On the size distribution of private microsatellite alleles

Private microsatellite alleles tend to be found in the tails rather than in the interior of the allele size distribution. To explain this phenomenon, we have investigated the size distribution of private alleles in a coalescent model of two populations, assuming the symmetric stepwise mutation model as the mode of microsatellite mutation. For the case in which four alleles are sampled, two from each population, we condition on the configuration in which three distinct allele sizes are present, one of which is common to both populations, one of which is private to one population, and the third of which is private to the other population. Conditional on this configuration, we calculate the probability that the two private alleles occupy the two tails of the size distribution. This probability, which increases as a function of mutation rate and divergence time between the two populations, is seen to be greater than the value that would be predicted if there was no relationship between privacy and location in the allele size distribution. In accordance with the prediction of the model, we find that in pairs of human populations, the frequency with which private microsatellite alleles occur in the tails of the allele size distribution increases as a function of genetic differentiation between populations.     

Coalescent process with fluctuating population size and its effective size

We consider a Wright-Fisher model whose population size is a finite Markov chain. We introduce a sequence of two-dimensional discrete time Markov chains whose components describe the coalescent process and the fluctuation of population size. For the limiting process of the sequence of Markov chains, the relationship of the expectation of coalescence time to the harmonic and the arithmetic means of population sizes is shown, and the Laplace transform of the distribution of coalescence time is calculated. We define the coalescence effective population size (cEPS) by the expectation of coalescence time. We show that cEPS is strictly larger (resp. smaller) than the harmonic (resp. arithmetic) mean. As the population size fluctuates more quickly (resp. slowly), cEPS is closer to the harmonic (resp. arithmetic) mean. For the case of a two-valued Markov chain, we show the explicit expression of cEPS and its dependency on the sample size.     

Are populations like a circuit? Comparing isolation by resistance to a new coalescent‐based method

A number of methods commonly used in landscape genetics use an analogy to electrical resistance on a network to describe and fit barriers to movement across the landscape using genetic distance data. These are motivated by a mathematical equivalence between electrical resistance between two nodes of a network and the ‘commute time’, which is the mean time for a random walk on that network to leave one node, visit the other, and return. However, genetic data are more accurately modelled by a different quantity, the coalescence time. Here, we describe the differences between resistance distance and coalescence time, and explore the consequences for inference. We implemented a Bayesian method to infer effective movement rates and population sizes under both these models, and found that inference using commute times could produce misleading results in the presence of biased gene flow. We then used forwards‐time simulation with continuous geography to demonstrate that coalescence‐based inference remains more accurate than resistance‐based methods on realistic data, but difficulties highlight the need for methods that explicitly model continuous, heterogeneous geography.     

New microsatellite loci for the European bushcricket,Ephippiger ephippiger (Orthoptera: Tettigoniidae)

Ephippiger ephippiger is a model organism for studies of sexual selection and phylogeography but little is known about fine‐scale population structure. Available microsatellite loci have null allele problems so we used an enrichment technique to isolate 21 new microsatellite loci for E. ephippiger. We present primer pairs for 10 polymorphic loci (3–11 alleles per locus). Observed heterozygosities at polymorphic loci ranged from 0.118 to 0.787, but several were significantly lower than expected.     

A simple derivation and classification of common probability distributions based on information symmetry and measurement scale

Commonly observed patterns typically follow a few distinct families of probability distributions. Over one hundred years ago, Karl Pearson provided a systematic derivation and classification of the common continuous distributions. His approach was phenomenological: a differential equation that generated common distributions without any underlying conceptual basis for why common distributions have particular forms and what explains the familial relations. Pearson's system and its descendants remain the most popular systematic classification of probability distributions. Here, we unify the disparate forms of common distributions into a single system based on two meaningful and justifiable propositions. First, distributions follow maximum entropy subject to constraints, where maximum entropy is equivalent to minimum information. Second, different problems associate magnitude to information in different ways, an association we describe in terms of the relation between information invariance and measurement scale. Our framework relates the different continuous probability distributions through the variations in measurement scale that change each family of maximum entropy distributions into a distinct family. From our framework, future work in biology can consider the genesis of common patterns in a new and more general way. Particular biological processes set the relation between the information in observations and magnitude, the basis for information invariance, symmetry and measurement scale. The measurement scale, in turn, determines the most likely probability distributions and observed patterns associated with particular processes. This view presents a fundamentally derived alternative to the largely unproductive debates about neutrality in ecology and evolution.     

