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G matrix The genetic additive variance/ covariance matrix that expresses the heritability of each trait (the diagonal elements) as well as the genetic covariance between each trait (the off-diagonal elements). P matrix The phenotypic variance/ covariance matrix that expresses the variance of each trait (the diagonal elements) as well as the covariance between each trait (the off-diagonal elements). s The selection differential, expressed as a vector of values when there are two or more traits.

z The phenotypic mean, expressed as a vector of values when there are two or more traits.

Dividing G by P expresses the additive genetic variance per unit of phenotypic variance. In this example, the heritabilities are h2 = 0.3 for trait one and h2 = 0.06 for trait two so clearly trait one would show a greater response to selection for a given value of the selection differential.

With the potential for different phenotypic variances for each trait, covariance between phenotypes, and genetic correlation, the breeder's equation needs to be extended. For two or more traits, the change in mean phenotype for each trait is predicted by

where P-1 indicates a matrix inverse (Lande 19 79; Lande & Arnold 1983). For more than one trait, the change in the mean trait value is represented by the vector z rather than the scalar R since there are as many means as there are traits. The selection differential is still symbolized as s even though it is now a vector.

Some examples will help illustrate how equation 9.21 serves to combine the direct effect of selection on each trait with the indirect effects of genetic and phenotypic correlations between traits to predict the total change in trait mean. In Table 9.4b, there is no genetic correlation and also no phenotypic correlation, making the traits completely independent. Both traits have heritabilities of 0.5 and selection differentials of 0.5. The response to selection is exactly as we would predict from the single-trait version of the breeder's equation for each trait separately. Note that trait B has a lower response to selection because it has more phenotypic variance, making the selection differential of 0.5 effectively weaker.

In the example of Table 9.4c, there is a fairly strong positive phenotypic correlation between the two traits and natural selection applied only to trait A but still no genetic correlation between traits. The mean of trait A is predicted to increase since it is experiencing natural selection and has a non-zero heritability. Trait B also shows response to selection, in this case a reduction in mean value. This change in mean is due to the correlation between the two traits alone and not due to any direct natural selection in trait B since its selection differential is zero.

In the final example in Table 9.4d, there is a strong positive genetic correlation between the two traits and natural selection is acting to increase the average of trait A. There is now no phenotypic correlation between the traits. The means of both traits are predicted to increase in this case. The mean of trait A will increase because of selection acting directly on it. At the same time the mean of trait B will also increase. This occurs not because selection is acting on trait B, after all it has a selection differential of zero, but rather because the two traits are genetically correlated. The change in genotype and allele

Interact box 9.2 Response to natural selection on two correlated traits

Solving the equation Az = GP~h requires the use of matrix algebra. For those who have access to the program Matlab, the text web page has a short program that can be used to define G, P, and s and then solve for Az.

As an exercise, change the sign of both the phenotypic and genetic covariances shown in the examples of Table 9.4. How does the predicted response to selection change?

frequency caused by response to selection on trait A has also changed genotype and allele frequencies that influence the mean of trait B. Direct selection for an increase in the mean of A indirectly causes an increase in the mean of B due to a genetic correlation between the traits. To distinguish direct and indirect effects of natural selection, the change in a trait mean due to a direct effect is called selection for while selection of describes a change in a trait mean caused by an indirect response to selection on a genetically correlated trait.

When traits are genetically correlated, natural selection and any response is potentially not as simple as it would be with a single trait, which would be completely independent of all other traits. This is particularly true of quantitative traits related to Darwinian fitness of individuals. First, natural selection experienced by one trait could lead to a response to selection in another trait that does not directly experience natural selection. This means that natural selection on correlated traits is capable of indirectly changing traits that have little or no relationship with fitness. Second, when two traits respond to selection over time it may lead to the evolution of a negative genetic correlation that prevents further response to selection. If two traits are both related to fitness and each experience natural selection, then we expect alleles at any loci that independently cause variation in either trait to become fixed by long-term response to selection. However, any loci that have opposite effects on the two traits will not experience fixation or loss caused by natural selection. For example, if increased frequencies of AA genotypes increase fitness of trait one but simultaneously lead to decreases fitness for trait two, natural selection should result in neither the A nor a allele fixing. Alleles at such loci with contrasting effects cannot be fixed by selection because changing the allele frequency to increase fitness for one trait simultaneously causes a decrease in fitness for the other trait. Genetic variation at loci with such contrasting effects on traits causes a negative genetic correlation and prevents further change in trait means by natural selection.

