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

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

n = 960 breeders Pedigree of breeding swans

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 Phil.Trans.R.Soc.Lond.B fixed effects (e.g. sex, age)

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

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

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

It’s just another mixed model… y = X β sex + Z a + e y = X β + Σ i Z i u i + e y = = 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 1 Fixed effect: sex 1 Random effect: additive genetic

y = = 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

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

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 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

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)

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 )

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

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

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

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

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

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

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

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 Journal Evolutionary Biology Comparing modelsUnivariateMutlivariateModel structurePitfalls

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

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

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