Estimation of the number of amino acid substitutions per site when the substitution rate varies among sites

Correspondence to: M. Nei 0871     

Estimation of the number of amino acid substitutions per site when the substitution rate varies among sites

A general model for estimating the number of amino acid substitutions per site (d) from the fraction of identical residues between two sequences (q) is proposed. The well-known Poisson-correction formula q = e ^–d corresponds to a site-independent and amino-acid-independent substitution rate. Equation q = (1 – e ^–2d )/2d, derived for the case of substitution rates that are site-independent, but vary among amino acids, approximates closely the empirical method, suggested by Dayhoff et al. (1978). Equation q = 1/(1 + d) describes the case of substitution rates that are amino acid-independent but vary among sites. Lastly, equation q = [ln(1 + 2d)]/2d accounts for the general case where substitution rates can differ for both amino acids and sites.     

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     

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

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

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.     

Inconsistency of the maximum parsimony method when the rate of nucleotide substitution is constant

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. Correspondence to: M. Nei     

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.