Statistics for Microsatellite Variation Based on Coalescence

The stepwise mutation model, which was at one time chiefly of interest in studying the evolution of protein charge-states, has recently undergone a resurgence of interest with the new popularity of microsatellites as phylogenetic markers. In this paper we describe a method which makes it possible to transfer many population genetics results from the standard infinite sites model to the stepwise mutation model. We study in detail the properties of pairwise differences in microsatellite repeat number between randomly chosen alleles. We show that the problem of finding the expected squared distance between two individuals and finding the variance of the squared distance can be reduced for a wide range of population models to finding the mean and mean square coalescence times. In many cases the distributions of coalescence times have already been studied for infinite site problems. In this study we show how to calculate these quantities for several population models. We also calculate the variance in mean squared pairwise distance (an estimator of mutation rate × population size) for samples of arbitrary size and show that this variance does not approach zero as the sample size increases. We can also use our method to study alleles at linked microsatellite loci. We suggest a metric which quantifies the level of association between loci—effectively a measure of linkage disequilibrium. It is shown that there can be linkage disequilibrium between partially linked loci at mutation–drift equilibrium.     

Likelihoods and simulation methods for a class of nonneutral population genetics models

Methods for simulating samples and sample statistics, under mutation-selection-drift equilibrium for a class of nonneutral population genetics models, and for evaluating the likelihood surface, in selection and mutation parameters, are developed and applied for observed data. The methods apply to large populations in settings in which selection is weak, in the sense that selection intensities, like mutation rates, are of the order of the inverse of the population size. General diploid selection is allowed, but the approach is currently restricted to models, such as the infinite alleles model and certain K-models, in which the type of a mutant allele does not depend on the type of its progenitor allele. The simulation methods have considerable advantages over available alternatives. No other methods currently seem practicable for approximating likelihood surfaces.     

Simulation results with stepwise mutation model and their interpretations

Summary Monte Carlo simulations are performed to compare the predictions based on the two presently used theoretical models for studying genetic variations in natural populations, the infinite allele model and the stepwise mutation model. Distribution of heterozygosity is noticed to be similar under these models until the product of population size and mutation rate is large. It is seen that electromorphs with high population frequency usually contain older alleles (at the codon level) than an electromorph of low population frequency. The interpretations of these results in explaining the allelic variations at electrophoretic level is also discussed.Research supported by U.S. Public Health Service General Research Support Grant 5 SO 5 RR 07148 from the University of Texas Health Science Center, Graduate School of Biomedical Sciences, Houston, Texas     

分子生态学重要概念——遗传距离及其测度的理论研究概况

综述了遗传距离的概念、背景,有关遗传距离的几种基本的突变模型以及和遗传距离有关的参量和几种常用统计量,指出在处理蛋白质数据、分子数据以及序列数据时,如何选择相应的统计量和可用的软件包,同时还着重指明了各种模型的假设前提,为处理实际的蛋白质或分子数据时选择合适的模型,和对数据的最终解释提供一些帮助。     

Evaluating ring-tailed lemurs (Lemur catta) from southwestern Madagascar for a genetic population bottleneck

In light of historical and recent anthropogenic influences on Malagasy primate populations, in this study ring-tailed lemur (Lemur catta) samples from two sites in southwestern Madagascar, Beza Mahafaly Special Reserve (BMSR) and Tsimanampetsotsa National Park (TNP), were evaluated for the genetic signature of a population bottleneck. A total of 45 individuals (20 from BMSR and 25 from TNP) were genotyped at seven microsatellite loci. Three methods were used to evaluate these populations for evidence of a historical bottleneck: M-ratio, mode-shift, and heterozygosity excess tests. Three mutation models were used for heterozygosity excess tests: the stepwise mutation model (SMM), two-phase model (TPM), and infinite allele model (IAM). M-ratio estimations indicated a potential bottleneck in both populations under some conditions. Although mode-shift tests did not strongly indicate a population bottleneck in the recent historical past when samples from all individuals were included, a female-only analysis indicated a potential bottleneck in TNP. Heterozygosity excess was indicated under two of the three mutation models (IAM and TPM), with TNP showing stronger evidence of heterozygosity excess than BMSR. Taken together, these results suggest that a bottleneck may have occurred among L. catta in southwestern Madagascar in the recent past. Given knowledge of how current major stochastic climatic events and human-induced change can negatively impact extant lemur populations, it is reasonable that comparable events in the historical past could have caused a population bottleneck. This evaluation additionally functions to highlight the continuing environmental and anthropogenic challenges faced by lemurs in southwestern Madagascar.     

