspects of rate variation along a DNA sequence and to assess the difference in the support of a tree along the sequence.     

Estimating the pattern of nucleotide substitution

Knowledge of the pattern of nucleotide substitution is important both to our understanding of molecular sequence evolution and to reliable estimation of phylogenetic relationships. The method of parsimony analysis, which has been used to estimate substitution patterns in real sequences, has serious drawbacks and leads to results difficult to interpret. In this paper a model-based maximum likelihood approach is proposed for estimating substitution patterns in real sequences. Nucleotide substitution is assumed to follow a homogeneous Markov process, and the general reversible process model (REV) and the unrestricted model without the reversibility assumption are used. These models are also applied to examine the adequacy of the model of Hasegawa et al. (J. Mol. Evol. 1985;22:160–174) (HKY85). Two data sets are analyzed. For the -globin pseudogenes of six primate species, the REV model fits the data much better than HKY85, while, for a segment of mtDNA sequences from nine primates, REV cannot provide a significantly better fit than HKY85 when rate variation over sites is taken into account in the models. It is concluded that the use of the REV model in phylogenetic analysis can be recommended, especially for large data sets or for sequences with extreme substitution patterns, while HKY85 may be expected to provide a good approximation. The use of the unrestricted model does not appear to be worthwhile.     

Relative efficiencies of the maximum-likelihood, neighbor-joining, and maximum-parsimony methods when substitution rate varies with site

The relative efficiencies of the maximum-likelihood (ML), neighbor- joining (NJ), and maximum-parsimony (MP) methods in obtaining the correct topology and in estimating the branch lengths for the case of four DNA sequences were studied by computer simulation, under the assumption either that there is variation in substitution rate among different nucleotide sites or that there is no variation. For the NJ method, several different distance measures (Jukes-Cantor, Kimura two- parameter, and gamma distances) were used, whereas for the ML method three different transition/transversion ratios (R) were used. For the MP method, both the standard unweighted parsimony and the dynamically weighted parsimony methods were used. The results obtained are as follows: (1) When the R value is high, dynamically weighted parsimony is more efficient than unweighted parsimony in obtaining the correct topology. (2) However, both weighted and unweighted parsimony methods are generally less efficient than the NJ and ML methods even in the case where the MP method gives a consistent tree. (3) When all the assumptions of the ML method are satisfied, this method is slightly more efficient than the NJ method. However, when the assumptions are not satisfied, the NJ method with gamma distances is slightly better in obtaining the correct topology than is the ML method. In general, the two methods show more or less the same performance. The NJ method may give a correct topology even when the distance measures used are not unbiased estimators of nucleotide substitutions. (4) Branch length estimates of a tree with the correct topology are affected more easily than topology by violation of the assumptions of the mathematical model used, for both the ML and the NJ methods. Under certain conditions, branch lengths are seriously overestimated or underestimated. The MP method often gives serious underestimates for certain branches. (5) Distance measures that generate the correct topology, with high probability, do not necessarily give good estimates of branch lengths. (6) The likelihood-ratio test and the confidence-limit test, in Felsenstein's DNAML, for examining the statistical of branch length estimates are quite sensitive to violation of the assumptions and are generally too liberal to be used for actual data. Rzhetsky and Nei's branch length test is less sensitive to violation of the assumptions than is Felsenstein's test. (7) When the extent of sequence divergence is < or = 5% and when > or = 1,000 nucleotides are used, all three methods show essentially the same efficiency in obtaining the correct topology and in estimating branch lengths.(ABSTRACT TRUNCATED AT 400 WORDS)      

Not so different after all: a comparison of methods for detecting amino acid sites under selection

