BOTTLENECK EFFECTS ON AVERAGE HETEROZYGOSITY AND GENETIC DISTANCE WITH THE STEPWISE MUTATION MODEL.

Abstract

When a population goes through a small bottleneck, the genetic variability of the population is expected to decline rapidly, but as soon as population size becomes large, it starts to increase owing to new mutations (Wright, 1931; Mayr, 1954). Recently, Nei et al. (1975) studied this problem mathematically and showed that the pattern of change in genetic variability largely depends on the size of bottleneck, rate of population growth, and mutation rate. They provided a general formula for computing the average heterozygosity per locus in a population of changing size. This study is based on the assumption that new mutations occurring in a population are always different from the preexisting ones (Kimura and Crow's (1964) infinite allele model). This assumption seems to be roughly correct if allelic differences are studied at the nucleotide or codon level. In practice, however, the genetic variability of natural populations is usually studied by electrophoretic mobility of proteins. Electrophoretic mobility of a protein is determined mainly by the net charge of the protein, and a positive or negative change in the net charge due to an amino acid substitution may be canceled by the second opposite charge change. Therefore, strictly speaking, the infinite allele model is not appropriate for the study of protein variation detected by electrophoresis. In view of this circumstance, Ohta and Kimura (1973) introduced the so-called stepwise mutation model. In this model each allele is represented as one of the infinite sequence of allelic states and mutation is assumed to produce a one-step change in either the positive or the negative direction. They derived a formula for average heterozygosity in an equilibrium population. Later, Wehrhahn (1975) and Li (1976) extended Ohta and Kimura's work to the case of nonequilibrium populations. On the other hand, Nei and Chakraborty (1973) studied the expected genetic distance between two populations when these are separated for an arbitrary number of generations, using a similar genetic model. The purpose of the present paper is to study the bottleneck effects on average heterozygosity and genetic distance by using the stepwise mutation model. Some studies on the bottleneck effect (the effect of reduction in population size) on genetic distance have been made by Chakraborty and Nei (1974) and Nei (1976), using the infinite allele model. These authors have shown that the reduction in population size results in an accelerated increase in genetic distance in the early generations. In the present paper we shall employ the method of generating function first used by Nei and Chakraborty (1973) in the study of protein variation and extended considerably by Wehrhahn ( 19 7 5).