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The reliability of a phylogenetic tree obtained from empirical data is usually measured by the bootstrap probability (Pb) of interior branches of the tree. If the bootstrap probability is high for most branches, the tree is considered to be reliable. If some interior branches show relatively low bootstrap probabilities, we are not sure that the inferred tree is really reliable. Here, we propose another quantity measuring the reliability of the tree called the stability of a subtree. This quantity refers to the probability of obtaining a subtree (Ps) of an inferred tree obtained. We then show that if the tree is to be reliable, both Pb and Ps must be high. We also show that Ps is given by a bootstrap probability of the subtree with the closest outgroup sequence, and computer program RESTA for computing the Pb and Ps values will be presented., (© The Author 2017. Published by Oxford University Press on behalf of the Society for Molecular Biology and Evolution.)

At the present time it is often stated that the maximum likelihood or the Bayesian method of phylogenetic construction is more accurate than the neighbor joining (NJ) method. Our computer simulations, however, have shown that the converse is true if we use p distance in the NJ procedure and the criterion of obtaining the true tree (Pc expressed as a percentage) or the combined quantity (c) of a value of Pc and a value of Robinson-Foulds' average topological error index (dT). This c is given by Pc (1 - dT/dTmax) = Pc (m - 3 - dT/2)/(m - 3), where m is the number of taxa used and dTmax is the maximum possible value of dT, which is given by 2(m - 3). This neighbor joining method with p distance (NJp method) will be shown generally to give the best data-fit model. This c takes a value between 0 and 1, and a tree-making method giving a high value of c is considered to be good. Our computer simulations have shown that the NJp method generally gives a better performance than the other methods and therefore this method should be used in general whether the gene is compositional or it contains the mosaic DNA regions or not., (© The Author 2016. Published by Oxford University Press on behalf of the Society for Molecular Biology and Evolution. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com.)

It is well known that the selection coefficient of a mutant allele varies from generation to generation, and the effect of this factor on genetic variation has been studied by many theoreticians. However, no consensus has been reached. One group of investigators believes that fluctuating selection has an effect of enhancing genetic variation, whereas the other group contends that it has a diversity-reducing effect. In recent years, it has become possible to study this problem by using single nucleotide polymorphisms (SNPs) as well as exome sequence data. In this article we present the theoretical distributions of mutant nucleotide frequencies for the two models of fluctuating selection and then compare the distributions with the empirical distributions obtained from SNP and exome sequence data in human populations. Interestingly, both SNP and exome sequence data showed that the neutral mutation model fits the empirical distribution quite well. Furthermore, the mathematical models with diversity-enhancing and diversity-reducing effects also fit the empirical distribution reasonably well. This result implies that there is no need of distinguishing between the diversity-enhancing and diversity-reducing models of fluctuating selection and the nucleotide polymorphism in human populations can be explained largely by neutral mutations when long-term evolution is considered.

Comparative analysis of molecular sequence data is essential for reconstructing the evolutionary histories of species and inferring the nature and extent of selective forces shaping the evolution of genes and species. Here, we announce the release of Molecular Evolutionary Genetics Analysis version 5 (MEGA5), which is a user-friendly software for mining online databases, building sequence alignments and phylogenetic trees, and using methods of evolutionary bioinformatics in basic biology, biomedicine, and evolution. The newest addition in MEGA5 is a collection of maximum likelihood (ML) analyses for inferring evolutionary trees, selecting best-fit substitution models (nucleotide or amino acid), inferring ancestral states and sequences (along with probabilities), and estimating evolutionary rates site-by-site. In computer simulation analyses, ML tree inference algorithms in MEGA5 compared favorably with other software packages in terms of computational efficiency and the accuracy of the estimates of phylogenetic trees, substitution parameters, and rate variation among sites. The MEGA user interface has now been enhanced to be activity driven to make it easier for the use of both beginners and experienced scientists. This version of MEGA is intended for the Windows platform, and it has been configured for effective use on Mac OS X and Linux desktops. It is available free of charge from http://www.megasoftware.net.

Microsatellite DNA loci or short tandem repeats (STRs) are abundant in eukaryotic genomes and are often used for constructing phylogenetic trees of closely related populations or species. These phylogenetic trees are usually constructed by using some genetic distance measure based on allele frequency data, and there are many distance measures that have been proposed for this purpose. In the past the efficiencies of these distance measures in constructing phylogenetic trees have been studied mathematically or by computer simulations. Recently, however, allele frequencies of 783 STR loci have been compiled from various human populations. We have therefore used these empirical data to investigate the relative efficiencies of different distance measures in constructing phylogenetic trees. The results showed that (1) the probability of obtaining the correct branching pattern of a tree (PC) is generally highest for DA distance; (2) FST*, standard genetic distance (DS), and FST/(1-FST) give similar PC-values, FST* being slightly better than the other two; and (3) (deltamu)2 shows PC-values much lower than the other distance measures. To have reasonably high PC-values for trees similar to ours, at least 30 loci with a minimum of 15 individuals are required when DA distance is used.

