Parametric identification of mathematical models of population genetics taking into account the geographical dispersion in finite samples

A method for parameter identification of population genetics' mathematical models, taking account of geographical disperse at limited samples of experimental data on mutant frequency values has been developed. The existence of the MLS (method of the least squares) estimations of the models' parameters studied has been proved, zero approach of the looked for estimations found and the iterative procedure of making them precise shown. A means of building up the a posteriori function of probability density of the zero and following approximations of the models' parameters is pointed out. The possibility of application of the proposed method to find estimations of mathematical models' parameters of population genetics, taking account of geographical disperse, has been shown on the particular example.     

Selection theory for selfed progenies

Summary The purpose of this article was to extend the model used to predict selection response with selfed progeny from 2 alleles per locus to a model which is general for number and frequency of alleles at loci. To accomplish this, 4 areas had to be dealt with: 1) simplification of the derivation and calculation of the condensed coefficients of identity; 2) presentation of the genetic variances expressed among and within selfed progenies as linear function of 5 population parameters; 3) presentation of selection response equations for selfed progenies as functions of these 5 population parameters; and 4) to identify a set of progeny to evaluate, such that one might be able to estimate these 5 population parameters.The five population parameters used in predicting gains were the additive genetic variance, the dominance variance, the covariance of additive and homozygous dominance deviations, the variance of the homozygous dominance deviations and a squared inbreeding depression term.Contribution from the Missouri Agricultural Experiment Station. Journal Series No. 9971     

Genetic study of the height and weight process during infancy.

Longitudinal height and weight data from 4649 Dutch twin pairs between birth and 2.5 years of age were analyzed. The data were first summarized into parameters of a polynomial of degree 4 by a mixed-effects procedure. Next, the variation and covariation in the parameters of the growth curve (size at one year of age, growth velocity, deceleration of growth, rate of change in deceleration [i.e., jerk] and rate of change in jerk [i.e., snap]) were decomposed into genetic and nongenetic sources. Additionally, the variation in the estimated size at birth and at 2 years of age interpolated from the polynomial was decomposed into genetic and nongenetic components. Variation in growth was best characterized by a genetic model which included additive genetic, common environmental and specific environmental influences, plus effects of gestational age. The effect of gestational age was largest for size at birth, explaining 39% of the variance. The differences between monozygotic and dizygotic twin correlations were largest for size at 1 and 2 years of age and growth velocity of weight, which suggests that these parameters are more influenced by heritability than size at birth, deceleration and jerk. The percentage of variance explained by additive genetic influences for height at 2 years of age was 52% for females and 58% for males. For weight at 2 years of age, heritability was approximately 58% for both sexes. Variation in snap height for males was also mainly influenced by additive genetic factors, while snap for females was influenced by both additive genetic and common environmental factors. The correlations for the additive genetic and common environmental factors for deceleration and snap are large, indicating that these parameters are almost entirely under control of the same additive genetic and common environmental factors. Female jerk and snap, and also female height at birth and height at 2 years of age, are mostly under control of the same additive genetic factor.     

Estimating Additive and Non-Additive Genetic Variances and Predicting Genetic Merits Using Genome-Wide Dense Single Nucleotide Polymorphism Markers

Non-additive genetic variation is usually ignored when genome-wide markers are used to study the genetic architecture and genomic prediction of complex traits in human, wild life, model organisms or farm animals. However, non-additive genetic effects may have an important contribution to total genetic variation of complex traits. This study presented a genomic BLUP model including additive and non-additive genetic effects, in which additive and non-additive genetic relation matrices were constructed from information of genome-wide dense single nucleotide polymorphism (SNP) markers. In addition, this study for the first time proposed a method to construct dominance relationship matrix using SNP markers and demonstrated it in detail. The proposed model was implemented to investigate the amounts of additive genetic, dominance and epistatic variations, and assessed the accuracy and unbiasedness of genomic predictions for daily gain in pigs. In the analysis of daily gain, four linear models were used: 1) a simple additive genetic model (MA), 2) a model including both additive and additive by additive epistatic genetic effects (MAE), 3) a model including both additive and dominance genetic effects (MAD), and 4) a full model including all three genetic components (MAED). Estimates of narrow-sense heritability were 0.397, 0.373, 0.379 and 0.357 for models MA, MAE, MAD and MAED, respectively. Estimated dominance variance and additive by additive epistatic variance accounted for 5.6% and 9.5% of the total phenotypic variance, respectively. Based on model MAED, the estimate of broad-sense heritability was 0.506. Reliabilities of genomic predicted breeding values for the animals without performance records were 28.5%, 28.8%, 29.2% and 29.5% for models MA, MAE, MAD and MAED, respectively. In addition, models including non-additive genetic effects improved unbiasedness of genomic predictions.     

