We would like to use maximum likelihood to estimate parameters such as the effective population size Ne, or, if we do not know mutation rates, the product 4Neμof mutation rate per site and effective population size. To compute the likelihood for a sample of unrecombined nucleotide sequences taken from a random-mating population it is necessary to sum over all genealogies that could have led to the sequences, computing for each one the probability that it would have yielded the sequences, and weighting each one by its prior probability. The genealogies vary in tree topology and in branch lengths. Although the likelihood and the prior are straightforward to compute, the summation over all genealogies seems at first sight hopelessly difficult. This paper reports that it is possible to carry out a Monte Carlo integration to evaluate the likelihoods pproximately. The method uses bootstrap sampling of sites to create data sets for each of which a maximum likelihood tree is estimated. The resulting trees are assumed to be sampled from a distribution whose height is proportional to the likelihood surface for the full data. That it will be so is dependent on a theorem which is not proven, but seems likely to be true if the sequences are not short. One can use the resulting estimated likelihood curve to make a maximum likelihood estimate of the parameter of interest, Ne or of 4Neμ. The method requires at least 100 times the computational effort required for estimation of a phylogeny by maximum likelihood, but is practical on today's work stations. The method does not at present have any way of dealing with recombination.

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 β-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.

It has been suggested that isochores are maintained by mutation biases, and that this leads to variation in the rate of mutation across the genome. A model of DNA replication is presented in which the probabilities of misincorporation and proofreading are affected by the composition and concentration of the free nucleotide pools. The relationship between sequence G + C content and the mutation rate is investigated. It is found that there is very little variation in the mutation rate between sequences of different G + C contents if the total concentration of the free nucleotides remains constant. However, variation in the mutation rate can be arbitrarily large if some mismatches are proofread and the total concentration of free nucleotides varies. Hence the model suggests that the maintenance of isochores by the replication of DNA in free nucleotide pools of biased composition does not lead per se to mutation rate variance. However, it is possible that changes in composition could be accompanied by changes in concentration, thus generating mutation rate variance. Furthermore, there is the possibility that germ-line selection could lead to alterations in the overall free nucleotide concentration through the cell cycle. These findings are discussed with reference to the variance in mammalian silent substitution rates.