It has been known that the conservation or diversity of homeobox genes is responsible for the similarity and variability of some of the morphological or physiological characters among different organisms. To gain some insights into the evolutionary pattern of homeobox genes in bilateral animals, we studied the change of the numbers of these genes during the evolution of bilateral animals. We analyzed 2,031 homeodomain sequences compiled from 11 species of bilateral animals ranging from Caenorhabditis elegans to humans. Our phylogenetic analysis using a modified reconciled-tree method suggested that there were at least about 88 homeobox genes in the common ancestor of bilateral animals. About 50-60 genes of them have left at least one descendant gene in each of the 11 species studied, suggesting that about 30-40 genes were lost in a lineage-specific manner. Although similar numbers of ancestral genes have survived in each species, vertebrate lineages gained many more genes by duplication than invertebrate lineages, resulting in more than 200 homeobox genes in vertebrates and about 100 in invertebrates. After these gene duplications, a substantial number of old duplicate genes have also been lost in each lineage. Because many old duplicate genes were lost, it is likely that lost genes had already been differentiated from other groups of genes at the time of gene loss. We conclude that both gain and loss of homeobox genes were important for the evolutionary change of phenotypic characters in bilateral animals.

Murphy and colleagues reported that the mammalian phylogeny was resolved by Bayesian phylogenetics. However, the DNA sequences they used had many alignment gaps and undetermined nucleotide sites. We therefore reanalyzed their data by minimizing unshared nucleotide sites and retaining as many species as possible (13 species). In constructing phylogenetic trees, we used the Bayesian, maximum likelihood (ML), maximum parsimony (MP), and neighbor-joining (NJ) methods with different substitution models. These trees were constructed by using both protein and DNA sequences. The results showed that the posterior probabilities for Bayesian trees were generally much higher than the bootstrap values for ML, MP, and NJ trees. Two different Bayesian topologies for the same set of species were sometimes supported by high posterior probabilities, implying that two different topologies can be judged to be correct by Bayesian phylogenetics. This suggests that the posterior probability in Bayesian analysis can be excessively high as an indication of statistical confidence and therefore Murphy et al.'s tree, which largely depends on Bayesian posterior probability, may not be correct.

Recently many investigators have used microsatellite DNA loci for studying the evolutionary relationships of closely related populations or species, and some authors proposed new genetic distance measures for this purpose. However, the efficiencies of these distance measures in obtaining the correct tree topology remains unclear. We therefore investigated the probability of obtaining the correct topology (PC) for these new distances as well as traditional distance measures by using computer simulation. We used both the infinite-allele model (IAM) and the stepwise mutation model (SMM), which seem to be appropriate for classical markers and microsatellite loci, respectively. The results show that in both the IAM and SMM CAVALLI-SFORZA and EDWARDS' chord distance (DC) and NEI et al.'s DA distance generally show higher PC values than other distance measures, whether the bottleneck effect exists or not. For estimating evolutionary times, however, NEI's standard distance and GOLDSTEIN et al.'s (delta mu)2 are more appropriate than other distances. Microsatellite DNA seems to be very useful for clarifying the evolutionary relationships of closely related populations.

The relative efficiencies of different protein-coding genes of the mitochondrial genome and different tree-building methods in recovering a known vertebrate phylogeny (two whale species, cow, rat, mouse, opossum, chicken, frog, and three bony fish species) was evaluated. The tree-building methods examined were the neighbor joining (NJ), minimum evolution (ME), maximum parsimony (MP), and maximum likelihood (ML), and both nucleotide sequences and deduced amino acid sequences were analyzed. Generally speaking, amino acid sequences were better than nucleotide sequences in obtaining the true tree (topology) or trees close to the true tree. However, when only first and second codon positions data were used, nucleotide sequences produced reasonably good trees. Among the 13 genes examined, Nd5 produced the true tree in all tree-building methods or algorithms for both amino acid and nucleotide sequence data. Genes Cytb and Nd4 also produced the correct tree in most tree-building algorithms when amino acid sequence data were used. By contrast, Co2, Nd1, and Nd41 showed a poor performance. In general, large genes produced better results, and when the entire set of genes was used, all tree-building methods generated the true tree. In each tree-building method, several distance measures or algorithms were used, but all these distance measures or algorithms produced essentially the same results. The ME method, in which many different topologies are examined, was no better than the NJ method, which generates a single final tree. Similarly, an ML method, in which many topologies are examined, was no better than the ML star decomposition algorithm that generates a single final tree. In ML the best substitution model chosen by using the Akaike information criterion produced no better results than simpler substitution models. These results question the utility of the currently used optimization principles in phylogenetic construction. Relatively simple methods such as the NJ and ML star decomposition algorithms seem to produce as good results as those obtained by more sophisticated methods. The efficiencies of the NJ, ME, MP, and ML methods in obtaining the correct tree were nearly the same when amino acid sequence data were used. The most important factor in constructing reliable phylogenetic trees seems to be the number of amino acids or nucleotides used.

