# 2006 Postgraduate course in Evolutionary Quantitative Genetics, Roenbjerg field station, University of Aarhus

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# Updates, and Quick download of the notes for the 18 lectures

### We will be using the R statistical software

R is free. You can download it (for almost all computers) at

## Course Schedule, Readings and Lecture Notes

Walsh and Lynch refers to on-line draft chapters from our (yet to be finished) volume, Evolution and Selection on Quantitative Traits.

## Day 1: (Tuesday, 6 June)

As the first day of class, we will start a little later (10am) to allow students a chance to arrive and settle in at the Field Station

• ### Lecture 2: Linear Algebra and Linear Models

• Background Reading: Lynch and Walsh: Chapter 8, Appendix A3
• Handouts
• Topics:
• Elementary Matrix Algebra
• Basic Notation
• Partitioned Matrices
• Addition And Subtraction
• Multiplication
• Dimensional Properties and Matrix Multiplication
• Transposition
• Inverses and Solutions to Systems of Equations
• Expectations Of Random Vectors And Matrices
• Covariance Matrices Of Transformed Vectors
• The Multivariate Normal Distribution
• Properties of the MVN
• Example: The Regression of Offspring Value on Parental Value
• Example: Regression of Offspring Breeding Value on Parental Breeding Values
• Overview Of Linear Models
• Examples of GLMs
• Ordinary Least Squares
• Example: Partial Regression
• Polynomial Regressions and Interaction Effects
• Fixed vs. Random Effect
• Example: Fixed vs. Random Effects in the Sire Model
• Generalized Least Squares
• Example: Weighted Least Squares
• Model Goodness-of-fit and Hypothesis Testing
• Chi-square and F-distributions
• Sums of Squares
• Hypothesis Testing
• Advanced matrix notes

## Day 2: (Wednesday 7 June)

• ### Lecture 3: Basic Concepts in Mendelian, Population and Quantitative Genetics

• Background Reading: Lynch and Walsh: Chapters 5, 6
• Handouts
• Topics:
• Overview
• A Tale of Two Papers: Darwin vs. Mendel
• Basic Mendelian Genetics
• Mendel's View of Inheritance: Single Locus
• The Genotype to Phenotype Mapping: Dominance and Epistasis
• Mendel's View of Inheritance: Multiple Loci
• Mendel was Wrong: Linkage
• Interlude: Chromosomal Theory of Inheritance
• Linkage
• Map Distances are Obtained from Recombination Frequencies via Mapping Functions
• The Prior Probability of Linkage and Morton's Posterior Error Rate
• Molecular Markers
• Basic Population Genetics
• Allele and Genotype Frequencies
• Gamete Frequencies, Linkage, and Linkage Disequilibrium
• The Effects of Population Structure
• Forces that Change Allele Frequencies: Genetic Drift
• Coalescence Theory
• Forces that Change Allele Frequencies: Mutation
• Forces that Change Allele Frequencies: Selection
• Interaction of Selection and Drift
• Basic Quantitative Genetics
• Dichotomous (Binary) Traits
• Contribution of a Locus to the Phenotypic Value of a Trait
• Example: Apolipoprotein E and Alzheimer' s age of onset
• Example: the Booroola (B) gene
• Fisher's Decomposition of the Genotypic Value
• Average Effects and Additive Genetic Values
• Genetic Variances
• Epistasis

• ### Lecture 4: Resemblances Between Relatives

• Background Reading: Lynch and Walsh: Chapter 7
• Handouts
• Topics:
• Phenotypic Resemblance Between Relatives
• Parent-offspring regressions
• Collateral relationships: ANOVA
• Causes of Phenotypic Covariance Among Relatives
• Genetic Covariance Between Relatives
• Offspring and one parent
• Half-sibs
• Full-Sibs
• General degree of relationship
• Environmental Causes of Relationship Between Relatives

## Day 3: (Thursday, 8 June)

• ### Lecture 5: Basic Designs for Estimation of Genetic Parameters

• Background Reading: Lynch and Walsh: Chapters 17, 18
• Handouts
• Topics:
• Heritability
• Why h2 instead of h?
• Heritabilities are Functions of a Population
• Increasing the Heritability
• Heritability and the Prediction of Breeding Values
• Heritability Values and Population Divergence
• Estimation: One-way ANOVA and the simple full-sib design
• Estimating Variances and Variance Components
• Estimating heritability
• Worked Example of a Full-sib Design
• Estimation: The Nested Full-sib, Half-sib Analysis
• Worked Example of a Nested Design
• Estimation: Parent-Offspring Regressions
• Broad-Sense Heritability H2 and Plant Breeding
• Estimation of Var(A) and breeding values in general pedigress
• The General Mixed Model
• Estimating Fixed Effects and Predicting Random Effects
• The animal model
• ANOVA vs. REML Variance Estimates
• ML Versus REML Variance estimation
• ### Lecture 6: Inbreeding and Heterosis