A population genetical model for sequence evolution under multiple types of mutation

DNA sequencing and restriction mapping provide us with information on DNA sequence evolution within populations, from which the phylogenetic relationships among the sequences can be inferred. Mutations such as base substitutions, deletions, insertions and transposable element insertions can be identified in each sequence. Theoretical study of this type of sequence evolution has been initiated recently. In this paper, population genetical models for sequence evolution under multiple types of mutation are developed. Models of infinite population size with neutral mutation, infinite population size with deleterious mutation and finite population size with neutral mutation are considered.     

梅花鹿的微卫星多态性及种群的遗传结构

梅花鹿是我国极度濒危的鹿科动物,其野生种群已濒临灭绝。为了探讨我国野生梅花鹿的保护和管理对策,我们选用了16 个微卫星位点检测来自东北、四川、江西和浙江种群的122 份样品,以此分析我国野生梅花鹿种群的遗传多样性和遗传结构。费希尔确切性检验表明,四个种群中均存在偏离哈迪- 温伯格平衡的现象。与其它的濒危动物相比,中国梅花鹿有着相对较高的遗传多样性:每个位点的平均等位基因数为4 个,平均期望杂合度为0. 559。四个种群中,东北种群拥有最高的平均等位基因数(n = 3. 688),东北种群、四川种群、江西种群以及浙江种群的平均期望杂合度分别为0. 584,0. 477,0. 585 和0.589,它们之间不存在显著的差异。同时,利用逐步突变模型、双相突变模型和无限等位基因突变模型检测了种群的瓶颈效应,结果表明:除四川种群外,其他种群在近期内都经历过遗传瓶颈。费希尔确切性检验及配对样品F ST 的结果均表明:四个梅花鹿种群间存在显著的遗传分化(P < 0. 001)。因此,我们建议将我国梅花鹿的野生种群划分为4 个管理单元进行保护和
管理。     

Microsatellite polymorphism and population subdivision in natural populations of European sea bass Dicentrarchus labrax (Linnaeus, 1758)

Polymorphism of microsatellite markers was used to study the genetic variability and structure in natural populations of European sea bass Dicentrarchus labrax. The data consisted of six microsatellite loci analysed for 172 individuals from three samples collected in the Golfe-du-Lion (France) and one sample collected in the Golfo-de-Valencia (Spain). Our goals were (i) to assess the level of genetic variability as revealed by these markers, (ii) to estimate the genetic differentiation among natural populations within a restricted area, and (iii) to evaluate how microsatellite loci fit the predictions of the two most widely used mutation models (the infinite allele model and the stepwise mutation model). As expected, our results indicate that the genetic polymorphism is very high when compared with previously used genetic markers, the mean expected heterozygosity per locus ranging between 0.69 and 0.93. We also found that all loci but one fitted the infinite allele model better. Using this model as a lower limit, we could extrapolate from the observed diversity effective population sizes on the order of 35 000 individuals. Our results also suggest that there may be a slight genetic differentiation between the two gulfs (F_ST= 0.007, P < 0.05), indicating that the corresponding populations are likely to be dynamically independent. This finding for a species with high dispersal abilities, if confirmed, has important beatings on fish-stock assessment.     

