Estimation of age and rate of increase of rare variants.

The problem considered is that of estimating the age or rate of increase of a variant on the basis of the present number of replicates observed in a population. In place of previous diffusion equation analyses of age probability distributions, the likelihood for the age is studied on the basis of a discrete branching process model. It is shown that variations inherent in the process of gene evolution in natural populations make it impossible to provide a reliable point estimate of the age of a specified variant, although the likelihood analysis provides a confidence interval which may place useful bounds on the period in which a variant originated. The observed distribution of numbers of several variants may also provide useful information. The problems of estimation are discussed with reference to rare variants arising in American Indian populations.     

On the Genealogy of a Rare Allele

The gene genealogy is derived for a rare allele that is descended from a mutant ancestor that arose at a fixed time in the past. Following Thompson (1976,Amer. J. Human Genet.28, 442–452), the fractional linear branching process is used as a model of the demography of a rare allele. The model does not require the total population size to be constant or the mutant class to be neutral; so long as individuals in the class are selectively equivalent, the class as a whole may have a selective advantage, or disadvantage, relative to other alleles in the population. An exact result is given for the joint probability distribution of the coalescence times among a sample of alleles descended from the mutant. A method is described for rapidly simulating these coalescence times. The relationship between the genealogical structure of a discrete generation branching process and a continuous generation birth–death process is elucidated. The theory may be applied to the problem of estimating the ages of rare nonrecurrent mutations.     

Rare and Low Frequency Variant Stratification in the UK Population: Description and Impact on Association Tests

Although variations in allele frequencies at common SNPs have been extensively studied in different populations, little is known about the stratification of rare variants and its impact on association tests. In this paper, we used Affymetrix 500K genotype data from the WTCCC to investigate if variants in three different frequency categories (below 1%, between 1 and 5%, above 5%) show different stratification patterns in the UK population. We found that these patterns are indeed different. The top principal component extracted from the rare variant category shows poor correlations with any principal component or combination of principal components from the low frequency or common variant categories. These results could suggest that a suitable solution to avoid false positive association due to population stratification would involve adjusting for the respective PCs when testing for variants in different allele frequency categories. However, we found this was not the case both on type 2 diabetes data and on simulated data. Indeed, adjusting rare variant association tests on PCs derived from rare variants does no better to correct for population stratification than adjusting on PCs derived from more common variants. Mixed models perform slightly better for low frequency variants than PC based adjustments but less well for the rarest variants. These results call for the need of new methodological developments specifically devoted to address rare variant stratification issues in association tests.     

Allele frequency distribution under recurrent selective sweeps

The allele frequency of a neutral variant in a population is pushed either upward or downward by directional selection on a linked beneficial mutation ("selective sweeps"). DNA sequences sampled after the fixation of the beneficial allele thus contain an excess of rare neutral alleles. This study investigates the allele frequency distribution under selective sweep models using analytic approximation and simulation. First, given a single selective sweep at a fixed time, I derive an expression for the sampling probabilities of neutral mutants. This solution can be used to estimate the time of the fixation of a beneficial allele from sequence data. Next, I obtain an approximation to mean allele frequencies under recurrent selective sweeps. Under recurrent sweeps, the frequency spectrum is skewed toward rare alleles. However, the excess of high-frequency derived alleles, previously shown to be a signature of single selective sweeps, disappears with recurrent sweeps. It is shown that, using this approximation and multilocus polymorphism data, genomewide parameters of directional selection can be estimated.     

The allele distribution in next-generation sequencing data sets is accurately described as the result of a stochastic branching process

With the availability of next-generation sequencing (NGS) technology, it is expected that sequence variants may be called on a genomic scale. Here, we demonstrate that a deeper understanding of the distribution of the variant call frequencies at heterozygous loci in NGS data sets is a prerequisite for sensitive variant detection. We model the crucial steps in an NGS protocol as a stochastic branching process and derive a mathematical framework for the expected distribution of alleles at heterozygous loci before measurement that is sequencing. We confirm our theoretical results by analyzing technical replicates of human exome data and demonstrate that the variance of allele frequencies at heterozygous loci is higher than expected by a simple binomial distribution. Due to this high variance, mutation callers relying on binomial distributed priors are less sensitive for heterozygous variants that deviate strongly from the expected mean frequency. Our results also indicate that error rates can be reduced to a greater degree by technical replicates than by increasing sequencing depth.     

