Arbitrary Number Of Alleles Research Articles

The pseudoentropy of allele frequency trajectories, the persistence of variation, and the effective population size

To concisely describe how genetic variation, at individual loci or across whole genomes, changes over time, and to follow transitory allelic changes, we introduce a quantity related to entropy, that we term pseudoentropy. This quantity emerges in a diffusion analysis of the mean time a mutation segregates in a population. For a neutral locus with an arbitrary number of alleles, the mean time of segregation is generally proportional to the pseudoentropy of initial allele frequencies. After the initial time point, pseudoentropy generally decreases, but other behaviours are possible, depending on the genetic diversity and selective forces present.For a biallelic locus, pseudoentropy and entropy coincide, but they are distinct quantities with more than two alleles. Thus for populations with multiple biallelic loci, the language of entropy suffices. Then entropy, combined across loci, serves as a concise description of genetic variation. We used individual based simulations to explore how this entropy behaves under different evolutionary scenarios. In agreement with predictions, the entropy associated with unlinked neutral loci decreases over time. However, deviations from free recombination and neutrality have clear and informative effects on the entropy’s behaviour over time.Analysis of publicly available data of a natural D. melanogaster population, that had been sampled over seven years, using a sliding-window approach, yielded considerable variation in entropy trajectories of different genomic regions. These mostly follow a pattern that suggests a substantial effective population size and a limited effect of positive selection on genome-wide diversity over short time scales.

Gene-mating dynamic evolution theory II: global stability of N-gender-mating polyploid systems.

Extending the previous 2-gender dioecious diploid gene-mating evolution model, we attempt to answer "whether the Hardy-Weinberg global stability and the exact analytic dynamical solutions can be found in the generalized N-gender N-polyploid gene-mating system with arbitrary number of alleles?" For a 2-gender gene-mating evolution model, a pair of male and female determines the trait of their offspring. Each of the pair contributes one inherited character, the allele, to combine into the genotype of their offspring. Hence, for an N-gender N-polypoid gene-mating model, each of N different genders contributes one allele to combine into the genotype of their offspring. We exactly solve the analytic solution of N-gender-mating $(n+1)$-alleles governing highly nonlinear coupled differential equations in the genotype frequency parameter space for any positive integer N and $n$. For an analogy, the 2-gender to N-gender gene-mating equation generalization is analogs to the 2-body collision to the N-body collision Boltzmann equations with continuous distribution functions of discretized variables instead of continuous variables. We find their globally stable solution as a continuous manifold and find no chaos. Our solution implies that the Laws of Nature, under our assumptions, provide no obstruction and no chaos to support an N-gender gene-mating stable system.

Genotype Calling from Population-Genomic Sequencing Data.

Genotype calling plays important roles in population-genomic studies, which have been greatly accelerated by sequencing technologies. To take full advantage of the resultant information, we have developed maximum-likelihood (ML) methods for calling genotypes from high-throughput sequencing data. As the statistical uncertainties associated with sequencing data depend on depths of coverage, we have developed two types of genotype callers. One approach is appropriate for low-coverage sequencing data, and incorporates population-level information on genotype frequencies and error rates pre-estimated by an ML method. Performance evaluation using computer simulations and human data shows that the proposed framework yields less biased estimates of allele frequencies and more accurate genotype calls than current widely used methods. Another type of genotype caller applies to high-coverage sequencing data, requires no prior genotype-frequency estimates, and makes no assumption on the number of alleles at a polymorphic site. Using computer simulations, we determine the depth of coverage necessary to accurately characterize polymorphisms using this second method. We applied the proposed method to high-coverage (mean 18×) sequencing data of 83 clones from a population of Daphnia pulex. The results show that the proposed method enables conservative and reasonably powerful detection of polymorphisms with arbitrary numbers of alleles. We have extended the proposed method to the analysis of genomic data for polyploid organisms, showing that calling accurate polyploid genotypes requires much higher coverage than diploid genotypes.

Mathematical models in genetics.

