I develop a mathematical model which can account for a distribution of the number of repeated genes per chromosome under the joint effects of sister chromatid exchange (SCE), inter-chromosomal crossing-over (ICC), and selection. The model can be applied not only to the cases of small gene clusters but also to multigene families. Based on this model, an appropriate mathematical formula is derived and used to obtain the equilibrium distribution. Assuming stabilizing selection and two simple schemes concerning SCE and ICC, I numerically calculate the equilibrium distribution and compare the result with observations on frequencies of single and triple α-haemoglobin genes in primates. It is also shown that if SCE and ICC occur according to the same probabilistic law, the distinction between them does not make much sense in the equilibrium distribution.

A single locus model of extranuclear genomes is developed under the assumption of the complete action of within-generation drift which is caused by random transmission of multiple copy genomes during cell division in a generation. Within-generation drift segregates different copy genomes in a cell into different cells, resulting in homoplasmic cells. Under some conditions, the present model reduces to that for haploid nuclear genomes. A point overlooked in previous models is that the multiplicity also admits of the possibility of selection occurring within a cell or between cells in an individual (within-generation selection). If there is selection mediated by, for instance, differential proliferation of genomes, then a haploid model no longer explains the dynamics of extranuclear genomes. Rather a model analogous to biased gene conversion at a single locus (Nagylaki, 1983; Walsh, 1983) is more appropriate. An application of this model to either the fixation probability or substitution rate of new mutations shows that strictly maternal inheritance does not allow the fullest use of mutations, as it obscures the effect of within-generation selection. But if there is appreciable paternal contribution, within-generation selection could be a strong evolutionary force to which nuclear genomes are never exposed.

Population genetics of extranuclear genomes is further developed under the neutral-mutation random-drift hypothesis, and the characteristic evolutionary aspects are summarized. Several formulae derived here are concerned with the variances of genetic variability (gene identity) at a single extranuclear locus and the evolutionary distance between two isolated populations which is estimated from a comparison of homologous linked nucleotide sites. Two types of variance are considered; one is the variance in the entire population (VQ) and the other is the variance within a single germ cell (VH). When compared with a Mendelian genetic system in a panmictic population, an extranuclear genetic system has the following equilibrium properties: (1) the mean genetic variability is low if, despite the high multiplicity of the genome in a cell, the proportion of the cytoplasmic contribution from the male's gamete is small, (2) the effect of recombination is small and a large amount of variance of linkage disequilibrium tends to be maintained, (3) the overall relationship between the mean and variance of genetic variability does not much differ but VQ (VH) is expected to be small if the paternal contribution is small, and (4) the evolutionary distance estimated depends on the extent of intrapopulational variation in a common ancestor population which in turn depends on within-cell variation. I argue that there is an analogy between the model of extranuclear genomes in a finite population and that of nuclear genes in a subdivided population. The analogy helps our understanding of some properties in an extranuclear genetic system.

A general model of linked genes or a part of a genome is proposed which enables us to study various problems in molecular population genetics in a unified way. Several formulae with special reference to the linkage disequilibrium and genetic distance are derived for neutral mutations in finite populations, based on the method of diffusion equations. It is argued that the model and formulae are useful particularly when observations are made in terms of DNA sequence.

When DNA sequence data on various kinds of homologous genes sampled from two related species are available, there is a way to infer the effective size of their ancestral species, which is a simple consequence of gene genealogical considerations. This method, when applied to the common ancestral species of human and rat, human and mouse, human and bovine, or rodents and bovine estimates their effective sizes all to be of the order of 107, supporting the view that these species indeed shared, around 75 million years ago, a common ancestral species from which they are descended. The effective size thus estimated would imply that the ancestral species was abundant enough to have ample opportunity for adaptive radiation. The extent of silent polymorphism in that species might have been very large, possibly comparable to the number of silent substitutions accumulated in a gene after the mammalian divergence. Some causes that may alter these results and require a more elaborated statistical analysis are discussed.

