Relationship Between DNA Polymorphism and Fixation Time Fumio Tajima

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. D NA polymorphism and fixation of mutants at nucleotide sites are not independent phenomena, but they are mutually related. For example, the fixation of selectively advantageous allele at one locus can reduce the amount of polymorphism at linked locus (KOJIMA and SCHAEFFER 1967; MAYNARD SMITH and HAIGH 1974; OHTA and KIMURA 1975). Recently KAPLAN, HUDSON and LANGLEY (1989) have examined this hitchhiking effect at the DNA level, and concluded that in the region of low crossing over the fixation of selectively advantageous mutant at one nucleotide site can substantially reduce the number of segregating or polymorphic nucleotide sites in a sample of DNA sequences from that expected under the neutral mutation model. The effect of the fixation of a mutant on the amount of polymorphism may occur even when the mutant is selectively neutral. WATTERSON (1 982a,b) has shown under the neutral mutation model that fixations tend to occur in clusters rather than behave as a Poisson process. This suggests that there might be some effect of fixation on DNA polymorphism even if all the mutants are selectively neutral. The purpose of this paper is to examine quantitatively the relationship between the DNA polymorphism and the fixation of a mutant nucleotide. The amount of DNA polymorphism can be measured by the average number of (pairwise) nucleotide Genetics 125: 447-454 (June, 1990) differences among a sample of DNA sequences or by the number of segregating (or polymorphic) sites in a sample of DNA sequences [for their statistical properties under the neutral mutation model, see WATTERSON (1 975) and TAJIMA (1 983)]. In this paper we use the expected number of nucleotide differences between two DNA sequences randomly sampled from a population as a measure of the amount of DNA polymorphism. This number equals not only the expectation of the average number of nucleotide differences among a sample of DNA sequences but also the expected number of segregating sites in a sample of two DNA sequences. The fixation time is one of the most important quantities that characterize the fixation. In this paper we study the relationship between the amount of DNA polymorphism at the time when a mutant at a particular nucleotide site has fixed and the fixation time for this mutant.