In this study, we present some of the basic ideas of population genetics. The founders of population genetics are R.A. Fisher, S. Wright, and J. B.S. Haldane. They, not only developed almost all the basic theory associated with genetics, but they also initiated multiple experiments in support of their theories. One of the first significant insights, which are a result of the Hardy–Weinberg law, is Mendelian inheritance preserves genetic variation on which the natural selection acts. We will limit to simple models formulated in terms of differential equations. Some of those differential equations are nonlinear and thus emphasize issues such as the stability of the fixed points and time scales on which those equations operate. First, we consider the classic case when selection acts on diploid locus at which wу can get arbitrary number of alleles. Then, we consider summaries that include recombination and selection at multiple loci. Also, we discuss the evolution of quantitative traits. In this case, the theory is formulated in respect of directly measurable quantities. Special cases of this theory have been successfully used for many decades in plants and animals breeding.

Inferring the evolutionary history of outcrossing populations through computing a multiallelic linkage–linkage disequilibrium map

Summary Linkage disequilibrium (LD), the non‐random association of alleles at different loci, has been used as an important parameter to study the genetic diversity and evolutionary history of natural populations. A joint analysis of LD with the linkage of the same marker pair has proven to gain more insight into the genetic signature of population diversification than LD analysis alone. We develop a unifying framework for simultaneously estimating the linkage and LD across pairs of multiallelic markers. The framework has particular power to construct the LD map from any markers with an arbitrary number of alleles per locus. We provide an efficient strategy to manipulate disequilibrium parameters whose number increases exponentially with the number of alleles. The model was tested through extensive simulation studies and validated by analysing a real marker data set from a population genetic research project of euphrates poplar, a desert tree, distributed in the north‐western China. For widespread undomesticated natural populations, compared with biallelic markers, multiallelic markers with a high level of polymorphism are more powerful to study their genetic structure and organization of an outcrossing population. The model developed will potentially have an immediate implication for population and evolutionary genetic studies.

Bayesian Mapping of Quantitative Trait Loci for Multiple Complex Traits with the Use of Variance Components

Complex traits important for humans are often correlated phenotypically and genetically. Joint mapping of quantitative-trait loci (QTLs) for multiple correlated traits plays an important role in unraveling the genetic architecture of complex traits. Compared with single-trait analysis, joint mapping addresses more questions and has advantages for power of QTL detection and precision of parameter estimation. Some statistical methods have been developed to map QTLs underlying multiple traits, most of which are based on maximum-likelihood methods. We develop here a multivariate version of the Bayes methodology for joint mapping of QTLs, using the Markov chain-Monte Carlo (MCMC) algorithm. We adopt a variance-components method to model complex traits in outbred populations (e.g., humans). The method is robust, can deal with an arbitrary number of alleles with arbitrary patterns of gene actions (such as additive and dominant), and allows for multiple phenotype data of various types in the joint analysis (e.g., multiple continuous traits and mixtures of continuous traits and discrete traits). Under a Bayesian framework, parameters--including the number of QTLs--are estimated on the basis of their marginal posterior samples, which are generated through two samplers, the Gibbs sampler and the reversible-jump MCMC. In addition, we calculate the Bayes factor related to each identified QTL, to test coincident linkage versus pleiotropy. The performance of our method is evaluated in simulations with full-sib families. The results show that our proposed Bayesian joint-mapping method performs well for mapping multiple QTLs in situations of either bivariate continuous traits or mixed data types. Compared with the analysis for each trait separately, Bayesian joint mapping improves statistical power, provides stronger evidence of QTL detection, and increases precision in estimation of parameter and QTL position. We also applied the proposed method to a set of real data and detected a coincident linkage responsible for determining bone mineral density and areal bone size of wrist in humans.

The exact test for cytonuclear disequilibria.