• Background Reading:
• Lynch and Walsh: Chapter 10
• Walsh and Lynch WL Chapter 12 [ link to pdf file (rough draft in places)]
• Handouts
• Topics:
• Changes in the Mean and Variance Under Inbreeding
• Inbreeding
• Inbreeding Depression in Fitness Traits
• Computing the Inbreeding Depression Coefficient, B
• Why do traits associated with fitness show inbreeding depression?
• Minimizing the Rate of Inbreeding
• Change of Variance With Inbreeding
• Example: Effect of Inbreeding on the Additive Genetic Variance
• Change of Variance with Inbreeding and Mutation
• Hybridization and Crossbreeding
• Types of Crosses
• Heterosis: Change in the Mean Under Crossbreeding
• Heterosis in F2
• Agricultural Importance of Heterosis
• Hybrid Corn
• Hybridization in Other Crops
• Crossbreeding in Animals: General Concepts
• Crossbreeding in Animals: Heterosis in { Bos indicus x Bos taurus Hybrids
• Synthetics and Rotational Crossbreeding
• Estimating the Amount of Heterosis in Maternal Effects

## Day 4: (Friday, 9 June)

• ### Lecture 7: Genetic Drift

• Handouts
• Topics:
• Single Locus Theory Of Random Genetic Drift
• The Wright-Fisher Model of Genetic Drift
• Inbreeding and Drift
• The Coalescent Process for Drift
• The Coalescent Process for a Sample
• Drift Generates Between-Population Variance in Replicate Lines
• Effective Population Size, Ne
• Ne With Unequal Sex Ratio
• Ne With Variable Population Size
• Ne With Unequal Offspring Contribution
• Mutation-Drift Equilibrium: Single Loci
• Mutation-Drift Equilibrium: Single Loci
• Divergence Between Populations
• Kimura's Neutral Theory of Molecular Evolution
• Mutation-Drift Equilibrium: Additive Variance
Equilibrium Additive Variance within a Population
• Divergence Between Populations
• Diffusion Theory
• Infinitesimal Means and Variances, m(x) and v(x)
• Stationary Distributions
• Probability of Fixation
• Diffusion Applications to Quantitative Characters
• Brownian Motion Model of Evolution
• Drift and Divergence in the Fossil Record
• Ornstein-Uhlenbeck Model of Evolution

• ### Lecture 8: Tests for Molecular Signatures of Selection

• Handouts
• Topics:
• Basic Logic of Sequence-Based Selection Tests
• Logic Behind Polymorphism-Based Tests
• Recombination and Polymorphism
• Logic Behind Divergence Tests
• Logic Behind Joint Polymorphism and Divergence Tests
• Tests Based Strictly on Within-Population Variation
• The Infinite Alleles Model: Ewen's Sampling Formula
• The Infinite Sites Model
• Tajma's D Test
• Fu and Li's D* and F* tests
• Depaulis and Veuille's K and H tests
• Fu's W and Fs Tests
• Fay and Wu's Test
• Genome-Wide Polymorphism Tests
• The Ghost of Lewontin-Krakauer: Genome Wide Fst-based Scans
• The Linkage Disequilibrium Decay (LLD) Test
• Joint Polymorphism and Divergence Tests
• McDonald-Kreitman Test
• Hudson-Kreitman-Aquade (HKA) Test
• Tests Based on Between-Population (Species) Divergence
• Parsimony-Based Ancestral Reconstruction Tests
• Likelihood-Based Codon Tests
• Bayesian Estimator of Sites Under Positive Selection

## Day 5: (Saturday, 10 June)

• ### Lecture 9: Short-term Response Selection

• Background Background Reading:
• Handouts
• Topics:
• Short Terms changes in the mean: the breeder's equation
• The Selection Differential S and Response R
• The Selection intensity i
• Truncation Selection
• Selection Intensities and Differentials Under Truncation Selection
• Correcting the Selection Intensity for Finite Samples
• Selection on Threshold Traits
• Permanent Versus Transient Response
• Response with Epistasis
• Maternal Effects: Response Under Falconer's Model
• Gene Frequency Changes Under Selection
• Short-term changes in the variance
• Bulmer's Equation for the Change in Variance
• Change in Variance Under Truncation Selection
• ### Lecture 10: Analysis of Short-term Selection Experiments

• Background Reading:
• Handouts
• Topics:
• Variance in response
• Realized heritabilitites
• Estimators for Several Generations of Selection
• Weighted Least-Squares Estimates of Realized Heritability
• Standard Errors for Realized Heritability Estimates
• Experimental Evaluation of the Breeders' Equation
• Asymmetric Selection Response
• Control Populations and Experimental Designs
• Basic Theory of Control Populations
• Divergent Selection Designs
• Variance in Response
• Control Populations and Variance in Response
• Optimal Experimental Design
• Nicholas' Criterion
• Mixed-model Estimation
• The Relationship Matrix Accounts for Drift and Disequilibrium
• Model Validation