We consider three approaches for estimating the rates of nonsynonymous and synonymous changes at each site in a sequence alignment in order to identify sites under positive or negative selection: (1) a suite of fast likelihood-based "counting methods" that employ either a single most likely ancestral reconstruction, weighting across all possible ancestral reconstructions, or sampling from ancestral reconstructions; (2) a random effects likelihood (REL) approach, which models variation in nonsynonymous and synonymous rates across sites according to a predefined distribution, with the selection pressure at an individual site inferred using an empirical Bayes approach; and (3) a fixed effects likelihood (FEL) method that directly estimates nonsynonymous and synonymous substitution rates at each site. All three methods incorporate flexible models of nucleotide substitution bias and variation in both nonsynonymous and synonymous substitution rates across sites, facilitating the comparison between the methods. We demonstrate that the results obtained using these approaches show broad agreement in levels of Type I and Type II error and in estimates of substitution rates. Counting methods are well suited for large alignments, for which there is high power to detect positive and negative selection, but appear to underestimate the substitution rate. A REL approach, which is more computationally intensive than counting methods, has higher power than counting methods to detect selection in data sets of intermediate size but may suffer from higher rates of false positives for small data sets. A FEL approach appears to capture the pattern of rate variation better than counting methods or random effects models, does not suffer from as many false positives as random effects models for data sets comprising few sequences, and can be efficiently parallelized. Our results suggest that previously reported differences between results obtained by counting methods and random effects models arise due to a combination of the conservative nature of counting-based methods, the failure of current random effects models to allow for variation in synonymous substitution rates, and the naive application of random effects models to extremely sparse data sets. We demonstrate our methods on sequence data from the human immunodeficiency virus type 1 env and pol genes and simulated alignments.     

Estimating phylogenetic trees from pairwise likelihoods and posterior probabilities of substitution counts

The field of phylogenetic tree estimation has been dominated by three broad classes of methods: distance-based approaches, parsimony and likelihood-based methods (including maximum likelihood (ML) and Bayesian approaches). Here we introduce two new approaches to tree inference: pairwise likelihood estimation and a distance-based method that estimates the number of substitutions along the paths through the tree. Our results include the derivation of the formulae for the probability that two leaves will be identical at a site given a number of substitutions along the path connecting them. We also derive the posterior probability of the number of substitutions along a path between two sequences. The calculations for the posterior probabilities are exact for group-based, symmetric models of character evolution, but are only approximate for more general models.     

Substitution rate variation among sites in mitochondrial hypervariable region I of humans and chimpanzees

Mitochondrial D-loop hypervariable region I (HVI) sequences are widely used in human molecular evolutionary studies, and therefore accurate assessment of rate heterogeneity among sites is essential. We used the maximum-likelihood method to estimate the gamma shape parameter alpha for variable substitution rates among sites for HVI from humans and chimpanzees to provide estimates for future studies. The complete data of 839 humans and 224 chimpanzees, as well as many subsets of these data, were analyzed to examine the effect of sequence sampling. The effects of the genealogical tree and the nucleotide substitution model were also examined. The transition/transversion rate ratio (kappa) is estimated to be about 25, although much larger and biased estimates were also obtained from small data sets at low divergences. Estimates of alpha were 0.28-0.39 for human data sets of different sizes and 0.20-0.39 for data sets including different chimpanzee subspecies. The combined data set of both species gave estimates of 0.42-0.45. While all those estimates suggest highly variable substitution rates among sites, smaller samples tend to give smaller estimates of alpha. Possible causes for this pattern were examined, such as biases in the estimation procedure and shifts in the rate distribution along certain lineages. Computer simulations suggest that the estimation procedure is quite reliable for large trees but can be biased for small samples at low divergences. Thus, an alpha of 0.4 appears suitable for both humans and chimpanzees. Estimates of alpha can be affected by the nucleotide sites included in the data, the overall tree length (the amount of sequence divergence), the number of rate classes used for the estimation, and to a lesser extent, the included sequences. The genealogical tree, the substitution model, and demographic processes such as population expansion do not have much effect.     

A simple method for estimating the parameter of substitution rate variation among sites

When the rate variation among sites is described by a gamma distribution, an important problem is how to estimate the shape parameter alpha, which is an index of the degree of among-site rate variation. The parsimony-based methods for estimating alpha are simple but biased, i.e., alpha tends to be overestimated. On the other hand, the likelihood-based methods are asymptotically unbiased but take a huge amount of computational time. In this paper, we have developed a new method to solve this problem: we first estimate the expected number of substitutions at each site, which is corrected for multiple hits, and then estimate the parameter alpha. Our method is computationally as fast as the parsimony method, and the estimation accuracy is much higher than that of parsimony and similar to that of the likelihood method.      