A mathematical theory of population genetics accounting for the genes transmitted through mitochondria or chloroplasts has been studied. In the model it is assumed that a population consists of Nm males and Nf females, the genetic contribution from a male is β and that from a female 1 – β, and each cell line contains n effective copies of a gene in its cytoplasm. Assuming selective neutrality and an infinite alleles model, it is shown that the sum (H) of squares of allelic frequencies within an individual and the corresponding sum (Q) for the entire population are, at equilibrium, given by

and

where ρ = 2β(1−β), Ne = {β2/Nm+(1−β)2 / Nf}−1, λ is the number of somatic cell divisions in one generation, and v is the mutation rate per cell division. If the genes are transmitted entirely through the female the formulae reduce to Ĥ ≃ 1/(1 + 2nv) and Q^ ≃ 1/{1 + (2Neλ+ 2n)υ}. Non-equilibrium behaviours of H and Q^ are also studied in the case of a panmictic population. These results are extended to geographically structured models, and applied to existing experimental data.

The n0 coalescent of Kingman (1982a, b) describes the family relationships among a sample of n0 individuals drawn from a panmictic species. It is a stochastic process resulting from n0 − 1 independent random events (coalescences) at each of which n (2 ≤ n ≤ n0) ancestral lineages of a sample are descended from n − 1 distinct ancestors for the first time. Here a similar genealogical process is studied for a species consisting of two populations with migration between them. The main interest is with the probability density of the time length between two successive coalescences and the spatial distribution of n − 1 ancestral lineages over two populations when n to n − 1 coalescence takes place. These are formulated based on a non-linear birth and death process with killing, and are used to derive several explicit formulae in selectively neutral population genetics models. To confirm and supplement the analytical results, a simulation method is proposed based on the underlying bivariate Markov chain. This method provides a general way for solving the present problem even when an analytical approach appears very difficult. It becomes clear that the effects of the present population structure are most conspicuous on 2 to 1 coalescence, with lesser extents on n to n − 1 (3 ≤ n) coalescence.This implies that in a more general model of population structure, the number of populations and the way in which a sample is drawn are important factors which determine the n0 coalescent.

DNA sequence comparison among homologous genes sampled at random from one or two populations allows one to estimate the ultimate amount of genetic variation maintained in a population and to construct the gene genealogy within and between populations. Moreover, if we use the finding of the molecular clock (Zuckerkandl & Pauling, 1965), it is also possible to estimate the divergence time of populations examined. Such an estimated divergence time is, however, intricately affected by samples and stochastic forces occurring in the course of evolution.

Within-line heterogeneity has been found in the mitochondrial DNA (mtDNA) in two isofemale lines of D. simulans. The co-existing types, S and M, were typical of the mtDNA in D. simulans and in D. mauritiana, respectively, their nucleotide divergence per site being ca. 2·1%. Segregation analysis confirmed that some individuals in these lines were heteroplasmic and suggested incomplete maternal inheritance of mtDNA in Drosophila. Examination of other lines of D. simulans revealed that the M type of D. mauritiana occurs at 71% in Réunion, 38% in Madagascar and 0% in Kenya. This finding and interspecific sequence comparisons of both M types indicate that D. mauritiana diverged from D. simulans probably less than 240000 years ago.

A ‘two locus two allele’ model is developed with special reference to the introgression of extranuclear genomes between two species of finite size. The model assumes that one locus, coded by a nuclear genome, causes the reproductive barrier while the other locus, coded by an extranuclear genome, causes nucleo-cytoplasmic incompatibility in particular genotypes. To fully study this model, simulations are conducted, and a diffusion equation is derived when introgression or extranuclear gene flow occurs in one direction. It is shown that although selection against the nuclear genome can reduce the levels of extranuclear gene flow and retard the introgression process, the dynamics are very similar to those without such selection. In contrast, the nucleo-cytoplasmic incompatibility directly affects the dynamics of introgressing extranuclear genomes: in large populations the ability of incompatibility to overcome extranuclear gene flow is conspicuous, but in small populations it is overshadowed by random sampling drift. Paternal leakage of extranuclear genomes, if present, is of evolutionary importance only when the male's migration rate is much larger than the female's. When the sizes of two populations are unequal, the introgression is most likely to occur from the larger population to the smaller one in the absence of mating preferences of backcross progeny. Recent observations on interspecific mitochondrial transfer in various species do not support the ubiquitousness of nucleo-cytoplasmic incompatibility as an efficient reproductive barrier.