R. C. Griffiths

Ancestral inference from samples of DNA sequences with recombination.

The sampling distribution of a collection of DNA sequences is studied under a model where recombination can occur in the ancestry of the sequences. The infinitely-many-sites model of mutation is assumed where there may only be one mutation at a given site. Ancestral inference procedures are discussed for: estimating recombination and mutation rates; estimating the times to the most recent common ancestors along the sequences; estimating ages of mutations; and estimating the number of recombination events in the ancestry of the sample. Inferences are made conditional on the configuration of the pattern of mutations at sites in observed sample sequences. A computational algorithm based on a Markov chain simulation is developed, implemented, and illustrated with examples for these inference procedures. This algorithm is very computationally intensive.

Ancestral Asian Source(s) of New World Y-Chromosome Founder Haplotypes

Haplotypes constructed from Y-chromosome markers were used to trace the origins of Native Americans. Our sample consisted of 2,198 males from 60 global populations, including 19 Native American and 15 indigenous North Asian groups. A set of 12 biallelic polymorphisms gave rise to 14 unique Y-chromosome haplotypes that were unevenly distributed among the populations. Combining multiallelic variation at two Y-linked microsatellites (DYS19 and DXYS156Y) with the unique haplotypes results in a total of 95 combination haplotypes. Contra previous findings based on Y- chromosome data, our new results suggest the possibility of more than one Native American paternal founder haplotype. We postulate that, of the nine unique haplotypes found in Native Americans, haplotypes 1C and 1F are the best candidates for major New World founder haplotypes, whereas haplotypes 1B, 1I, and 1U may either be founder haplotypes and/or have arrived in the New World via recent admixture. Two of the other four haplotypes (YAP+ haplotypes 4 and 5) are probably present because of post-Columbian admixture, whereas haplotype 1G may have originated in the New World, and the Old World source of the final New World haplotype (1D) remains unresolved. The contrasting distribution patterns of the two major candidate founder haplotypes in Asia and the New World, as well as the results of a nested cladistic analysis, suggest the possibility of more than one paternal migration from the general region of Lake Baikal to the Americas.

The age of a mutation in a general coalescent tree

Kimura and Ohta showed that the expected age of a neutral mutation observed to be of frequency x in a population is We put this classical result in a general coalescent process context that allows questions to be asked about mutations in a sample, as well as in the population. In the general context the population size may vary back in time. Assuming an infmitely-many-sites model of mutation, we find the distribution of the number of mutant genes at a particular site in a sample; the probability that an allele at that site of a given frequency is ancestral; the distribution of the age of a mutation given its frequency in a sample, or population; and the distribution of the time to the most recent common ancestor, given the frequency of a mutation in a sample, or in the population

Lines of descent in the diffusion approximation of neutral Wright-Fisher models.

Abstract This paper studies lines of descent in the diffusion approximation of neutral Wright-Fisher models where the mutation rate away from each gene per generation is the same. Here a line of descent begins with a single gene and has branches at each generation where genes are reproduced from a parent in the line. New mutations are not included in a line of descent but are considered to begin a new line. The joint distribution of the number of lines of descent surviving in a population from time 0 to time t and the frequencies in these lines is derived. Expected times between loss of lines of descent are found. The distribution of the number of lines of descent in a sample from the population is derived. This leads to the distribution of the number of types in a sample from a nonstationary infinite alleles population.

Importance sampling on coalescent histories. I

Stephens and Donnelly (2000) constructed an efficient sequential importance-sampling proposal distribution on coalescent histories of a sample of genes for computing the likelihood of a type configuration of genes in the sample. In the current paper a characterization of their importance-sampling proposal distribution is given in terms of the diffusion-process generator describing the distribution of the population gene frequencies. This characterization leads to a new technique for constructing importance-sampling algorithms in a much more general framework when the distribution of population gene frequencies follows a diffusion process, by approximating the generator of the process.

Unrooted Genealogical Tree Probabilities in the Infinitely-Many-Sites Model

The infinitely-many-sites process is often used to model the sequence variability observed in samples of DNA sequences. Despite its popularity, the sampling theory of the process is rather poorly understood. We describe the tree structure underlying the model and show how this may be used to compute the probability of a sample of sequences. We show how to produce the unrooted genealogy from a set of sites in which the ancestral labeling is unknown and from this the corresponding rooted genealogies. We derive recursions for the probability of the configuration of sequences (equivalently, of trees) in both the rooted and unrooted cases. We give a computational method based on Monte Carlo recursion that provides approximates to sampling probabilities for samples of any size. Among several applications, this algorithm may be used to find maximum likelihood estimators of the substitution rate, both when the ancestral labeling of sites is known and when it is unknown.

Neutral two-locus multiple allele models with recombination☆

Abstract General formulae for the homozygosity and variance of linkage disequilibrium are derived for neutral, stationary, two-locus multiple allele models where there is a symmetric type of mutation at each locus. Particular cases examined are K allele models, the infinite alleles model, and the stepwise mutation model. The two-locus infinite allele model is examined at the molecular level and a joint probability generating function is found for the number of heterozygous sites at each locus in two randomly chosen gametes.

A transition density expansion for a multi-allele diffusion model

An expansion in orthogonal polynomials is found for the transition density in a neutral multi-allele diffusion model where the mutation rates of allele types Ai -A, are assumed to be u.(>i0). The density is found when the mutation rate is positive for all allele types, and when some or all have zero mutation. The asymptotic conditional density is found for a mixture of positive and zero mutation rates. The infinite alleles limit with equal mutation is studied. Eigenfunctions of the process are derived and the frequency spectrum found. An important result is that the first eigenfunction depends only on the homozygosity. A density for the time to fixation with zero mutation is found for the K allele, and infinite alleles model. ALLELE FREQUENCY; DIFFUSION; DIRICHLET DISTRIBUTION; ORTHOGONAL POLYNOMIALS; TRANSITION DENSITY

The frequency spectrum of a mutation, and its age, in a general diffusion model.

General formulae are derived for the probability density and expected age of a mutation of frequency x in a population, and similarly for a mutation with b copies in a sample of n genes. A general formula is derived for the frequency spectrum of a mutation in a sample. Variable population size models are included. Results are derived in two frameworks: diffusion process models for the frequency of the mutation; and birth and death process models. The coalescent structure within the mutant gene group and the non-mutant group is considered.

Characterization of infinitely divisible multivariate gamma distributions

A particular class of p-dimensional exponential distributions have Laplace transforms I + VT-1, V positive definite or positive semi-definite and T = diagonal (t1,..., tp). A characterization is given of when these Laplace transforms are infinitely divisible.