Maximum likelihood estimation of the heterogeneity of substitution rate among nucleotide sites

This paper presents a maximum likelihood approach to estimating the variation of substitution rate among nucleotide sites. We assume that the rate varies among sites according to an invariant+gamma distribution, which has two parameters: the gamma parameter alpha and the proportion of invariable sites theta. Theoretical treatments on three, four, and five sequences have been conducted, and computer program have been developed. It is shown that rho = (1 + theta alpha)/(1 + alpha) is a good measure for the rate heterogeneity among sites. Extensive simulations show that (1) if the proportion of invariable sites is negligible, i.e., theta = 0, the gamma parameter alpha can be satisfactorily estimated, even with three sequences; (2) if the proportion of invariable sites is not negligible, the heterogeneity rho can still be suitably estimated with four or more sequences; and (3) the distances estimated by the proposed method are almost unbiased and are robust against violation of the assumption of the invariant + gamma distribution.      

The Impact of Modelling Rate Heterogeneity among Sites on Phylogenetic Estimates of Intraspecific Evolutionary Rates and Timescales

Phylogenetic analyses of DNA sequence data can provide estimates of evolutionary rates and timescales. Nearly all phylogenetic methods rely on accurate models of nucleotide substitution. A key feature of molecular evolution is the heterogeneity of substitution rates among sites, which is often modelled using a discrete gamma distribution. A widely used derivative of this is the gamma-invariable mixture model, which assumes that a proportion of sites in the sequence are completely resistant to change, while substitution rates at the remaining sites are gamma-distributed. For data sampled at the intraspecific level, however, biological assumptions involved in the invariable-sites model are commonly violated. We examined the use of these models in analyses of five intraspecific data sets. We show that using 6–10 rate categories for the discrete gamma distribution of rates among sites is sufficient to provide a good approximation of the marginal likelihood. Increasing the number of gamma rate categories did not have a substantial effect on estimates of the substitution rate or coalescence time, unless rates varied strongly among sites in a non-gamma-distributed manner. The assumption of a proportion of invariable sites provided a better approximation of the asymptotic marginal likelihood when the number of gamma categories was small, but had minimal impact on estimates of rates and coalescence times. However, the estimated proportion of invariable sites was highly susceptible to changes in the number of gamma rate categories. The concurrent use of gamma and invariable-site models for intraspecific data is not biologically meaningful and has been challenged on statistical grounds; here we have found that the assumption of a proportion of invariable sites has no obvious impact on Bayesian estimates of rates and timescales from intraspecific data.     

Inferring complex DNA substitution processes on phylogenies using uniformization and data augmentation

A new method is developed for calculating sequence substitution probabilities using Markov chain Monte Carlo (MCMC) methods. The basic strategy is to use uniformization to transform the original continuous time Markov process into a Poisson substitution process and a discrete Markov chain of state transitions. An efficient MCMC algorithm for evaluating substitution probabilities by this approach using a continuous gamma distribution to model site-specific rates is outlined. The method is applied to the problem of inferring branch lengths and site-specific rates from nucleotide sequences under a general time-reversible (GTR) model and a computer program BYPASSR is developed. Simulations are used to examine the performance of the new program relative to an existing program BASEML that uses a discrete approximation for the gamma distributed prior on site-specific rates. It is found that BASEML and BYPASSR are in close agreement when inferring branch lengths, regardless of the number of rate categories used, but that BASEML tends to underestimate high site-specific substitution rates, and to overestimate intermediate rates, when fewer than 50 rate categories are used. Rate estimates obtained using BASEML agree more closely with those of BYPASSR as the number of rate categories increases. Analyses of the posterior distributions of site-specific rates from BYPASSR suggest that a large number of taxa are needed to obtain precise estimates of site-specific rates, especially when rates are very high or very low. The method is applied to analyze 45 sequences of the alpha 2B adrenergic receptor gene (A2AB) from a sample of eutherian taxa. In general, the pattern expected for regions under negative selection is observed with third codon positions having the highest inferred rates, followed by first codon positions and with second codon positions having the lowest inferred rates. Several sites show exceptionally high substitution rates at second codon positions that may represent the effects of positive selection.     