Dissimilarity of individual microsatellite profiles under different mutation models: Empirical approach

Microsatellites (simple sequence repeats, SSRs) still remain popular molecular markers for studying neutral genetic variation. Two alternative models outline how new microsatellite alleles evolve. Infinite alleles model (IAM) assumes that all possible alleles are equally likely to result from a mutation, while stepwise mutation model (SMM) describes microsatellite evolution as stepwise adding or subtracting single repeat units. Genetic relationships between individuals can be analyzed in higher precision when assuming the SMM scenario with allele size differences as a proxy of genetic distance. If population structure is not predetermined in advance, an empirical data analysis usually includes (a) estimating proximity between individual SSR profiles with a selected dissimilarity measure and (b) determining putative genetic structure of a given set of individuals using methods of clustering and/or ordination for the obtained dissimilarity matrix. We developed new dissimilarity indices between SSR profiles of haploid, diploid, or polyploid organisms assuming different mutation models and compared the performance of these indices for determining genetic structure with population data and with simulations. More specifically, we compared SMM with a constant or variable mutation rate at different SSR loci to IAM using data from natural populations of a freshwater bryozoan Cristatella mucedo (diploid), wheat leaf rust Puccinia triticina (dikaryon), and wheat powdery mildew Blumeria graminis (monokaryon). We show that inferences about population genetic structure are sensitive to the assumed mutation model. With simulations, we found that Bruvo's distance performs generally poorly, while the new metrics are capturing the differences in the genetic structure of the populations.     

Differential patterns of male and female mtDNA exchange across the Atlantic Ocean in the blue mussel, Mytilus edulis

Comparisons among loci with differing modes of inheritance can reveal unexpected aspects of population history. We employ a multilocus approach to ask whether two types of independently assorting mitochondrial DNAs (maternally and paternally inherited: F- and M-mtDNA) and a nuclear locus (ITS) yield concordant estimates of gene flow and population divergence. The blue mussel, Mytilus edulis, is distributed on both North American and European coastlines and these populations are separated by the waters of the Atlantic Ocean. Gene flow across the Atlantic Ocean differs among loci, with F-mtDNA and ITS showing an imprint of some genetic interchange and M-mtDNA showing no evidence for gene flow. Gene flow of F-mtDNA and ITS causes trans-Atlantic population divergence times to be greatly underestimated for these loci, although a single trans-Atlantic population divergence time (1.2 MYA) can be accommodated by considering all three loci in combination in a coalescent framework. The apparent lack of gene flow for M-mtDNA is not readily explained by different dispersal capacities of male and female mussels. A genetic barrier to M-mtDNA exchange between North American and European mussel populations is likely to explain the observed pattern, perhaps associated with the double uniparental system of mitochondrial DNA inheritance.     

Isolation and characterization of microsatellites in Sparganium emersum and cross‐species amplification in the related species S. erectum

We developed seven novel polymorphic microsatellite loci for the aquatic macrophyte Sparganium emersum (Sparganiaceae). These were characterized on 62 individuals collected from nine different populations. In this set of individuals, seven to 20 alleles per locus were detected and observed heterozygosity ranged between 0.16 and 0.95. Cross‐species amplification was tested in the related species Sparganium erectum, and was successful for five of the seven microsatellite loci.     

The coalescent in two colonies with symmetric migration

Kingman's coalescent process is extended to two colonies with symmetric migration. The mean waiting time until a sample of genes taken from two colonies coalesces to a common ancestor is obtained. The final step in the waiting time before the process is absorbed at 1 is observed to have an intriguing behaviour. The distribution of this final waiting time converges to the known distribution of the corresponding waiting time in the case of a single population as the migration rate tends to zero. The mean, however, does not converge. The waiting time until a sample has two common ancestors is modeled as a function of the migration rate. Finally bounds for the expected waiting time for the two colonies to have j > 1 ancestors are derived.     

Which locus has the oldest allele?