A branching process, its application in biology: influence of demographic parameters on the social structure in mammal groups

Branching processes are widely used in biology. This theoretical tool is used in cell dynamics, epidemics and population dynamics. In population dynamics, branching processes are mainly used to access extinction probabilities of populations, groups or families, with the Galton-Watson branching process. Many mammal species live in socially-structured groups, and the smallest units of these groups are lineages (or families) of kin-related individuals. In many primate species, these lineages are matrilines, as females remain in their natal groups most of the time, whereas males generally disperse. Lineage parameters, such as numbers of matrilines, size of each matriline and average degree of relatedness, could strongly influence the genetic composition of groups. Evidence indicates that division along matrilines could induce substantial differentiation among fission groups. Here, we develop a novel mathematical model based on the branching process theory describing demographic dynamics of groups. The main result of this model is an explicit analytical expression of the joint distribution of numbers of lineages and sizes of socially-structured groups. We investigated the influence of parameters such as natality and mortality on the outcome of the process, including extinction probability. Finally, we discuss this theoretical result with respect to biological significance.     

Global population genetic structure and male-mediated gene flow in the green sea turtle (Chelonia mydas): analysis of microsatellite loci

We assessed the degree of population subdivision among global populations of green sea turtles, Chelonia mydas, using four microsatellite loci. Previously, a single-copy nuclear DNA study indicated significant male-mediated gene flow among populations alternately fixed for different mitochondrial DNA haplotypes and that genetic divergence between populations in the Atlantic and Pacific Oceans was more common than subdivisions among populations within ocean basins. Even so, overall levels of variation at single-copy loci were low and inferences were limited. Here, the markedly more variable microsatellite loci confirm the presence of male-mediated gene flow among populations within ocean basins. This analysis generally confirms the genetic divergence between the Atlantic and Pacific. As with the previous study, phylogenetic analyses of genetic distances based on the microsatellite loci indicate a close genetic relationship among eastern Atlantic and Indian Ocean populations. Unlike the single-copy study, however, the results here cannot be attributed to an artifact of general low variability and likely represent recent or ongoing migration between ocean basins. Sequence analyses of regions flanking the microsatellite repeat reveal considerable amounts of cryptic variation and homoplasy and significantly aid in our understanding of population connectivity. Assessment of the allele frequency distributions indicates that at least some of the loci may not be evolving by the stepwise mutation model.     

Genetic variation atapolipoprotein E locus in Ethiopia: an E5 variant corresponds to two different mutant alleles: E*5 (Glu212Lys) and E*5 (Gln204Lys; Cys112Arg)

A previous investigation on apolipoprotein E polymorphism in the Ethiopian population highlighted the presence of a further variant allele named E*5 in addition to the three common alleles. The variant is considered rare elsewhere but has a frequency of more than 1% in this population. Now characterized by gene sequencing and restriction isotyping in many members of the families of the original carriers, the variant isoform has actually been found to be determined by two different gene mutations. Effectively rare in Ethiopians, one of the two, E5 (Gln204Lys, Cys 112Arg), has never been described before. The other, E5 (Glu212Lys), previously described in a subject of Turkish origin, is present at the polymorphic level only in the Ethiopian population. No subjects bearing these variants had anomalous lipid or apolipoprotein patterns. In the course of the present investigation both have been found to occur as rare variants in the southern Italian population as well. The occurrence of the two variants in the populations of Ethiopia and of the Mediterranean basin could be explained by taking into account the relevant Caucasoid contribution to the Ethiopian gene pool.     

Bisexual branching processes to model extinction conditions for Y-linked genes

In a two-sex monogamic population, the evolution of the number of carriers of the two alleles of a Y-linked gene is considered. To this end, a multitype bisexual branching model is presented in which it is assumed that the gene has no influence on the mating process. It is deduced from this model that the average numbers of female and male descendants per mating unit constitute the key to determining the extinction or survival of each allele. Moreover, the destiny of each allele in the population is found not to depend on the behavior of the other.     

