UE FMOV309 Génétique quantitative évolutive – 12 Nov 2013 Le modèle animal Génétique quantitative en populations naturelles.

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1 UE FMOV309 Génétique quantitative évolutive – 12 Nov 2013 Le modèle animal Génétique quantitative en populations naturelles

2 Partie 1. Modèle univarié But: décomposer la variance phénotypique et estimer l’héritabilité d’un caractère

3 n = 960 breeders Pedigree of breeding swans 1979 - 2003

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5 The ‘animal model’ y i = μ + b i + a i + e i random residual error additive genetic effects population mean phenotype of individual i Pro:Exploits all pedigree information (more powerful) Accommodates unbalanced datasets Can include repeated measures on the same individual Non-genetic resemblance can be controlled for and estimated Con: Computational complexity Requires long-term datasets It combines pedigree information + measures on phenotypes for the partition of individual phenotypes: Simplest form of animal model: Kruuk. 2004. Phil.Trans.R.Soc.Lond.B fixed effects (e.g. sex, age)

6 y i = μ + b i + a i + e i breeding value of individual i a i is ADDITIVE GENETIC MERIT or BREEDING VALUE of individual i Breeding value of i Is the sum of the additive effects of its genes on trait y Is twice the expected deviation of its offspring phenotype from the population mean (under random breeding). Cannot be measured, but can be estimated or predicted

7 Phenotypic data Pedigree information Animal Model 1. (Co)variance components including V A 2. Estimated (predicted) breeding values, a i Test biological hypotheses ! ! INPUT DATA OUTPUT TOOL: Statistical model used by animal breeders (and evolutionary biologists) OBJECTIVE

8 The univariate ‘animal model’ The animal model is a form of mixed model solved by REML or a Bayesian approach X is a design matrix of 0s and 1s relating each observation to corresponding fixed effects (such as population mean) given in the vector β Each Z i is a design matrix for a corresponding vector of random terms u i Matrix form : y = X β + Σ i Z i u i + e vector of measures of the trait y on all individuals vector of fixed effects vectors of random effects vector of residual errors

9 It’s just another mixed model… y = X β sex + Z a + e y = X β + Σ i Z i u i + e y = = + + 1 1 1 1 1 1 1 1 1 1 17.8 16.4 17.0 16.8 15.5 17.1 15.7 15.9 16.4 17.2 0 1 0 1 1 0 1 1 0 0 μβMμβM 1000000000 0100000000 0010000000 0001000000 0000100000 0000010000 0000001000 0000000100 0000000010 0000000001 a 1 a 2 a 3 a 4 a 5 a 6 a 7 a 8 a 9 a 10 e 1 e 2 e 3 e 4 e 5 e 6 e 7 e 8 e 9 e 10 1 Fixed effect: sex 1 Random effect: additive genetic

10 y = = + + 1 1 1 1 1 1 1 1 1 1 17.8 16.4 17.0 16.8 15.5 17.1 15.7 15.9 16.4 17.2 0 1 0 1 1 0 1 1 0 0 μβMμβM a 1 a 2 a 3 a 4 a 5 a 6 a 7 a 8 a 9 a 10 e 1 e 2 e 3 e 4 e 5 e 6 e 7 e 8 e 9 e 10 BLUPs: predictors of random effects BLUEs: estimates of fixed effects Step 2 σ2Aσ2A Step 1 σ2Eσ2E

11 Using the pedigree to estimate var(a i ) = V A a 1 a 2 a 3 a 4 a 5 a 6 a 7 a 8 a 9 a 10 - For any pair of individuals i and j, Additive genetic covariance for a trait = 2 Θ ij V A Θ ij : the coefficient of coancestry the probability that an allele drawn randomly from individual i will be identical by descent to an allele drawn randomly from individual j. => e.g. Θ ij = 0.25 for parent and offspring, so additive genetic covariance between parent and offspring is ½ V A => the pedigree specifies Θij for all i,j in a population => A is the additive genetic relationship matrix with individual elements: Aij = 2 Θij. V A = var(a i ) : the additive genetic variance for the trait

12 The G matrix The variance-covariance matrix G for the vector a is the square n x n matrix of genetic variances and covariances for n traits G = V A.1 COV A.12... COV A.1n COV A.12 V A.2... COV A.2n... COV A.1n COV A.2n... V A.n... - For 1 trait - For 2 traits - For n traits

