Probability of fixation of a mutant gene in a finite population when selective advantage decreases with time.

Abstract

ROBABILITY of gene fixation, that is to say, the probability by which a muPtant allele becomes eventually established in a population is a subject of considerable interest both in population and evolutionary genetics. In his pioneering work, HALDANE (1927) showed that in an infinite population an individual mutant gene having selective advantage s can reach fixation with the probability of about 2s. Later, more general results were obtained by KIMURA (1957) for finite populations based on diffusion models (see also KIMURA 1964). His formulae were used by ROBERTSON (1960) to develop a theory of limits in artificial selection. A still more general but quite simple formula for the probability of fixation was also obtained by &MUM (1962) as a function of the mean (Mao) and the variance (V,,) of the rate of change in gene frequency per generation. The formula is quite general, and as far as a single locus with a pair of alleles is concerned, it can cope with any degree of dominance and also random fluctuation in selection intensity. However, there are still restrictions in using the formula, the most serious of which is that the process of change in mutant gene frequency must be time homogeneous. In other words, the selection coefficients of mutant homoand heterozygotes have to remain constant with time. When we consider the fate of a new mutant (including chromosome mutant) in natural populations, there are numerous situations for which time nonhomogeneity has to be taken into account because of changing environment as well as alteration of genetic background with time. For example, consider the fate of a chromosome with a new inversion. If it happens to have a good combination of genes at the beginning, it will spread in the population reaching fixation or leading to inversion polymorphism. The fitness of the inverted chromosome segment in such a process can best be expressed by the exponential function of time, since deleterious mutants will accumulate in the inverted segment with time as shown by MUKAI ( 1964). This may also apply to a mutant gene that is itself neutral but happens to be included in a chromosome segment which does not contain deleterious genes. It is also possible that a mutant gene which is originally advantageous gradually loses its advantage due to deterioration of environment. The fixation probability in such a time nonhomogeneous process was investi-