We extend the analysis of the statistical properties of cytonuclear disequilibria in two major ways. First, we develop the asymptotic sampling theory for the nonrandom associations between the alleles at a haploid cytoplasmic locus and the alleles and genotypes at a diploid nuclear locus, when there are an arbitrary number of alleles at each marker. This includes the derivation of the maximum likelihood estimators and their sampling variances for each disequilibrium measure, together with simple tests of the null hypothesis of no disequilibrium. In addition to these new asymptotic tests, we provide the first implementation of Fisher's exact test for the genotypic cytonuclear disequilibria and some approximations of the exact test. We also outline an exact test for allelic cytonuclear disequilibria in multiallelic systems. An exact test should be used for data sets when either the marginal frequencies are extreme or the sample size is small. The utility of this new sampling theory is illustrated through applications to recent nuclear-mtDNA and nuclear-cpDNA data sets. The results also apply to population surveys of nuclear loci in conjunction with markers in cytoplasmically inherited microorganisms.

Constraints and normalized measures for cytonuclear disequilibria

The full bounds are derived for cytonuclear disequilibria in two-locus systems with an arbitrary number of alleles at the cytoplasmic and nuclear markers. The associated marginal frequencies constrain the nonrandom associations between cytoplasmic alleles and nuclear genotypes in the same way that the allele frequencies constrain the linkage disequilibrium between two nuclear loci. Additional constraints are imposed on the nonrandom associations between cytoplasmic and nuclear alleles, however, by the marginal frequencies of nuclear genotypes carrying either two or no copies of the associated nuclear allele. These bounds are analysed and used to define normalized measures of cytonuclear disequilibria, whose practical utility is illustrated through applications to two sets of recent nuclear-mitochondrial data.

A deterministic analysis of self-incompatibility alleles

A number of plant species have a self-incompatibility locus that prevents self-fertilization. We 'analyse a deterministic model with an arbitrary number of alleles. We prove that the only polymorphic equilibrium is the one for which all (heterozygous) genotypes are equally frequent, and we prove that all (initially) polymorphic populations converge to this equilibrium.

Inter‐ and Intracultivar Effects of Selection on Heterosis

The effect of selection on heterosis depends on the gene action and the type of selection. Intracultivar (W) selection, full‐sib family selection within each of two cultivars, and intercultivar (R) selection, reciprocal recurrent selection between the two cultivars, were contrasted for their effects on heterosis. The cultivars were considered to be random replicate populations from an ancestral population. The relatedness of individuals within cultivars and the degree of divergence between cultivars were formalized in terms of identity by descent (IBD) measures for two, three, two‐pair, and four genes a locus and joint IBD measures for genes at each of two loci. Quantitative genetic components for a model that is general for additive, dominance, and additive by additive (a ✕ a) effects with an arbitrary number of alleles were defined for the ancestral population, and selection responses were formulated for the cultivars and the hybrid for each selection method with the aid of the IBD measures. A further analysis was made utilizing a biological building block model of gene effects. The contribution of a ✕ a effects was such that they reduced heterosis from W selection, but slightly increased heterosis from R selection. It was concluded that R selection always increased heterosis while W selection could increase or decrease heterosis, but decreased heterosis for a preponderance of the genetic parameters considered.

Separable models for age-structured population genetics

This paper is concerned with the applications of nonlinear age-dependent dynamics to population genetics. Age-structured models are formulated for a single autosomal locus with an arbitrary number of alleles. The following cases are considered: (a) haploid populations with selection and mutation; (b) monoecious diploid populations with or without mutation reproducing by self-fertilization of by two types of random mating. The diploid models do not deal with selection. For these cases the genic and genotypic frequencies evolve towards time-persistent forms, whether the total population size tends towards exponential growth or not.

Separable models for age-structured population genetics

This paper is concerned with the applications of nonlinear age-dependent dynamics to population genetics. Age-structured models are formulated for a single autosomal locus with an arbitrary number of alleles. The following cases are considered: a) haploid populations with selection and mutation; b) monoecious diploid populations with or without mutation reproducing by self-fertilization or by two types of random mating. The diploid models do not deal with selection. For these cases the genic and genotypic frequencies evolve towards time-persistent forms, whether the total population size tends towards exponential growth or not.