Expected Shannon Entropy and Shannon Differentiation between Subpopulations for Neutral Genes under the Finite Island Model

Shannon entropy H and related measures are increasingly used in molecular ecology and population genetics because (1) unlike measures based on heterozygosity or allele number, these measures weigh alleles in proportion to their population fraction, thus capturing a previously-ignored aspect of allele frequency distributions that may be important in many applications; (2) these measures connect directly to the rich predictive mathematics of information theory; (3) Shannon entropy is completely additive and has an explicitly hierarchical nature; and (4) Shannon entropy-based differentiation measures obey strong monotonicity properties that heterozygosity-based measures lack. We derive simple new expressions for the expected values of the Shannon entropy of the equilibrium allele distribution at a neutral locus in a single isolated population under two models of mutation: the infinite allele model and the stepwise mutation model. Surprisingly, this complex stochastic system for each model has an entropy expressable as a simple combination of well-known mathematical functions. Moreover, entropy- and heterozygosity-based measures for each model are linked by simple relationships that are shown by simulations to be approximately valid even far from equilibrium. We also identify a bridge between the two models of mutation. We apply our approach to subdivided populations which follow the finite island model, obtaining the Shannon entropy of the equilibrium allele distributions of the subpopulations and of the total population. We also derive the expected mutual information and normalized mutual information (“Shannon differentiation”) between subpopulations at equilibrium, and identify the model parameters that determine them. We apply our measures to data from the common starling (Sturnus vulgaris) in Australia. Our measures provide a test for neutrality that is robust to violations of equilibrium assumptions, as verified on real world data from starlings.     

Neutral two-locus multiple allele models with recombination

General formulae for the homozygosity and variance of linkage disequilibrium are derived for neutral, stationary, two-locus multiple allele models where there is a symmetric type of mutation at each locus. Particular cases examined are K allele models, the infinite alleles model, and the stepwise mutation model. The two-locus infinite allele model is examined at the molecular level and a joint probability generating function is found for the number of heterozygous sites at each locus in two randomly chosen gametes.     

New Statistical Tests of Neutrality for DNA Samples from a Population

The purpose of this paper is to develop statistical tests of the neutral model of evolution against a class of alternative models with the common characteristic of having an excess of mutations that occurred a long time ago or a reduction of recent mutations compared to the neutral model. This class of population genetics models include models for structured populations, models with decreasing effective population size and models of selection and mutation balance. Four statistical tests were proposed in this paper for DNA samples from a population. Two of these tests, one new and another a modification of an existing test, are based on EWENS'' sampling formula, and the other two new tests make use of the frequencies of mutations of various classes. Using simulated samples and regression analyses, the critical values of these tests can be computed from regression equations. This approach for computing the critical values of a test was found to be appropriate and quite effective. We examined the powers of these four tests using simulated samples from structured populations, populations with linearly decreasing sizes and models of selection and mutation balance and found that they are more powerful than existing statistical tests of the neutral model of evolution.     

A synthetic theory of molecular evolution

According to the neo-Darwinian view of evolution evolution rate nu depends solely on the environment variation rate gamma, whereas in the non-Darwinian view evolution rate is determined mainly by the mutation rate mu. We have studied two kinds of population genetics models which exhibit both types of evolution in different parametric regions: one is a dynamical model representing infinite population, and the other is a Markov process model representing a nearly monomorphic finite population. In the infinite population model, after proving general time-derivative and mu-derivative formulas for the population average of quantitative traits, we show that if the mutation rate is adaptively determined, mu must be larger than nu in the stationary state. Loads of evolution are obtained in both regions. A high evolution rate such as nu = 1 per genome per generation is consistent with Haldane's value of tolerable load if and only if the functional constraint is not large and selection is weak, independent of whether the evolution is neo-Darwinian or non-Darwinian. As the selection intensity increases, nu is shown to change discontinuously from nearly mu to gamma at the transition point. In the finite population model, the transition of v is not discontinuous, but is very steep. On the other hand, no steep change of polymorphism takes place at the transition point. The steepness of the transition in our model suggests that real molecular evolution can be divided into either neo-Darwinian or non-Darwinian,and that the intermediate type of evolution is rather rare.     

