Estimation of average number of nucleotide substitutions when the rate of substitution varies with nucleotide

Summary A formal mathematical analysis of Kimura's (1981) six-parameter model of nucleotide substitution for the case of unequal substitution rates among different pairs of nucleotides is conducted, and new formulae for estimating the number of nucleotide substitutions and its standard error are obtained. By using computer simulation, the validities and utilities of Jukes and Cantor's (1969) one-parameter formula, Takahata and Kimura's (1981) four-parameter formula, and our sixparameter formula for estimating the number of nucleotide substitutions are examined under three different schemes of nucleotide substitution. It is shown that the one-parameter and four-parameter formulae often give underestimates when the number of nucleotide substitutions is large, whereas the six-parameter formula generally gives a good estimate for all the three substitution schemes examined. However, when the number of nucleotide substitutions is large, the six-parameter and four-parameter formulae are often inapplicable unless the number of nucleotides compared is extremely large. It is also shown that as long as the mean number of nucleotide substitutions is smaller than one per nucleotide site the three formulae give more or less the same estimate regardless of the substitution scheme used.On leave of absence from the Department of Biology, Faculty of Science, Kyushu University 33, Fukuoka 812, Japan     

Estimation of evolutionary distance between nucleotide sequences

A mathematical formula for estimating the average number of nucleotide substitutions per site (delta) between two homologous DNA sequences is developed by taking into account unequal rates of substitution among different nucleotide pairs. Although this formula is obtained for the equal-input model of nucleotide substitution, computer simulations have shown that it gives a reasonably good estimate for a wide range of nucleotide substitution patterns as long as delta is equal to or smaller than 1. Furthermore, the frequency of cases to which the formula is inapplicable is much lower than that for other similar methods recently proposed. This point is illustrated using insulin genes. A statistical method for estimating the number of nucleotide changes due to deletion and insertion is also developed. Application of this method to globin gene data indicates that the number of nucleotide changes per site increases with evolutionary time but the pattern of the increase is quite irregular.      

Relative efficiencies of the maximum parsimony and distance-matrix methods in obtaining the correct phylogenetic tree

The relative efficiencies of the maximum parsimony (MP) and distance-matrix methods in obtaining the correct tree (topology) were studied by using computer simulation. The distance-matrix methods examined are the neighbor-joining, distance-Wagner, Tateno et al. modified Farris, Faith, and Li methods. In the computer simulation, six or eight DNA sequences were assumed to evolve following a given model tree, and the evolutionary changes of the sequences were followed. Both constant and varying rates of nucleotide substitution were considered. From the sequences thus obtained, phylogenetic trees were constructed using the six tree-making methods and compared with the model (true) tree. This process was repeated 300 times for each different set of parameters. The results obtained indicate that when the number of nucleotide substitutions per site is small and a relatively small number of nucleotides are used, the probability of obtaining the correct topology (P1) is generally lower in the MP method than in the distance-matrix methods. The P1 value for the MP method increases with increasing number of nucleotides but is still generally lower than the value for the NJ or DW method. Essentially the same conclusion was obtained whether or not the rate of nucleotide substitution was constant or whether or not a transition bias in nucleotide substitution existed. The relatively poor performance of the MP method for these cases is due to the fact that information from singular sites is not used in this method. The MP method also showed a relatively low P1 value when the model of varying rate of nucleotide substitution was used and the number of substitutions per site was large. However, the MP method often produced cases in which the correct tree was one of several equally parsimonious trees. When these cases were included in the class of "success," the MP method performed better than the other methods, provided that the number of nucleotide substitutions per site was small.     