Evolution and maintenance of quantitative genetic variation by mutations.

The genotypic variance within, sigma 2w, and between, sigma 2b, random mating populations and rates and times for convergence to equilibrium values from different founder populations are formulated for an additive genetic model with an arbitrary number of alleles k, number of loci m, population size N, and mutation rate u, with unequal mutation rates for alleles. As a base of reference, the additive variance sigma 2a in an infinite equilibrium population is used. sigma 2a increases as k increases and decreases with variation in the mutation rates. Both transitional and equilibrium values of the variance within populations could be expressed as sigma 2w = (1 - theta)sigma 2a, where theta is the coancestry with mutations of individuals within populations. Thus, rates of convergence and evolutionary times are a function of those for theta, which involves both N and u. When the founder population is fixed, very long times are required to obtain a perceptible increase in sigma 2w and equilibrium values of sigma 2w are very small when 4Nu less than or equal to 10(-1). The variance between populations can be expressed as sigma 2b = 2 theta sigma 2a when the founder population is an infinite equilibrium population, and as sigma 2b = 2(theta - alpha)sigma 2a when the founder population is fixed, where alpha is a function only of u. In this latter case, rates of divergence, while affected by both N and u, are dominated by u and asymptotically a function of u only. With u = 10(-5), very long times (10(3) generations) are required for any perceptible divergence, even for N = 1-10. At equilibrium, most of the variance is between small populations and within very large populations. Migration increases the variance within populations and decreases the variance between populations.

On the maintenance of gynodioecy: Lewis' result extended

The classical one-locus model of the maintenance of male sterile plants among hermaphrodites, first published by Lewis (1941) , is slightly modified and generalized to the case with an arbitrary number of sex determining loci, each with an arbitrary numbers of alleles. An essential feature of the model is that only fitness differences between sex phenotypes are considered. It is shown that the results of our population-genetic model differ from those of a phenotypic model, applied to this situation by Lloyd (1977) . Further, Lewis' model was extended to include overlapping generations so that the impact of differences between sexes in adult survival, i.e. in the number of seasons available for reproduction, on the frequency of male steriles could be studied. Small differences in adult survival may greatly affect sex ratio, and frequencies of male steriles higher than 50% are possible. The possibility of such differences has seldom been taken into account in experimental studies, although the majority of gynodioecious species have a perennial life habit. The problem in detecting survival rate differences among adults, especially in long lived species, is discussed.

Modifiers of mutation rate: A general reduction principle

A deterministic two-locus population genetic model with random mating is studied. The first locus, with two alleles, is subject to mutation and arbitrary viability selection. The second locus, with an arbitrary number of alleles, controls the mutation at the first locus. A class of viability-analogous Hardy-Weinberg equilibria is analyzed in which the selected gene and the modifier locus are in linkage equilibrium. It is shown that at these equilibria a reduction principle for the success of new mutation-modifying alleles is valid. A new allele at the modifier locus succeeds if its marginal average mutation rate is less than the mean mutation rate of the resident modifier allele evaluated at the equilibrium. Internal stability properties of these equilibria are also described.

An evolutionary reduction principle for genetic modifiers.

The joint evolution of major genes under viability selection and a modifier locus that controls recombination between the major genes, mutation at the major gene, or migration between two demes is studied. The modifying locus is selectively neutral and may have an arbitrary number of alleles. For each case a class of polymorphic equilibria exists in which the frequencies of the modifying alleles are those computed by assuming that the recombination, mutation, or migration rates were viabilities and in which the major and modifier loci are not statistically associated. These are called viability-analogous Hardy-Weinberg (VAHW) equilibria. A new allele introduced near these equilibria will enter the population if its marginal average rate of recombination, mutation, or migration (whichever applies) is less than the population average prior to its introduction. Stability properties of these VAHW equilibria are also reported.

