Statistical properties of the Jukes-Holmquist method of estimating the number of nucleotide substitutions: Reply to Holmquist and Conroy's criticism

Summary Conducting computer simulations, Nei and Tateno (1978) have shown that Jukes and Holmquist's (1972) method of estimating the number of nucleotide substitutions tends to give an overestimate and the estimate obtained has a large variance. Holmquist and Conroy (1980) repeated some parts of our simulation and claim that the overestimation of nucleotide substitutions in our paper occurred mainly because we used selected data. Examination of Holmquist and Conroy's simulation indicates that their results are essentially the same as ours when the Jukes-Holmquist method is used, but since they used a different method of computation their estimates of nucleotide substitutions differed substantially from ours. Another problem in Holmquist and Conroy's Letter is that they confused the expected number of nucleotide substitution with the number in a sample. This confusion has resulted in a number of unnecessary arguments. They also criticized ourX ₂ measure, but this criticism is apparently due to a misunderstanding of the assumptions of our method and a failure to use our method in the way we described. We believe that our earlier conclusions remain unchanged.     

Simple methods for estimating the numbers of synonymous and nonsynonymous nucleotide substitutions

Two simple methods for estimating the numbers of synonymous and nonsynonymous nucleotide substitutions are presented. Although they give no weights to different types of codon substitutions, these methods give essentially the same results as those obtained by Miyata and Yasunaga's and by Li et al.'s methods. Computer simulation indicates that estimates of synonymous substitutions obtained by the two methods are quite accurate unless the number of nucleotide substitutions per site is very large. It is shown that all available methods tend to give an underestimate of the number of nonsynonymous substitutions when the number is large.      

Nonrandom amino acid substitution and estimation of the number of nucleotide substitutions in evolution

Summary A method of estimating the number of nucleotide substitutions from amino acid sequence data is developed by using Dayhoff's mutation probability matrix. This method takes into account the effect of nonrandom amino acid substitutions and gives an estimate which is similar to the value obtained by Fitch's counting method, but larger than the estimate obtained under the assumption of random substitutions (Jukes and Cantor's formula). Computer simulations based on Dayhoff's mutation probability matrix have suggested that Jukes and Holmquist's method of estimating the number of nucleotide substitutions gives an overestimate when amino acid substitution is not random and the variance of the estimate is generally very large. It is also shown that when the number of nucleotide substitutions is small, this method tends to give an overestimate even when amino acid substitution is purely at random.     

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     

Accuracy of phylogenetic trees estimated from DNA sequence data

The relative merits of four different tree-making methods in obtaining the correct topology were studied by using computer simulation. The methods studied were the unweighted pair-group method with arithmetic mean (UPGMA), Fitch and Margoliash's (FM) method, thd distance Wagner (DW) method, and Tateno et al.'s modified Farris (MF) method. An ancestral DNA sequence was assumed to evolve into eight sequences following a given model tree. Both constant and varying rates of nucleotide substitution were considered. Once the DNA sequences for the eight extant species were obtained, phylogenetic trees were constructed by using corrected (d) and uncorrected (p) nucleotide substitutions per site. The topologies of the trees obtained were then compared with that of the model tree. The results obtained can be summarized as follows: (1) The probability of obtaining the correct rooted or unrooted tree is low unless a large number of nucleotide differences exists between different sequences. (2) When the number of nucleotide substitutions per sequence is small or moderately large, the FM, DW, and MF methods show a better performance than UPGMA in recovering the correct topology. The former group of methods is particularly good for obtaining the correct unrooted tree. (3) When the number of substitutions per sequence is large, UPGMA is at least as good as the other methods, particularly for obtaining the correct rooted tree. (4) When the rate of nucleotide substitution varies with evolutionary lineage, the FM, DW, and MF methods show a better performance in obtaining the correct topology than UPGMA, except when a rooted tree is to be produced from data with a large number of nucleotide substitutions per sequence.(ABSTRACT TRUNCATED AT 250 WORDS)      

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.     