A simple statistical test for comparing three alternative phylogenetic hypotheses for four monophyletic groups is presented. This test is based on the minimum-evolution principle, and it does not require any information regarding the branching order within each monophyletic group. It is computationally efficient and can be easily extended to five or more monophyletic groups.

Using linear invariants for various models of nucleotide substitution, we developed test statistics for examining the applicability of a specific model to a given dataset in phylogenetic inference. The models examined are those developed by Jukes and Cantor (1969), Kimura (1980), Tajima and Nei (1984), Hasegawa et al. (1985), Tamura (1992), Tamura and Nei (1993), and a new model called the eight-parameter model. The first six models are special cases of the last model. The test statistics developed are independent of evolutionary time and phylogeny, although the variances of the statistics contain phylogenetic information. Therefore, these statistics can be used before a phylogenetic tree is estimated. Our objective is to find the simplest model that is applicable to a given dataset, keeping in mind that a simple model usually gives an estimate of evolutionary distance (number of nucleotide substitutions per site) with a smaller variance than a complicated model when the simple model is correct. We have also developed a statistical test of the homogeneity of nucleotide frequencies of a sample of several sequences that takes into account possible phylogenetic correlations. This test is used to examine the stationarity in time of the base frequencies in the sample. For Hasegawa et al.'s and the eight-parameter models, analytical formulas for estimating evolutionary distances are presented. Application of the above tests to several sets of real data has shown that the assumption of stationarity of base composition is usually acceptable when the sequences studied are closely related but otherwise it is rejected. Similarly, the simple models of nucleotide substitution are almost always rejected when actual genes are distantly related and/or the total number of nucleotides examined is large.

The inconsistency of the maximum parsimony method is known to occur even when the rate of nucleotide substitution is constant. To understand why this inconsistency occurs, a mathematical study was conducted for the cases of five, six, and seven sequences. The results obtained indicate that this inconsistency occurs because the probability of occurrence of nucleotide configurations generated by one substitution on a short interior branch is often lower than that of configurations generated by more substitutions on other longer branches. The chance of occurrence of this event--or, the inconsistency of the maximum parsimony method--apparently increases as the number of sequences increases. The inconsistency may occur even when the extent of sequence divergence is relatively small.

Nei and Gojobori (1986) developed a simple method to estimate the numbers of synonymous (ds) and nonsynonymous (dN) substitutions per site. In the present paper, we have developed a method for computing variances and covariances of ds's and dN's and of the proportions of synonymous (ps) and nonsynonymous (pN) differences. We also have developed a method for computing the variances of mean dS, dN, pS, pN, without constructing a phylogenetic tree of the genes. We have conducted computer simulations based on simple evolutionary models and have shown that the new method gives good estimates of variances and covariances.

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.

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)

The minimum-evolution (ME) method of phylogenetic inference is based on the assumption that the tree with the smallest sum of branch length estimates is most likely to be the true one. In the past this assumption has been used without mathematical proof. Here we present the theoretical basis of this method by showing that the expectation of the sum of branch length estimates for the true tree is smallest among all possible trees, provided that the evolutionary distances used are statistically unbiased and that the branch lengths are estimated by the ordinary least-squares method. We also present simple mathematical formulas for computing branch length estimates and their standard errors for any unrooted bifurcating tree, with the least-squares approach. As a numerical example, we have analyzed mtDNA sequence data obtained by Vigilant et al. and have found the ME tree for 95 human and 1 chimpanzee (outgroup) sequences. The tree was somewhat different from the neighbor-joining tree constructed by Tamura and Nei, but there was no statistically significant difference between them.