This paper studies aspects of the distribution of non-mutant ancestors of a sample of gametes in a two-locus infinitely-many-alleles model. The ancestral process of two gametes is considered in detail. Included are algorithms for calculating the probability that the oldest allele is from the first locus, and the expected age of the oldest allele. Extensions to an r-locus model in the cases of complete linkage and independence are also studied.     

Genetic and environmental factors affecting mating type frequency in natural isolates of Tetrahymena thermophila

In Tetrahymena thermophila mating type alleles specify temperature sensitive frequency distributions of multiple mating types. A-like alleles specify mating types I, II, III, V and VI, whereas B-like alleles specify mating types II through VII. We have characterized the mating type distributions specified by several A- and B-like genotypes segregated by genomic exclusion from cells isolated from a pond in northwestern Pennsylvania. The B-like genotypes are alike in specifying very low frequencies of mating type III, but differ with respect to the frequencies of other mating types, particularly II and VII. An A-like genotype specifies a high frequency of mating type III and is unstable in successive generations for the expression of mating type II, suggesting a possible modifier. Inter se crosses performed at 18 degrees C, 28 degrees C and 34 degrees C showed that each genotype specifies a frequency distribution that is uniquely affected by temperature. No mating type was affected the same way by temperature in all genotypes. In A/B heterozygotes, the B-like genotype exhibited partial dominance. The genotypes described here differ significantly from previously described genotypes from the same pond, indicating that there are numerous mating type alleles. For frequency-dependent selection to equalize mating type frequencies, it must act not only on complex multiple alleles but also on the response of mating type alleles to temperature.     

Genetic effective size of a wild primate population: influence of current and historical demography

A comprehensive assessment of the determinants of effective population size (N(e)) requires estimates of variance in lifetime reproductive success and past changes in census numbers. For natural populations, such information can be best obtained by combining longitudinal data on individual life histories and genetic marker-based inferences of demographic history. Independent estimates of the variance effective size (N(ev), obtained from life-history data) and the inbreeding effective size (N((eI), obtained from genetic data) provide a means of disentangling the effects of current and historical demography. The purpose of this study was to assess the demographic determinants of N(e) in one of the most intensively studied natural populations of a vertebrate species: the population of savannah baboons (Papio cynocephalus) in the Amboseli Basin, southern Kenya. We tested the hypotheses that N(eV) < N < N(eI) (where N = population census number) due to a recent demographic bottleneck. N(eV) was estimated using a stochastic demographic model based on detailed life-history data spanning a 28-year period. Using empirical estimates of age-specific rates of survival and fertility for both sexes, individual-based simulations were used to estimate the variance in lifetime reproductive success. The resultant values translated into an N(eV)/N estimate of 0.329 (SD = 0.116, 95% CI = 0.172-0.537). Historical N(eI), was estimated from 14-locus microsatellite genotypes using a coalescent-based simulation model. Estimates of N(eI) were 2.2 to 7.2 times higher than the contemporary census number of the Amboseli baboon population. In addition to the effects of immigration, the disparity between historical N(eI) and contemporary N is likely attributable to the time lag between the recent drop in census numbers and the rate of increase in the average probability of allelic identity-by-descent. Thus, observed levels of genetic diversity may primarily reflect the population's prebottleneck history rather than its current demography.     

Dynamics of neutral and selected alleles when the offspring distribution is skewed

We analyze the dynamics of two alternative alleles in a simple model of a population that allows for large family sizes in the distribution of offspring number. This population model was first introduced by Eldon and Wakeley, who described the backward-time genealogical relationships among sampled individuals, assuming neutrality. We study the corresponding forward-time dynamics of allele frequencies, with or without selection. We derive a continuum approximation, analogous to Kimura's diffusion approximation, and we describe three distinct regimes of behavior that correspond to distinct regimes in the coalescent processes of Eldon and Wakeley. We demonstrate that the effect of selection is strongly amplified in the Eldon-Wakeley model, compared to the Wright-Fisher model with the same variance effective population size. Remarkably, an advantageous allele can even be guaranteed to fix in the Eldon-Wakeley model, despite the presence of genetic drift. We compute the selection coefficient required for such behavior in populations of Pacific oysters, based on estimates of their family sizes. Our analysis underscores that populations with the same effective population size may nevertheless experience radically different forms of genetic drift, depending on the reproductive mechanism, with significant consequences for the resulting allele dynamics.     

