Peter Beerli

Maximum likelihood estimation of a migration matrix and effective population sizes in n subpopulations by using a coalescent approach

A maximum likelihood estimator based on the coalescent for unequal migration rates and different subpopulation sizes is developed. The method uses a Markov chain Monte Carlo approach to investigate possible genealogies with branch lengths and with migration events. Properties of the new method are shown by using simulated data from a four-population n-island model and a source–sink population model. Our estimation method as coded in migrate is tested against genetree; both programs deliver a very similar likelihood surface. The algorithm converges to the estimates fairly quickly, even when the Markov chain is started from unfavorable parameters. The method was used to estimate gene flow in the Nile valley by using mtDNA data from three human populations.

The Strength of Phenotypic Selection in Natural Populations

How strong is phenotypic selection on quantitative traits in the wild? We reviewed the literature from 1984 through 1997 for studies that estimated the strength of linear and quadratic selection in terms of standardized selection gradients or differentials on natural variation in quantitative traits for field populations. We tabulated 63 published studies of 62 species that reported over 2,500 estimates of linear or quadratic selection. More than 80% of the estimates were for morphological traits; there is very little data for behavioral or physiological traits. Most published selection studies were unreplicated and had sample sizes below 135 individuals, resulting in low statistical power to detect selection of the magnitude typically reported for natural populations. The absolute values of linear selection gradients |β| were exponentially distributed with an overall median of 0.16, suggesting that strong directional selection was uncommon. The values of |β| for selection on morphological and on life‐history/phenological traits were significantly different: on average, selection on morphology was stronger than selection on phenology/life history. Similarly, the values of |β| for selection via aspects of survival, fecundity, and mating success were significantly different: on average, selection on mating success was stronger than on survival. Comparisons of estimated linear selection gradients and differentials suggest that indirect components of phenotypic selection were usually modest relative to direct components. The absolute values of quadratic selection gradients |γ| were exponentially distributed with an overall median of only 0.10, suggesting that quadratic selection is typically quite weak. The distribution of γ values was symmetric about 0, providing no evidence that stabilizing selection is stronger or more common than disruptive selection in nature.

PERSPECTIVE: GENE DIVERGENCE, POPULATION DIVERGENCE, AND THE VARIANCE IN COALESCENCE TIME IN PHYLOGEOGRAPHIC STUDIES

Molecular methods as applied to the biogeography of single species (phylogeography) or multiple codistributed species (comparative phylogeography) have been productively and extensively used to elucidate common historical features in the diversification of the Earths biota. However, only recently have methods for estimating population divergence times or their confidence limits while taking into account the critical effects of genetic polymorphism in ancestral species become available, and earlier methods for doing so are underutilized. We review models that address the crucial distinction between the gene divergence, the parameter that is typically recovered in molecular phylogeographic studies, and the population divergence, which is in most cases the parameter of interest and will almost always postdate the gene divergence. Assuming that population sizes of ancestral species are distributed similarly to those of extant species, we show that phylogeographic studies in vertebrates suggest that divergence of alleles in ancestral species can comprise from less than 10% to over 50% of the total divergence between sister species, suggesting that the problem of ancestral polymorphism in dating population divergence can be substantial. The variance in the number of substitutions (among loci for a given species or among species for a given gene) resulting from the stochastic nature of DNA change is generally smaller than the variance due to substitutions along allelic lines whose coalescence times vary due to genetic drift in the ancestral population. Whereas the former variance can be reduced by further DNA sequencing at a single locus, the latter cannot. Contrary to phylogeographic intuition, dating population divergence times when allelic lines have achieved reciprocal monophyly is in some ways more challenging than when allelic lines have not achieved monophyly, because in the former case critical data on ancestral population size provided by residual ancestral polymorphism is lost. In the former case differences in coalescence time between species pairs can in principle be explained entirely by differences in ancestral population size without resorting to explanations involving differences in divergence time. Furthermore, the confidence limits on population divergence times are severely underestimated when those for number of substitutions per site in the DNA sequences examined are used as a proxy. This uncertainty highlights the importance of multilocus data in estimating population divergence times; multilocus data can in principle distinguish differences in coalescence time (T) resulting from differences in population divergence time and differences in T due to differences in ancestral population sizes and will reduce the confidence limits on the estimates.