Steady state of homozygosity and Gst for the island model

We examine homozygosity and G(st) for a subdivided population governed by the finite island model. Assuming an infinite allele model and strong mutation we show that the steady state distributions of G(st) and homozygosity have asymptotic expansions in the mutation rate. We use this observation to derive asymptotic expansions for various moments of homozygosity and to derive rigorous formulas for the mean and variance of G(st). We show that G(st) approximately 1/(1+2Nm), similarly to the well known formula of Wright for the infinite island model, and that the variance of G(st) goes to zero as mutation increases.     

The Structure of Genealogies and the Distribution of Fixed Differences between DNA Sequence Samples from Natural Populations

When two samples of DNA sequences are compared, one way in which they may differ is in the presence of fixed differences, which are defined as sites at which all of the sequences in one sample are different from all of the sequences in a second sample. The probability distribution of the number of fixed differences is developed. The theory employs Wright-Fisher genealogies and the infinite sites mutation model. For the case when both samples are drawn randomly from the same population it is found that genealogies permitting fixed differences are very unlikely. Thus the mere presence of fixed differences between samples is statistically significant, even for small samples. The theory is extended to samples from populations that have been separated for some time. The relationship between a simple Poisson distribution of mutations and the distribution of fixed differences is described as a function of the time since populations have been isolated. It is shown how these results may contribute to improved tests of recent balancing or directional selection.     

A Computer Simulation Study of Vntr Population Genetics: Constrained Recombination Rules Out the Infinite Alleles Model

Extensive allelic diversity in variable numbers of tandem repeats (VNTRs) has been discovered in the human genome. For population genetic studies of VNTRs, such as forensic applications, it is important to know whether a neutral mutation-drift balance of VNTR polymorphism can be represented by the infinite alleles model. The assumption of the infinite alleles model that each new mutant is unique is very likely to be violated by unequal sister chromatid exchange (USCE), the primary process believed to generate VNTR mutants. We show that increasing both mutation rates and misalignment constraint for intrachromosomal recombination in a computer simulation model reduces simulated VNTR diversity below the expectations of the infinite alleles model. Maximal constraint, represented as slippage of single repeats, reduces simulated VNTR diversity to levels expected from the stepwise mutation model. Although misalignment rule is the more important variable, mutation rate also has an effect. At moderate rates of USCE, simulated VNTR diversity fluctuates around infinite alleles expectation. However, if rates of USCE are high, as for hypervariable VNTRs, simulated VNTR diversity is consistently lower than predicted by the infinite alleles model. This has been observed for many VNTRs and accounted for by technical problems in distinguishing alleles of neighboring size classes. We use sampling theory to confirm the intrinsically poor fit to the infinite alleles model of both simulated VNTR diversity and observed VNTR polymorphisms sampled from two Papua New Guinean populations.     

Generating samples under a Wright-Fisher neutral model of genetic variation

A Monte Carlo computer program is available to generate samples drawn from a population evolving according to a Wright-Fisher neutral model. The program assumes an infinite-sites model of mutation, and allows recombination, gene conversion, symmetric migration among subpopulations, and a variety of demographic histories. The samples produced can be used to investigate the sampling properties of any sample statistic under these neutral models.     

Equilibrium Values of Measures of Population Subdivision for Stepwise Mutation Processes

Expected values of WRIGHT's F-statistics are functions of probabilities of identity in state. These values may be quite different under an infinite allele model and under stepwise mutation processes such as those occurring at microsatellite loci. However, a relationship between the probability of identity in state in stepwise mutation models and the distribution of coalescence times can be deduced from the relationship between probabilities of identity by descent and the distribution of coalescence times. The values of F(IS) and F(ST) can be computed using this property. Examination of the conditional probability of identity in state given some coalescence time and of the distribution of coalescence times are also useful for explaining the properties of F(IS) and F(ST) at high mutation rate loci, as shown here in an island model of population structure.     