Accuracy of estimated phylogenetic trees from molecular data

Summary The accuracies and efficiencies of four different methods for constructing phylogenetic trees from molecular data were examined by using computer simulation. The methods examined are UPGMA, Fitch and Margoliash's (1967) (F/M) method, Farris' (1972) method, and the modified Farris method (Tateno, Nei, and Tajima, this paper). In the computer simulation, eight OTUs (32 OTUs in one case) were assumed to evolve according to a given model tree, and the evolutionary change of a sequence of 300 nucleotides was followed. The nucleotide substitution in this sequence was assumed to occur following the Poisson distribution, negative binomial distribution or a model of temporally varying rate. Estimates of nucleotide substitutions (genetic distances) were then computed for all pairs of the nucleotide sequences that were generated at the end of the evolution considered, and from these estimates a phylogenetic tree was reconstructed and compared with the true model tree. The results of this comparison indicate that when the coefficient of variation of branch length is large the Farris and modified Farris methods tend to be better than UPGMA and the F/M method for obtaining a good topology. For estimating the number of nucleotide substitutions for each branch of the tree, however, the modified Farris method shows a better performance than the Farris method. When the coefficient of variation of branch length is small, however, UPGMA shows the best performance among the four methods examined. Nevertheless, any tree-making method is likely to make errors in obtaining the correct topology with a high probability, unless all branch lengths of the true tree are sufficiently long. It is also shown that the agreement between patristic and observed genetic distances is not a good indicator of the goodness of the tree obtained.     

Unbiased estimation of evolutionary distance between nucleotide sequences

A new algorithm for estimating the number of nucleotide substitutions per site (i.e., the evolutionary distance) between two nucleotide sequences is presented. This algorithm can be applied to many estimation methods, such as Jukes and Cantor's method, Kimura's transition/transversion method, and Tajima and Nei's method. Unlike ordinary methods, this algorithm is always applicable. Numerical computations and computer simulations indicate that this algorithm gives an almost unbiased estimate of the evolutionary distance, unless the evolutionary distance is very large. This algorithm should be useful especially when we analyze short nucleotide sequences. It can also be applied to amino acid sequences, for estimating the number of amino acid replacements.      

Biases of the estimates of DNA divergence obtained by the restriction enzyme technique

Summary A mathematical formula for the relationship between the average number of nucleotide substitutions per site and the proportion of shared restriction sites between two homologous nucleons is developed by taking into account the unequal rates of substitution among different pairs of nucleotides. Using this formula, the possible amount of bias of the estimate of the number of nucleotide substitutions obtained by the Upholt-Nei-Li formula for restriction site data is investigated. The results obtained indicate that the bias depends upon the nucleotides in the recognition sequence of the restriction enzyme used, the unequal rates of substitution among different nucleotides, and the unequal nucleotide frequencies, but the primary factor is the unequal rates of nucleotide substitution. The amount of bias is generally larger for four-base enzymes than for six-base enzymes. However, when many restriction enzymes are used for the study of DNA divergence, the bias is unlikely to be very large unless the rate of substitution greatly varies from nucleotide to nucleotide.     

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.     

K-Estimator: calculation of the number of nucleotide substitutions per site and the confidence intervals.

SUMMARY: K-Estimator 4.5 is a Windows program that estimates the number of nucleotide substitutions per site (divergence) when comparing two aligned nucleotide sequences, both protein-coding and non-coding. Confidence intervals of the divergence estimates are obtained by Monte Carlo simulation. AVAILABILITY: The program is available for non-profit use via anonymous ftp at ftp.bio.indiana. edu/molbio/mswin. CONTACT: jcomeron@midway.uchicago.edu     

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.     

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.     

A comparison of two methods for constructing evolutionary distances from a weighted contribution of transition and transversion differences

Since the initial work of Jukes and Cantor (1969), a number of procedures have been developed to estimate the expected number of nucleotide substitutions corresponding to a given observed level of nucleotide differentiation assuming particular evolutionary models. Unlike the proportion of different sites, the expected number of substitutions that would have occurred grows linearly with time and therefore has had great appeal as an evolutionary distance. Recently, however, a number of authors have tried to develop improved statistical approaches for generating and evaluating evolutionary distances (Schoniger and von Haeseler 1993; Goldstein and Polock 1994; Tajima and Takezaki 1994). These studies clearly show that the estimated number of nucleotide substitutions is generally not the best estimator for use in reconstruction of phylogenetic relationships. The reason for this is that there is often a large error associated with the estimation of this number. Therefore, even though its expectation is correct (i.e., on average the expected number of substitutions is proportional to time- -but see Tajima 1993), it is not expected to be as useful as estimators designed to have a lower variance.      

