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Stanford 30 m

Mather 1400 m

Timberline 3050 m

Stanford 30 m

Mather 1400 m Growing location

Timberline 3050 m

Figure 9.7 The longest stem and number of stems phenotypes for seven Achillea genotypes originally sampled at Aspen Valley, California, cloned from vegetative clippings, and then transplanted at three elevations. Achillea phenotypes show that genetic ( vy, environmental (FE), and genotype-by-environment (VGxE) interaction contribute to total phenotypic variation (Fp). Data are from Table 11 and the photograph from Figure 17 of Clausen et al. (1948). Used with permission of Carnegie Institution of Washington.

phenotypic value of a genotype and any environmental impact on the phenotype of that genotype are completely independent. The absence of a correlation between genotypes and effects of the environment is illustrated in Fig. 9.3, where phenotypic values are as likely to increase as to decrease due to their environment. In contrast, a correlation between genotypic value and environment is common in agricultural contexts. For example, domestic animals are often fed in proportion to their individual size or productivity, such as adjusting the amount of feed given to individual cows according to their production of milk. This feeding practice introduces a correlation between genotypic value and environmental variation that then has an impact on the total phenotypic variation. In the case of cows being fed based on milk production, total phenotypic variation should increase since individuals producing more milk (because of their geno-typic value) are fed more (a non-random environment), which in turn makes their milk production even greater due to the impact of having more to eat.

### Additional sources of phenotypic variance

In many organisms, progeny share an environment with their mother, their father or both parents for some period of time as embryos or during their development and growth. This common environment, sometimes symbolized VEc for common environmental variance, can cause parents and offspring to resemble each other to a greater degree than they would if they each inhabited randomly sampled environments. Maternal effects, the correlation between environment and genotype for mothers and their offspring, are common in mammals since mothers supply the pre-natal environment and often provide extensive postnatal care for young. As an example, consider mammals that nurse their progeny. A mother living in an environment rich with food is likely to have a greater body mass since she is well fed. Because the mother receives ample nutrition, she will also be able to produce ample milk for her offspring. If the offspring grow faster and larger due to their mother's ample milk supply, then there will be positive covariance between the mother's body size and their own body size. This positive covariance in body size is caused by the rich environment shared by the mother and her offspring. A shared common environment can also lead to an increase in phenotypic resemblance among full or half siblings since they too share the same environment. Thinking again of a mother providing an ample supply of milk due to a rich environment, the progeny of such a mother could exhibit positive covariance among their phenotypes

(a similar large body size or fast growth rate) due to their common milk supply environment compared to siblings raised in different random environments.

The total phenotypic variation is modified for non-independence of genotypic value and environmental variation by

where covGE is the covariance between genotypic value and environmental variation in phenotype. Since a covariance can be either positive or negative, the total phenotypic variance can be either increased or decreased by a correlation between the genotypic value and the impact of the environment. The covGE is almost never quantified in practice and is assumed to be zero. The contribution of covGE (if any) to VP is therefore lumped in with non-additive causes of VG. Despite this, it is important to remember that any covariance between genotypes and environments is a potential cause of increased or decreased variance in phenotype.

### Math box 9.1 Summing two variances

In quantitative genetics, it is common to sum variance components to obtain a total variance. For example, the total phenotypic variance is the sum of the genotypic variance and the environmental variance according to VP = VG + VE. When summing variances of two variables, it is possible that they are not completely independent of each other. Therefore, it is necessary to account for the possibility of covariance between the variables and to adjust the total variance. When summing the variance of two variables X and Y to obtain the total variance, var(X + Y) = var(X) + var(Y) + 2covar(X, Y)

A covariance can either be positive (e.g. when the value of X is large the value of Y tends to be large) or negative (when the value of X is large the value of Y tends to be small). Unless X and Y are independent (for any value of X the value of Y is random), the sum of two variances may be increased or decreased by any covariance.

