Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
What matters for evolution is the additive genetic variance VA V P = VA + VD + VI + VE Why? The additive genetic variance component is not affected by interactions within and between loci, and therefore the genetic part that predictably passed on from parent to offspring. Some Definitions • Genotypic value: the phenotypic value a genotype would produce in the absence of environmental effects. A property of the diploid genotype. Not (fully) passed on to next generation. OR (working definition): The average phenotype of a genotype across all environments. • Average effect: the mean deviation from the population mean of individuals which received that gene from one parent, the gene received from the other parent having come at random from the population (Falconer & Mackay 1996). Not easily measurable. • Breeding value: mean genotypic value of an individual‘s offspring (determined by the average effect of the gene). Also termed „additive genotype“, or „additive genetic value“. A property of the haploid genotype. Passed on to next generation! The „variance in breeding values“ determines the response to selection and hence evolutionary change. 1 From Values to Means and Variances: The Normal (Gaussian Distribution) A: small mean, large variance B: small mean, small variance C: large mean, large variance Density = (2πVP ) e 0.5 ⎡ ( P − P )2 ⎤ ⎢− i ⎥ 2VP ⎥⎦ ⎢⎣ The standard normal distribution is defined by two parameters, the mean and the variance From Values to Means and Variances: The Normal Distribution Phenotypic Mean 1 P= N Phenotypic Variance ∑ Pi i =1 ∑ (P − P ) N N Vp = i =1 The The standardized standardized normal normal distribution: distribution: Values Values minus minus mean, mean, devided devided by by standard standard deviation deviation → → mean mean of of distribution distribution == 0, 0, Variance Variance == 1; 1; → → N(0,1). N(0,1). 2 i N −1 P −P Pˆi = i Vp xˆi = xi − x sd x 2 From Gene-Frequencies to Population Means Genotype Frequency Genotypic Value Mean value per genotype AA p2 2pq q2 +a d -a p2a 2pqd Aa aa Population mean genotypic/phenotypic value (sum of products) -q2a G = P = p 2 a + 2 pqd − q 2 a = a( p − q )( p + q ) + 2 pqd = a( p − q ) + 2 pqd From Gene-Frequencies to Population Means m Loci (no epistasis): m m j =1 j =1 P = G = ∑ a j ( p j − q j ) + 2 ∑ d j p jq j j: index for locus identity 3 From Gene-Frequencies to Population Variances Additive Variance (variance of breeding values) VA = 2 pq[a + d (q − p )] 2 Note: The additive genetic variance partially contains variance due to dominance effects! Dominance Variance (Variance of dominance deviations) VD = (2 pqd ) 2 Note: The dominance genetic variance is purely based on variation from dominance effects! From Gene-Frequencies to Population Variances VA = 2 pq[a + d (q − p )] 2 VD = (2 pqd ) 2 Pure Additivity (d=0) Complete Dominance (a=d) Genotypic Variance (VG) Additive Variance (VA) Complete Overdominance (a=0) Dominance Variance (VA) 4 From Gene-Frequencies to Population Variances Level: Population Variance components: VP = VA + VD + VI + VE Phenotypic Additive Dominance Interaction (epistatic) Environmental Genotypic Summary We built now a distribution of phenotypes in a population by simulating genetic and environmental effects on the phenotypes. We used a “bottom-up” approach, starting with assumed gene action patterns and predicting the phenotypic distribution as determined by genotype frequencies and gene action effects. In applied quantitative genetics, we usually do the opposite – measuring the distribution of phenotypes in a population and designing experiments to separate statistically the different genetic and environmental components of variation. This can be said to be a “topdown” approach to inferring genetic components involved in generating heritable trait variation. 5 Heritability Non-additive Variance Broadsense V VA + VD + VI h2 = G = VP VA + VD + VI + VE Broad-sense heritability: „degree of genetic determination“ Narrowsense V VA h2 = A = VP VA + VD + VI + VE = regression of breeding value on phenotypic value = phenotype – additive genotype covariance Heritability 6 The importance of heritability The „Breeders Equation“ R = h2S Va = S Vp R R: response to selection S: selection differential; the difference between the mean trait value of selected individuals and the whole population before selection; can also be estimated by the covariance of fitness with the trait. h2 1 S Empirical Estimation of Variance Components 1. 2. 3. Statistical background Experimental design Statistical models 7 1. Statistical background Variance / Covariance Definitions Y - + Covariance(X,X) = Variance Covariance(X,Y) = Covariance Y - + 0 ∑ (X n 0 X Cov ( X , Y ) = i =1 i − X )(Yi − Y ) n −1 X n: sample size, number of replicates 8 Regression Regression y = a + bx Y a = y − bx b= b Cov( X , Y ) Var ( X ) a: intercept b: slope a 0 0 Correlation X r ( X , Y )= The regression line is the line through a cloud of datapoints that minimizes the summ of squared deviations of Y from a fitted line. Cov( X , Y ) Var ( X )Var (Y ) Check Check out out Appendix Appendix in in Conner Conner and and Hartl! Hartl! 4-offspring mean pistil length 2-offspring mean pistil length Heritability, Regression and Correlation Slope=h2=0.79(0.12) R2=0.47 Slope=h2=0.74(0.09) R2=0.59 Midparent pistil length Slope of regression stays quite stable, Correlation becomes stronger Pistil: Stempel 9 Y ANOVA (Analysis of Variance) Y1 Y2 Comparison of two groups •• •• •• •• •• V : Variance between Vmeans means: Variance between group group means means n: n: number number of of groups groups SS: SS: sum sum of of squared squared deviations deviations d.f.: d.f.: degrees degrees of of freedom freedom n-1: n-1: because because 11 d.f. d.f. is is „used „used up“ up“ for for estimating estimating the the grand grand mean mean ANOVA (Analysis of Variance) Comparison of multiple groups •• •• •• MS: MS: Mean Mean squares squares F: F: F-statistics, F-statistics, MS MS among among devided devided by by MS MS within within P: P: statistical statistical significance, significance, calculated calculated based based on on FFstatistics statistics 10 ANOVA (Example) 11