Relationship between DNA Polymorphism and Fixation Time

When there is no recombination among nucleotide sites in DNA sequences, DNA polymorphism and fixation of mutants at nucleotide sites are mutually related. Using the method of gene genealogy, the relationship between the DNA polymorphism and the fixation of mutant nucleotide was quantitatively investigated under the assumption that mutants are selectively neutral, that there is no recombination among nucleotide sites, and that the population is a random mating population with N diploid individuals. The results obtained indicate that the expected number of nucleotide differences between two DNA sequences randomly sampled from the population is 42% less when a mutant at a particular nucleotide site reaches fixation than at a random time, and that heterozygosity is also expected to be less when fixation takes place than at a random time, but the amount of reduction depends on the value of 4Nv in this case, where v is the mutation rate per DNA sequence per generation. The formula for obtaining the expected number of nucleotide differences between the two DNA sequences for a given fixation time is also derived, and indicates that, even when it takes a large number of generations for a mutant to reach fixation, this number is 33% less than at a random time. The computer simulation conducted suggests that the expected number of nucleotide differences between the two DNA sequences at the time when an advantageous mutant becomes fixed is essentially the same as that of neutral mutant if the fixation time is the same. The effect of recombination on the amount of DNA polymorphism was also investigated by using computer simulation.     

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.     

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.     

Low diversity and biased substitution patterns in the mitochondrial DNA control region of sperm whales: implications for estimates of time since common ancestry

The mitochondrial DNA (mtDNA) control region was sequenced in 37 sperm whales from a large part of the global range of the species. Nucleotide diversity was several-fold lower than that reported for control regions of abundant and outbred mammals, but similar to that for populations known to have experienced bottlenecks. Relative neck tests did not suggest that the low diversity is due to a lower substitution rate in sperm whale mtDNA. Rather, it is more likely that demographic factors have reduced diversity. The pattern of nucleotide substitutions was examined by cladistic methods, facilitated by the apparent monophyly of lineages from the Southern Hemisphere, as defined by a single base pair deletion. Substitutions were nonrandom in nature, confined to a few "hot spots," and parallel substitutions constituted a majority of the inferred changes. The substitution pattern fitted a negative binomial distribution better than a Poisson distribution, and the bias in number of substitutions among sites was considerably higher than previously reported for the mtDNA control region of any species. A novel method of estimating time since common ancestry was developed, which utilizes the transition/transversion ratio R and the number of substitutions inferred from a parsimony analysis. Using this method, we estimated the age of sperm whale mtDNA diversity to be about 6,000-25,000 years, and when the uncertainty of R was accounted for, a range of about 1,000- 100,000 years was obtained.      

On investigating the statistical properties of the populous path algorithm by computer simulation

Summary Goodman et al.'s (1974) populous path algorithm for estimating hidden mutational change in protein evolution is designed to be used as an adjunct to the maximum parsimony method. When the algorithm is so used, the augmented maximum parsimony distances, far from being overestimates, are underestimates of the actual number of nucleotide substitutions which occur in Tateno and Nei's (1978) computer simulation by the Poisson process model, even when the simulation is carried out at two and a half times the sequence density. Although underestimates, our evidence shows that they are nevertheless more accurate than estimates obtained by a Poisson correction. In the maximum parsimony reconstruction, there is a bias towards overrepresenting the number of shared nucleotide identities between adjacent ancestral and descendant nodal sequences with the bias being stronger in those portions of the evolutionary tree sparser in sequence data. Because of this particular property of maximum parsimony reconstructed sequences, the conclusions of Tateno and Nei concerning the statistical properties of the populous path algorithm are invalid. We conclude that estimates of protein evolutionary rates by the maximum parsimony - populous path approach will become more accurate rather than less as larger numbers of closely related species are included in the analysis.