Combining capture–recapture data and pedigree information to assess heritability of demographic parameters in the wild

Quantitative genetic analyses have been increasingly used to estimate the genetic basis of life‐history traits in natural populations. Imperfect detection of individuals is inherent to studies that monitor populations in the wild, yet it is seldom accounted for by quantitative genetic studies, perhaps leading to flawed inference. To facilitate the inclusion of imperfect detection of individuals in such studies, we develop a method to estimate additive genetic variance and assess heritability for binary traits such as survival, using capture–recapture (CR) data. Our approach combines mixed‐effects CR models with a threshold model to incorporate discrete data in a standard ‘animal model’ approach. We employ Markov chain Monte Carlo sampling in a Bayesian framework to estimate model parameters. We illustrate our approach using data from a wild population of blue tits (Cyanistes caeruleus) and present the first estimate of heritability of adult survival in the wild. In agreement with the prediction that selection should deplete additive genetic variance in fitness, we found that survival had low heritability. Because the detection process is incorporated, capture–recapture animal models (CRAM) provide unbiased quantitative genetics analyses of longitudinal data collected in the wild.     

Consequences of paternally inherited effects on the genetic evaluation of maternal effects

Background

Mixed models are commonly used for the estimation of variance components and genetic evaluation of livestock populations. Some evaluation models include two types of additive genetic effects, direct and maternal. Estimates of variance components obtained with models that account for maternal effects have been the subject of a long-standing controversy about strong negative estimates of the covariance between direct and maternal effects. Genomic imprinting is known to be in some cases statistically confounded with maternal effects. In this study, we analysed the consequences of ignoring paternally inherited effects on the partitioning of genetic variance.

Results

We showed that the existence of paternal parent-of-origin effects can bias the estimation of variance components when maternal effects are included in the evaluation model. Specifically, we demonstrated that adding a constraint on the genetic parameters of a maternal model resulted in correlations between relatives that were the same as those obtained with a model that fits only paternally inherited effects for most pairs of individuals, as in livestock pedigrees. The main consequence is an upward bias in the estimates of the direct and maternal additive genetic variances and a downward bias in the direct-maternal genetic covariance. This was confirmed by a simulation study that investigated five scenarios, with the trait affected by (1) only additive genetic effects, (2) only paternally inherited effects, (3) additive genetic and paternally inherited effects, (4) direct and maternal additive genetic effects and (5) direct and maternal additive genetic plus paternally inherited effects. For each scenario, the existence of a paternally inherited effect not accounted for by the estimation model resulted in a partitioning of the genetic variance according to the predicted pattern. In addition, a model comparison test confirmed that direct and maternal additive models and paternally inherited models provided an equivalent fit.

Conclusions

Ignoring paternally inherited effects in the maternal models for genetic evaluation can lead to a specific pattern of bias in variance component estimates, which may account for the unexpectedly strong negative direct-maternal genetic correlations that are typically reported in the literature.

Electronic supplementary material

The online version of this article (doi:10.1186/s12711-015-0141-5) contains supplementary material, which is available to authorized users.     

The effect of index selection on allele frequencies and future genetic gains when traits are correlated

The application of the selection index in the case of an additive two-trait model in which the genetic effect on each trait is determined by a finite number of loci is examined. Simulation results indicate that the direction of change in the frequency of favourable alleles is not necessarily in the positive direction at all loci when index selection is used as the basis for truncation selection. When the genetic correlation was positive (or favourable with respect to the economic weights), there was little difference (<5%) in genetic gain over 20 generations and no difference in the direction of change in allele frequencies or genetic correlation whether or not updated values for the genetic (co)variances were used in constructing the selection index. However, when the genetic correlation was negative or unfavourable, the effect of using genetic parameters which were not updated had unexpected effects on the allele frequencies and genetic correlation and reduced the genetic gain by a greater amount (< 12%).     