Codon Substitution in Evolution and the "Saturation" of Synonymous Changes

A mathematical model for codon substitution is presented, taking into account unequal mutation rates among different nucleotides and purifying selection. This model is constructed by using a 61 X 61 transition probability matrix for the 61 nonterminating codons. Under this model, a computer simulation is conducted to study the numbers of silent (synonymous) and amino acid-altering (nonsynonymous) nucleotide substitutions when the underlying mutation rates among the four kinds of nucleotides are not equal. It is assumed that the substitution rates are constant over evolutionary time, the codon frequencies being in equilibrium, and, thus, the numbers of synonymous and nonsynonymous substitutions both increase linearly with evolutionary time. It is shown that, when the mutation rates are not equal, the estimate of synonymous substitutions obtained by F. Perler, A. Efstratiadis, P. Lomedico, W. Gilbert, R. Kolodner and J. Dodgson's "Percent Corrected Divergence" method increases nonlinearly, although the true number of synonymous substitutions increases linearly. It is, therefore, possible that the "saturation" of synonymous substitutions observed by Perler et al. is due to the inefficiency of their method to detect all synonymous substitutions.     

Human Mitochondrial DNA Variation and Evolution: Analysis of Nucleotide Sequences from Seven Individuals

We have analyzed nucleotide sequence variation in an approximately 900-base pair region of the human mitochondrial DNA molecule encompassing the heavy strand origin of replication and the D-loop. Our analysis has focused on nucleotide sequences available from seven humans. Average nucleotide diversity among the sequences is 1.7%, several-fold higher than estimates from restriction endonuclease site variation in mtDNA from these individuals and previously reported for other humans. This disparity is consistent with the rapidly evolving nature of this noncoding region. However, several instances of convergent or parallel gain and loss of restriction sites due to multiple substitutions were observed. In addition, other results suggest that restriction site (as well as pairwise sequence) comparisons may underestimate the total number of substitutions that have occurred since the divergence of two mtDNA sequences from a common ancestral sequence, even at low levels of divergence. This emphasizes the importance of recognizing the large standard errors associated with estimates of sequence variability, particularly when constructing phylogenies among closely related sequences. Analysis of the observed number and direction of substitutions revealed several significant biases, most notably a strand dependence of substitution type and a 32-fold bias favoring transitions over transversions. The results also revealed a significantly nonrandom distribution of nucleotide substitutions and sequence length variation. Significantly more multiple substitutions were observed than expected for these closely related sequences under the assumption of uniform rates of substitution. The bias for transitions has resulted in predominantly convergent or parallel changes among the observed multiple substitutions. There is no convincing evidence that recombination has contributed to the mtDNA sequence diversity we have observed.     

Evolutionary change of restriction sites under unequal rates of nucleotide substitution among the three positions of codons

Summary Under the assumption of unequal rates of nucleotide substitution among the three positions of codons, mathematical formulas are derived for the probability that a restriction site observed at time 0 in a DNA sequence will be present at time t, the probability that a new restriction site will emerge at a particular place in two descendant sequences, and the proportion of identical restriction sites between two such sequences. All three quantitites are shown to be larger for the case of unequal rates than for the case of equal rates. As a consequence, estimates of nucleotide divergence (2t) between two sequences based on restriction-site data tend to be lower than the actual values of the assumption of equal rates is made but actually does not hold. However, the degree of underestimation is slight if 2t is 0.10 or smaller. The underestimation can be serious when 2t becomes 0.20 or larger, particularly if the substitution rates at the three codon positions are very different or if transitional substitution occurs more frequently than transversional substitution. The underestimation is more serious for four-base enzymes than for sixbase enzymes.     

Relative efficiencies of the maximum-parsimony and distance-matrix methods of phylogeny construction for restriction data.