Bisexual branching processes in a genetic context: the extinction problem for Y-linked genes

In this paper the inheritance of a Y-linked gene with alleles R and r in a population with both females and males is modelled using a two-type bisexual branching process. It is assumed that the reproductive distribution associated with the R allele can differ from the associated with the r allele and that females prefer to mate with a male having the R allele rather than with a male with the r allele. Under these assumptions, we provide some conditions for the extinction and/or survival of both alleles in the population. These conditions depend on the magnitudes of the average number of females and males per mating unit. Moreover, the almost sure extinction of the r allele is independent of the behaviour of the R allele. On the other hand, the survival of the R allele with positive probability may depend strongly on the reproductive behaviour of the other allele. Theoretical results are illustrated by means of simulated examples and some open problems are proposed to the reader as conjectures.     

A logistic branching process for population genetics

A logistic (regulated population size) branching process population genetic model is presented. It is a modification of both the Wright-Fisher and (unconstrained) branching process models, and shares several properties including the coalescent time and shape, and structure of the coalescent process with those models. An important feature of the model is that population size fluctuation and regulation are intrinsic to the model rather than externally imposed. A consequence of this model is that the fluctuation in population size enhances the prospects for fixation of a beneficial mutation with constant relative viability, which is contrary to a result for the Wright-Fisher model with fluctuating population size. Explanation of this result follows from distinguishing between expected and realized viabilities, in addition to the contrast between absolute and relative viabilities.     

ON PHENOTYPIC EXPRESSION,MORPHOGENETIC PATTERN AND SYNANGIUM EVOLUTION IN PSILOTUM

Both the typical form and the appendageless variant of Psilotum nudum produce terminal synangia at the ultimate tips of the aerial axes. One clone in particular of the typical appendaged form produced synangia entirely at the tips of the aerial branches, as in the appendageless variant, and also developed occasional lateral transitional entities on the upper aerial axis displaying appendagelike and axislike morphological qualities. A developmental comparison of synangium development at the ultimate tips of aerial branches and of unusually elongating and normal sized fertile-appendages showed that the morphogenetic pattern of synangium development was similar. Anatomical and morphological evidence showed the synangium to be derived from terminal subdivisions or bifurcations of the apical meristem of each structure studied. This supports the phyletic concept that the synangium of the Psilotaceae is basically terminal to an axis or an axis homologue, and that it probably evolved from terminal bifurcative branching. Occasional multiple sporangium lobes may be formed on a P. nudum synangium which may not be represented by vascular bundles. Two hypothetical phyletic models of synangium evolution are proposed that could be used to explain this phenomenon and which should be tested by further evidence. Typical and appendageless P. nudum were compared in their morphogenetic pattern developed at the upper axis vegetative apical meristem, and a reconciliation was made between the structuring of the apparently disparate forms, which involved the presence or absence of serial ordering in apical derivatives. It is suggested that this could serve as a model for appendage evolution in the family Psilotaceae.     

Time-continuous branching walk models of unstable gene amplification

We consider a stochastic mechanism of the loss of resistance of cancer cells to cytotoxic agents, in terms of unstable gene amplification. Two models being different versions of a time-continuous branching random walk are presented. Both models assume strong dependence in replication and segregation of the extrachromosomal elements. The mathematical part of the paper includes the expression for the expected number of cells with a given number of gene copies in terms of modified Bessel functions. This adds to the collection of rare explicit solutions to branching process models. Original asymptotic expansions are also demonstrated. Fitting the model to experimental data yields estimates of the probabilities of gene amplification and deamplification. The thesis of the paper is that purely stochastic mechanisms may explain the dynamics of reversible drug resistance of cancer cells. Various stochastic approaches and their limitations are discussed.     

On the fixation process of a beneficial mutation in a variable environment

A population that adapts to gradual environmental change will typically experience temporal variation in its population size and the selection pressure. On the basis of the mathematical theory of inhomogeneous branching processes, we present a framework to describe the fixation process of a single beneficial allele under these conditions. The approach allows for arbitrary time-dependence of the selection coefficient s(t) and the population size N(t), as may result from an underlying ecological model. We derive compact analytical approximations for the fixation probability and the distribution of passage times for the beneficial allele to reach a given intermediate frequency. We apply the formalism to several biologically relevant scenarios, such as linear or cyclic changes in the selection coefficient, and logistic population growth. Comparison with computer simulations shows that the analytical results are accurate for a large parameter range, as long as selection is not very weak.     