A formula for the effective population size for the finite island model of subdivided populations is derived. The formula indicates that the effective size can be substantially greater than the actual number of individuals in the entire population when the migration rate among subpopulations is small. It is shown that the mean nucleotide diversity, coalescence time, and heterozygosity for genes sampled from the entire population can be predicted fairly well from the theory for randomly mating populations if the effective population size for the finite island model is used.

Lake's evolutionary parsimony (EP) method of constructing a phylogenetic tree is primarily applied to four DNA sequences. In this method, three quantities--X, Y, and Z--that correspond to three possible unrooted trees are computed, and an invariance property of these quantities is used for choosing the best tree. However, Lake's method depends on a number of unrealistic assumptions. We therefore examined the theoretical basis of his method and reached the following conclusions: (1) When the rates of two transversional changes from a nucleotide are unequal, his invariance property breaks down. (2) Even if the rates of two transversional changes are equal, the invariance property requires some additional conditions. (3) When Kimura's two-parameter model of nucleotide substitution applies and the rate of nucleotide substitution varies greatly with branch, the EP method is generally better than the standard maximum-parsimony (MP) method in recovering the correct tree but is inferior to the neighbor-joining (NJ) and a few other distance matrix methods. (4) When the rate of nucleotide substitution is the same or nearly the same for all branches, the EP method is inferior to the MP method even if the proportion of transitional changes is high. (5) When Lake's assumptions fail, his chi2 test may identify an erroneous tree as the correct tree. This happens because the test is not for comparing different trees. (6) As long as a proper distance measure is used, the NJ method is better than the EP and MP methods whether there is a transition/transversion bias or whether there is variation in substitution rate among different nucleotide sites.

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.

A new method called the neighbor-joining method is proposed for reconstructing phylogenetic trees from evolutionary distance data. The principle of this method is to find pairs of operational taxonomic units (OTUs [= neighbors]) that minimize the total branch length at each stage of clustering of OTUs starting with a starlike tree. The branch lengths as well as the topology of a parsimonious tree can quickly be obtained by using this method. Using computer simulation, we studied the efficiency of this method in obtaining the correct unrooted tree in comparison with that of five other tree-making methods: the unweighted pair group method of analysis, Farris's method, Sattath and Tversky's method, Li's method, and Tateno et al.'s modified Farris method. The new, neighbor-joining method and Sattath and Tversky's method are shown to be generally better than the other methods.

The genetic differentiation of populations is usually studied by using the equilibrium theory of Wright's infinite island model. In practice, however, populations are not always in equilibrium, and the number of subpopulations is often very small. To get some insight into the dynamics of genetic differentiation of these populations, numerical computations are conducted about the expected gene diversities within and between subpopulations by using the finite island model. It is shown that the equilibrium values of gene diversities (HS and HT) and the coefficient of genetic differentiation (GST) depend on the pattern of population subdivision as well as on migration and that the GST value is always smaller than that for the infinite island model. When the number of migrants per subpopulation per generation is greater than 1, the equilibrium values of HS and HT are close to those for panmictic populations, as noted by previous authors. However, the values of HS, HT, and GST in transient populations depend on the pattern of population subdivision, and it may take a long time for them to reach the 95 per cent range of the equilibrium values. The implications of the results obtained for the conservation of genetic variability in small populations are discussed. It is argued that any single principle should not be imposed as a general guideline for the management of small populations.

A mathematical theory is developed for computing the probability that m genes sampled from one population (species) and n genes sampled from another are derived from l genes that existed at the time of population splitting. The expected time of divergence between the two most closely related genes sampled from two different populations and the time of divergence (coalescence) of all genes sampled are studied by using this theory. It is shown that the time of divergence between the two most closely related genes can be used as an approximate estimate of the time of population splitting (T) only when T identical to t/(2N) is small, where t and N are the number of generations and the effective population size, respectively. The variance of Nei and Li's estimate (d) of the number of net nucleotide differences between two populations is also studied. It is shown that the standard error (Sd) of d is larger than the mean when T is small (T much less than 1). In this case, Sd is reduced considerably by increasing sample size. When T is large (T greater than 1), however, a large proportion of the variance of d is caused by stochastic factors, and increase in the sample size does not help to reduce Sd. To reduce the stochastic variance of d, one must use data from many independent unlinked gene loci.

