Genetics and Analysis of Quantitative Traits 01

Genetics and Analysis of Quantitative Traits Michael Lynch Bruce Walsh Contents CONTENTS i FREFACE xiii I. FOUNDATIONS OF QUANTITATIVE GENETICS i 1. AN OVERVIEW OF QUANTITATIVE GENETICS 3 The Adaptationist Approach to Phenotypic Evolution 3 Quantitative Genetics and Phenotypic Evolution 4 Histórica! Background 7 The Major Goals of Quantitative Genetics 13 The nature of quantitative-trait variation 13 The consequences of inbreeding and outcrossing 14 The constraints on the evolutionary process 15 The estimation of breeding valúes 15 The developrnent of predictive models for evolutionary change 16 Mathematics in Biology 16 2. PROPERTIES OF DISTRIBUTIONS 19 Parameters of Univariate Distributions 19 The Normal Distribution 26 The truncated normal distribution 29 Confídence Intervals 32 3. COVARIANCE, REGRESSION, AND CORRELATION 35 Jointly Distributed Random Variables 35 Expectations of jointly distributed variables 36 Covariance 36 Useful identities for variances and covariances 38 Regression 39 Derivation of the least-squares linear regression 39 Properties of least-squares regressions 41 Correlation 43 A Taste of Quantitative-Genetic Theory 45 Directional selection differentials and the Robertson-Price identity 45 The correlation between genotypic and phenotypic valúes 47 Regression of offspring phenotype on parental phenotype 48 ii CONTENTS 4. PROPERTIES OF SINGLE LOCI 51 Alíele and Genotype Frequencies 52 The Transmission of Genetic Information 54 The Hardy-Weinberg principie 54 Sex-linked loci 56 Polyploidy 57 Age structure 60 Testing for Hardy-Weinberg proportions 60 Characterizing the Influence of a Locus on the Phenotype 61 The Basis of Dominance 63 Fisher's Decomposition of the Genotypic Valué 65 Partitioning the Genetic Va nance 69 Additive Effects, Average Excesses, and Breeding Valúes 71 Extensions to Múltiple Alíeles and Nonrandom Mating 74 Average excess 74 Additive effects 75 Additive genetic variance 76 5. SOURCES OF GENETIC VARIATION FOR MULTILOCUS TRAITS 81 Epistasis 82 A General Least-Squares Model for Genetic Effects 85 Extensión to haploids and polyploids 92 Linkage 94 Estimation of gametic phase disequilibrium 97 Effect of Disequilibrium on the Genetic Variance 100 The evidence 103 6. COMPONENTS OF ENVIRONMENTAL VARIATION 107 Extensión of the Linear Model to Phenotypes 108 Special Environmental Effects 111 Within-individual variation 112 Developmental horneostasis and homozygosity 116 Repeatability 121 General Environmental Effects of Maternal Influence 123 Genotype x Environment Interaction 127 7. RESEMBLANCE BETWEEN RELATIVES 131 Measures of Relatedness 132 Coefficients of identity 133 Coeffícients of coancestry and inbreeding 135 The coefficient of fraternity 140 The Genetic Covariance Between Relatives 141 The Effects of Linkage and Gametic Phase Disequilibrium 146 CONTENTS iii Linkage 146 Gametic phase disequilibrium 150 Assortative Mating 153 Polyploidy 161 Environmental Sources of Covariance Between Relativas 162 The Heritability Concept 170 Evolvability 175 8. INTRODUCTION OF MATRIX ALGEBRA AND LINEAR MODELS 177 Múltiple Regression 177 An application to multivariate selection 180 Elementary Matrix Algebra 182 Basic notation 182 Partitioned matrices 183 Addition and subtraction 183 Multiplicador! 184 Transposition 186 Inverses and solutions to systems of equations 187 Determinante and minors 189 Computing inverses 190 Expectations of Random Vectors and Matrices 192 Covariance Matrices of Transformed Vectors 193 The Multivariate Normal Distribution 194 Properties of the MVN 195 Overview of Linear Models 198 Ordinary least squares 200 Generalized least squares 202 9. ANALYSIS OF LINE CROSSES 205 Expectations for Line-cross Means 206 Estimación of Composite Effects 213 Hypothesis testing 215 Line crosses in Nicotiana rustica 219 Additional data 221 The Genetic Interpretation of Heterosis and Outbreeding Depression 222 Variance of Line-cross Derivatives 226 Biometrical