Evolution and Selection of Quantitative Traits

Transcription

1 Evolution and Selection of Quantitative Traits B. Walsh and M. Lynch Draft version 30 April 2010 Contents for Chapters Covered in Wageningen University Course, June 2010 i 7. THE POPULATION GENETICS OF SELECTION 201 Single-locus Selection: Two Alleles Viability Selection Expected time for allele frequency change Complications: Differential Viability Selection on the Sexes Complications: Frequency-dependent Selection Complications: Fertililty Selection Complications: Sexual Selection Wright s Formula Adapative topographies and Wright s formula Single-locus Selection: Multiple Alleles Marginal fitnesses and average excesses Changes in genotypic fitnesses, W ij Changes in mean fitness and equilibrium values Internal, corner, and edge equilibrium; Basins of attraction Dynamics of response under viability and fertility selection Wright s Formula With Multiple Alleles Selection on Two Loci Dynamics of gamete frequency change Gametic equilibrium values, linkage disequilibrium, and mean fitness Particular fitness models Phenotypic stabilizing selection and the maintenance of genetic variation Theorems of Natural Selection: Fundamental and Otherwise The classical interpretation of Fishers fundamental theorem What did Fisher really mean? Mean fitness, Wright s adaptive topography, and Fisher s fundamental theorem Heritability values for characters correlated with fitness Non-additive genetic variances and traits under selection Robertson s Secondary Theorem of Natural Selection The secondary theorem under arbitary epistasis Selection on a Quantitative Trait Locus A single gene underlies the character Many loci of small effect underlying the character A population-genetics derivation of the breeders equation Correct quadratic terms for s i SHORT-TERM CHANGES IN THE MEAN: 1. THE BREEDERS EQUATION 101 i

2 ii Single-generation Response: The Breeders Equation Response is the Change in Mean Breeding Value The importance of linearity Response under More General Parent-offspring Regressions The Selection Intensity The Robertson-Price Identity, S = σ(w, z) Correcting for Reproductive Differences: Effective Selection Differentials Expanding the Basic Breeders Equation Accuracy Reducing Environmental Noise: Stratified Mass Selection Reducing Environmental Noise: Repeated-measures Selection Adjustments for Non-overlapping Generations Maximizing Response Under the Breeders Equation Maximizing the Economic Rate of Response Prelude to the multivariate breeders equation Truncation Selection Selection intensities and differentials under truncation selection Correcting the selection intensity for finite samples Response in Discrete Traits: Binary Traits The Threshold/Liability model Logistic Regressions and the Logistic Distribution Direct Selection on the Threshold T Response in Discrete Traits: Poission-distributed Characters Summary: Limitations of the Breeders Equation SHORT-TERM CHANGES IN THE MEAN: 2. PERMANENT VERSUS TRANSIENT RESPONSE 139 Permanent Versus Transient Response Response with epistasis Selection on autotetraploids Ancestrial Regressions Response due to Environmental Correlations Maternal Effects Response Under Falconer s Dilution Model Other Models of Maternal Effects SHORT-TERM CHANGES IN THE VARIANCE 179 Changes in Variance Due to Linkage Disequilibrium Changes in Variance Under the Infinitesimal Model Within- and between-family variance under the infinitesimal model Accounting for inbreeding and drift Change in Variance Under Truncation Selection Changes in correlated traits Directional Truncation Selection: Theory Directional Truncation Selection: Experimental results Effects of epistasis: Does the Griffing effect overpower the Bulmer effect? Double Truncation Selection: Theory Double Truncation Selection: Experimental Results Response under Normalizing Selection Selection with Assortative Mating

