Estimators for QST and coalescence times

Comparisons of to can provide insights into the evolutionary processes that lead to differentiation, or lack thereof, among the phenotypes of different groups (e.g., populations, species), and these comparisons have been performed on a variety of taxa, including humans. Here, I show that for neutrally evolving (i.e., by genetic drift, mutation, and gene flow alone) quantitative characters, the two commonly used estimators have somewhat different interpretations in terms of coalescence times, particularly when the number of groups that have been sampled is small. A similar situation occurs for estimators. Consequently, when observations come from only a small number of groups, which is not an unusual situation, it is important to match estimators appropriately when comparing to .     

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.     

Sex-specific genetic differentiation and coalescence times: estimating sex-biased dispersal rates

I derive the equilibrium values of sex-specific FST parameters, in an island model for a dioecious species with sex-biased dispersal and binomial distribution of family size before dispersal (as assumed in a Wright-Fisher population). I show that FST may take different values among males and among females whenever dispersal is a trait conditioned on gender. This has not always been recognized, because some models assumed that genes are sampled before dispersal. In particular, the ratios of sex-specific FST parameters evaluated after dispersal over FST evaluated before dispersal are simple functions of sex-specific dispersal rates. Therefore, a simple moment-based estimator of sex-specific dispersal rate is proposed. This method is based on the comparison of FST estimated before and after dispersal and assumes equilibrium between migration and drift. I evaluate this method through stochastic simulations for a range of sex-specific dispersal rates and sampling effort (sample size, number of loci scored).     

Estimation of coalescence probabilities and population divergence times from SNP data

We present a method called the G(A|B) method for estimating coalescence probabilities within population lineages from genome sequences when one individual is sampled from each population. Population divergence times can be estimated from these coalescence probabilities if additional assumptions about the history of population sizes are made. Our method is based on a method presented by Rasmussen et al. (2014) to test whether an archaic genome is from a population directly ancestral to a present-day population. The G(A|B) method does not require distinguishing ancestral from derived alleles or assumptions about demographic history before population divergence. We discuss the relationship of our method to two similar methods, one introduced by Green et al. (2010) and called the F(A|B) method and the other introduced by Schlebusch et al. (2017) and called the TT method. When our method is applied to individuals from three or more populations, it provides a test of whether the population history is treelike because coalescence probabilities are additive on a tree. We illustrate the use of our method by applying it to three high-coverage archaic genomes, two Neanderthals (Vindija and Altai) and a Denisovan.Subject terms: Rare variants, Evolutionary genetics

One of the goals of population genetics is to estimate the divergence time of isolated populations. We will review several methods that have been proposed and present a new method that is closely related to two existing methods. We will emphasize the assumptions made when using different methods. It will be useful to make the distinction between estimating coalescence probabilities within populations and estimating population divergence times. We will also introduce a test for a treelike population history based on our method.For distantly related populations, the numbers of mutational differences between sequences indicate relative times of divergence. Relative times are converted to absolute times by assuming a mutation rate. This method traces to Zuckerkandl and Pauling (1962, 1965) and has been used and refined extensively. This class of methods estimates genomic divergence times. Using it to estimate population or species divergence times assumes that those times are so large that the difference between them can be ignored.For recently diverged populations, the numbers of mutational differences probably do not provide a reliable estimate of population divergence times both because there may be too few mutations that differentiate populations and because the difference between the genomic and population divergence times may be substantial. To overcome this problem, Green et al. (2010) (in Supplement 14) introduced a method that accounts for the difference between genomic and population divergence. This method was used in later papers from the same group (Meyer et al. 2012; Prüfer et al. 2014, 2017).The Green et al. (2010) method is applicable when one genome is sampled from each of two populations. It depends on the statistic F(A|B), which is the fraction of sites in population A that carry the derived allele when that site is heterozygous in population B. Green et al. (2010) showed by simulation that the expectation of F(A|B) decreases roughly exponentially with the separation time of A and B. The rate of decrease depends on the history of population sizes both in B and in the population ancestral to A and B. Green et al. (2010) estimated population divergence times by interpolating their simulation results.More recently, Schlebusch et al. (2017), in Section 9.1 of their supplementary materials, introduced a similar method, called the TT method. Their method is based on analytic expressions for the configuration probabilities of SNPs that are polymorphic in the two populations. The TT method assumes that ancestral and derived alleles can be distinguished and the population before divergence was of constant size. The TT method is developed and elaborated on by Sjödin et al. (2020).In the present paper, we present a new method that is closely related to the F(A|B) and TT methods. We call it the G(A|B) method to emphasize its similarity to F(A|B). Our method is based on a method presented by Rasmussen et al. (2014) to test whether an ancient DNA sequence is from a population directly ancestral to a present-day population. We will show that our method provides a way to test whether the history of three or more populations is accurately represented by a population tree even if the demographic histories of those populations are not known.     

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.     

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.     

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.