Many examples of correlated responses to natural selection have been observed in agricultural organisms since artificial selection is routinely practiced to alter quantitative traits to increase yield and improve growth and harvest phenotypes. In one experiment, pigs were subjected to one generation of artificial selection for increased litter size in sows (Estany et al. 2002). During the course of this artificial selection experiment, the progeny of the selected females were measured for a number of morphological and behavioral phenotypes between the ages of 75 and 165 days old. A direct response to artificial selection was observed, increasing the litter size of the selected sows by an average of 0.46 piglets per litter compared to unselected controls. At the same time, the progeny of the selected sows showed a number of phenotypic differences from the control progeny of unselected sows. The pigs from selected sows deposited fat more rapidly, grew at different rates, and gained less weight per kilogram of feed. The pigs in the selected and control populations also showed behavioral differences, with the progeny of selected sows visiting feeding stations less frequently but spending more time eating and eating more per feeding bout. Since only the number of piglets per litter was under artificial selection, all of the other changes observed in quantitative traits were the result of correlated responses to selection caused by genetic correlations.

Long-term response to selection

The breeder's equation predicts response to selection over single-generation intervals. Extrapolating the breeder's equation to longer time periods implicitly assumes that h2 and s remain constant through time. However, when natural or artificial selection continues for many generations, the assumptions of the breeder's equation may no longer hold. In particular, additive genetic variation may be consumed by response to selection over time as allele frequencies at the multiple loci that cause quantitative trait variation change over time. How rapidly additive genetic variation is exhausted by selection depends critically on the number of loci that underlie a quantitative trait as well as the percentage of trait variation that is caused by each locus (Fig. 9.12).

Genotypic variation in quantitative traits can sometimes be caused by the alleles segregating at a relatively small number of loci. When quantitative trait variation is caused by a small number of loci, additive genetic variance for a trait is depleted over time by response to selection. The decline in herit-ability occurs because response to selection causes allele frequencies at the loci that cause genotypic variation to move toward fixation and loss. When only a few loci explain genetic variation in a quantitative trait, changes in the trait mean from one generation to the next come about due to substantial changes in the allele frequencies of those few loci. For example, when there is artificial selection for an increased trait mean as in Fig. 9.10, alleles that confer higher values of the trait at each locus are increased in frequency each generation. The greater a response to selection that has occurred, due to either continued selection over time or stronger selection, the more likely that allele frequencies at each locus will have been altered toward fixation and loss (Fig. 9.12a). In addition, genotypic variation in all traits will decrease over time in finite populations due to genetic drift, a process that is accelerated by selection since selection itself leads to reduced effective population sizes because not all individuals in the population contribute alleles to the next generation of progeny.

An alternative model is that the additive genetic variation in a quantitative trait is caused by a very large number of individual loci and all of these many loci have equal very small effects on a quantitative trait. Under these assumptions, response to selection may continue for a long time before changes occur in the amount of additive genetic variance (Fig. 9.12b). Under this infinitesimal model, as the number of loci grows very large then the amount of trait variation explained by any one locus approaches zero. Under the many loci assumption, when a quantitative trait responds to selection the trait mean changes but there is almost no change in the allele frequencies of the individual loci that cause genotypic variation because each locus explains such a small fraction of the additive genetic variation. Under the infinitesimal model then, response to selection can occur for many generations without causing substantial changes in the heritability required for response to selection. Note that, even under the infinitesimal model, response to selection acts to increase gametic disequilibrium for the loci that cause genotypic variation, potentially slowing response to selection over time.

Infinitesimal model A model of the genetic basis of quantitative traits that assumes that a very large number of independent loci contribute equally to trait genotypic variation, so that the impact of each locus on trait genotypic variation approaches zero.

The amount of additive genetic variation for a quantitative trait over many generations is not simply a function of the amount of standing genetic variation before natural selection starts. Rather, the amount of additive genetic variation over time is the net outcome of natural selection and genetic drift working

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