Lecture 13: Quantitative Genetics: the classics
Contents
Lecture 13: Quantitative Genetics: the classics¶
The next section of the course focuses on understanding the genetic bases of phenotypic variation. Quantitative genetics deals with the genetics of continuously varying characters. Rather than considering changes in the frequencies of specific alleles of genotypes, quantitative genetics seeks to “quantify” changes in the frequency distribution of traits that cannot easily be placed in discrete phenotypic classes. The reason for the continuous variation is usually that the traits are polygenic (controlled by many genes) and there are environmental effects that alter the phenotypic state of each individual.
Partitioning Phenotypic Variance and Heritability¶
Consider two inbred strains that represent “extremes” of a phenotypic distribution: high and low oil content in corn for example or long and short carrots. We will assume that the plants of each type are homozygous at all loci. Under this assumption the variation we see within each group is entirely environmental variation and the variation we see between the two groups is mostly (but not entirely) genetic variation. If we then cross an individual from the high group (ABCD) with an individual from the low group (abcd) we would get \(F_1\) hybrids (ABCD/abcd) that are intermediate in phenotype. We would notice that each individual is not identical in phenotype even though each is identical in genotype (all \(F_1\)’s). We would then attribute all the variation in phenotype to an environmental component, \(V_E\). If we than crossed all the F1’s with each other, we would get an \(F_2\) distribution that would have a wider distribution. Because of independent assortment of chromosomes and recombination in the \(F_1\)’s each \(F_2\) is likely to have a unique multilocus genotype. Thus the total phenotypic variance in the \(F_2\) distribution will have both a genetic component, \(V_G\) and an environmental component (\(V_E\)). In simple terms, these are related by the expression
If you were given a bunch of plants with a smooth continuous distribution of phenotypes, how would you determine if there was a genetic basis to the variation? Simply select individuals from the distribution with distinct phenotypes, breed them (=parents) and compare the phenotypes of these parents to that of their offspring. If the mean phenotype of offspring was close to the mean of the parents this would be evidence for a genetic basis for the phenotype and the trait would be identified as heritable. If on the other hand, the offspring produced from two “high” parents were extremely variable in phenotype and offspring produced from two “low” parents were extremely variable there would be a weak genetic component to the trait. The heritability in a “broad” sense can be expressed as the proportion of the total phenotypic variance that has a genetic component,
This is known as broad sense heritability because we have not yet partitioned the genetic variance component into additive and non-additive components itself. We’ll get to this in a second.
This correlation between parent and offspring can serve as a simple means of quantifying the heritability of the trait: if there is a 1:1 correlation of phenotype between parents and offspring (e.g., a 45 degree slope of the regression of offspring phenotype vs. parent phenotype) then the trait has the maximal heritability. With no relation between parents and offspring (a slope of zero) the heritability would be zero.
The genetic component of the variation can be broken up into different sub-components. Consider a simple additive model of height in corn plants where the number of \(A_1\) alleles you have determines your height: \(A_1A_1\) = 3m, \(A_1A_2\) = 2m and \(A_2A_2\) = 1m. The mean of these phenotypes is 2m; if you subtract this mean from each of the three phenotypes you get 1m, 0m and -1m as the difference. These values describe the additive effect of replacing one \(A_2\) allele with one \(A_1\) allele. A single \(A_1\) allele has one half the effect of two \(A_1\) alleles, so our additive effect, \(a = 1\). If we cross a \(A_1A_1 \times A_2A_2\) we would get only \(A_1A_2\) with a height of 2m. Crosses between these \(F_1\)s would result in a \(1:2:1\) ratio of 3m:2m:1m plant heights and the mean of the \(F_2\)s (\(2\)) would be the same as the mean of the \(F_1\)s and the mean of the two parents.
