Broad-sense heritability tells us the proportion of the variance in a population that is due to genetic factors. Broad-sense heritability expresses the degree to which the differences in the phenotypes among individuals in a population are determined by differences in their genotypes. However, even when there is genetic variation in a population as measured by broad-sense heritability, it may not be transmissible to the next generation in a predictable way. In this section, we will explore how genetic variation comes in two forms—additive and dominance (nonadditive) variation. Whereas additive variation is predictably transmitted from parent to offspring, dominance variation is not. We will also define another form of heritability called narrow-sense heritability, which is the ratio of the additive variance to the phenotypic variance. Narrow-sense heritability provides a measure of the degree to which the genetic constitution of individuals determines the phenotypes of their offspring.

The different modes of gene action (interaction among alleles at a locus) are at the heart of understanding narrow-sense heritability, so we will briefly review them. Consider a locus, B, that controls the number of flowers on a plant. The locus has two alleles B₁ and B₂ and three genotypes— B₁/B₁, B₁/B₂, and B₂/B₂. As diagrammed in Figure 19-6a, plants with the B₁/B₁ genotype have 1 flower, B₁/B₂ plants have 2 flowers, and B₂/B₂ plants have 3 flowers. In a case like this, when the heterozygote’s trait value is midway between those of the two homozygous classes, gene action is defined as additive. In Figure 19-6b, the heterozygote has 3 flowers, the same as the B₂/B₂ homozygote. Here, the B₂ allele is dominant to the B₁ allele. In this case, the gene action is defined as dominant. (We could also define this gene action as recessive with the B₁ allele being recessive to the B₂ allele.) Gene action need not be purely additive or dominant but can show partial dominance. For example, if B₁/B₂ heterozygotes had 2.5 flowers on average, then we would say that the B₂ allele shows partial dominance.

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Let’s work through a simple example to show how the mode of gene action influences heritability. Suppose a plant breeder wants to create an improved plant population with more flowers per plant. Flower number is controlled by the B locus, which has two alleles, B₁ and B₂, as diagrammed in Figure 19-6a. The frequencies of the B₁ and B₂ alleles are both 0.5, and the frequencies of the B₁/B₁,B₁/B₂, and B₂/B₂ genotypes are 0.25, 0.50, and 0.25, respectively. Plants with the B₁/B₁ genotype have 1 flower, B₁/B₂ plants have 2 flowers, and B₂/B₂ plants have 3 flowers. The mean number of flowers per plant in the population is 2.0. (Remember that we can calculate the mean as the sum of the products of frequency of each class times the value for that class.)

Since the heterozygote has a phenotype that is midway between the two homozygous classes, gene action is additive. There are no environmental effects, and the genotype alone determines the number of flowers, so H² is 1.0. If the plant breeder selects 3-flowered plants (B₂/B₂), intermates them, and grows the offspring, then all the offspring will be B₂B₂ and the mean number of flowers per plant among the offspring will be 3.0. When gene action is completely additive and there are no environmental effects, the phenotype is fully heritable. Selection as practiced by the plant breeder works perfectly.

Now let’s consider the case diagrammed in Figure 19-6b, in which the B₂ allele is dominant to the B₁. In this case, the B₁B₂ heterozygote is 3-flowered. The frequency of the B₁ and B₂ alleles are both 0.5, and the frequencies of the B₁/B₁, B₁/B₂, and B₂/B₂ genotypes are 0.25, 0.50, and 0.25, respectively. Again, there is no environmental contribution to the differences among individuals, so H² is 1.0. The mean number of flowers per plant in the starting population is 2.5.

If the plant breeder selects a group of 3-flowered plants, 2/3 will be B₁/B₂ and 1/3 B₂/B₂. When the breeder intermates the selected plants, 0.44 (2/3 × 2/3) of the crosses would be between heterozygotes, and 1/4 of the offspring from these crosses would be B₁/B₁ and thus 1-flowered. The remainder of the offspring would be either B₁/B₂ or B₂/B₂ and thus 3-flowered. The overall mean for the offspring would be 2.78, although the mean of their parents was 3.0. Hence, when there is dominance, the phenotype is not fully heritable. Selection as practiced by the plant breeder worked but not perfectly because some of the differences among individuals are due to dominance.

In conclusion, when there is dominance, we cannot strictly predict the offspring’s phenotypes from the parents’ phenotypes. Some of the differences (variation) among the individuals in the parental generation are due to the dominance interactions between alleles. Since parents transmit their genes but not their genotypes to their offspring, these dominance interactions are not transmitted to the offspring.

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As described above, traits controlled by genes with additive gene action will respond very differently to selection than those with dominance. Thus, geneticists need to quantify the degree of dominance and additivity. In this section, we will see how this is done. Let’s again consider the B locus that controls the number of flowers on a plant (see Figure 19-6). The additive effect (A) provides a measure of the degree of change in the phenotype that occurs with the substitution of one B₂ allele for one B₁ allele. The additive effect is calculated as the difference between the two homozygous classes divided by 2. For example, as shown in Figure 19-6a, if the trait value of the B₁/B₁ genotype is 1 and the trait value of the B₂/B₂ genotype is 3, then

The dominance effect (D) is the deviation of the heterozygote (B₁/B₂)from the midpoint of the two homozygous classes. As shown in Figure 19-6b, if the trait value of the B₁/B₁ genotype is 1, of the B₁/B₂ genotype, 3, and of the B₂/B₂ genotype, 3, then

If you calculate D for the situation depicted in Figure 19-6a, you’ll find D = 0; that is, no dominance.

