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     

Molecular evolutionary rates of oncogenes

Summary Using nine sets of viral and cellular oncogenes, the rates of nucleotide substitutions were computed by using Gojobori and Yokoyama's (1985) method. The results obtained confirmed our previous conclusion that the rates of nucleotide substitution for the viral oncogenes are about a million times higher than those for their cellular counterparts. For cellular oncogenes and most viral oncogenes, however, the rate of synonymous substitution is higher than that of nonsynonymous substitution. Moreover, the pattern of nucleotide substitutions for viral oncogenes is more similar to that for functional genes (such as cellular oncongenes) than for pseudogenes. This implies that nucleotide substitutions in viral oncogenes may be functionally constrained. Thus, our observation supports that nucleotide substitutions for the oncogenes in those DNA and RNA genomes are consistent with Kimura's neutral theory of molecular evolution (Kimura 1968, 1983).     

A simple method for estimating evolutionary rates of base substitutions through comparative studies of nucleotide sequences

Summary Some simple formulae were obtained which enable us to estimate evolutionary distances in terms of the number of nucleotide substitutions (and, also, the evolutionary rates when the divergence times are known). In comparing a pair of nucleotide sequences, we distinguish two types of differences; if homologous sites are occupied by different nucleotide bases but both are purines or both pyrimidines, the difference is called type I (or transition type), while, if one of the two is a purine and the other is a pyrimidine, the difference is called type II (or transversion type). Letting P and Q be respectively the fractions of nucleotide sites showing type I and type II differences between two sequences compared, then the evolutionary distance per site is K = — (1/2) ln {(1 — 2P — Q) }. The evolutionary rate per year is then given by k = K/(2T), where T is the time since the divergence of the two sequences. If only the third codon positions are compared, the synonymous component of the evolutionary base substitutions per site is estimated by K'_S = — (1/2) ln (1 — 2P — Q). Also, formulae for standard errors were obtained. Some examples were worked out using reported globin sequences to show that synonymous substitutions occur at much higher rates than amino acid-altering substitutions in evolution.Contribution No. 1330 from the National Institute of Genetics, Mishima, 411 Japan     

Estimation of evolutionary distances from protein spatial structures

New equations are derived to estimate the number of amino acid substitutions per site between two homologous proteins from the root mean square (RMS) deviation between two spatial structures and from the fraction of identical residues between two sequences. The equations are based on evolutionary models, analyzing predominantly structural changes and not sequence changes. Evolution of spatial structure is treated as a diffusion in an elastic force field. Diffusion accounts for structural changes caused by amino acid substitutions, and elastic force reflects selection, which preserves protein fold. Obtained equations are supported by analysis of protein spatial structures. Received: 21 September 1995 / Accepted: 19 May 1997     

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.      

Assessing Substitution Variation Across Sites in Grass Chloroplast DNA

We assess the similarity of base substitution processes, described by empirically derived 4 × 4 matrices, using chi-square homogeneity tests. Such significance analyses allow us to assess variation in sequence evolution across sites and we apply them to matrices derived from noncoding sites in different contexts in grass chloroplast DNA. We show that there is statistically significant variation in rates and patterns of mutation among noncoding sites in different contexts and then demonstrate a similar and significant influence of context on substitutions at fourfold degenerate sites of coding regions from grass chloroplast DNA. These results show that context has the same general effect on substitution bias in coding and noncoding DNA: the A+T content of flanking bases is correlated with rate of substitution, transition bias, and GC → AT pressure, while the number of flanking pyrimidines on a single strand is correlated with a mutational bias, or skew, toward pyrimidines. Despite the similarity in general trends, however, when we compare coding and noncoding matrices we find that there is a statistically significant difference between them even when we control for context. Most noticeably, fourfold degenerate sites in coding sequences are undergoing substitution at a higher rate and there are also significant differences in the relationship between pyrimidines skew and the number of flanking pyrimidines. Possible reasons for the differences between coding and noncoding sites are discussed. Furthermore, our analysis illustrates a simple statistical way for comparing substitution processes across sites allowing us to better study variation in evolutionary processes across a genome. [Reviewing Editor: Dr. Martin Kreitman]     

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.     