3 iii Results using the infinitesimal model Assortative mating and enhanced response Disruptive selection, assortative mating, and reproductive isolation Selection in the Presence of Heritable Variation in σe Micro-environmental variance, developmental noise and canalization Evidence for heritable variation in the environmental variance Modeling genetic variation in σe h 2 v, the heritability of the environmental variance Translating the response in A v into response in σe Selection response in σe THE INFINITESIMAL MODEL AND ITS EXTENSIONS 100 The Infinitesimal Model Allele frequencies do not change under the infinitesimal model Disequilibrium under the infinitesimal model Dominance Gaussian features of the infinitesimal Not all limits are Gaussian Modifications of the infinitesimal model Gaussian Continuum-Of-Alleles Models Infinite alleles and continuum-of-alleles models Drift Drift and a finite number of loci Effective number of loci, n e Dynamics: σa 2 and d change on different time scales Response in stabilizing selection experiments: selection or drift? How robust is the continuum-of-alleles model? The Bulmer Effect Under LInkage An approximate treatment A more careful treatment Response Under Non-Gaussian Distributions Describing the genotypic distribution: Moments Describing the genotypic distribution: Cumulants and Gram-Charlier series Application: Departure from normality under truncation selection Short-term response ignoring linkage disequilibrium Short-term response ignoring allele frequency change Summary: Where Does All This Modeling Leave Us? LONG-TERM RESPONSE: DETERMINISTIC ASPECTS 134 Idealized Long-term Response in a Large Population Deterministic Single-locus Theory Lande s model: a major gene in an infinitesmal background Are major genes or polygenes more important for long-term response? An Overview of Long-term Selection Experiments Estimating selection limits and half-lives General features of long-term selection experiments Increases in variances and accelarated responses Linkage effects Conflicts Between Natural and Artificial Selection Accumulation of lethals in selected lines

4 iv Expected equilibrium frequency of recessive lethals Lerner s model of genetic homoestasis Characterizing the Nature of Selection Limits LONG-TERM RESPONSE: FINITE POPULATIONS 291 Population Genetics of Selection and Drift Drift and selection at a single locus Fixation probabilities for alleles at QTL Expected allele frequency in a particular generation The Cohan effect: increased divergence under uniform selection Results for two loci: the Hill-Robertson effect The Effect of Selection on Effective Population Size The expected reduction in N e from directional selection Molecular variation is reduced in regions of low recombination Drift and Long-term Selection Response Basic theory Robertson s theory of selection limits Tests of Robertson s theory of selection limits Weber s selection experiments on Drosophila flight speed The Effects of Linkage on the Selection Limit Optimal Selection Intensities for Maximizing Long-term Response Effects of Population Structure on Long-Term Response Founder effects and population bottlenecks Population subdivision Within-family selection Asymptotic Response due to Mutational Input Results for the infinitesimal model Expected asymptotic response under more general conditions Optimizing Asymptotic Selection Response VII. MEASURING SELECTION INDIVIDUAL FITNESS AND THE MEASUREMENT OF UNIVARIATE SELECTON 301 Episodes of Selection and the Assignment of Fitness Fitness components Assigning fitness components Potential issues with assigning discrete fitness values Assigning components of offspring fitness to their mothers Concurrent episodes, reproductive timing and individual fitness λ ind Variance in Individual Fitness Partitioning I across episodes of selection Correcting lifetime reproductive success for random offspring mortality Variance in mating success: Bateman s principles Some caveats in using opportunity of selection Descriptions of Phenotypic Selection: Introductory Remarks Fitness surfaces Descriptions of Phenotypic Selection: Changes in Phenotypic Moments Directional selection

5 v Quadratic selection Gradients describe the local geometry of the fitness surface Gradients appear n selection response equations Partitioning changes in means and variances into episodes of selection Choice of the reference population: independent partitioning Standard errors for estimates of differentials and gradients Descriptions of Phenotypic Selection: Individual Fitness Surfaces Linear and quadratic approximations of W (z) Hypothesis testing and approximate confidence intervals Power Quadratic surfaces can be very misleading Fitting other parameteric fitness functions Nonparametric approachers: Schluter s cublic-spline estimate The importance of experimental manipulation MEASURING MULTIVARIATE SELECTION 370 Selection on Multivariate Phenotypes: Differentials and Gradients Changes in the mean vector: the directional selection differential S The directional selection gradient β Directional gradients, fitness surface geometry and selection response Changes in the covariance matrix: the quadratic selection differential C The quadratic selection gradient γ Quadratic gradients, fitness surface geometry and selection response Fitness surface curvature and within-generation changes in variances and covariances 379 Multivariate Quadratic Fitness Regressions Estimation, hypothesis testing and confidence intervals Regression packages and coefficients of γ Geometric aspects A brief digression: orthonormal and diagonalized matrices Canonical transformation of γ Are traits based on canonical axes real? Strength of selection: γ ii versus λ Significance and confidence regions for a stationary point Multivariate Nonparametric Fitness Surface Estimation Projection pursuit regression Thin-plate splines Strenght of Selection in Natural Populations Kingsolver s meta-analysis Directional selection: strong or weak? Quadratic selection: strong or weak? Directional selection on body size and Cope s law Unmeasured Characters and Other Biological Caveats Path Analysis and Fitness Estimation VIII. SELECTION ON MULTIPLE CHARACTERS MULTIVARIATE RESPONSE: CHANGES IN MEANS 1 The Multivariate Breeders Equation... 1