An evaluation of power and type I error of single-nucleotide polymorphism transmission/disequilibrium-based statistical methods under different family structures, missing parental data, and population stratification

Researchers conducting family-based association studies have a wide variety of transmission/disequilibrium (TD)-based methods to choose from, but few guidelines exist in the selection of a particular method to apply to available data. Using a simulation study design, we compared the power and type I error of eight popular TD-based methods under different family structures, frequencies of missing parental data, genetic models, and population stratifications. No method was uniformly most powerful under all conditions, but type I error was appropriate for nearly every test statistic under all conditions. Power varied widely across methods, with a 46.5% difference in power observed between the most powerful and the least powerful method when 50% of families consisted of an affected sib pair and one parent genotyped under an additive genetic model and a 35.2% difference when 50% of families consisted of a single affection-discordant sibling pair without parental genotypes available under an additive genetic model. Methods were generally robust to population stratification, although some slightly less so than others. The choice of a TD-based test statistic should be dependent on the predominant family structure ascertained, the frequency of missing parental genotypes, and the assumed genetic model.     

Development and Application of Genomic Control Methods for Genome-Wide Association Studies Using Non-Additive Models

Genome-wide association studies (GWAS) comprise a powerful tool for mapping genes of complex traits. However, an inflation of the test statistic can occur because of population substructure or cryptic relatedness, which could cause spurious associations. If information on a large number of genetic markers is available, adjusting the analysis results by using the method of genomic control (GC) is possible. GC was originally proposed to correct the Cochran-Armitage additive trend test. For non-additive models, correction has been shown to depend on allele frequencies. Therefore, usage of GC is limited to situations where allele frequencies of null markers and candidate markers are matched. In this work, we extended the capabilities of the GC method for non-additive models, which allows us to use null markers with arbitrary allele frequencies for GC. Analytical expressions for the inflation of a test statistic describing its dependency on allele frequency and several population parameters were obtained for recessive, dominant, and over-dominant models of inheritance. We proposed a method to estimate these required population parameters. Furthermore, we suggested a GC method based on approximation of the correction coefficient by a polynomial of allele frequency and described procedures to correct the genotypic (two degrees of freedom) test for cases when the model of inheritance is unknown. Statistical properties of the described methods were investigated using simulated and real data. We demonstrated that all considered methods were effective in controlling type 1 error in the presence of genetic substructure. The proposed GC methods can be applied to statistical tests for GWAS with various models of inheritance. All methods developed and tested in this work were implemented using R language as a part of the GenABEL package.     

Methodology and Accuracy of Estimation of Quantitative Trait Loci Parameters in a Half-Sib Design Using Maximum Likelihood

Maximum likelihood methods were developed for estimation of the six parameters relating to a marker-linked quantitative trait locus (QTL) segregating in a half-sib design, namely the QTL additive effect, the QTL dominance effect, the population mean, recombination between the marker and the QTL, the population frequency of the QTL alleles, and the within-family residual variance. The method was tested on simulated stochastic data with various family structures under two genetic models. A method for predicting the expected value of the likelihood was also derived and used to predict the lower bound sampling errors of the parameter estimates and the correlations between them. It was found that standard errors and confidence intervals were smallest for the population mean and variance, intermediate for QTL effects and allele frequency, and highest for recombination rate. Correlations among standard errors of the parameter estimates were generally low except for a strong negative correlation (r = -0.9) between the QTL's dominance effect and the population mean, and medium positive and negative correlations between the QTL's additive effect and, respectively, recombination rate (r = 0.5) and residual variance (r = -0.6). The implications for experimental design and method of analysis on power and accuracy of marker-QTL linkage experiments were discussed.     

Fast combinatorial methods to estimate the probability of complex temporal patterns of spikes

 This study presents two efficient algorithms – combinatorial and probabilistic combinatorial methods (CM and PCM) – for estimation of a number of precise patterns of discharges that occur by chance in records of multiple single-unit spike trains. The confidence limits estimated by these methods are in good agreement with different sets of simulated test data as well as with the ad-hoc method. Both combinatorial methods provided a better accuracy than the bootstrap algorithm and in most cases of nonstationary data PCM provided better estimations than the ad-hoc method. Introduction of a jitter for searching patterns with a precision of a few milliseconds and burst filtering may introduce biases in the estimations. Comparison of a new filtering procedure based upon a filtering frequency with previously described schemes of filtering indicates the possibility of using a simple setting which remains accurate over a wide range of parameters. We aim to implement a combination of PCM for estimations of the number of patterns formed by three to seven spikes and CM for higher-order complexities for estimations during experiments in progress. Received: 12 June 1995 / Accepted in revised form: 5 February 1997     

Genetic parameters for growth traits of a Brazilian Bos taurus x Bos indicus beef composite