## Day 6: (Monday, 12 June)

• ### Lecture 11: Long-Term Response and Selection Limits

• Background Reading:
• Walsh. 2003. Population- and Quantitative-Genetic Models of Selection Limits: Plant breeding Reviews [ pdf file of reprint ]
• Walsh and Lynch : Chapter 16 [ link to pdf file ]
• Handouts
• Topics
• Idealized Long-Term Response in a Large Population
• Deterministic Single-Locus Theory
• An Overview of Long-term Selection Experiments
• Estimating Selection Limits and Half-Lives
• General Features of Long-Term Selection Experiments
• Increases In Variances And Accelerated Responses
• Conflicts Between Natural And Artificial Selection
• Characterizing The Nature Of Selection Limits
• Long-Term Response in Finite Populations
• Fixation Probabilities of Favorable QTL Alleles
• Limits Under Drift and Selection
• Variance In Response
• Robertson's Theory of Selection Limits
• Tests of Robertson's Theory
• Weber's Selection Experiment on Drosphilia Flight Speed
• Response from Mutational Input
• Contributions from New Mutation
• Mutational Response Under the Infinitesimal Model

• ### Lecture 12: Individual Fitness and Measures of Univariate Selection

• Background Reading:
• Handouts
• Topics
• Episodes of selection
• Fitness Components
• Assigning Fitness Components
• Variance in Individual Fitness
• Some caveats in using opportunity of selection
• Descriptions of Phenotypic Selection: Introductory Remarks
• Fitness surfaces
• Descriptions of Phenotypic Selection: Changes in Phenotypic Moments
• The Price-Robertson Identity
• Direction Selection: Differentials (S) and Gradients (beta)
• Quadratic Selection: Differentials (C) and Gradients (gamma)
• beta and gamma Measure the Geometery of the Fitness Surface
• beta and gamma Describe the Selection Dynamics
• Descriptions of Phenotypic Selection: Individual Fitness Surfaces
• Linear and quadratic approximations of w(z)
• Schluter's cublic-spline estimate
• Complications from Unmeasured Correlated Variables
• Strenght of Selection in Natural Populations
• The importance of experimental manipulation

## Day 7: (Tuesday, 13 June)

• ### Lecture 13: Genetic Correlations and Multivariate Selection Response

• Background Reading:
• Walsh and Lynch : Chapter 20 [ link to pdf file ]
• Blows, M. W. and A. Hoffman. 2005. A reasessment of genetic limits to evolutionary change. Ecology 86: 1371--1384. [ pdf file of reprint ]
• Pigliucci, M. 2006. Genetic variance-covariance matrices: a critique of the evolutionary quantitative genetics research program. Biology and Philosophy 2: 1-13. [ pdf file of reprint ]
• Walsh and Lynch : Chapter 23, Theory of Index Selection [ link to pdf file ]
• Walsh and Lynch : Chapter 24, Applications of Index Selection [ link to pdf file ]
• Deckers and Hospital, 2002. The use of molecular genetics in the improvement of agricultural populations. Nature Reviews Genetics 3: 22-32 [ pdf file of reprint ]
• Handouts
• Topics:
• Phenotypic and Genetic Correlations
• Estimating the Genetic Correlation
• Half sibs
• Parent-Offspring
• Correlated Response to Selection
• Correlated Selection Differentials
• Indirect Selection
• General Multivariate Selection Response
• The Directional Selection Gradient
• Derviation of the Multivariate Breeders' Equation
• Realized Selection Gradients
• The Effects of Genetic Correlations: Direct and Correlated Responses
• Realized Genetic Correlations
• Evolutionary Constraints Imposed by Genetic Correlations
• A Short Diversion: The Geometry of a Matrix
• Comparing Vectors: Lengths and Angles
• Matrices Describe Vector Transformations
• Eigenvalues and Eigenvectors
• Quantifying Multivariate Constraints to Response
• Is There Genetic Variation in the Multivariate Direction of Selection?
• Schluter's Genetic Line of Least Resistance
• Blow's Matrix Subspace Projection
• General Theory of Selection on a Linear Index
• MAS -- Marker Assisted Selection
• Indirect Selection on Marker Score: Applications to Sex-limited Traits

