Your DNA extractions! 10 kb

Your DNA extractions! 10 kb

Quantitative characters: polygenes and environment Most ecologically important quantitative traits (QTs) vary. Distributions are often unimodal and approximately normal. Offspring and parents are correlated. What s the explanation? Independent contributions by genotypes at many loci, and by random environmental influences. Biol 2005, 6 October 2014

The simplest QT model: independent loci with + and - alleles Assume each individual s trait value is the sum of its + alleles at all loci. That is, a + allele at locus A has the same effect as a + at locus B. Then with random mating, we get quasi-binomial distributions of the number of +. As the number of loci increases, these distributions become smooth and normal. aa aa AA aabb aabb + aabb aabb + AAbB AABB aabb AAbb aabb

Also, as the number of loci affecting the trait increases 6 loci 25 loci short tall short tall And the sum of effects (e.g., on height) becomes very 100 loci close to normally distributed, even if 400 the loci individual effects are very non- normal (e.g., + and - with different effect sizes and allele frequencies). short tall short its variance relative to the potential range decreases. As a result, quantitative traits can easily evolve to new mean values far outside their old range, if they re controlled by many loci, each of which has an individually small effect on the trait value. tall

The simplest model: genomic and environmental causes add up Mom makes a genomic contribution X m. Its variance (over moms) is V(X m ). The environment makes a contribution ε. Its variance (over offspring) is V(ε). Dad makes a genomic contribution X p Its variance (over dads) is V(X p ) For any given offspring, its phenotype (quantitative character state) is the sum of these three contributions. And over the population as a whole, the variance of the phenotypic values is the sum of the variances of the three contributions: V(P) = V(X m ) + V(X p ) + V(ε) = V G + V E (This assumes that the parents are uncorrelated with each other, and with the environment.)

QTs are normally distributed because each of the three contributions is itself the sum of many independent genetic or environmental causes. Offspring are correlated with their parents (and siblings) because their genes are half identical to those of each parent.

V E is the phenotypic variation induced by environmental factors Even clones and identical twins differ from each other! Homozygous (inbred) shortflowered parents Homozygous (inbred) tallflowered parents From tree #1 From tree #2 Heterozygous but genetically uniform F1 offspring Clones (cuttings) of Achillea grown at three different elevations where the species normally occurs in California. Edward East s Nicotiana plants growing in the same garden plots. Leaves from a natural clone of quaking aspen (Populus tremuloides) growing at the top of Millcreek Canyon.

October 5, 2014

October 6, 2013

September 29, 2014

September 29, 2013

How the components of the phenotypic variance add 1. Genetic variance is the variance among phenotypes caused by genotypic differences among individuals (holding their environments constant). 2. Environmental variance is the variance among phenotypes caused by differences in the experiences of individuals (holding genotypes constant). Example: Suppose the average trait values of AA, Aa and aa individuals are -1, 0, and +1 units, and p = q = 0.5. Then the genetic variance (average squared deviation from the population mean) is 0.5. But suppose 25% of each genotype deviates one unit above or below its average trait value, because of the environment. Then the environmental variance is also 0.5. The resulting phenotypic variance is 0.5 + 0.5 = 1.0. V P = V G + V E [= (V A + V D ) + (V E )] -1 0 +1 AA Aa aa -2-1 0 AA AA AA V G V E V P Broad-sense heritability: H 2 = V G /V P. Narrow-sense heritability: h 2 = V A /V P. -2-1 0 +1 +2 AA AA AA Aa Aa Aa aa aa aa

h 2 is the regression (slope) of offspring on parents offspring h 2 0 offspring h 2 ½ offspring h 2 1 parents parents parents Definition of the regression coefficient (slope): b yx = cov(x,y)/var(x) Here x is the midparent value (parental mean), y is the offspring value. The higher the slope, the better offspring resemble their parents. In other words, the higher the heritability, the better offspring trait values are predicted by parental trait values (because genotypic effects are large, and predominantly additive).