The relative efficiencies of the maximum-parsimony (MP), UPGMA, and neighbor-joining (NJ) methods in obtaining the correct tree (topology) for restriction-site and restriction-fragment data were studied by computer simulation. In this simulation, six DNA sequences of 16,000 nucleotides were assumed to evolve following a given model tree. The recognition sequences of 20 different six-base restriction enzymes were used to identify the restriction sites of the DNA sequences generated. The restriction-site data and restriction-fragment data thus obtained were used to reconstruct a phylogenetic tree, and the tree obtained was compared with the model tree. This process was repeated 300 times. The results obtained indicate that when the rate of nucleotide substitution is constant the probability of obtaining the correct tree (Pc) is generally higher in the NJ method than in the MP method. However, if we use the average topological deviation from the model tree (dT) as the criterion of comparison, the NJ and MP methods are nearly equally efficient. When the rate of nucleotide substitution varies with evolutionary lineage, the NJ method is better than the MP method, whether Pc or dT is used as the criterion of comparison. With 500 nucleotides and when the number of nucleotide substitutions per site was very small, restriction-site data were, contrary to our expectation, more useful than sequence data. Restriction-fragment data were less useful than restriction-site data, except when the sequence divergence was very small. UPGMA seems to be useful only when the rate of nucleotide substitution is constant and sequence divergence is high.     

Unbiased estimates of the number of nucleotide substitutions when substitution rate varies among different sites

When the number of nucleotides examined is relatively small, the estimators of nucleotide substitutions between DNA sequences often introduce systematic error even if the data used fit the mathematical model underlying the estimation formula. The systematic error of this kind is especially large for models that allow variation in substitution rate among different sites. In the present paper we present a number of formulas that produce virtually bias-free estimates of evolutionary distances for these models. Correspondence to: M. Nei     

Empirical relationship between the number of nucleotide substitutions and interspecific identity of amino acid sequences in some proteins

There are three different methods of estimating the number of nucleotide substitutions between a pair of species from amino acid sequence data, i.e. the Poisson correction method, random evolutionary hit method, and counting the actual but minimum number of nucleotide substitutions. In this paper the relationships among the estimates obtained by these methods are studied empirically. The results obtained indicate that there is a high correlation among these estimates and in practice any of the three methods may be used for constructing evolutionary trees or relating nucleotide substitutions to evolutionary time. The effects of varying rates of nucleotide substition among different sites on the Poisson correction and random evolutionary hit methods are also studied mathematically. It is shown that these two methods are quite insensitive to the variation of the rate of nucleotide substitution.     

Estimation of the number of nucleotide substitutions in the control region of mitochondrial DNA in humans and chimpanzees

Examining the pattern of nucleotide substitution for the control region of mitochondrial DNA (mtDNA) in humans and chimpanzees, we developed a new mathematical method for estimating the number of transitional and transversional substitutions per site, as well as the total number of nucleotide substitutions. In this method, excess transitions, unequal nucleotide frequencies, and variation of substitution rate among different sites are all taken into account. Application of this method to human and chimpanzee data suggested that the transition/transversion ratio for the entire control region was approximately 15 and nearly the same for the two species. The 95% confidence interval of the age of the common ancestral mtDNA was estimated to be 80,000-480,000 years in humans and 0.57-2.72 Myr in common chimpanzees.      

A Simple Method for Estimating Average Number of Nucleotide Substitutions within and between Populations from Restriction Data

A simple method is proposed for estimating the average number of nucleotide substitutions per site within and between populations for the case where a large number of individuals are examined for many restriction enzymes. This method gives essentially the same results as those obtained by Nei and Li's method but saves a large amount of computer time. The variances of the quantities estimated can be obtained by the jackknife method, and these variances are very similar to those obtained by Nei and Jin's more sophisticated method. A similar method can also be applied to DNA sequence data.     

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.     

Substantial Regional Variation in Substitution Rates in the Human Genome: Importance of GC Content, Gene Density, and Telomere-Specific Effects

This study presents the first global, 1-Mbp-level analysis of patterns of nucleotide substitutions along the human lineage. The study is based on the analysis of a large amount of repetitive elements deposited into the human genome since the mammalian radiation, yielding a number of results that would have been difficult to obtain using the more conventional comparative method of analysis. This analysis revealed substantial and consistent variability of rates of substitution, with the variability ranging up to twofold among different regions. The rates of substitutions of C or G nucleotides with A or T nucleotides vary much more sharply than the reverse rates, suggesting that much of that variation is due to differences in mutation rates rather than in the probabilities of fixation of C/G vs. A/T nucleotides across the genome. For all types of substitution we observe substantially more hotspots than coldspots, with hotspots showing substantial clustering over tens of Mbp’s. Our analysis revealed that GC-content of surrounding sequences is the best predictor of the rates of substitution. The pattern of substitution appears very different near telomeres compared to the rest of the genome and cannot be explained by the genome-wide correlations of the substitution rates with GC content or exon density. The telomere pattern of substitution is consistent with natural selection or biased gene conversion acting to increase the GC-content of the sequences that are within 10–15 Mbp away from the telomere.Reviewing Editor: Dr. Jerzy Jurka

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This revised version was published online in July 2005 with corrected page numbers.     