• ### Lecture 14: Measuring Multivariate Selection

• Handouts
• Topics:
• Selection On Multivariate Phenotypes: Differentials And Gradients
• Changes in the Mean Vector: The Directional Selection Differential, S
• The Directional Selection Gradient, Beta
• Directional Gradients, Fitness Surface Geometry and Selection Response
• Changes in the Covariance Matrix: The Quadratic Selection Differential, C
• The Quadratic Selection Gradient, Gamma
• Quadratic Gradients, Fitness Surface Geometry and Selection Response
• Summary
• Multidimensional Quadratic Fitness Regressions
• Estimation, Hypothesis Testing and Confidence Intervals
• Geometric Aspects
• A Brief Digression: Orthonormal and Diagonalized Matrices
• Canonical Transformation of Gamma
• Strength of Selection: gammaii versus lambdai
• Canonical Subspaces of Gamma
• Unmeasured Characters and Other Biological Caveats
• The Bias due to Environmental Correlations Between Fitness and Characters

## Day 8: (Wednesday, 14 June)

• ### Lecture 15: Phenotypic Evolution Models

• Handouts
• Topics:
• Univariate Guassian Fitness Functions
• Response Under Normalizing Selection
• Application: Charlesworth's Model of the Cost of Phenotypic Selection
• Application: Drift vs. Selection in the Fossil Record
• Multivariate Gaussian Fitness Functions
• Phenotypes and Breeding Values Remain MVN After Selection
• The Selection Differential and Gradient
• The Within-Generation Change in G
• Changes in G Under the Infinitesimal Model
• Long-term Response: Balance Between Directional and Stabilizing Selection
• Long-term Response: The Infinitesimal Model with Drift and Mutation
• Derivatives Of Vectors And Vector-Valued Functions
• The Hessian Matrix, Local Maxima/Minima, and Multidimensional Taylor Series

• ### Lecture 16: Major Genes, Polygenes, and QTLs

• Background Reading: Lynch and Walsh: Chapters 13
• Handouts
• Topics:
• Major and Minor Genes
• Major Genes and Isoalleles
• Polygenic Mutation and the Mutational Variance
• Simple Tests for Detecting Major Genes
• Mixture Models
• Complex Segregation Analysis
• Likelihood Functions Assuming a Single Major Gene
• Common-family Effects
• Genetic Maps and Candidate Genes
• Map Distances vs. Recombination Frequencies
• Linkage Disequilibrium Mapping
• Fine-mapping Major Genes Using LD
• The Transmission/Disequilibrium Test
• Example: Mapping Type 1 Diabetes
• Linkage vs. Association
• Dense SNP Association Mapping
• Genomic control
• Structured Association Analysis

## Day 9: (Thursday, 15 June)

• ### Lecture 17: QTL Mapping

• BackgroundReading: Lynch and Walsh: Chapters 15, 16
• Handouts
• Topics:
• Mapping Using inbred line crosses
• Experimental designs
• Conditional Probabilities of QTL Genotypes
• Example: Conditional Probabilities for an F2
• Expected Marker Means
• Linear Models for QTL Detection
• Maximum Likelihood Methods for QTL Mapping and Detection
• Likelihood Maps
• Precision of ML Estimates of QTL Position
• Interval Mapping with Marker Cofactors
• Power and Repeatability: The Beavis Effect
• Mapping in Outbred Populations
• QTL Mapping Using Sib Families
• General Pedigree Methods
• Haseman-Elston regression
• Affected sib pair methods

• ### Lecture 18: Gene Expression analysis: microarrays and eQTLs

• Background Reading:
• Walsh and Henderson. 2004. Microarrays and beyond: What potential do current and future genomics tools have for breeders? Anim Sci. 82: E292-299E. [ pdf file of reprint ]
• Handouts
• Topics:
• Gene Regulation is a Complex Trait
• QTLs Involved in Protein Regulation
• Microarrays
• A Brief Overview Of The Technology
• Analysis of Microarray Data
• Microarray Analysis Is Best Regarded As An EDA Approach
• Problems (and Pitfalls) of Gene Discovery via Microarray Analysis
• General Patterns of Transcriptional Variation
• Gene Expression Levels are Typically Highly Heritable
• Correlations Between Rates of Regulatory (Transcriptional) and Sequence Divergence
• Correlations Between Regulatory Divergence and Expression Level/Pattern
• Does Divergence in Expression Follow a Neutral Model?
• Analysis of Pathways
• Two-hybrid screen: Construction Protein-protein interaction maps
• Kascer-Burns Metabolic Flux Theory
• Regulatory Neworks and Graph Theory
• Erdos-Renyi Random Graphs and Random Boolean Networks
• Graphs: Small Worlds, Scale-Free, and Power Laws
• Advanced notes Sequential Bonferroni corrections and the False Discovery Rate . (pdf, 17 pages)

## Day 10 (Friday, 16 June)

• ### Lecture 19: Wrap-up: Quantitative genetics and genomics

• Background Reading:
• Walsh, 2001. Quantitative Genetics in the Age of Genomics, in Theoretical Population Biology 59: 175--184. [ pdf file of reprint ]