Comparison of Bayesian and maximum-likelihood inference of population genetic parameters

UNLABELLED Comparison of the performance and accuracy of different inference methods, such as maximum likelihood (ML) and Bayesian inference, is difficult because the inference methods are implemented in different programs, often written by different authors. Both methods were implemented in the program MIGRATE, that estimates population genetic parameters, such as population sizes and migration rates, using coalescence theory. Both inference methods use the same Markov chain Monte Carlo algorithm and differ from each other in only two aspects: parameter proposal distribution and maximization of the likelihood function. Using simulated datasets, the Bayesian method generally fares better than the ML approach in accuracy and coverage, although for some values the two approaches are equal in performance. MOTIVATION The Markov chain Monte Carlo-based ML framework can fail on sparse data and can deliver non-conservative support intervals. A Bayesian framework with appropriate prior distribution is able to remedy some of these problems. RESULTS The program MIGRATE was extended to allow not only for ML(-) maximum likelihood estimation of population genetics parameters but also for using a Bayesian framework. Comparisons between the Bayesian approach and the ML approach are facilitated because both modes estimate the same parameters under the same population model and assumptions.

The utility of single nucleotide polymorphisms in inferences of population history

Single nucleotide polymorphisms (SNPs) represent the most widespread type of sequence variation in genomes, yet they have only emerged recently as valuable genetic markers for revealing the evolutionary history of populations. Their occurrence throughout the genome also makes them ideal for analyses of speciation and historical demography, especially in light of recent theory suggesting that many unlinked nuclear loci are needed to estimate population genetic parameters with statistical confidence. In spite of having lower variation compared with microsatellites, SNPs should make the comparison of genomic diversities and histories of different species (the core goal of comparative biogeography) more straightforward than has been possible with microsatellites. The most pervasive, but correctable, complication to SNP analysis is a bias towards analyzing only the most variable loci, an artifact that is usually introduced by the limited number of individuals used to screen initially for polymorphisms. Although the use of SNPs as markers in population studies is still new, innovative methods for SNP identification, automated screening, haplotype inference and statistical analysis might quickly make SNPs the marker of choice. Traditionally, phylogeography has used gene trees of nonrecombining, uniparentally inherited LOCI (see Glossary), such as mitochondrial DNA or the vertebrate Y chromosome, to study the geographical distribution of genetic variation within species [1]. As evolutionary biologists have started to examine variation in recombining, biparentally inherited loci, a natural outgrowth of phylogeography is a shift from gene trees to analyses, based on COALESCENT THEORY, of multi-locus, recombining histories. This new discipline, dubbed historical demography [2,3] or statistical phylogeography [4], is concerned less with gene trees than with estimating population parameters such as genetic diversities, divergence times, growth rates and gene flow between populations. The shift in focus is, in part, a result of recent advances in population genetics, which suggest that, from a statistical standpoint, the ability of single-locus phylogeography to determine the timing of speciation events and the historical demography of populations has been overestimated [3‐7]. The errors surrounding estimates of divergence times, rates of gene flow and population-size changes during speciation are all reduced substantially when information from multiple unlinked loci is combined [8,9]. With the move to analyses of multiple loci, phylogeographers must re-learn an old lesson: that the number of loci required to estimate the preceding parameters with statistical confidence can be soberingly large because of the high stochasticity of the gene tree of any single locus [10]. What is required is a suite of unlinked nuclear genetic markers that can capture a genome-wide picture of the population history [3,11‐14]. The polymerase chain reaction (PCR) as well as fluorescent sequencing and fragment analysis technologies have catalyzed a revolution in the development of genetic markers for the analysis of natural populations. Emphasizing discoveries in nonmodel species, we discuss one emerging marker of great relevance to historical demography: single nucleotide polymorphisms (SNPs).

Unified Framework to Evaluate Panmixia and Migration Direction Among Multiple Sampling Locations

For many biological investigations, groups of individuals are genetically sampled from several geographic locations. These sampling locations often do not reflect the genetic population structure. We describe a framework using marginal likelihoods to compare and order structured population models, such as testing whether the sampling locations belong to the same randomly mating population or comparing unidirectional and multidirectional gene flow models. In the context of inferences employing Markov chain Monte Carlo methods, the accuracy of the marginal likelihoods depends heavily on the approximation method used to calculate the marginal likelihood. Two methods, modified thermodynamic integration and a stabilized harmonic mean estimator, are compared. With finite Markov chain Monte Carlo run lengths, the harmonic mean estimator may not be consistent. Thermodynamic integration, in contrast, delivers considerably better estimates of the marginal likelihood. The choice of prior distributions does not influence the order and choice of the better models when the marginal likelihood is estimated using thermodynamic integration, whereas with the harmonic mean estimator the influence of the prior is pronounced and the order of the models changes. The approximation of marginal likelihood using thermodynamic integration in MIGRATE allows the evaluation of complex population genetic models, not only of whether sampling locations belong to a single panmictic population, but also of competing complex structured population models.