Determining evolutionary distances from highly diverged nucleic acid sequences: Operator metrics

Summary Operator metrics are explicity designed to measure evolutionary distances from nucleic acid sequences when substitution rates differ greatly among the organisms being compared, or when substitutions have been extensive. Unlike lengths calculated by the distance matrix and parsimony methods, in which substitutions in one branch of a tree can alter the measured length of another branch, lengths determined by operator metrics are not affected by substitutions outside the branch.In the method, lengths (operator metrics) corresponding to each of the branches of an unrooted tree are calculated. The metric length of a branch reconstructs the number of (transversion) differences between sequences at a tip and a node (or between nodes) of a tree. The theory is general and is fundamentally independent of differences in substitution rates among the organisms being compared. Mathematically, the independence has been obtained becuase the metrics are eigen vectors of fundamental equations which describe the evolution of all unrooted trees.Even under conditions when both the distance matrix method or a simple parsimony length method are show to indicate lengths than are an order of magnitude too large or too small, the operator metrics are accurate. Examples, using data calculated with evolutionary rates and branchings designed to confuse the measurement of branch lengths and to camouflage the topology of the true tree, demonstrate the validity of operator metrics. The method is robust. Operator metric distances are easy to calculated, can be extended to any number of taxa, and provide a statistical estimate of their variances.The utility of the method is demonstrated by using it to analyze the origins and evolutionary of chloroplasts, mitochondria, and eubacteria.     

Reconstruction of ancestral nucleotide sequences and estimation of substitution frequencies in a star phylogeny

Maximum likelihood phylogeny reconstruction methods are widely used in uncovering and assessing the evolutionary history and relationships of natural systems. However, several simplifying assumptions commonly made in this analysis limit the explanatory power of the results obtained. We present an algorithm that performs the phylogenetic analysis without making the common assumptions for sequence data from at least three leaf nodes in a star phylogeny. In particular, the underlying nucleotide substitution model does not have to be reversible and may include neighbor-dependent processes like the CpG methylation deamination process (CpG-effect). The base composition of the sequences at the external nodes and the one of the ancestral sequence may be different from each other and they do not have to be stationary state distributions of the corresponding substitution model. The algorithm is able to reconstruct the ancestral base composition and accurately estimate substitution frequencies in the branches of the star phylogeny. Extensive tests on simulated data validate the very favorable performance of the algorithm. As an application we present the analysis of aligned genomic sequences from human, mouse, and dog. Different substitution pattern can be observed in the three lineages.     

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.      

A method of estimating from two aligned present-day DNA sequences their ancestral composition and subsequent rates of substitution,possibly different in the two lineages,corrected for multiple and parallel substitutions at the same site

Summary The course of evolutionary change in DNA sequences has been modeled as a Markov process. The Markov process was represented by discrete time matrix methods. The parameters of the Markov transition matrices were estimated by least-squares direct-search optimization of the fit of the calculated divergence matrix to that observed for two aligned sequences. The Markov process corrected for multiple and parallel substitutions of bases at the same site. The method avoided the incorrect assumption of all previously described methods that the divergence between two present-day sequences is twice the divergence of either from the common and unknown ancestral sequence. The three previous methods were shown to be equivalent. The present method also avoided the undesirable assumptions that sequence composition has not changed with time and that the substitution rates in the two descendant lineages were the same. It permitted simultaneous estimation of ancestral sequence composition and, if applicable, of different substitution rates for the two descendant lineages, provided the total number of estimated parameters was less than 16. Properties of the Markov chain were discussed. It was proved for symmetric substitution matrices that all elements of the equilibrium divergence matrix equal 1/16, and that the total difference in the divergence matrix at epoch k equals the total change in the common substitution matrix at epoch 2k for all values of k. It was shown how to resolve an ambiguity in the assignment of two different substitution rates to the two descendant lineages when four or more similar sequences are available. The method was applied to the divergence matrix for codon site 3 for the mouse and rabbit beta-globins. This observed divergence matrix was significantly asymmetric and required at least two different substitution rates. This result could be achieved only by using different asymmetric substitution matrices for the two lineages.     