Approaches to the Estimation of Gene Number 231 The Castle-Wright estimator 233 Effect of the leading factor 238 Extensions to haploids 241 Other Biometrical Approaches to Gene Number Estimation 244 The inbred-backcross technique 244 Genotype assay 246 iv CONTENTS 10. INBREEDING DEPRESSION 251 The Genetic Basis of Inbreeding Depression 252 A more general model 256 Methodologjcal Considerations 259 Single-generation analysis 260 Multigenerational analyses 262 Ritland's method 266 Epistasis and inbreeding depression 267 Variance in inbreeding depression 268 The Evidente 269 Purging Inbreeding Depression 274 Nurnber of Lethal Equivalents 276 Rcsults from vertebrares 278 Results from Dmsopliüa 279 Results from plants 281 Partial Recessives vs. Overdominance 283 The (A+B)/A ratio 283 Estimating the average dcgree of dominance 284 Inferences from molecular markers 287 11. MATTERS OF SCALE 293 Transformations to Achieve Normality 293 Log-normal distributions and the log transform 294 Tests for normality 295 Stabilizing the Variance 300 Kleckowski's transformation 300 General variance stabilizing-transformations 301 The Roginskii-Yablokov effect 302 The Kluge-Kerfoot phenomenon 305 Allometry: the Scaling Implications of Body Size 305 Removing Interaction Effects 307 Developmental Maps, Canalization, and Genetic Assimilation 309 Estimating developmental maps 310 Selection and canalization 314 Genetic assimilation 316 11. QUANTITATIVE TRAIT LOCI 319 12. FOLYGENES AND POLYGENIC MUTATION 321 The Gene tic Basis of Quantitative-Genetic Variation 322 Major genes and isoalleles 322 The molecular nature of QTL variation 323 The Mutational Rate of Production of Quantitative Variation 328 CONTENTS v Estirnation from divergente experiments 330 Bristle numbers in Dwsophila 333 Additional data 335 The Deleterious Effects of New Mutations 340 The Bateman-Mukai technique 341 Results from flies, plants, and bacteria 343 Analysis of natural populations 348 The persistente of new mutations 351 13. DETECTING MAJOR GENES 353 Elementar}' Tests 354 Departures from normality 354 Tests based on sibship variances 355 Major-gene índices (MGI) 357 Konpararnetric line-cross tests 358 Mixture Models 359 The distribution under a mixture model 360 Parameter estimation 360 Hypothesis testing 361 Complex Segregation Analysis 364 Likelihood functions assuming a single major gene 366 Common-family effects 370 Polygenic background 371 Other extensions 373 Ascertainment bias 374 Estimating individual genotypes 374 Analysis of Discrete Characters 375 Single-locus penetrante model 376 Major gene plus a polygenic background 377 14. PRINCIPLES OF MARKER-BASED ANALYSIS 379 Classical Approaches 379 Chromosomal assays 380 Thoday's method 381 Genetics o£ Drosophila bristle number 385 Genetics of Dwsophila speciation 387 Molecular Markers 390 Genetic Maps 393 Map distance vs. recombination frequencies 394 How many markers are needed? 397 Marker-trait Associations 398 Selective genotyping and progeny testing 401 Recornbinant inbred lines (RILs) 401 Bulked segregant analysis 402 vi CONTENTS QTL mapping by marker changes in populations under selection 404 Marker-based Analysis Using Nearly Isogenic Lines (NILs) 405 Marker-based introgressions 407 Fine Mapping of Major Genes Using Population-level Disequilibrium 413 LD mapping in expanding populations 414 Candidate Loci 418 The transmission/disequilibrium test 419 Estimating effects of candidate loci 422 Templeton and Sing's method: Using the histórica! information in haplotypes 424 Cloning QTLs 425 Transposon tagging 426 Positional cloning and comparative mapping 426 15. MAPPING AND CHARACTERIZING QTLS: INBRED LINE CROSSES 431 Foundations of Line-Cross Mapping 431 Experimental designs 432 Conditional probabilities of QTL genotypes 433 Expected marker-class means 437 Marker variance and higher-order moments 439 Overall significance leve! with múltiple tests 441 QTL Detection and Estimation Using Linear Models 442 QTL Detection and Estimation via Máximum Likelihood 445 Likelihood maps 446 Precisión of ML estimates of QTL position 448 ML interval mapping 450 Approximating ML interval mapping by Haley-Knott regressions 453 Dealing with Múltiple QTLs 457 Marker-difference regression 459 Interval rnapping with marker cofactors 463 Detecting múltiple linked QTLs using standard marker-trait regressions .. 