Mathematical models are presented for the evolution of postmating and premating reproductive isolation. In the case of postmating isolation it is assumed that hybrid sterility or inviability is caused by incompatibility of alleles at one or two loci, and evolution of reproductive isolation occurs by random fixation of different incompatibility alleles in different populations. Mutations are assumed to occur following either the stepwise mutation model or the infinite-allele model. Computer simulations by using Itô's stochastic differential equations have shown that in the model used the reproductive isolation mechanism evolves faster in small populations than in large populations when the mutation rate remains the same. In populations of a given size it evolves faster when the number of loci involved is large than when this is small. In general, however, evolution of isolation mechanisms is a very slow process, and it would take thousands to millions of generations if the mutation rate is of the order of 10(-5) per generation. Since gene substitution occurs as a stochastic process, the time required for the establishment of reproductive isolation has a large variance. Although the average time of evolution of isolation mechanisms is very long, substitution of incompatibility genes in a population occurs rather quickly once it starts. The intrapopulational fertility or viability is always very high. In the model of premating isolation it is assumed that mating preference or compatibility is determined by male- and female-limited characters, each of which is controlled by a single locus with multiple alleles, and mating occurs only when the male and female characters are compatible with each other. Computer simulations have shown that the dynamics of evolution of premating isolation mechanism is very similar to that of postmating isolation mechanism, and the mean and variance of the time required for establishment of premating isolation are very large. Theoretical predictions obtained from the present study about the speed of evolution of reproductive isolation are consistent with empirical data available from vertebrate organisms.

The extent and pattern of protein and DNA polymorphisms are discussed with emphasis on the mechanism of maintenance of the polymorphisms. Statistical studies suggest that a large proportion of genetic variability at the molecular level is maintained by a mutation-drift balance. At some loci, such as those for histocompatibility in mammals, however, a form of overdominant selection seems to be involved. In the presence of overdominant selection, polymorphic alleles may be maintained for tens of millions of years, so that the number of nucleotide differences between alleles is often very large, as in the case of self-incompatibility alleles in plants. There are also an increasing number of examples in which an adaptive change of a morphological or physiological character is caused by a single nucleotide substitution. Nevertheless, these mutations seem to be a small proportion of the total nucleotide changes that contribute to genetic variability and evolution. Although there are many examples of frequency-dependent selection, this form of selection is apparently unimportant for the maintenance of genetic variability except in some special cases. Observations on the evolutionary change of DNA suggest that the driving force of evolution is mutation rather than selection.

The accuracies and efficiencies of three different methods of making phylogenetic trees from gene frequency data were examined by using computer simulation. The methods examined are UPGMA, Farris' (1972) method, and Tateno et al.'s (1982) modified Farris method. In the computer simulation eight species (or populations) were assumed to evolve according to a given model tree, and the evolutionary changes of allele frequencies were followed by using the infinite-allele model. At the end of the simulated evolution five genetic distance measures (Nei's standard and minimum distances, Rogers' distance, Cavalli-Sforza's f theta, and the modified Cavalli-Sforza distance) were computed for all pairs of species, and the distance matrix obtained for each distance measure was used for reconstructing a phylogenetic tree. The phylogenetic tree obtained was then compared with the model tree. The results obtained indicate that in all tree-making methods examined the accuracies of both the topology and branch lengths of a reconstructed tree (rooted tree) are very low when the number of loci used is less than 20 but gradually increase with increasing number of loci. When the expected number of gene substitutions (M) for the shortest branch is 0.1 or more per locus and 30 or more loci are used, the topological error as measured by the distortion index (dT) is not great, but the probability of obtaining the correct topology (P) is less than 0.5 even with 60 loci. When M is as small as 0.004, P is substantially lower. In obtaining a good topology (small dT and high P) UPGMA and the modified Farris method generally show a better performance than the Farris method. The poor performance of the Farris method is observed even when Rogers' distance which obeys the triangle inequality is used. The main reason for this seems to be that the Farris method often gives overestimates of branch lengths. For estimating the expected branch lengths of the true tree UPGMA shows the best performance. For this purpose Nei's standard distance gives a better result than the others because of its linear relationship with the number of gene substitutions. Rogers' or Cavalli-Sforza's distance gives a phylogenetic tree in which the parts near the root are condensed and the other parts are elongated. It is recommended that more than 30 loci, including both polymorphic and monomorphic loci, be used for making phylogenetic trees. The conclusions from this study seem to apply also to data on nucleotide differences obtained by the restriction enzyme techniques.

Statistical methods for computing the variances of nucleotide diversity within populations and of nucleotide divergence between populations are developed. Both variances are computed by finding the phylogenetic relationships of the DNA sequences studied through the unweighted pair-group method or some other tree-making method. The methods developed are applicable to both DNA sequence and restriction-site map data.

Considering the multinomial sampling of genotypes, unbiased estimators of various gene diversity measures in subdivided populations are presented. Using these quantities, formulae for estimating Wright's fixation indices (FIS, FIT, and FST) from a finite sample are developed.