Nucleic acid composition,codon usage,and the rate of synonymous substitution in protein-coding genes

Summary Based on the rates of synonymous substitution in 42 protein-codin gene pairs from rat and human, a correlation is shown to exist between the frequency of the nucleotides in all positions of the codon and the synonymous substitution rate. The correlation coefficients were positive for A and T and negative for C and G. This means that AT-rich genes accumulate more synonymous substitutions than GC-rich genes. Biased patterns of mutation could not account for this phenomenon. Thus, the variation in synonymous substitution rates and the resulting unequal codon usage must be the consequence of selection against A and T in synonymous positions. Most of the varition in rates of synonymous substitution can be explained by the nucleotide composition in synonymous positions. Codon-anticodon interactions, dinucleotide frequencies, and contextual factors influence neither the rates of synonymous substitution nor codon usage. Interestingly, the nucleotide in the second position of codons (always a nonsynonymous position) was found to affect the rate of synonymous substitution. This finding links the rate of nonsynonymous substitution with the synonymous rate. Consequently, highly conservative proteins are expected to be encoded by genes that evolve slowly in terms of synonymous substitutions, and are consequently highly biased in their codon usage.     

Maximum Likelihood Estimation of the Number of Nucleotide Substitutions from Restriction Sites Data

A simple method of the maximum likelihood estimation of the number of nucleotide substitutions is presented for the case where restriction sites data from many different restriction enzymes are available. An iteration method, based on nucleotide counting, is also developed. This method is simpler than the maximum likelihood method but gives the same estimate. A formula for computing the variance of a maximum likelihood estimate is also presented.     

Statistical Properties of a DNA Sample under the Finite-Sites Model

Statistical properties of a DNA sample from a random-mating population of constant size are studied under the finite-sites model. It is assumed that there is no migration and no recombination occurs within the locus. A Markov process model is used for nucleotide substitution, allowing for multiple substitutions at a single site. The evolutionary rates among sites are treated as either constant or variable. The general likelihood calculation using numerical integration involves intensive computation and is feasible for three or four sequences only; it may be used for validating approximate algorithms. Methods are developed to approximate the probability distribution of the number of segregating sites in a random sample of n sequences, with either constant or variable substitution rates across sites. Calculations using parameter estimates obtained for human D-loop mitochondrial DNAs show that among-site rate variation has a major effect on the distribution of the number of segregating sites; the distribution under the finite-sites model with variable rates among sites is quite different from that under the infinite-sites model.     

Neighboring-nucleotide effects on the rates of germ-line single-base-pair substitution in human genes.

The spectrum of single-base-pair substitutions logged in The Human Gene Mutation Database (HGMD), comprising 7,271 different lesions in the coding regions of 547 different human genes, was analyzed for nearest-neighbor effects on relative mutation rates. Owing to its retrospective nature, HGMD allows mutation rates to be estimated only in relative terms. Therefore, a novel methodology was devised in order to obtain these estimates in iterative fashion, correcting, at the same time, for the confounding effects of differential codon usage and for the fact that different types of amino acid replacement come to clinical attention with different probabilities. Over and above the hypermutability of CpG dinucleotides, reflected in transition rates five times the base mutation rate, only a subtle and locally confined influence of the surrounding DNA sequence on relative single-base-pair substitution rates was observed, which extended no farther than 2 bp from the substitution site. A disparity between the two DNA strands was evidenced by the fact that, when substitution rates were estimated conditional on the 5' and 3' flanking nucleotides, a significant rate difference emerged for 10 of 96 possible pairs of complementary substitutional events. Mutational bias, favoring substitutions toward flanking bases, a phenomenon reminiscent of misalignment mutagenesis, was apparent and exhibited both directionality and reading-frame sensitivity. No specific preponderance of repeat-sequence motifs was observed in the vicinity of nucleotide substitutions, but a moderate correlation between the relative mutability and thermodynamic stability of DNA triplets emerged, suggesting either inefficient DNA replication in regions of high stability or the transient stabilization of misaligned intermediates.     

