Relative-Rate Test for Nucleotide Substitutions between Two Lineages

The molecular-clock hypothesis, i.e., the rate constancy of molecular evolution (Zuckerkandl and Pauling 1965), has been a cornerstone for several issues in molecular evolution, such as the neutral mutation hypothesis (Kimura 1983) and the estimation of phylogenies (Nei 1987, pp. 309-3 13 ) and divergence times (Li and Graur 199 1, pp. 117-l 18). For the purpose of assessing the constancy of molecular evolution, a relative-rate test was developed to determine whether protein replacements are the same in two taxa (Sarich and Wilson 1973), and this test has been extended to nucleotide substitutions ( Wu and Li 1985; Li and Tanimura 1987). In the latter test, the evolutionary distance between taxon 1 and a reference taxon is compared with that between taxon 2 and the reference taxon, so that only two taxa could be simultaneously compared. However, it is of interest in some cases to compare numbers of nucleotide substitutions between two lineages, with each lineage including several taxa. One solution is to perform a series of relative-rate tests between taxa in a pairwise fashion, with one taxon from each of the two lineages. These tests, however, are not independent, so that statistical inferences drawn are not clear. A more rational solution would be to construct a test by using all the sequences from the two lineages. In this note, we extend the relative-rate test of Wu and Li ( 1985) to cases where more than one taxon are sampled for one nucleotide sequence in two lineages compared. Suppose that a phylogenetic tree consists of two lineages, with lineage 1 having n taxa and with lineage 2 having m taxa (fig. 1). First, we present formulas for calculating both the weighted number of substitutions in each lineage as compared with a common reference taxon and the variance of the difference in the weighted number of nucleotide substitutions between the two lineages. Then we assume that the number of nucleotide substitutions follows a Poisson process, so that the standardized normal distribution could be used to test whether the numbers of nucleotide substitutions are the same in the two lineages ( Wu and Li 1985). We define the following terms: N, = the number of nucleotides compared between the ith sequence in lineage 1 and the reference sequence, i = 1, . . . , n. Nj = the number of nucleotides compared between the jth sequence in lineage 2 and the reference sequence, j = 1, . . . , m.