6 vi Overview of key features and concepts... 1 Derivation of the multivariate breeders equation... 4 The multivariate secondary theorem of natural selection... 7 Response when the parent-offspring regression is multivariate linear... 8 Bivariate Selection... 9 Correlated response to selection Indirect selection may give a larger response than direct selection Realized genetic correlations General bivariate selection Realized genetic correlations with bivariate selection Asymmetric correlated responses are frequently seen Comparison of Multivariate Responses Standardization of response Mean standarization and evolvability Realized selection gradients Evolutionary Constraints Imposed by Genetic Correlations Dynamics of quantitative traits on an adaptive topography What happens to mean fitness W? Constraints are given by the eigenstructure of G Trade-offs, developmental constraints, genetic correlations, and the Lande equation 30 Multivariate measures of evolvability Schluter s genetic line of least resistance, g max Is there genetic variation in the direction of response? Blow s matrix subspace projection Conditional genetic variance and conditional evolvability MULTIVARIATE RESPONSE: CHANGES IN COVARIANCES 1 Changes in G Under the Infinitesimal Model... 1 The dynamics of the disequilibrium matrix D... 2 The proportional change model for P... 3 Within-generation changes G due to selection on variances and covariances... 5 Asymmetric correlated responses occurs under the infinitesimal model... 6 Response in G under a multivariate Gaussian fitness function... 8 Allele Frequency Changes and Instability of Genetic Covariances Pleiotropic-based genetic correlations may become more negative over time Genetic covariances are more fragile than genetic variances It is difficult for antagonistic pleiotropy to maintain variation Hidden Pleiotropy: A zero genetic covariance can still harbor many pleiotropic alleles 15 Experimental Studies of the Response to Selection to Change Covariances Genetic Models of Covariances Resource partitioning models: background James analysis of changes in covariances under resource partitioning models Tradeoffs can lead to positive, as well as negative, covariances Björklund s analysis Optimization models, Functional Constraints, and G Long-Term Directional Selection The infinitesimal model with drift The infinitesimal model with drift and mutation The balance between directional and stabilizing selection: infinitesimal model results 29 Long-term response is a function of the distribution of allelic effects... 30

7 vii The balance between directional and stabilizing selection: Finite locus models Long-term Quadartic Selection Lande s multivariate model of pleiotropic mutation-selection balance γ and G Model assumptions, genetic correlations, and hidden pleiotropy Stability of G COMPARISONS OF G AND ITS STABILITY 1 Changes in G Under Drift... 1 Under additivity, G shows an expected proportional decrease... 1 The experimental results of Phillips, Whitlock, and Fowler... 2 Changes in G when non-additive genetic variance is present... 5 The eigenstructure of G under drift and mutation... 6 Comparing Covariance Matrices: Methodology... 7 General issues of inference on G using a population sample... 8 Identity, proportionality, common orientation, common scaling... 9 Element-by-element tests Roff s jackknife MANOA approach Roff s T test Mantel s test and other matrix correlation approaches Regression methods: tests of proportionality Likelihood-based tests assuming multivariate normality: variance components Likelihood-based tests assuming multivariate normality: Bertlett s modified test.. 17 Random skewers: probing the geometry of G with responses to selection response 18 Comparison of shared geometry: the Flury hierarchy Comparison of shared geometry: Krzanowski subspace compairson Still no ideal solution Comparing Covariance Matrices: Data Conclusions Estimating the Dimension of a Covariance Matrix Leading eigenvalues are overestimated, smaller eigenvalues underestimated Problems with bootstrap-based confidence intervals for eigenvalues and rank Canonical decomposition of the estimated covariance matrix Amemiya s rank test Reduced Rank estimates of G Factor-analytic approaches for building reduced-rank estimates Dimensionality of G: data Eigenvalue-based measures of effective dimensionality THEORY OF INDEX SELECTION 400 General Theory Selection on a Linear Index Genetic variance, heritability, and response of an index Response in the individual components of the index The retrospective index The selection and response indices may contain different traits Changes in the additive variance of I due to index selection Changes in G and P under index selection Optimizing the Expected Response of a Linear Index The index of selection usually does not equal the index of responsse Selection and response indices with non-overlapping traits