A new method for estimating synonymous and nonsynonymous rates of nucleotide substitution considering the relative likelihood of nucleotide and codon changes

A new method is proposed for estimating the number of synonymous and nonsynonymous nucleotide substitutions between homologous genes. In this method, a nucleotide site is classified as nondegenerate, twofold degenerate, or fourfold degenerate, depending on how often nucleotide substitutions will result in amino acid replacement; nucleotide changes are classified as either transitional or transversional, and changes between codons are assumed to occur with different probabilities, which are determined by their relative frequencies among more than 3,000 changes in mammalian genes. The method is applied to a large number of mammalian genes. The rate of nonsynonymous substitution is extremely variable among genes; it ranges from 0.004 X 10(-9) (histone H4) to 2.80 X 10(-9) (interferon gamma), with a mean of 0.88 X 10(-9) substitutions per nonsynonymous site per year. The rate of synonymous substitution is also variable among genes; the highest rate is three to four times higher than the lowest one, with a mean of 4.7 X 10(-9) substitutions per synonymous site per year. The rate of nucleotide substitution is lowest at nondegenerate sites (the average being 0.94 X 10(-9), intermediate at twofold degenerate sites (2.26 X 10(-9)). and highest at fourfold degenerate sites (4.2 X 10(-9)). The implication of our results for the mechanisms of DNA evolution and that of the relative likelihood of codon interchanges in parsimonious phylogenetic reconstruction are discussed.     

γ-MYN: a new algorithm for estimating Ka and Ks with consideration of variable substitution rates

Background  

Over the past two decades, there have been several approximate methods that adopt different mutation models and used for estimating nonsynonymous and synonymous substitution rates (Ka and Ks) based on protein-coding sequences across species or even different evolutionary lineages. Among them, MYN method (a Modified version of Yang-Nielsen method) considers three major dynamic features of evolving DNA sequences–bias in transition/transversion rate, nucleotide frequency, and unequal transitional substitution but leaves out another important feature: unequal substitution rates among different sites or nucleotide positions.     

A Space-Time Process Model for the Evolution of DNA Sequences

We describe a model for the evolution of DNA sequences by nucleotide substitution, whereby nucleotide sites in the sequence evolve over time, whereas the rates of substitution are variable and correlated over sites. The temporal process used to describe substitutions between nucleotides is a continuous-time Markov process, with the four nucleotides as the states. The spatial process used to describe variation and dependence of substitution rates over sites is based on a serially correlated gamma distribution, i.e., an auto-gamma model assuming Markov-dependence of rates at adjacent sites. To achieve computational efficiency, we use several equal-probability categories to approximate the gamma distribution, and the result is an auto-discrete-gamma model for rates over sites. Correlation of rates at sites then is modeled by the Markov chain transition of rates at adjacent sites from one rate category to another, the states of the chain being the rate categories. Two versions of nonparametric models, which place no restrictions on the distributional forms of rates for sites, also are considered, assuming either independence or Markov dependence. The models are applied to data of a segment of mitochondrial genome from nine primate species. Model parameters are estimated by the maximum likelihood method, and models are compared by the likelihood ratio test. Tremendous variation of rates among sites in the sequence is revealed by the analyses, and when rate differences for different codon positions are appropriately accounted for in the models, substitution rates at adjacent sites are found to be strongly (positively) correlated. Robustness of the results to uncertainty of the phylogenetic tree linking the species is examined.