Maximum-likelihood estimation of coalescence times in genealogical trees

We develop a method for maximum-likelihood estimation of coalescence times in genealogical trees, based on population genetics data. For this purpose, a Viterbi-type algorithm is constructed to maximize the joint likelihood of the coalescence times. Marginal confidence intervals for the coalescence times based on the profile likelihoods are also computed. Our method of finding MLEs and calculating C.I.'s appears to be more accurate than alternative numerical maximization methods, and maximum-likelihood inference appears to be more accurate than other existing model-free approaches to estimating coalescent times. We demonstrate the method on two different data sets: human Y chromosome DNA data and fungus DNA data.     

Expected coalescence times and segregating sites in a model of glacial cycles

The climatic fluctuations of the Quaternary have influenced the distribution of numerous plant and animal species. Several species suffer population reduction and fragmentation, becoming restricted to refugia during glacial periods and expanding again during interglacials. The reduction in population size may reduce the effective population size, mean coalescence time and genetic variation, whereas an increased subdivision may have the opposite effect. To investigate these two opposing forces, we proposed a model in which a panmictic and a structured phase alternate, corresponding to interglacial and glacial periods. From this model, we derived an expression for the expected coalescence time and number of segregating sites for a pair of genes. We observed that increasing the number of demes or the duration of the structured phases causes an increase in coalescence time and expected levels of genetic variation. We compared numerical results with the ones expected for a panmictic population of constant size, and showed that the mean number of segregating sites can be greater in our model even when population size is much smaller in the structured phases. This points to the importance of population structure in the history of species subject to climatic fluctuations, and helps explain the long gene genealogies observed in several organisms.     

On the use of star-shaped genealogies in inference of coalescence times

Genealogies from rapidly growing populations have approximate "star" shapes. We study the degree to which this approximation holds in the context of estimating the time to the most recent common ancestor (T(MRCA)) of a set of lineages. In an exponential growth scenario, we find that unless the product of population size (N) and growth rate (r) is at least approximately 10(5), the "pairwise comparison estimator" of T(MRCA) that derives from the star genealogy assumption has bias of 10-50%. Thus, the estimator is appropriate only for large populations that have grown very rapidly. The "tree-length estimator" of T(MRCA) is more biased than the pairwise comparison estimator, having low bias only for extremely large values of Nr.     

Estimation of parameters in large offspring number models and ratios of coalescence times

The ratio of singletons to the total number of segregating sites is used to estimate a reproduction parameter in a population model of large offspring numbers without having to jointly estimate the mutation rate. For neutral genetic variation, the ratio of singletons to the total number of segregating sites is equivalent to the ratio of total length of external branches to the total length of the gene genealogy. A multinomial maximum likelihood method that takes into account more frequency classes than just the singletons is developed to estimate the parameter of another large offspring number model. The performance of these methods with regard to sample size, mutation rate, and bias, is investigated by simulation. The expected value of the ratio of the total length of external branches to the total length of the whole tree is, using simulation, shown to decrease for the Kingman coalescent as sample size increases, but can increase or decrease, depending on parameter values, for Λ coalescents. Considering ratios of tree statistics, as opposed to considering lengths of various subtrees separately, can yield better insight into the dynamics of gene genealogies.     

Inbreeding coefficients in six species of Ainsliaea and two species of Pertya (Asteraceae)

As a basis for comparative studies of mating systems, we estimated genetic diversities and inbreeding coefficients for six perennial species of Ainsliaea and two shrubby species of Pertya (Asteraceae). These related species are known to vary from frequent cleistogamy to xenogamy by bumblebee pollination, and we examined allozyme variation in these species to describe mating system variation in further detail. Significant heterozygote deficit was found in five species, while no heterozygote deficit was found in three species. A cleistogamous species showed no genetic polymorphism. Multilocus average of inbreeding coefficients among seven species ranged from –0.12 to 0.78. These findings indicated that related species can vary from predominant selfers to predominant outcrossers under perennial life cycles.     

A note on distributions of times to coalescence,under time-dependent population size

Expressions for marginal distributions of times in the time-varying coalescence process are derived. The proposed method allows also for computation of joint probability distribution for pairs, triples, etc. of coalescence times. The expressions derived are useful for (1) extending several statistics from time constant to time-varying case, (2) increasing efficiency and accuracy of simulations in time-varying evolution, and (3) debugging coalescence simulation software.     