The genetic analysis of composite data is very complicated, mainly because it is necessary to adjust data to the effects of heterosis and breed complementarity, and because there is usually considerable confounding of these data with several other effects, such as contemporary group effects, breed composition of the animal and maternal breed composition, among others. Data on birth weight (n = 151,083), weaning weight adjusted to 205 days (n = 137,257), yearling weight adjusted to 390 days (n = 61,410), weight gain from weaning to yearling (n = 56,653), and scrotum circumference (n = 23,323) and muscle score (n = 54,770), both adjusted to 390 days, from Bos taurus x Bos indicus composite beef calves born from 1994 to 2003 were analyzed to estimate (co)variance components and genetic parameters of growth traits. The animals belonged to the Montana Tropical program. Estimation was made by three models that approach adjustment to heterozygosis in order to suggest the best model. The RM model included contemporary groups, class of age of dam, outcrossing percentages for direct and maternal effects, and direct and maternal additive genetic breed effects as covariates; the R model was the same as RM, but without additive maternal breed effects, and H was the same as RM, but not considering any additive breed effect. Both R2 values and consistency of genetic parameters indicate that the more complex model (RM), which considers maternal and individual additive genetic breed effect, produces the best estimates when compared to other models. The R model seems to overestimate (co)variance components. The magnitudes of direct and maternal heritability estimates, obtained in this study, would permit genetic improvement for weight and growth traits, as much by selection of direct genetic effects for weight and growth as for the improvement of maternal performance, but in different lineages. Therefore, the correlations between these effects were unfavorable.     

Heritability of Wing Morphology in a Natural Population of Drosophila gouveai

The natural and laboratory heritabilities of a series of parameters related to wing size and shape were estimated in a population of Drosophila gouveai (repleta group) under field and laboratory conditions. A morphometric analysis was done using 17 wing parameters related to wing landmark positions obtained using the method of the best adjustment of an ellipse to the wing edge. Three parameters (thetaA, thetaC and thetaD) showed highly significant heritability in the wild (average 0.61), whereas only wing size (W(SI)) had significant heritability in the laboratory (0.71). The additive genetic variance of most parameters was greater in the wild than in the laboratory. These results showed that some parameters possessed a substantial genetic additive component in their phenotypic variance, and that morphometric parameters of D. gouveai wings are appropriate quantitative markers for assessing morphological differentiation among populations.     

Bayesian inference of genetic parameters for ultrasound scanning traits of Kivircik lambs

Ultrasound scanning traits have been adapted in selection programs in many countries to improve carcass traits for lean meat production. As the genetic parameters of the traits interested are important for breeding programs, the estimation of these parameters was aimed at the present investigation. The estimated parameters were direct and maternal heritability as well as genetic correlations between the studied traits. The traits were backfat thickness (BFT), skin+backfat thickness (SBFT), eye muscle depth (MD) and live weights at the day of scanning (LW). The breed investigated was Kivircik, which has a high quality of meat. Six different multi-trait animal models were fitted to determine the most suitable model for the data using Bayesian approach. Based on deviance information criterion, a model that includes direct additive genetic effects, maternal additive genetic effects, direct maternal genetic covariance and maternal permanent environmental effects revealed to be the most appropriate for the data, and therefore, inferences were built on the results of that model. The direct heritability estimates for BFT, SBFT, MD and LW were 0.26, 0.26, 0.23 and 0.09, whereas the maternal heritability estimates were 0.27, 0.27, 0.24 and 0.20, respectively. Negative genetic correlations were obtained between direct and maternal effects for BFT, SBFT and MD. Both direct and maternal genetic correlations between traits were favorable, whereas BFT–MD and SBFT–MD had negligible direct genetic correlation. The highest direct and maternal genetic correlations were between BFT and SBFT (0.39) and between MD and LW (0.48), respectively. Our results, in general, indicated that maternal effects should be accounted for in estimation of genetic parameters of ultrasound scanning traits in Kivircik lambs, and SBFT can be used as a selection criterion to improve BFT.     

Evaluation of the Best Linear Unbiased Prediction in Mixed Linear Models with Estimated Variance Components by Means of the MSE of Prediction and the Genetic Selection Differential

Prediction in mixed linear models by Henderson 's (1972) BLUP (Best Linear Unbiased Prediction) requires knowledge of the underlying variance/covariance components to have the property ‘best’. In breeding value prediction these parameters are not known, generally. They have to be replaced by estimations and BLUP becomes estimated BLUP (EBLUP). The aim of this investigation was the evaluation of EBLUP with help of a designed simulation experiment. Criteria used for the evaluation were the mean squared error (MSE) and the (genetic) selection differential (GSD). Besides, an idea of the overestimation of the accuracy of EBLUP by the naive MSE approximation based on the MSE formulas of BLUP with variance component estimations instead of unknown parameters is given.     