13 Software 1. Estimation of variance components ASReml(Gilmour) ASReml-R (Gilmour) WOMBAT (Meyer) * VCE (Groeneveld) DFREML Genstat, SAS.. Kinship MCMCglmm (Hadfield) 2. BLUP runs for breeding values (requires prior estimates of variance components) * PEST (Parameter ESTimation: Groeneveld) Pedigree Viewer (Kinghorn)

14 MCMCglmm : inférence Bayésienne 1.Le principe VraisemblanceBayésien Utilise les données observées + une distribution à priori (expertise, étude préliminaire…) Suppose que les paramètres sont fixes avec une valeur inconnue à estimer Suppose que les paramètres suivent une distribution inconnue Estimer des paramètres θ (ici par exemple σ 2 A ou σ 2 E )

15 MCMCglmm : inférence Bayésienne 1.Le principe Estimer des paramètres θ (ici par exemple V A et V E ) Théorème de Bayes : π(θ | data) = L(data | θ) p(θ) distribution à priori des paramètres vraisemblance des données sachant les paramètres Distribution à posteriori des paramètres sachant les données

16 MCMCglmm : inférence Bayésienne 2. Comment comparer les modèles Ajustement du modèle : plus il y a de paramètres, plus la déviance est petite (ou la vraisemblance est grande) VraisemblanceBayésien Critère d’Information d’Akaïke DIC AIC = −2log(L(θ))+ 2×np np est le nombre de paramètres compromis entre qualité de l’ajustement et complexité d’un modèle

17 Partie 2. Modèle multivarié But: décomposer la variance phénotypique de plusieurs caractères et estimer la covariance (ou corrélation) génétique entre les traits

18 Two traits can covary because: 1) Influenced by same (or linked) genetic loci (i.e., pleiotropy, linkage) 2) Influenced by same environmental effects (e.g. more food bigger size and larger clutch size) => Covariance can be partitioned as variance hence for 2 traits 1 & 2: V P1 = V A1 + V E1 V P2 = V A2 + V E2 COV P.12 = COV A.12 + COV E.12

19 Animal model framework extends to multiple traits If traits are correlated, measurements on one trait are informative for the other (even if unmeasured) Additive genetic covariance : covariance between two traits that is due to additive genetic effects Often rescaled to genetic correlation -1< r G <1 Example: Estimating genetic correlations between sexes r G = COV A12 √(V A1.V A2 ) Advantages of multivariate models

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21 Any questions..? « L’Ecoute », Victorien Bastet

22 FIXED effect  When the levels of interest are those of the study  Tells you the impact of each factor on the mean  Costly in DF  E.g. sex, categorical, 2 levels = male / female age, continuous, 4 levels = 1-4 yrs old RANDOM effect  When the levels of the study are a random sample from a larger population  Tells you the part of the variance explained by each factor  Uses 1 DF  E.g. Maternal identity FIXED or RANDOM ? Comparing modelsUnivariateMutlivariateModel structurePitfalls

23 If a variable can be expected to have V A ≠ 0, do not include it as fixed effect, but rather as covariate in a multivariate animal model Provide V P(obs) from raw data as well as V P = V A + V E When implementing R = h 2 S, use similar fixed effects for estimations of h 2 and S Prefer V A or CV A to h 2 when comparing studies Rules of thumb on fixed effects Wilson. 2008. Journal Evolutionary Biology Comparing modelsUnivariateMutlivariateModel structurePitfalls

24 YEAR as FIXED effect  Removes variance due to between year environmental differences, e.g. population density, climate  Estimate impact of the mean of each year (BLUEs) YEAR as RANDOM effect  Estimates how much of the total phenotypic variance is explained by between year environmental differences, e.g. population density, climate  Typically if n > 20 FIXED or RANDOM ? The example of year Comparing modelsUnivariateMutlivariateModel structurePitfalls

25 Accounting for non-genetic causes of resemblance also relies on data => Best way to distinguish gene/environment effects on phenotypic similarities is to combine animal model + cross-fostering The delicate use of fixed effects Misassigned paternities Difficulty to analyse non-gaussian traits (e.g. survival) Computational complexity: beware of the black box!! Pitfalls Comparing modelsUnivariateMutlivariateModel structurePitfalls

26 Suitable for complex and incomplete pedigrees => ideal for the available long-term dataset Not restricted to one level of relatedness  Makes full use of all data simultaneously  Estimates are more precise  Higher power (to detect low h 2 ) Less susceptible to environmental biases BUT this all depends on the quality of the data Advantages of the animal model Comparing modelsUnivariateMutlivariateModel structurePitfalls