Coalescent theory for a completely random mating monoecious population

Consider a large random mating monoecious diploid population that has N individuals in each generation. Let us assume that at time 0 a random sample of ninfinity. It is then possible to obtain a generalization of coalescent theory for haploid populations if the distribution of G1 has a finite second moment and E[G(1)(3)]/N-->0 as N-->infinity.     

The sampling theory of neutral alleles and an urn model in population genetics

The behaviour of a Pólya-like urn which generates Ewens' sampling formula in population genetics is investigated. Connections are made with work of Watterson and Kingman and to the Poisson-Dirichlet distribution. The order in which novel types occur in the urn is shown to parallel the age distribution of the infinitely many alleles diffusion model and consequences of this property are explored. Finally the urn process is related to Kingman's coalescent with mutation to provide a rigorous basis for this parallel.This research was partially supported by the Sloan Foundation under Grant 85-6-14 and by the National Science Foundation     

Cross‐species amplification of microsatellite loci in Aquilegia and Semiaquilegia (Ranunculaceae)

We developed 16 microsatellite loci from an F₂ hybrid between Aquilegia formosa and Aquilegia pubescens. In samples of 28 individuals, we found an average of 14 alleles per locus from each parental species. We tested these loci for cross‐amplification in 10 additional species of Aquilegia and found that all 16 loci amplified in other North American species and 12 consistently amplified in European or Asian species. Nine loci amplified in the sister species to Aquilegia, Semiaquilegia adoxoides. The success of cross‐species amplification suggests that these microsatellites should prove useful for studies in a broad range of Aquilegia species.     

Allele frequency distributions of Apo B VNTR locus in Cukurova, Turkey

The highly polymorphic minisatellites contain a variable number of tandemly repeated (VNTR) DNA sequences. They are extremely useful and informative markers to study genetic variation among human populations. We have analysed the allele frequency distribution at the highly polymorphic apolipoprotein B (Apo B) VNTR locus in order to obtain the population data for the Cukurova region in Turkey by using the polymerase chain reaction and polyacrylamide gel electrophoresis. We observed 10 different alleles and 21 genotypes in a sample of 100 unrelated individuals. The allele frequencies ranged from 0.01 to 0.4, with an expected heterozygosity of 0.69 for the Apo B locus. Alleles 37 (frequency = 0.4) and 35 (frequency = 0.17) were the most common in the Cukurova population. There was a significant deviation from the Hardy-Weinberg equilibrium (HWE) for genotype frequencies (chi2 = 29.12; df = 1; p = 0.000). This study possesses novelty as it is the first DNA polymorphism study conducted at the Cukurova population using an Apo B minisatellite locus.     

Molecular evidence for multiple introductions of garlic mustard (Alliaria petiolata, Brassicaceae) to North America

Invasive species offer excellent model systems for studying rapid evolutionary change. In this context, molecular markers play an important role because they provide information about pathways of introduction, the amount of genetic variation introduced, and the extent to which founder effects and inbreeding after population bottlenecks may have contributed to evolutionary change. Here, we studied microsatellite variation in eight polymorphic loci among and within 27 native and 26 introduced populations of garlic mustard (Alliaria petiolata), a European herb which is a current serious invader in North American deciduous forests. Overall, introduced populations were genetically less diverse. However, considerable variability was present and when compared to the probable source regions, no bottleneck was evident. Observed heterozygosity was very low and resulted in high inbreeding coefficients, which did not differ significantly between native and introduced populations. Thus, selfing seems to be equally dominant in both ranges. Consequently, there was strong population differentiation in the native (F(ST) = 0.704) and the introduced (F(ST) = 0.789) ranges. The high allelic diversity in the introduced range strongly suggests multiple introductions of Alliaria petiolata to North America. Out of six European regions, the British Isles, northern Europe, and central Europe had significantly higher proportions of alleles, which are common to the introduced range, and are therefore the most probable source regions. The genetic diversity established by multiple introductions, and the lack of inbreeding depression in this highly selfing species, may have contributed to the invasion success of Alliaria petiolata.