So far in this course we have dealt entirely with the evolution of characters that are controlled by simple Mendelian inheritance at a single locus. There are notes on the course website about gametic disequilibrium and how allele frequencies change at two loci simultaneously, but we didn’t discuss them. In every example we’ve considered we’ve imagined that we could understand something about evolution by examining the evolution of a single gene. That’s the domain of classical population genetics. For the next few weeks we’re going to be exploring a field that’s actually older than classical population genetics, although the approach we’ll be taking to it involves the use of population genetic machinery. If you know a little about the history of evolutionary biology, you may know that after the rediscovery of Mendel’s work in 1900 there was a heated debate between the “biometricians” (e.g., Galton and Pearson) and the “Mendelians” (e.g., de Vries, Correns, Bateson, and Morgan). Biometricians asserted that the really important variation in evolution didn’t follow Mendelian rules. Height, weight, skin color, and similar traits seemed to

Human biochemical genetics

Recent advances in molecular genetic techniques will make dense marker maps available and genotyping many individuals for these markers feasible. Here we attempted to estimate the effects of ∼50,000 marker haplotypes simultaneously from a limited number of phenotypic records. A genome of 1000 cM was simulated with a marker spacing of 1 cM. The markers surrounding every 1-cM region were combined into marker haplotypes. Due to finite population size (Ne = 100), the marker haplotypes were in linkage disequilibrium with the QTL located between the markers. Using least squares, all haplotype effects could not be estimated simultaneously. When only the biggest effects were included, they were overestimated and the accuracy of predicting genetic values of the offspring of the recorded animals was only 0.32. Best linear unbiased prediction of haplotype effects assumed equal variances associated to each 1-cM chromosomal segment, which yielded an accuracy of 0.73, although this assumption was far from true. Bayesian methods that assumed a prior distribution of the variance associated with each chromosome segment increased this accuracy to 0.85, even when the prior was not correct. It was concluded that selection on genetic values predicted from markers could substantially increase the rate of genetic gain in animals and plants, especially if combined with reproductive techniques to shorten the generation interval.

https://www.genetics.org/content/genetics/157/4/1819.full.pdf

Prediction of Total Genetic Value Using Genome-Wide Dense Marker Maps

For most human complex diseases and traits, SNPs identified by genome-wide association studies (GWAS) explain only a small fraction of the heritability. Here we report a user-friendly software tool called genome-wide complex trait analysis (GCTA), which was developed based on a method we recently developed to address the “missing heritability” problem. GCTA estimates the variance explained by all the SNPs on a chromosome or on the whole genome for a complex trait rather than testing the association of any particular SNP to the trait. We introduce GCTA's five main functions: data management, estimation of the genetic relationships from SNPs, mixed linear model analysis of variance explained by the SNPs, estimation of the linkage disequilibrium structure, and GWAS simulation. We focus on the function of estimating the variance explained by all the SNPs on the X chromosome and testing the hypotheses of dosage compensation. The GCTA software is a versatile tool to estimate and partition complex trait variation with large GWAS data sets.

https://www.cell.com/ajhg/pdf/S0002-9297(10)00598-7.pdf

GCTA: a tool for genome-wide complex trait analysis.

Summary: Association analyses that exploit the natural diversity of a genome to map at very high resolutions are becoming increasingly important. In most studies, however, researchers must contend with the confounding effects of both population and family structure. TASSEL (Trait Analysis by aSSociation, Evolution and Linkage) implements general linear model and mixed linear model approaches for controlling population and family structure. For result interpretation, the program allows for linkage disequilibrium statistics to be calculated and visualized graphically. Database browsing and data importation is facilitated by integrated middleware. Other features include analyzing insertions/deletions, calculating diversity statistics, integration of phenotypic and genotypic data, imputing missing data and calculating principal components. Availability: The TASSEL executable, user manual, example data sets and tutorial document are freely available at http://www. maizegenetics.net/tassel. The source code for TASSEL can be found at http://sourceforge.net/projects/tassel.

TASSEL: software for association mapping of complex traits in diverse samples

Generalized linear mixed models provide a flexible framework for modeling a range of data, although with non-Gaussian response variables the likelihood cannot be obtained in closed form. Markov chain Monte Carlo methods solve this problem by sampling from a series of simpler conditional distributions that can be evaluated. The R package MCMCglmm implements such an algorithm for a range of model fitting problems. More than one response variable can be analyzed simultaneously, and these variables are allowed to follow Gaussian, Poisson, multi(bi)nominal, exponential, zero-inflated and censored distributions. A range of variance structures are permitted for the random effects, including interactions with categorical or continuous variables (i.e., random regression), and more complicated variance structures that arise through shared ancestry, either through a pedigree or through a phylogeny. Missing values are permitted in the response variable(s) and data can be known up to some level of measurement error as in meta-analysis. All simu- lation is done in C/ C++ using the CSparse library for sparse linear systems.

https://cran.r-project.org/web//packages/MCMCglmm/vignettes/Overview.pdf

MCMC Methods for Multi-Response Generalized Linear Mixed Models: The MCMCglmm R Package

Mixed linear models are assumed in most animal breeding applications. Convenient methods for computing BLUE of the estimable linear functions of the fixed elements of the model and for computing best linear unbiased predictions of the random elements of the model have been available. Most data available to animal breeders, however, do not meet the usual requirements of random sampling, the problem being that the data arise either from selection experiments or from breeders' herds which are undergoing selection. Consequently, the usual methods are likely to yield biased estimates and predictions. Methods for dealing with such data are presented in this paper.

Best linear unbiased estimation and prediction under a selection model.

Applications of linear models in animal breeding

The theory of variance component analysis has been discussed recently by Crump (1946, 1951) and by Eisenhart (1947). These papers and, indeed, most of the published works on estimating variance components deal with the one-way classification, with "nested" classifications, and with factorial classifications having equal subclass numbers. Also most papers on this subject are concerned with what Eisenhart (1947) has called Model II; that is, all elements of the linear model save gi are regarded as random variables. In the above cases, estimation of variance components is usually accomplished by computing the mean squares in the standard analysis of variance, equating these mean squares to their expectations, and solving for the unknown variances. These techniques are described in many statistical textbooks. Unfortunately, research workers in some of those fields in which much use is made of variance component estimates are unable to obtain data which have the above described characteristics. This is particularly true in those fields in which survey data must be used or where, even in a well-planned experiment, the subclasses are of quite unequal size due, for example, to differences in litter numbers. Also,

Estimation of variance and covariance components

Sire evaluation and genetic trends

The inverse of a numerator relationship matrix is needed for best linear unbiased prediction of breeding values. The purpose of this paper to is present a rapid and simple method for computation of the elements of this inverse without computing the relationship matrix itself. The method is particularly useful in noninbred populations but is much faster than the conventional method in the presence of inbreeding.

C.R. Henderson

Papers

Best linear unbiased estimation and prediction under a selection model.

Applications of linear models in animal breeding

Estimation of variance and covariance components

Sire evaluation and genetic trends

A simple method for computing the inverse of a numerator relationship matrix used in prediction of breeding values