Analysis of single gene multitrait effects in livestock by the use of Gibbs sampling

The paper presents a method of multivariate data analysis described by a model which involves fixed effects, additive polygenic individual effects and the effects of a major gene. To find the estimates of model parameters, the maximization of likelihood function method is applied. The maximum of likelihood function is computed by the use of the Gibbs sampling approach. In this approach, following the conditional posterior distributions, values of all unknown parameters are generated. On the basis of the obtained samples the marginal posterior densities as well as the estimates of fixed effects, gene frequency, genotypic values, major gene, polygenic and error (co)variances are calculated. A numerical example, supplemented to theoretical considerations, deals with data simulated according to the considered model.     

Covariance between relatives in multibreed populations: additive model

Covariance between relatives in a multibreed population was derived for an additive model with multiple unlinked loci. An efficient algorithm to compute the inverse of the additive genetic covariance matrix is given. For an additive model, the variance for a crossbred individual is a function of the additive variances for the pure breeds, the covariance between parents, and segregation variances. Provided that the variance of a crossbred individual is computed as presented here, the covariance between crossbred relatives can be computed using formulae for purebred populations. For additive traits the inverse of the genotypic covariance matrix given here can be used both to obtain genetic evaluations by best linear unbiased prediction and to estimate genetic parameters by maximum likelihood in multibreed populations. For nonadditive traits, the procedure currently used to analyze multibreed data can be improved using the theory presented here to compute additive covariances together with a suitable approximation for nonadditive covariances.Supported in part by the Illinois Agricultural Experiment Station, Hatch Projects 35-0345 (RLF) and 35-0367 (MG)     

非交叉配子形成体的连锁图谱构建方法

根据非交叉(achiasmatic)遗传模型,提出采用最大似然法计算遗传交换率的方法,同时开发了构建非交叉生物(F2群体)连锁图谱的计算机软件。通过卡方验检可测性连锁分子标记。对于无交叉生物现象,采用蒙特卡洛模拟技术,对交叉(chiasmatic)和非交叉两个遗传模型遗传交换率的估计值和作图效率进行了比较。模拟结果表明,非交叉模型能提供无偏的估计值,而交叉模型则只有实际值的一半。在所有同等的条件下,基于非交叉模型的作图效率均高于基于交叉模型(无校正)的作图效率。对于非交叉配子形成体,采用基于非交叉模型的交换率计算方法能获得理想的作图效率。     

Influence of the Interaction for Fitness on the Population Structure

The incidence of non-allelic interaction for fitness in a diallelic digenic model is analyzed. Three types of interaction between the fitnesses are considered, namely: i) additive, ii) exponential and, iii) logarithmic. The model of MATHER and JINKS (1970) is applied in the computation of the interaction parameters. The theory of finite absorbing Markov chains (FELLER, 1968) is utilized in the representation of the genotypic structure of the selfing population over time. Expressions depending on the parameters of interaction are obtained for the mean number of generations that the process of genetic information flow delays in transitions among transient states. Also, a numerical example is studied in which the mean absorption times and the variances are calculated. Finally, the impact of fitness interaction on the genetic diversity of some populations is analyzed through the calculation of the genetic information neg-entropy (BRILLOUIN, 1959).     

Redistribution of gene frequency and changes of genetic variation following a bottleneck in population size

Although the distribution of frequencies of genes influencing quantitative traits is important to our understanding of their genetic basis and their evolution, direct information from laboratory experiments is very limited. In theory, different models of selection and mutation generate different predictions of frequency distributions. When a large population at mutation-selection balance passes through a rapid bottleneck in size, the frequency distribution of genes is dramatically altered, causing changes in observable quantities such as the mean and variance of quantitative traits. We investigate the gene frequency distribution of a population at mutation-selection balance under a joint-effect model of real stabilizing and pleiotropic selection and its redistribution and thus changes of the genetic properties of metric and fitness traits after the population passes a rapid bottleneck and expands in size. If all genes that affect the trait are neutral with respect to fitness, the additive genetic variance (VA) is always reduced by a bottleneck in population size, regardless of their degree of dominance. For genes that have been under selection, VA increases following a bottleneck if they are (partially) recessive, while the dominance variance increases substantially for any degree of dominance. With typical estimates of mutation parameters, the joint-effect model can explain data from laboratory experiments on the effect of bottlenecking on fitness and morphological traits, providing further support for it as a plausible mechanism for maintenance of quantitative genetic variation.