Strength and tempo of directional selection in the wild

Directional selection is a major force driving adaptation and evolutionary change. However, the distribution, strength, and tempo of phenotypic selection acting on quantitative traits in natural populations remain unclear across different study systems. We reviewed the literature (1984–1997) that reported the strength of directional selection as indexed by standardized linear selection gradients (β). We asked how strong are viability and sexual selection, and whether strength of selection is correlated with the time scale over which it was measured. Estimates of the magnitude of directional selection (|β|) were exponentially distributed, with few estimates greater than 0.50 and most estimates less than 0.15. Sexual selection (measured by mating success) appeared stronger than viability selection (measured by survival). Viability selection that was measured over short periods (days) was typically stronger than selection measured over longer periods (months and years), but the strength of sexual selection did not vary with duration of selection episodes; as a result, sexual selection was stronger than viability selection over longer time scales (months and years), but not over short time scales (days).

BEAGLE: An Application Programming Interface and High-Performance Computing Library for Statistical Phylogenetics

Abstract Phylogenetic inference is fundamental to our understanding of most aspects of the origin and evolution of life, and in recent years, there has been a concentration of interest in statistical approaches such as Bayesian inference and maximum likelihood estimation. Yet, for large data sets and realistic or interesting models of evolution, these approaches remain computationally demanding. High-throughput sequencing can yield data for thousands of taxa, but scaling to such problems using serial computing often necessitates the use of nonstatistical or approximate approaches. The recent emergence of graphics processing units (GPUs) provides an opportunity to leverage their excellent floating-point computational performance to accelerate statistical phylogenetic inference. A specialized library for phylogenetic calculation would allow existing software packages to make more effective use of available computer hardware, including GPUs. Adoption of a common library would also make it easier for other emerging computing architectures, such as field programmable gate arrays, to be used in the future. We present BEAGLE, an application programming interface (API) and library for high-performance statistical phylogenetic inference. The API provides a uniform interface for performing phylogenetic likelihood calculations on a variety of compute hardware platforms. The library includes a set of efficient implementations and can currently exploit hardware including GPUs using NVIDIA CUDA, central processing units (CPUs) with Streaming SIMD Extensions and related processor supplementary instruction sets, and multicore CPUs via OpenMP. To demonstrate the advantages of a common API, we have incorporated the library into several popular phylogenetic software packages. The BEAGLE library is free open source software licensed under the Lesser GPL and available from http://beagle-lib.googlecode.com. An example client program is available as public domain software.

Effect of unsampled populations on the estimation of population sizes and migration rates between sampled populations

Current estimators of gene flow come in two methods; those that estimate parameters assuming that the populations investigated are a small random sample of a large number of populations and those that assume that all populations were sampled. Maximum likelihood or Bayesian approaches that estimate the migration rates and population sizes directly using coalescent theory can easily accommodate datasets that contain a population that has no data, a so‐called ‘ghost’ population. This manipulation allows us to explore the effects of missing populations on the estimation of population sizes and migration rates between two specific populations. The biases of the inferred population parameters depend on the magnitude of the migration rate from the unknown populations. The effects on the population sizes are larger than the effects on the migration rates. The more immigrants from the unknown populations that are arriving in the sample populations the larger the estimated population sizes. Taking into account a ghost population improves or at least does not harm the estimation of population sizes. Estimates of the scaled migration rate M (migration rate per generation divided by the mutation rate per generation) are fairly robust as long as migration rates from the unknown populations are not huge. The inclusion of a ghost population does not improve the estimation of the migration rate M; when the migration rates are estimated as the number of immigrants Nm then a ghost population improves the estimates because of its effect on population size estimation. It seems that for ‘real world’ analyses one should carefully choose which populations to sample, but there is no need to sample every population in the neighbourhood of a population of interest.

GEOLOGICALLY DATED SEA BARRIERS CALIBRATE A PROTEIN CLOCK FOR AEGEAN WATER FROGS

Reliable estimates of phylogenetic relationships and divergence times are a crucial requirement for many evolutionary studies, but are usually difficult because fossils are scarce and their interpretation is often uncertain. Frogs are fresh water animals that generally are unable to cross salt water barriers (their skin is readily permeable to both salt and water). The geologically determined ages of salt water barriers that isolate related frog populations thus provide an independent measure of the minimum date of genetic divergence between pairs of such populations. For the genetically well‐studied western Palearctic water frogs (Rana esculenta group), the Aegean region provides an ideal area for determining the relationship between genetic divergence and time of spatial isolation, using a nested set of geologically determined isolation times (12,000 yr, 200,000 yr, 1.8 Myr, 2–3 Myr, and 5.2 Myr).