Determination and evolutionary significance of nucleotide sequences near to the 3''-end of 16S ribosomal RNA of mycobacteria

The nucleotide sequence at the 3'-end of 16Sr-RNA (nucleotides 1305-1508) was determined, by the primer extension method, for Mycobacterium smegmatis, Mycobacterium tuberculosis and Mycobacterium vaccae, in addition to Mycobacterium leprae. No differences in nucleotide sequence were detected, indicating that this region of 16SrRNA is highly conserved among mycobacteria. The nucleotide sequence common to the four above-mentioned mycobacteria differs from that reported for species of other genera. For example, for helix 39 (nucleotides 1408-1491) the mycobacterial sequence has 58% similarity with the Escherichia coli sequence, 74% similarity with the Bacillus subtilis sequence and 93% similarity with the Streptomyces sequence. The observations support the assignment of M. leprae to the genus Mycobacterium.     

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.     

Goodman et al.'s method for augmenting the number of nucleotide substitutions

Summary Statistical properties of Goodman et al.'s (1974) method of compensating for undetected nucleotide substitutions in evolution are investigated by using computer simulation. It is found that the method tends to overcompensate when the stochastic error of the number of nucleotide substitutions is large. Furthermore, the estimate of the number of nucleotide substitutions obtained by this method has a large variance. However, in order to see whether this method gives overcompensation when applied together with the maximum parsimony method, a much larger scale of simulation seems to be necessary.     

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.     

Simulation studies on the evolution of amino acid sequences

Summary A model of molecular evolution in which the parameter (intrinsic rate of amino acid substitution) fluctuates from time to time was investigated by simulating the process. It was found that the usual method of estimation such as Poisson fitting underestimates this variation of the parameter when remote comparisons are made. At the same time, four distance measures (minimum base difference, Poisson fitting, random nucleotide substitutions and negative binomial fitting) were tested for their accuracy. When the substitution rate is not uniform among the amino acid sites, the negative binomial fitting gives most satisfactory results, however, one needs to know the parameter beforehand in order to use this method. It was pointed out that the fluctuation of the evolutionary rate is expected if the nearly neutral but very slightly deleterious mutations play an important role on molecular evolution.Contribution No. 1087 from the National Institute of Genetics, Mishima, Shizuoka-ken, 411 Japan.     

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.     

Estimation of amino acid residue substitution rates at local spatial regions and application in protein function inference: a Bayesian Monte Carlo approach

The amino acid sequences of proteins provide rich information for inferring distant phylogenetic relationships and for predicting protein functions. Estimating the rate matrix of residue substitutions from amino acid sequences is also important because the rate matrix can be used to develop scoring matrices for sequence alignment. Here we use a continuous time Markov process to model the substitution rates of residues and develop a Bayesian Markov chain Monte Carlo method for rate estimation. We validate our method using simulated artificial protein sequences. Because different local regions such as binding surfaces and the protein interior core experience different selection pressures due to functional or stability constraints, we use our method to estimate the substitution rates of local regions. Our results show that the substitution rates are very different for residues in the buried core and residues on the solvent-exposed surfaces. In addition, the rest of the proteins on the binding surfaces also have very different substitution rates from residues. Based on these findings, we further develop a method for protein function prediction by surface matching using scoring matrices derived from estimated substitution rates for residues located on the binding surfaces. We show with examples that our method is effective in identifying functionally related proteins that have overall low sequence identity, a task known to be very challenging.     

Modeling DNA Base Substitution in Large Genomic Regions from Two Organisms

We studied the substitution patterns in 7661 well-conserved human–mouse alignments corresponding to the intergenic regions of human chromosome 22. Alignments with a high average GC content tend to have a higher human GC content than mouse GC content, indicating a lack of stationarity. Segmenting the alignments into four groups of GC content and fitting the general reversible substitution model (REV) separately gave significantly better fits than the overall fit and the levels of fit are close to that expected under an REV model. In addition, most of the fitted rate matrices are not of the HKY type but are remarkably strand-symmetric, and we constructed a number of substitution matrices that should be useful for genomic DNA sequence alignment. We did not find obvious signs of temporal inhomogeneity in the substitution rates and concluded that the conserved intergenic regions in human chromosome 22 and mouse appear to have evolved from their common ancestors via a process that is approximately reversible and strand-symmetric, assuming site homogeneity and independence.