Selection Response Based on Selfed Progenies1

A quantitative genetic modelf or arbitrary numbers of alleles and loci with additive, dominance, and additive by additive epistatle effects was employed. Utilizing descent measures and the genetic components, the selection response from selecting among self‐fertilized progenies in generation g, and measuredin selffertilized relatives in generation g', where the last common ancestor was in generation t, was formulated. Permanent response, g' = ∞, led to considerable simplifications. The effect of the number of generations, t, of selfing before testing on the permanent response relative to the response from selecting among homozygotes,t = ∞, was evaluated numerically for a model with two loci, two alleles each. The result varied within a narrow range depending on gene frequencies and effects with a mean relative response of 0.68, 0.83, 0.92, 0.96, 0.98 for t = 1,2,3,4,5, respectively. Selection response based on self‐progeny evaluation was also formulated for the outbred performance of the offspring of randomly mated g's. On a per generation basis, the numerical evaluations gave about the same overall picture for t = 1 and t = 2 and less response for t > 2. If time is of no importance, the response generally increased with t, but there are parameter combinations of gene frequencies and effects for which the response is slightly negative. These responses were compared to those from evaluating full‐sib progenies from crossing pairs of self progenies, which give exactly the same result for t = 1 and t = 2 on a per generation basis. Self‐progeny evaluations generally lead to more selection response than fullsib progenies except for those parameter combinations for which the response is slightly negative, and then it is very small for the full‐sib progeny evaluation. Permanent response is slightly less than immediate response for inbred performance, and varies more with gene frequencies and effects when t is small. Permanent response for outbred performancec an be muchl ess than immediate response depending on the additive by additive variance.

Population genetic theory of kin selection: Multiple alleles at one locus

Exact population genetic models of one-locus sib-to-sib kin selection with an arbitrary number of alleles are studied. First, a natural additive scaling is established for the genotypic value associated with probabilities of performance of altruism. Two classes of polymorphic equilibria are possible, one corresponding to the usual one-locus viability equilibria and the other reflecting the kin-selection assumptions of the model. At both, the covariance between additive genotypic value and genotypic fitness vanish. Further, the sign of this covariance determines the fate of rare alleles introduced near the first class of equilibria. In addition, the covariance explains the differences between Hamilton's rule, which results from Hardy-Weinberg assumptions, and exact initial increase conditions.

Selection in plant populations of effectively infinite size: I. realized genotypic fitnesses

The meaning of genotypic fitness in Mendelian populations consisting of unisexual and/or bisexual individuals is discussed for a model which is applicable toplant populations but not only to these. In order to encourage general application the model is formulated for arbitrary numbers of loci, sex-linked or autosomal, and arbitrary numbers of alleles per locus. The necessity of expressing fitness values in terms of the number of successful gametes is demonstrated. To prevent confusion with other frequently used fitness concepts, the term “realized genotypic fitness” is introduced. Such fitness values have the general advantage that they determine directly the gametic frequencies in the next generation. It is shown that realized genotypic fitness is frequency-independent only for exceptional cases. Necessary and sufficient conditions for the occurrence of such casesare derived. The mode of dependence of realized fitness on genotypic frequencies is illustrated: along straight lines passing through the state space of genotypic structures, realized genotypic fitness changes monotonically. Hence the conclusion that extreme fitness values are not assumed when all genotypic frequencies are positive. It was possible to find necessary and sufficient conditions for one genotype to be superior in realized fitness to another genotype for all genotypic frequencies (frequency-independent ranking).

Exclusion probabilities obtainable by biochemical polymorphisms in dogs

SummaryGeneral formulae are given to calculate the exclusion probabilities in false paternity and parentage cases by means of gene loci with an arbitrary number of alleles whereas in paternity cases an arbitrary number of offspring per litter is considered additionally.By aid of these formulae and on the basis of the allele frequencies of four blood protein and enzyme systems the probabilities of excluding incorrect paternity and parentage are calculated in seven German dog breeds. The results are tabulated and discussed.It can be shown that the exclusion probability in false paternity cases increases distinctly with an increasing number of offspring per litter and its maximum is nearly attained if 5 offspring are examined. Therefore it is of value to consider entire litters in paternity controls in dogs.