Frequentist estimation of coalescence times from nucleotide sequence data using a tree-based partition

This article proposes a method of estimating the time to the most recent common ancestor (TMRCA) of a sample of DNA sequences. The method is based on the molecular clock hypothesis, but avoids assumptions about population structure. Simulations show that in a wide range of situations, the point estimate has small bias and the confidence interval has at least the nominal coverage probability. We discuss conditions that can lead to biased estimates. Performance of this estimator is compared with existing methods based on the coalescence theory. The method is applied to sequences of Y chromosomes and mtDNAs to estimate the coalescent times of human male and female populations.     

The distribution of coalescence times and distances between microsatellite alleles with changing effective population size

We investigate the effects of past changes of the effective population size on the present allelic diversity at a microsatellite marker locus. We first derive the analytical expression of the generating function of the joint probabilities of the time to the Most Recent Common Ancestor for a pair of alleles and of their distance (the difference in allele size). We give analytical solutions in the case of constant population size and the geometrical mutation model. Otherwise, numerical inversion allows the distributions to be calculated in general cases. The effects of population expansion or decrease and the possibility to detect an ancient bottleneck are discussed. The method is extended to samples of three and four alleles, which allows investigating the covariance structure of the frequencies f(k) of pairs of alleles with a size difference of k motifs, and suggesting some approaches to the estimation of past demography.     

Inbreeding coefficients for stochastically varying small population sizes-bias of calculation based on effective numbers

The commonly used procedure to calculate inbreeding coefficients by effective population numbers (Ne) by the harmonic mean of generation-by-generation population sizes involves a computational bias. If the individual population sizes are considered as realizations of a binomially distributed random variable with sample size N and probability p, this bias can be investigated for the two cases p = constant and p = variable (Markov chain). The bias is of practical relevance only for small probabilities p, short period of initial successive generations, and small population sizes. The largest values for this computational bias are in the range of 0.05-0.06. It is concluded that for most practical purposes the approximate procedure is appropriate.     

Rotational correlation times and partition coefficients of a spin label solute in lecithin vesicles.

Di-tert-butylnitroxide dissolved in an aqueous suspension of egg yolk lecithin vesicles is distributed between the two phases. Partition coefficients of the nitroxide between the lipid and the water, calculated from the nitroxide electron paramagnetic resonance (EPR) spectra, decrease with decreasing temperature until approximately the freezing point of the solvent. Below this temperature the nitroxide is detected only in the lecithin. The rotational correlation times of the spin label present in the lecithin were calculated for the temperature range from +45 to -60 degrees C. At low temperatures, the EPR spectra are characteristic of a superposition of two spectra resulting from the nitroxide dissolved in the lipid in two environments with different rotational correlation times.     

Compartmental models with Erlang distributed residence times and random rate coefficients

In this paper a stochastic model for a two-compartment system which incorporates Erlang residence time distributions (i.e. the residence times have the gamma distribution where the shape parameters assume integer values only) into each compartment is generalized to include random rate coefficients. Analytical forms of the model are derived for the case where the rate coefficients have gamma densities. A relationship is established between the new models and existing models that are in current practical usage.     

Inbreeding

Abstract

Data on inbreeding in several contemporary human populations are compared, showing the highest local rates of inbreeding to be in Brazil, Japan, India, and Israel. American populations are noteworthy for their extremely low inbreeding rates, with the mean frequency of first cousin marriages in the United States probably lower than 0.1 per cent. Some localities in Puerto Rico and Sweden show inbreeding levels half‐way between the extremes found in Brazil and Japan and in the United States. For European countries, Denmark, France, Italy, and North Ireland have a mean frequency of first cousin marriages of less than 1 per cent, while England, Wales, Germany, and Holland have a frequency probably lower than 0.5 per cent. The frequency of first cousin marriages in Portugal may lie between 1 and 2 per cent.     

Population differentiation and migration: coalescence times in a two-sex island model for autosomal and X-linked loci

Evolutionists have debated whether population-genetic parameters, such as effective population size and migration rate, differ between males and females. In humans, most analyses of this problem have focused on the Y chromosome and the mitochondrial genome, while the X chromosome has largely been omitted from the discussion. Past studies have compared F_ST values for the Y chromosome and mitochondrion under a model with migration rates that differ between the sexes but with equal male and female population sizes. In this study we investigate rates of coalescence for X-linked and autosomal lineages in an island model with different population sizes and migration rates for males and females, obtaining the mean time to coalescence for pairs of lineages from the same deme and for pairs of lineages from different demes. We apply our results to microsatellite data from the Human Genome Diversity Panel, and we examine the male and female migration rates implied by observed F_ST values.