8 viii The Smith-Hazel index Properties of the Smith-Hazel Index Other useful results for the Smith-Hazel index Estimated, base, and Elston indices The Hayes-Hill transformation: detecting flaws in the estimated index Bending and rounding corrections of the estimated index Constraints on R and S given a specified selection intensity Restricted and Desired-gains Indices Restricted indicies Desired-gains indicies Summary of Linear Selection Indices Nonlinear Selection Indices Specific issues with nonlinear indices Quadratic indices Linear indices for nonlinear merit Exact optimization of nonlinear indices Optimal weights depend on the lenght of the experiment Sequential Approaches: Tandem Selection and Independent Culling Tandem selection Independent culling Selection of extremes Relative Efficiencies of Index Selection, Independent Culling, and Tandem Selection Theory Data Multistage Selection Optimal values for multistage cullings Cotterill and James approximately optimal two-stage selection Multistage index selection Xu and Muir s method of transformed culling and orthogonal index selection SOME APPLICATIONS OF INDEX AND MULTIPLE-TRAIT SELECTION 455 Improving the Response of a Single Character Using a Selection Index General theory More detailed analysis of two special cases Repeated measures of a character Using Information From Relatives General Theory Information from a single relative Constructing selection indices when the individual itself is not measured Within and Between Family Selection Lush s index Osborne s index Selection on a Ratio Approximate linear indices for ratio selection Other linear-based indicies for ratio selection Which method is best? Selection directly on a ratio: selection differentials and response Selection and Sexual Dimorphic Traits

9 ix Components of the genotype sex interaction variance Selection in sex-limited traits Differential selection across the sexes Sex-specific transmission differences The joint response for a single dimorphic trait Response with sex-linkage Sexual dimorphism: a correlated or direct response? Sexual dimorphism in size: Rensch s rule Selection on a vector of sexually dimorphic traits Selection of the Environmental Variance σe The bivariate Mulder-Bijima-Hill Model: Estimation The bivariate Mulder-Bijima-Hill Model: Response in σe Extensions of the Mulder-Bijima-Hill Model: Accounting for skew Extensions of the Mulder-Bijima-Hill Model: Family and Sire-slection Changes in the genetic variances and covariance for A m,a v A2. INTRODUCTION TO BAYESIAN STATISTICS 21 Bayes theorem From Likelihood to Bayesian Analysis Marginal Posterior Distributions Summarizing the Posterior Distribution Highest Density Regions (HDRs) Bayes Factors and Hypothesis Testing The Choice of a Prior Diffuse Priors Sufficient Statistics and Data-transformed likelhoods The Jeffreys Prior Posterior Distirbutiosn Under Normality Assumptions Known Variance and Unknown Mean Gamma, Inverse-Gamma, χ 2 and χ 2 distributions Unknown Variance, Inverse-χ 2 priors General case: unknown mean and variance Conjugate Priors The Beta and Dirichlet Distributions Wishart and Inverse-Wishart Distributions Conjugate Priors for the Exponential Family of Distributions A3. MCMC METHODS AND BAYESIAN ANALYSIS 39 Monte Carlo Integration Importance Sampling Introduction to Markov Chains The Metropolis-Hastings Algorithm Metropolis-Hasting Sampling as a Markov Chain Burning-in the Sampler Simulated Annealing Choosing a Jumping (Proposla) Distribution Convergence Diasgonistics Autocorrelation and Sample Size Inflation Tests for Convergence One Long Chain or Many Smaller Chains?... 51

10 x The Gibbs Sampler Using the Gibbs Sampler to Approximate Marginal Distributions The Monte Carlo Variance of a Gibbs-Sampled Based Estimate Convergence Diagonistics: The Gibbs Sampler A4. THE GEOMETRY OF VECTORS AND MATRICES: EIGENVALUES AND EIGENVECTORS The Geometry of Vectors and Matrices Comparing vectors: lengths and angles Matrices describe vector transformations Orthonormal matrices Eigenvalues and eigenvectors Properties of symmetric matrices Correlations can be removed by a matrix transformation Canonical axes of quadratic forms Implications for the multivariate normal distribution Principal components of the variance-covariance matrix Testing for Multivariate Normality Graphical tests: Chi-square plots Mardina s test: Multivariate skewness and kurtosis A5. DERIVATIVES OF VECTORS AND VECTOR-VALUED FUNCTIONS Derivatives of Vectors and Vector-valued Functions The hessian matrix, local maxima/minima, and multidimensional Taylor series Optimization under constraints... 86