Another possible source of genotypic variance (VG) comes from gametic disequilibrium between loci as well as autozygosity within loci. As detailed in Chapter 2, physical linkage, finite population size, admixture of genetically diverged populations, mutation, and natural selection can all produce gametic disequilibrium. Additionally, consanguineous mating also causes an increase in gametic disequilibrium. When consanguineous mating occurs, one result is increased homozygosity for individual loci. An increase in homozygosity leads to gametic disequilibrium because genotypes at two or more loci within an individual will not be a random combination of all possible genotypes but will tend to be combinations of homozygous genotypes.

All forms of gametic disequilibrium contribute to the disequilibrium covariance of genotypic values that increases or decreases the total pheno-typic variance (Cockerham 1956; Weir et al. 1980). The degree of gametic disequilibrium can be measured as a covariance between individuals of the correlation between genotypic values at two loci within an individual. To visualize this covariance, first imagine the possible correlations between the genotypic values at two loci within a single individual. For example, consider the two locus genotype AABB in a population. If the A and B alleles both contribute to larger phenotypic values, the AA genotypic value and the BB genotypic value are positively correlated in an individual with the AABB genotype. Under random mating and free recombination, the AA genotypic value and the BB genotypic value are found together in the same individual only occasionally since the two loci are independent. Free recombination and random mating mean that AA and BB single locus genotypes co-occur at random. In contrast, imagine the AABB genotype is common in the population due to consanguineous mating or strong gametic disequilibrium that causes high frequencies of AB gametes. In that case there will be a positive correlation of genotypic values between the A and B loci within individuals because the AA and BB single locus genotypes tend to occur together more frequently than expected under independent assortment.

Under random mating in a large population there should be little gametic disequilibrium for a trait not under selection and the fixation index (F) should be approximately zero. This leads to a disequilibrium covariance of zero. Since the genotypic disequilibrium covariance is not estimated in practice, these conditions become implicit assumptions in the decomposition of VP = VG + VE. As with correlations between genotypic value and environment, the contribution (if any) of genotypic covariance to VP is lumped in with non-additive causes of VG.

9.2 Evolutionary change in quantitative traits

• Heritability.

• Changes in quantitative trait mean and variance due to natural selection.

• Estimating heritability by parent-offspring regression.

• Response to selection on correlated traits.

• Long-term response to selection.

• Neutral evolution of quantitative traits.

Evolutionary change in quantitative traits is caused by the processes that reduce, shape, or increase variation in the genotypes that underlie the geno-typic portion of the total phenotypic variance. Just like the individual loci considered in earlier chapters, the multiple loci that compose a quantitative trait will experience genetic drift and mutation and will also be subject to natural selection. Change in quantitative trait means and variances will occur to the extent that these processes change the allele and genotypes frequencies at those loci that contribute to a quantitative trait. Because it is usually not possible to track the individual loci that underlie a quantitative trait, the genetic basis of quantitative traits is tracked by summary measures that describe a population. A critical distinction that we need to bear in mind is the difference between additive genetic variation (VA) that leads to resemblance between relatives, and dominance and interaction genetic variation (VD and Vj) that is not inherited. Because additive genetic variation does lead to parent-offspring resemblance, it is the basis of the action of natural selection on quantitative traits.

### Heritability

The heritability is used to express the proportion of the total phenotypic variance (VP) that is caused by either all types of genotypic variance (VG) or by only the additive genetic variance (VA). Utilizing the equation for the components of the total phenotypic variance VP = VG + VE, both sides of the equation can be multiplied by 1/ VP to obtain

This divides the total phenotypic variance, or 100% of Vp, into the proportion caused by genotypic variation and the proportion caused by environmental variation. The proportion of the total phenotypic variance caused by genotypic variance defines the broad-sense heritability:

Since genotypic variance is composed of the separate components VA + VD + Vj, the proportion of the total phenotypic variance caused by only the additive genetic variance defines the narrow-sense heritability:

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