Pairwise statistical significance of local sequence alignment using sequence-specific and position-specific substitution matrices

Pairwise sequence alignment is a central problem in bioinformatics, which forms the basis of various other applications. Two related sequences are expected to have a high alignment score, but relatedness is usually judged by statistical significance rather than by alignment score. Recently, it was shown that pairwise statistical significance gives promising results as an alternative to database statistical significance for getting individual significance estimates of pairwise alignment scores. The improvement was mainly attributed to making the statistical significance estimation process more sequence-specific and database-independent. In this paper, we use sequence-specific and position-specific substitution matrices to derive the estimates of pairwise statistical significance, which is expected to use more sequence-specific information in estimating pairwise statistical significance. Experiments on a benchmark database with sequence-specific substitution matrices at different levels of sequence-specific contribution were conducted, and results confirm that using sequence-specific substitution matrices for estimating pairwise statistical significance is significantly better than using a standard matrix like BLOSUM62, and than database statistical significance estimates reported by popular database search programs like BLAST, PSI-BLAST (without pretrained PSSMs), and SSEARCH on a benchmark database, but with pretrained PSSMs, PSI-BLAST results are significantly better. Further, using position-specific substitution matrices for estimating pairwise statistical significance gives significantly better results even than PSI-BLAST using pretrained PSSMs.     

Taking Variation of Evolutionary Rates Between Sites into Account in Inferring Phylogenies

As methods of molecular phylogeny have become more explicit and more biologically realistic following the pioneering work of Thomas Jukes, they have had to relax their initial assumption that rates of evolution were equal at all sites. Distance matrix and likelihood methods of inferring phylogenies make this assumption; parsimony, when valid, is less limited by it. Nucleotide sequences, including RNA sequences, can show substantial rate variation; protein sequences show rates that vary much more widely. Assuming a prior distribution of rates such as a gamma distribution or lognormal distribution has deservedly been popular, but for likelihood methods it leads to computational difficulties. These can be resolved using hidden Markov model (HMM) methods which approximate the distribution by one with a modest number of discrete rates. Generalized Laguerre quadrature can be used to improve the selection of rates and their probabilities so as to more nearly approach the desired gamma distribution. A model based on population genetics is presented predicting how the rates of evolution might vary from locus to locus. Challenges for the future include allowing rates at a given site to vary along the tree, as in the ``covarion' model, and allowing them to have correlations that reflect three-dimensional structure, rather than position in the coding sequence. Markov chain Monte Carlo likelihood methods may be the only practical way to carry out computations for these models. Received: 8 February 2001 / Accepted: 20 May 2001     

Genealogical evidence for positive selection in the nef gene of HIV-1.

The pattern and process of evolution in the nef gene of HIV-1 was analyzed within and among patients. Using a maximum likelihood method that allows for variable intensity of selection pressure among codons, strong positive selection was detected in a hemophiliac patient over 30 mo of infection. By reconstructing the process of allele substitution in this patient using parsimony, the synapomorphic amino acid changes separating each time point were found to have high probabilities of being under positive selection, with selective coefficients of at least 3.6%. Positive selection was also detected among 39 nef sequences from HIV-1 subtype B. In contrast, multiple pairwise comparisons of nonsynonymous and synonymous substitution rates provided no good evidence for positive selection and sliding window analyses failed to detect most positively selected sites. These findings demonstrate that positive selection is an important determinant of nef gene evolution and that genealogy-based methods outperform pairwise methods in the detection of adaptive evolution. Mapping the locations of positively selected sites may also be of use in identifying targets of the immune response and hence aid vaccine design.     

Patterns of Nucleotide Substitution in Mitochondrial Protein Coding Genes of Vertebrates

Maximum likelihood methods were used to study the differences in substitution rates among the four nucleotides and among different nucleotide sites in mitochondrial protein-coding genes of vertebrates. In the 1st+2nd codon position data, the frequency of nucleotide G is negatively correlated with evolutionary rates of genes, substitution rates vary substantially among sites, and the transition/transversion rate bias (R) is two to five times larger than that expected at random. Generally, largest transition biases and greatest differences in substitution rates among sites are found in the highly conserved genes. The 3rd positions in placental mammal genes exhibit strong nucleotide composition biases and the transitional rates exceed transversional rates by one to two orders of magnitude. Tamura-Nei and Hasegawa-Kishino-Yano models with gamma distributed variable rates among sites (gamma parameter, α) adequately describe the nucleotide substitution process in 1st+2nd position data. In these data, ignoring differences in substitution rates among sites leads to largest biases while estimating substitution rates. Kimura's two-parameter model with variable-rates among sites performs satisfactorily in likelihood estimation of R, α, and overall amount of evolution for 1st+2nd position data. It can also be used to estimate pairwise distances with appropriate values of α for a majority of genes.