Bayesian models for the analysis of genetic structure when populations are correlated

MOTIVATION: Population allele frequencies are correlated when populations have a shared history or when they exchange genes. Unfortunately, most models for allele frequency and inference about population structure ignore this correlation. Recent analytical results show that among populations, correlations can be very high, which could affect estimates of population genetic structure. In this study, we propose a mixture beta model to characterize the allele frequency distribution among populations. This formulation incorporates the correlation among populations as well as extending the model to data with different clusters of populations. RESULTS: Using simulated data, we show that in general, the mixture model provides a good approximation of the among-population allele frequency distribution and a good estimate of correlation among populations. Results from fitting the mixture model to a dataset of genotypes at 377 autosomal microsatellite loci from human populations indicate high correlation among populations, which may not be appropriate to neglect. Traditional measures of population structure tend to overestimate the amount of genetic differentiation when correlation is neglected. Inference is performed in a Bayesian framework. CONTACT: fur@ohsu.edu.     

A note on modelling the dynamics of budding yeast populations using branching processes

A multitype branching process is proposed as a model for the behaviour of populations of the budding yeast Saccharomyces Cerevisiae. Using the idea of branching processes counted by random characteristics, we are able to obtain explicit expressions describing different aspects of the asymptotic composition of such populations. The main purpose of this note is to show that the branching process approach is an alternative to deterministic population models based on differential equation methods.Supported by the Swedish Natural Science Research Council     

A stochastic model of the distribution of unequal competitors between resource patches

We present a stochastic model of individuals' movements between two patches of resources. The population is made up of two types of individual with differing competitive abilities, and two types of movements occur, with individuals moving either to increase their intake rate or at random. Several previous models have used simulations to evaluate the likely distribution of individuals. We instead derive equations for the equilibrium distribution of the population, which can be solved numerically. This avoids the need to choose an initial distribution for the population, and enables us to obtain the probability with which rare events occur. This may not be possible when simulations are used, since a rare event may not occur at all. We find that when random movements are rare, an increase in the rate of random movements out of a patch can increase the number of individuals on that patch. We consider an approximation to the model with rare random movements, which provides an explanation for this phenomenon.     

中国广东人红细胞PGM_1亚型遗传性多态现象

采用薄层聚丙烯酰胺凝胶等电聚焦技术,调查了中国(广东)406名无亲缘关系的正常人红细胞磷酸葡萄糖变位酶-1(PGM_1)亚型的遗传多态性。除了常见的10种亚型外,还发现了由一个新的变异型等位基因和常见的4个等位基因杂合产生的9例变异型。PGM_1位点的等位基因频率PGM_1~(1+)、PGM_1~(1-)、PGM_1~(2+)、PGM-1~(2-)和PGM_1~(V丰)(变异型等位基因)分别为0.5973、0.1256、0.1724、0.0936和0.0111;群体处于Hardy-Weinberg式平衡状态。变异型等位基因以多态频率出现,可能成为该群体的一个重要的遗传性特征。     

Adaptive dynamics in diploid, sexual populations and the evolution of reproductive isolation

Evolutionary branching is the process whereby an initially monomorphic population evolves to a point where it undergoes disruptive selection and splits up into two phenotypically diverging lineages. We studied evolutionary branching in three models that are ecologically identical but that have different genetic systems. The first model is clonal, the second is sexual diploid with additive genetics on a single locus and the third is like the second but with an additional locus for mate choice. Evolutionary branching occurred under exactly the same ecological circumstances in all three models. After branching the evolutionary dynamics may be qualitatively different. In particular, in the diploid, sexual models there can be multiple evolutionary outcomes whereas in the corresponding clonal model there is only one. We showed that evolutionary branching favours the evolution of (partial) assortative mating and that this in turn effectively restores the results from the clonal model by rendering the alternative outcomes unreachable except for the one that also occurs in the clonal model. The evolution of assortative mating during evolutionary branching can be interpreted as the initial phase of sympatric speciation with phenotypic divergence and partial reproductive isolation.