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

Statistics for Spatial Data.

Genetics and analysis of quantitative traits

To feed the several billion people living on this planet, the production of high-quality food must increase with reduced inputs, but this accomplishment will be particularly challenging in the face of global environmental change. Plant breeders need to focus on traits with the greatest potential to increase yield. Hence, new technologies must be developed to accelerate breeding through improving genotyping and phenotyping methods and by increasing the available genetic diversity in breeding germplasm. The most gain will come from delivering these technologies in developing countries, but the technologies will have to be economically accessible and readily disseminated. Crop improvement through breeding brings immense value relative to investment and offers an effective approach to improving food security.

https://science.sciencemag.org/content/sci/327/5967/818.full.pdf

Breeding Technologies to Increase Crop Production in a Changing World

LEARNING The objectives of BIOS 781 are to present: OBJECTIVES: 1. basic population and quantitative genetic principles, including classical genetics, chromosomal theory of inheritance, and meiotic recombination 2. an exposure to QTL mapping methods of complex quantitative traits and linkage methods to detect co-segregation with disease 3. methods for assessing marker-disease linkage disequilibrium, including case-control approaches 4. methods for genome-wide association and stratification control.

Genetic Data Analysis

Best linear unbiased prediction (BLUP) is a standard method for estimating random effects of a mixed model. This method was originally developed in animal breeding for estimation of breeding values and is now widely used in many areas of research. It does not, however, seem to have gained the same popularity in plant breeding and variety testing as it has in animal breeding. In plants, application of mixed models with random genetic effects has up until recently been mainly restricted to the estimation of genetic and non- genetic components of variance, whereas estimation of genotypic values is mostly based on a model with fixed effects. This paper reviews recent developments in the application of BLUP in plant breeding and variety testing. These include the use of pedigree information to model and exploit genetic correlation among relatives and the use of flexible variance-covariance structures for genotype-by-environment interaction. We demonstrate that BLUP has good predictive accuracy compared to other procedures. While pedi- gree information is often included via the so-called numerator relationship matrix ðAÞ, we stress that it is frequently straightforward to exploit the same infor- mation by a simple mixed model without explicit reference to the A-matrix.

https://www.esalq.usp.br/departamentos/lce/arquivos/aulas/2012/LCE5872/Piepho%20et%20al.,%202008.pdf

BLUP for phenotypic selection in plant breeding and variety testing

Heritability is often used by plant breeders and geneticists as a measure of precision of a trial or a series of trials. Its main use is for computing the response to selection. Most formulas proposed for calculating heritability implicitly assume balanced data and independent genotypic effects. Both of these assumptions are often violated in plant breeding trials. This article proposes a simulation-based approach to tackle the problem. The key idea is to directly simulate the quantity of interest, e.g., response to selection, rather than trying to approximate it using some ad hoc measure of heritability. The approach is illustrated by three examples.

https://www.genetics.org/content/genetics/177/3/1881.full.pdf

Computing Heritability and Selection Response From Unbalanced Plant Breeding Trials

Association-mapping methods promise to overcome the limitations of linkage-mapping methods. The main objectives of this study were to (i) evaluate various methods for association mapping in the autogamous species wheat using an empirical data set, (ii) determine a marker-based kinship matrix using a restricted maximum-likelihood (REML) estimate of the probability of two alleles at the same locus being identical in state but not identical by descent, and (iii) compare the results of association-mapping approaches based on adjusted entry means (two-step approaches) with the results of approaches in which the phenotypic data analysis and the association analysis were performed in one step (one-step approaches). On the basis of the phenotypic and genotypic data of 303 soft winter wheat ( Triticum aestivum L.) inbreds, various association-mapping methods were evaluated. Spearman's rank correlation between P -values calculated on the basis of one- and two-stage association-mapping methods ranged from 0.63 to 0.93. The mixed-model association-mapping approaches using a kinship matrix estimated by REML are more appropriate for association mapping than the recently proposed QK method with respect to (i) the adherence to the nominal α-level and (ii) the adjusted power for detection of quantitative trait loci. Furthermore, we showed that our data set could be analyzed by using two-step approaches of the proposed association-mapping method without substantially increasing the empirical type I error rate in comparison to the corresponding one-step approaches.

https://www.genetics.org/content/genetics/178/3/1745.full.pdf

Comparison of mixed-model approaches for association mapping.

Series of plant breeding trials are often unbalanced and have a complex genetic structure. To reduce computing cost, it is common practice to employ a two-stage approach, where adjusted means per location are estimated and then a mixed model analysis of these adjusted means is performed. An important question is how means from the first step should be weighted in the second step. Our objective therefore was the comparison of different weighting methods in the analysis of four typical series of plant breeding trials using mixed models with fixed or random genetic effects. We used four published weighting methods and proposed three new methods. Four evaluation criteria were computed to compare methods, using one-stage analysis as benchmark. We found that the two-stage analysis gave acceptable results with fixed genetic effects. When genetic effects were taken as random in stage two, in three of four datasets the two-stage analysis gave acceptable results. In both cases differences between weighting methods were small and the best weighting method depended on the dataset but not on the evaluation criteria. A two-stage analysis without weighting also produced acceptable results, but weighting mostly performed better. In the fourth dataset the missing data pattern was informative, resulting in violation of the missing-at-random (MAR) assumption in one- and two-stage analysis. In this case both analyses were not strictly valid.

Comparison of Weighting in Two-Stage Analysis of Plant Breeding Trials

Plant breeders and variety testing agencies routinely test candidate genotypes (crop varieties, lines, test hybrids) in multiple environments. Such multi-environment trials can be efficiently analysed by mixed models. A single-stage analysis models the entire observed data at the level of individual plots. This kind of analysis is usually considered as the gold standard. In practice, however, it is more convenient to use a two-stage approach, in which experiments are first analysed per environment, yielding adjusted means per genotype, which are then summarised across environments in the second stage. Stage-wise approaches suggested so far are approximate in that they cannot fully reproduce a single-stage analysis, except in very simple cases, because the variance-covariance matrix of adjusted means from individual environments needs to be approximated by a diagonal matrix. This paper proposes a fully efficient stage-wise method, which carries forward the full variance-covariance matrix of adjusted means from the individual environments to the analysis across the series of trials. Provided the variance components are known, this method can fully reproduce the results of a single-stage analysis. Computations are made efficient by a diagonalisation of the residual variance-covariance matrix, which necessitates a corresponding linear transformation of both the first-stage estimates (e.g. adjusted means and regression slopes for plot covariates) and the corresponding design matrices for fixed and random effects. We also exemplify the extension of the general approach to a three-stage analysis. The method is illustrated using two datasets, one real and the other simulated. The proposed approach has close connections with meta-analysis, where environments correspond to centres and genotypes to medical treatments. We therefore compare our theoretical results with recently published results from a meta-analysis.

Jens Möhring

Papers

BLUP for phenotypic selection in plant breeding and variety testing

Computing Heritability and Selection Response From Unbalanced Plant Breeding Trials

Comparison of mixed-model approaches for association mapping.

Comparison of Weighting in Two-Stage Analysis of Plant Breeding Trials

A stage-wise approach for the analysis of multi-environment trials.