Do Eve's alleles live on?

Consider a random sample of genes at a locus, drawn from a population evolving according to the infinitely many, neutral, alleles model. The sample will have a most recent common ancestor gene, which we shall call 'Eve'. The probability distribution, for the number of genes of oldest allelic type in a sample, is known and has a neat form. Rather less is known about the distribution for the number of genes in the sample which are of the same allelic type as Eve possessed. If the latter number is positive, then these genes are automatically of the oldest type in the sample. But Eve may have no non-mutant descendants in the sample; then, the oldest allele will be a mutant arising in a line of descent after Eve. The paper studies the number of non-mutant descendants from Eve, its distribution and moments. It seems that there may be few neat results. In large samples, the proportion of genes of Eve's type has an approximate beta-like density, together with a discrete probability atom at zero, if the mutation rate parameter is low. Extinction of the allele of even the population's common ancestor is possible, but not certain, and bounds are obtained for its probability. Some comments are made about the applications and implications of the results for human mitochondrial DNA.

A stochastic analysis of three viral sequences.

This paper analyzes the nucleotide sequences of three viruses: Kunjin, west Nile, and yellow fever. Each virus has one long open reading frame of greater than 10,200 nucleotides that codes for four structural and seven nonstructural genes. The Kunjin and west Nile viruses are the most closely related pair, when assessed on the basis of matches between their nucleotide sequences. As would be expected, the matching is least for bases at third-position codon sites and is greatest for second-position sites. Statistics are presented for the numbers of mismatches that are transitions or transversions. Nucleotide base usage is also reported. To each of the 33 virus-gene segments, nonhomogeneous Markov chain models have been fitted to describe the sequences of nucleotide bases. The models allow for different transition probabilities ("transition" is used in the mathematical sense here) and for different degrees of dependency, at the three sites in the codons. Reasonably satisfactory fits can be obtained for many of the genes by using models that are first order for both first- and second-position sites in the codon but that are second order for third-position sites. One consequence of such a model is that the correlation between one amino acid and the next is limited to the correlation of the last base of the former with the first base of the latter. Other consequences are that the model can (and does) prohibit the occurrence of stop codons within a gene and that subsequences of only first-position bases, or only third-position bases, are also first-order Markov chains. In theory, second-position subsequences may not be Markov chains at all. In practice, the data suggest that each of these subsequences is effectively a zero-order Markov chain, i.e., bases spaced three apart are statistically independent. Stationarity of nucleotide base distributions can be interpreted in either of two ways: (1) spatially along the sites or (2) temporally at each site. These interpretations must often be inconsistent, when the former allows for Markov dependence between adjacent sites whereas the latter assumes independence between sites. The inconsistency can be overcome, for these viruses, if subsequences at different codon positions are analyzed separately.

The number of alleles in multigene families.

The probability distribution and moments of the number of alleles present in a sample of homologous chromosomes are studied. It is assumed that there are multiple copies of the gene on each chromosome. When there are only two copies per chromosome or when there are only two or three chromosomes, it is possible to use analytic methods to tackle the problem. Otherwise, a simulation method is suggested.

Allele frequencies in multigene families. I. Diffusion equation approach.

Simulation studies by Ohta, and theoretical studies by Shimizu, indicated that the alleles in one chromosome multigene family follow Ewens' sampling distribution at stationarity. The reasons for this are explored, and the theory is extended to obtain transient and stationary results for the sampling distribution. Expressions are obtained for the amount of variation of numbers of alleles across the chromosomes in a population. The population model includes mutation, and gene conversion, but not recombination.

Allele frequencies in multigene families. II. Coalescent approach.

Shimizu's model for the evolution of multigene families, subject to gene conversion and mutation, is studied. The probability distribution for the allelic configuration of two chromosomes is found, by using "coalescent" methods. For the two chromosomes, moments are found for the number of alleles they have in common and for their total number of alleles, together with the probability that there is only one allele present. Some numerical examples are given.

The homozygosity test after a change in population size.

The homozygosity of a population will be influenced by any recent change in the population size. The "homozygosity test" of the neutral mutation hypothesis might also be influenced by a population change. A computer simulation method is described that establishes the significance levels of observed homozygosities after a change in population size. Some numerical examples are given.

On the time for gene silencing at duplicate Loci.

A simple approximate formula is found for the mean time for mutant genes to first fix at one or another of two duplicate loci. The fixation is a result of one-way mutation from the wild-type to the mutant alleles, but is retarded by strong selection against the doubly homozygous mutants. The two-dimensional diffusion that models the evolution of the mutant allele frequencies at the two loci is studied by means of a transformation to two univariate diffusions having different time scales. Most of the results, but not all, agree well with the numerical studies by previous authors. Some new simulation results are presented, which agree well with the analytical results and, therefore, cast doubt on some previous simulations.

Testing selection at a single locus.

In this paper some methods for testing for the presence of selection at a single locus in a single population are discussed. It is assumed that at that locus only two alleles are present, and that the allele frequencies are observed for a number of generations. Tests are derived for detecting heterozygote advantage, genic selection and more general selection. Some simulation studies illustrate the power, or lack of power, of the tests, and show the connection between on the tests and the theory of Watterson (1979, Advances in Applied Probability 11, 14-30).

The homozygosity test of neutrality.

An earlier paper showed that the homozygosity (of a population or sample) was a good statistic for testing departures from selective neutrality in the direction of heterozygote advantage or disadvantage. It is here shown that homozygosity is also influenced by the presence of deleterious alleles and by other departures from neutrality, but at a lower order of magnitude of effect if the selection coefficients are of the same small order of magnitude. Tables are provided for the significance points and moments of the homozygosity, under the null hypothesis of neutrality.

An analysis of multi-allelic data.

Some multi-allelic data obtained by Coyne (1976) and by Singh, Lewontin and Felton (1976) are analyzed for their compatibility with the neutral alleles theory. It is found that strict neutrality appears not to be the case, but that if further alleles were to be distinguished in the samples, neutrality could become a possible explanation.

Heterosis or neutrality?

Various statistics have been proposed on an ad hoc basis to test whether alleles at a locus are selectively neutral. By considering population models in which selection operates, this paper shows that the population homozygosity is a powerful test statistic for testing departures from neutrality, in the direction of heterozygote advantage or disadvantage. The sample homozygosity plays a similar role when only sample data are available. Some numerical examples are included, showing the application of the test.--An analysis is made of the effect of heterosis on such quantities as the expected number of alleles in the population or sample, the effective number of alleles, the expected homozygosity, and on the population and sample allele frequency distributions generally.