Simultaneous estimation of all the parameters of a stepwise mutation model.

Minisatellite and microsatellite are short tandemly repetitive sequences dispersed in eukaryotic genomes, many of which are highly polymorphic due to copy number variation of the repeats. Because mutation changes copy numbers of the repeat sequences in a generalized stepwise fashion, stepwise mutation models are widely used for studying the dynamics of these loci. We propose a minimum chi-square (MCS) method for simultaneous estimation of all the parameters in a stepwise mutation model and the ancestral allelic type of a sample. The MCS estimator requires knowing the mean number of alleles of a certain size in a sample, which can be estimated using Monte Carlo samples generated by a coalescent algorithm. The method is applied to samples of seven (CA)n repeat loci from eight human populations and one chimpanzee population. The estimated values of parameters suggest that there is a general tendency for microsatellite alleles to expand in size, because (1) each mutation has a slight tendency to cause size increase and (2) the mean size increase is larger than the mean size decrease for a mutation. Our estimates also suggest that most of these CA-repeat loci evolve according to multistep mutation models rather than single-step mutation models. We also introduced several quantities for measuring the quality of the estimation of ancestral allelic type, and it appears that the majority of the estimated ancestral allelic types are reasonably accurate. Implications of our analysis and potential extensions of the method are discussed.SINCE the discovery that a large number of loci with tandemly repeated sequences in human and many eukaryote species are highly polymorphic because of copy number variation of the repeats in different individuals (Jeffreys 1985; Litt and Luty 1989; Weber and May 1989), allele size data from such loci are rapidly becoming the dominant source of genetic markers for genome mapping, forensic testing, and population studies. Loci with repeat sequences longer than 5 bp are generally referred to as minisatellite or variable number tandem repeat loci, and those with repeat sequences between 2 to 5 bp are referred to as microsatellite or short tandem repeat loci (Tautz 1993). Because mutations change the copy number of such loci in a stepwise fashion, rapid accumulation of population samples from minisatellite and microsatellite loci has resurrected the interest of the stepwise mutation model (SMM), which was popular in the 1970s.     

Selection for Increased Mutation Rates with Fertility Differences between Matings

Previous studies of mutation modification have considered models in which selection is a result of viability differences that are sex symmetric. The results of a numerical study of a model in which selection is a result of fertility differences between mated pairs demonstrate that the type of selection to which a population is subject can have a significant impact on the evolution of various parameters of the genetic system. When the fertility of matings between individuals with different genotypes exceeds the fertility of at least some of the matings between individuals with the same genotype, selection may favor increased rates of mutation, in contrast to the results from all existing constant viability models with random mating and infinite population size. Increased mutation rates are most frequently favored when forward and back mutation occur at approximately equal rates and when the modifying locus is loosely linked to the selected locus. We present one example in which selection favors increased rates of mutation even though the selection scheme is reducible to one of differential viability between the sexes.     

On some principles of evolution viewed as a stochastic process

From current stochastic models of population evolution it may be concluded that all populations of limited size will asymptotically die out. This conclusion underestimates the power of genetic evolution. In the model described in this paper, the infinitely long existence of a population of limited size is possible through its increased fitness due to the infinite perfectability of its genetic structure. Definitions of some population features, which generalize variability of known evolving units (mutation), inheritance, selection and ability to widen the range of environmental conditions where existence is possible, are introduced. It is shown that the presence of each of these features in a population of limited size is the condition necessary for its asymptotic non-extinction with positive probability. The necessary and sufficient condition that there be a positive probability that a population will not ultimately become extinct is also obtained.