467 Sample Size Required for QTL Detection 469 Power under selective genotyping 474 Power and repeatabilty of mapping experiments 474 Selected Applications 477 The nature of transgressive segregation 477 QTLs involved in reproductive isolation in Minndus 478 QTLs involved in protein regulation 478 QTLs in the Illinois long-term selection maize lines 479 QTLs involved in the differences between maize and teosinte 481 QTLs for age-specific growth in mice 484 Summary oí QTL mapping experiments 484 CONTENTS vii 16. MAPPING AND CHARACTERIZING QTLS: OUTBRED POPULATIONS 491 Measures of Informativeness 492 Sib Analysis: Linear Models 495 A single half-sib farníly 496 Several half-sib families 498 Power of Nested ANOVA Designs 501 A single full-sib family 502 Several full-sib families 504 Sib Analysis: Máximum Likelihood 505 Constructing likelihood functions 507 Máximum Likelihood over General Pedigrees: Variance Componente 510 Estimating QTL position 512 The Haseman-Elston Regression 513 Derivation of the Haseman-Elston regression 513 Estimating the number of marker genes ibd 516 Power and improvements 517 Interval mapping by a modified Haseman-Elston regression 518 Mapping Dichotomous Characters 521 Recurrent and relative risks of pairs of relatives 523 Affected sib-pair tests 525 Power of ASP tests and related issues 527 Genomic scanning 529 Exclusión mapping and information content mapping 530 Affected pedigree member tests 532 III. ESTIMATION PROCEDURES 535 17. PARENT-OFFSPRING REGRESSION 537 Estimation Procedures 538 Balanced data 538 Unequal family sizes 539 Standardizaron of data from the different sexes 542 Precisión of Estimates 542 Optimum Experimental Designs 543 Assortative mating 547 Estimation of Heritability in Natural Populations 548 Linearity of the Parent-Offspring Regression 550 18. SIB ANALYSIS 553 Half-sib Analysis 554 One-way analysis of variance 556 Hypothesis testing 560 viii CONTENTS Sampling variance and standard errors 561 Confidence ínter vals 562 Negativo estimares of heritability 563 Optimal experimental design 564 Unbalanced data 566 Resampling procedures 569 Full-sib Analvsis 570 Nested analvsis of variance 573 Hypothesis testing 574 Sampling error 576 Optimal design 577 19. TWINS AND CLONES 581 The Classical Approach 582 Heritability estimation 584 The Monozygotic-Twin Half-sib Method 587 Clonal Analysis 592 20. CROSS-CLASSIFIED DESIGNS 597 North Carolina Design TI 598 The average degree of dominance 603 The Cockerham-Weir model 605 Diallels 610 Pooled reciprocáis, no self crosses 611 Reciprocáis, no self crosses 614 Complete diallels 618 Partial diallels 618 Hayman-Jinks analysis 619 North Carolina Design III and the Triple Test Cross 625 Some Closing Statistical Considerations 627 21. CORRELATIONS BETWEEN CHARACTERS 629 Theoretical Composition of the Genetic Covariance 630 Estimation of the Genetic Covariance 632 Pairwise comparison of relatives 632 Nested analvsis of variance and covariance 633 Regression of family means 636 Components of Phenotypic Correlation 637 Phenotypic correlations as surrogate estímales of genetic correlations . . . 