Now consider that \(A_1A_1\) and \(A_1A_2\) have the same phenotype (i.e., there is dominance): \(A_1A_1\) = 3m, \(A_1A_2\) = 3m and \(A_2A_2\) = 1m. A cross between \(A_1A_1\) and \(A_2A_2\) would produce \(A_1A_2\) \(F_1\)s all 3m tall . An \(F_1\) cross \(A_1A_2 \times A_1A_2\) would produce \(F_2\)s with a 3:1 ratio of 3m:1m. In these the mean of the two parents would be 2, the mean of the \(F_1\)s would be 3 and the mean of the \(F_2\)s would be 2.5. Thus, dominance would affect the variation in phenotypes and there is said to be a dominance component to the variance. The genetic variance can be partitioned into additive and dominance components (and an interaction component which we will ignore)
Now the total phenotypic variance can be rewritten as a function of our individual variance components
The point of this is that we want to know the additive genetic component of the total phenotypic variance since this is what makes parent and offspring look alike and is what selection can act upon. Note also that interactions between genes can be non-additive: we call such interactions epistasis. Let’s now refine our description of heritability to mean the proportion of the total phenotypic variance that is due to additive genetic effects
where \(h^2_N\) is called narrow sense heritability. This allows us to define what kind of response to selection we would get if we imposed a specific intensity of selection on a phenotypic trait. The selection differential (\(S\)) is the difference between the mean of the parents selected to produce the next generation and the mean of all individuals in the population. The response to selection (\(R\)) is the change in the mean phenotype after selection. The response to selection depends of the heritability of the trait such that
An important consequence of this is that as selection proceeds, the additive genetic variation will be reduced (e.g. “low” alleles removed). As the \(V_A\) decreases, the heritability decreases (see equation for heritability above). Will selection come to a halt?? Probably no because mutation is constantly introducing a trickle of new alleles each with different additive effects. Under this view the gradual changes in phenotype seen over long evolutionary times might be explained by a continual mutation-selection balance.
What are the heritabilities of various traits in nature? They vary a lot One trend is that fitness-related traits tend to have lower heritabilities than other traits. Why? In natural populations fitnesses determine our “selection differentials” so selection should remove genetic variation for fitness traits and heritabilities will drop. Why then are not the heritabilities of fitness related traits zero? One answer: genetic correlations.
Due to the linkage of genes along chromosomes (or epistatic interactions among genes) selection of one trait can lead to selection for another trait. If a viability gene is linked between genes for bristle number, selection for high bristle number could lead to low viability if the low viability allele became associated with the high oil content alleles. If this were the case, there would be a negative genetic correlation between bristle number and viability. As it turns out, many fitness related traits have negative genetic correlations (e.g., size of eggs negatively correlated with number of eggs). Thus if fitness determines selection differential, selection in natural populations could not remove all the additive genetic variation for two fitness traits that are negatively correlated.
Plasticity and norms of reaction¶
Before we dive further in to understanding the genetic basis of variation (i.e. \(V_A\)) lets take a look at the environmental component of phenotypic variance (\(V_E\)).
Phenotypic plasticity is the ability of individuals to alter its physiology, morphology and/or behavior in response to a change in the environmental conditions. This is clearly demonstrated by the appearance of plants grown at different densities: crowded plants look spindly and lanky, uncrowded plants look healthy and robust. In the context of evolution, phenotypic plasticity demonstrates the two meanings of adaptation: the plastic response is itself an example of a physiological adaptation and it is widely held that the ability to be plastic is adaptive in the sense of increasing fitness.
In thinking about phenotypic plasticity as a evolutionary adaptation it is important to separate the trait in question from the plasticity for that trait. For example: growing taller in response to plant crowding is adaptive in the sense that it increases an individual’s competitive ability for sunlight (lower fitness when shaded by other plants). The “normal” height for a plant (lets assume there is such a thing) may have evolved in response to pressures to allocate resources to growth versus reproduction in a particular way. Thus there is a genetic basis for plasticity of plant height, and a genetic basis for plant height itself. The point is that different genes probably control these processes so the trait and its plasticity can (as opposed to must) evolve independently.
Now consider the environment: certain physical properties of the environment can be described by the mean (average) value or the range of values (highest - lowest). Which aspect of an organism (the trait itself or the plasticity for that trait) will evolve in response to which measure? It may be that the plasticity for a trait will evolve in response to the range of values the environment throws at an organism (e.g., coldest - hottest, driest-wettest days), whereas the trait itself (e.g., thickness of fur) will evolve in response to the mean. This is not a rule! but would be an interesting thing to test and/or think about.
The idea of plasticity is interwoven with the notion of canalization. In light of the ball rolling down the trough of a developmental pathway, one can consider the width of the trough as an indication of the amount of plasticity “tolerated” in the organism in question. A highly canalized organism (or developmental program) would have low plasticity.