How to estimate the variance components (classical approach) 1. Measure phenotypes (trait values) in a large random sample of the population. 2. Calculate the mean and variance: the variance is V P. 3. Estimate the heritability by regressing offspring on midparent values. 4. The additive variance (V A ) is the heritable fraction of the total: V A = h 2 V P. 5. The remainder is environmental (V E ) and dominance variance (and other minor stuff). 6. If we can clone or closely inbreed members of the species, or find identical twins, then we can directly estimate the environmental variance (V E ).

Leaf shape within and among six quaking aspen clones mean variance East Canyon Upper Millcreek Clone 1 Clone 2 Clone 3 Clone 1 Clone 2 Clone 3 0.902 0.00351 0.992 0.00237 1.075 0.00271 0.861 0.00552 1.028 0.00200 0.918 0.00947 All 0.963 0.00990

Analysis of variance (ANOVA) East Canyon Upper Millcreek Clone 1 Clone 2 Clone 3 Clone 1 Clone 2 Clone 3 All mean variance 0.902 0.00351 0.992 0.00237 1.075 0.00271 0.861 0.00552 1.028 0.00200 0.918 0.00947 0.963 0.00990 R.A. Fisher literally invented ANOVA in his 1918 paper on The resemblance between relatives on the supposition of Mendelian inheritance. Variance among clones = var(0.902, 0.992,, 0.918) = 0.00564 Variance within clones = mean(0.00351,, 0.00947) = 0.00426 Total variance = 0.00564 + 0.00426 = 0.00990 Fraction explained by clones = 0.00564 / 0.00990 = 0.57 = H 2

This analysis focuses on the variation of individual leaves, not on the variation of individual trees (i.e., their mean leaf shapes). mean variance East Canyon Upper Millcreek Clone 1 Clone 2 Clone 3 Clone 1 Clone 2 Clone 3 0.902 0.00351 0.992 0.00237 1.075 0.00271 0.861 0.00552 1.028 0.00200 0.918 0.00947 All 0.963 0.00990

For the seven clones with more than two sampled trees, clone membership explains 75% of the variation in the trees mean W/L ratios. Within clones, mean W/L increases going west (i.e., trees with larger west longitudes tend to have broader leaves, on average).

111.593 111.592 The east-west distance between longitudinal lines (e.g., 111.591 and 111.592) is roughly 300 feet (the length of a football field). 111.591 111.590

The relationships among genotypes often differ among environments Clausen, Keck & Heisey grew the same genotypes of Achillea at different elevations in California. Plant height was highly variable and heritable in each environment, and all genotypes were shorter at high elevation (Mather). However, some plants that were relatively tall in one environment were relatively short in the other. Thus the genetic variance depends on the population s environment! And the environmental variance depends on the population s genes! This situation-dependence of the variance components is called genotype by environment (G x E) interaction.

Upshot: the heritability of a trait can be defined only for a given population (gene pool) in a given environment. In C, K & H s experiment with Achillea, height was highly heritable within each environment. But height in each environment was a poor predictor of height in the others, in two ways: (1) Relative heights differed between the environments, and (2) All genotypes grew taller at Stanford. Differences within populations were largely determined by genes, but with different outcomes in each environment. And the large average difference between populations was entirely non- genetic!

Summary The narrow-sense heritability of a trait is the fraction of the total phenotypic variance that is caused by the additive effects of genes. There can be much non- additive genetic variance caused by dominance (V D ) and epistasis (V I ), but this does not contribute to the resemblance between parents and offspring, or to the response to selection. (But the dominance variance increases the resemblance of full siblings.) There can also be a lot of environmental variance (that is, variance of the trait values that is caused by effects of the environment). These three components of the phenotypic variance literally add up to the total: V P = [V A + (V D + V I )] + [V E ] = [V G ] + [V E ] The analysis of variance (ANOVA) was invented by R.A. Fisher to allow these components to be estimated separately! High heritability within local populations does not imply that average differences between populations are genetic!