639 Statistical Issuos 639 Hypothesis tests 641 Standard errors 642 Bias due to selection 644 Applications 648 CONTENTS ix Genetic basis of population differentiation 648 The homogeneity of genetic covariance matrices among species 650 Evolutíonary allometry 653 Evolution of life-history characters 655 22. GENOTYPE x ENVIRONMENT INTERACTION 657 Genetic Correlation Across Two Environments 660 Estimation procedures 663 Two-way Analysis of Varíance 666 Relationship to Falconer's correlation across environments 671 Further Characterization of Interaction Effects 672 Joint-regression analysis 672 Testing for Cross-over Interaction 678 Concepts of Stability and Plasticity 680 Additional issues 682 The Quantitative Genetics of Genotype x Environrnent Interaction 683 23. MATERNAL EFFECTS 687 Components of Variance and Covariance 689 Cytoplasmic transmission 693 Postpollination reproductive traits in plants 695 Cross-fostering experiments 696 Body weight in mice 700 Eisen's Approach 703 Bondari's experiment 703 Falconer's Approach 706 Extensión to Other Types of Relatives 711 24. SEX LINKAGE AND SEXUAL DIMORPHISM 715 Sex-Hnked Loci and Dosage Compensation 715 Sex-modified Expression of an Autosomal Locus 718 Gametic imprinting 718 Extensión to Múltiple Loci and the Covariance Between Relatives 719 Variation for Sexual Dimorphism 724 25. THRESHOLD CHARACTERS 727 Heritability on the Underlying Scale 730 Múltiple Thresholds 736 Genetic Correlation Among Threshold Traits 739 Heritability on the Observed Scaie 741 26. ESTIMATION OF BREEDING VALÚES 745 The General Mixed Model 746 Estimating Fixed Factors and Predicting Random Effects 748 x CONTENTS Estimability o£ fíxed factors 753 Standard errors 754 Models for íhe Estimation of Breeding Valúes 755 The animal model 755 The gametic model 758 The reduced animal model 759 Simple Rules for Computing A and A"1 762 Allowing for mutation when computing A 766 Joint Estimation of Several Vectors of Random Effects 767 BLUP estímales of dominance valúes 767 Repeated records 769 Maternal effects 773 Múltiple traite 774 27. VARIANCE-COMPONENT ESTIMATION WITH COMPLEX PEDIGREES 779 ME versus REME Estimates of Variance Components 780 A simple exaniple of ML versus REML 781 ME Estimates of Variance Components in the General Mixed Model 784 Standard errors of ML estimates 788 Restricted Máximum Likelihood 789 Multivariate analysis 792 ML/REML estimation in populations undcr selection 792 Solving ML/REML Equations 793 Derivan've-based methods 794 EM methods 797 Additional approaches 799 A Molecular-marker Based Method for Inferring Variance Components . . . . 800 IV. APPENDICES 805 Al. EXPECTATIONS, VARIANCES, AND COVARIANCES OF COMPOUND VARIABLES 807 The Delta Method 807 Expectations of complex variables 808 Variances of complex variables 810 Covariances of complex variables 813 Variance of Variances and Covariances 813 Expectations and Variances of Products 817 Expectations and Variances of Ratios 818 Sampling variance of regression and correlations coefficients 818 Sampling variance of a coefficient of variation 819 CONTENT5 xi A2. PATH ANALYSIS 823 Univariate Analysis 823 Bivariate Analysis 826 Applications 826 Phenotypic correlation between parents and offspring 827 Correlations between characters 829 Growth analysis 831 A3. FURTHER TOPICS IN MATRIX ALGEBRA AND LINEAR MODELS 835 Generalized Inverses and Solutions to Singular Systems of Equations 835 Generalized inverses 836 Consistency and solutions to consistent systems 836 Estímability of fixed factors 839 The Square Root of a Matrix 841 Derivation of the GLS Estimators 842 Quadratic Forms and Sums of Squares 843 Moments of quadratic forms 843 The sarnple variance as a quadratic form 844 Sums of squares expressed as a quadratic form 846 Testing Hypotheses About Linear Models 848 Equivalent Linear Models 849 Derivativas of Vectors and Matrices 851 A4. MÁXIMUM LIKELIHOOD ESTIMATION AND LIKELIHOOD-RATIO TESTS 853 Likelihood, Support, and Score Functions 853 Large-sample properties of MLEs 854 The Fisher information matrix 855 Likelihood-ratio tests 857 The G-test 859 Likelihood-ratio tests for the general linear model 860 Iterativa Methods for Solving ML Equations 861 Newton-Raphson methods 861 Expectation-maximization methods 863 EM for mixture model likelihoods 863 EM rnodifications for QTL mapping 865 A5. COMPUTING THE POWER OF STATISTICAL TESTS 869 Power of Normally Distributed Test Statistics 870 One-sided tests 870 Two-sided tests 872 Applications: Parent-offspring regressions 874