The ratio of D/A provides a measure of the degree of dominance. For Figure 19-6a, D/A = 0.0, indicating pure additivity or no dominance. For Figure 19-6b, D/A = 1.0, indicating complete dominance. A D/A ratio of −1 would indicate a complete recessive. (The distinction between dominance and recessivity depends on how the phenotypes are coded and is in this sense arbitrary.) Values that are greater than 0 and less than 1 represent partial dominance, and values that are less than 0 and greater than −1 represent partial recessivity.

Here is an example of calculating additive and dominance effects at a single locus. Three-spined sticklebacks (Gasterosteus aculeatus) have marine populations with long pelvic spines and populations that live near the bottoms of freshwater lakes with highly reduced pelvic spines (Figure 19-7a). The spines are thought to play a role in defense against predation. The bottom-dwelling freshwater populations are derived from the ancestral marine populations. A change in predation between the marine and freshwater environments may explain the loss of spines in the freshwater environments (see Chapter 20).

Pitx1 is one of several genes that contributes to pelvic-spine length in sticklebacks. This gene encodes a transcription factor that regulates the development of the pelvis in vertebrates, including the growth of pelvic spines in sticklebacks. Michael Shapiro and his colleagues at Stanford University measured the pelvic-spine length in an F₂ population that segregated for the marine or long (l) allele and freshwater or short (s) allele of Pitx1. They recorded the following mean values (in units of proportion of body length) for pelvic-spine length for the three genotypic classes:

Using these values and the formulas above, we can calculate the additive and dominance effects. The additive effect (A) is

(0.148 – 0.068)/2 = 0.04

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or 4 percent of body length. The dominance effect (D) is

0.132 – [(0.148 + 0.068)/2] = 0.024

The dominance/additivity ratio is

0.024/0.04 = 0.6

The 0.6 value for the ratio indicates that the long (l) allele of Pitx1 is partially dominant to the short (s) allele.

One can also calculate additive and dominance effects averaged over all the genes in the genome that affect the trait. Here is an example using cave fish (Astyanax mexicanus) and their surface relatives (Figure 19-7b). The cave populations have highly reduced (small-diameter) eyes compared to the surface populations. Populations colonizing lightless caves do not benefit from having eyes. Since there are physiological and neurological costs to forming and maintaining eyes, evolution may have favored a reduction in the size of the eye in cave populations.

Horst Wilkins at the University of Hamburg measured mean eye diameter (in mm) for the cave and surface populations and their F₁ hybrid:

Using the formulas above, we calculate that A = 2.48, D = 0.52, and D/A = 0.21. In this case, gene action is closer to a purely additive state, although the surface genome is slightly dominant.

The example above with the B locus and flower number shows that we cannot accurately predict offspring phenotypes from parental phenotypes when there is dominance, although we can do so in cases of pure additivity. When predicting the phenotypes of offspring, we need to separate the additive and dominance contributions. To do this, we need to modify the simple model introduced in Section 19.2, x = g + e.

Let’s begin by looking more closely at the situation depicted in Figure 19-6b. Individuals with the B₁/B₂ and B₂/B₂ genotypes have the same phenotype, 3 flowers. If we subtract the population mean (2.5) from their trait value (3), we see that they have the same genotypic deviation (g):

Now let’s calculate the mean phenotypes of their offspring. If we self-pollinate a B₁/B₂ individual, the offspring will be B₁/B₁, B₁/B₂, and B₂/B₂, and the mean trait value of these offspring would be 2.75. However, if we self-pollinate a B₂/B₂ individual, the offspring will all be B₂/B₂, and the mean trait value of these offspring would be 3.0. Even though the B₁/B₂ and B₂/B₂ individuals have the same trait value and the same value for their genotypic deviation (g), they do not produce the equivalent offspring because the underlying basis of their phenotypes is different. The phenotype of the B₁/B₂ individual depends on the dominance effect (D), while that of the B₂/B₂ individual does not involve dominance.

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We can expand the simple model (x = g + e) to incorporate the additive and dominance contributions. The genotypic deviation (g) is the sum of two components—a the additive deviation, which is transmitted to offspring, and d the dominance deviation, which is not transmitted to offspring. We can rewrite the simple model and separate out these two components as follows:

The additive deviation is transmitted from parent to offspring in a predictable way. The dominance deviation is not transmitted from parent to offspring since new genotypes and thus new interactions between alleles are created each generation.

Let’s look at how the genetic deviation is decomposed into the additive and dominance deviations for the case shown in Figure 19-6b.

The genotypic deviations (g) are simply calculated by subtracting the population mean (2.5) from the trait value for each genotype. Each genotypic deviation is then decomposed into the additive (a) and dominance (d) deviations using formulas that are beyond the scope of this book. These formulas include the additive (A) and dominance (D) effects as well as the frequencies of the B₁ and B₂ allele in the population. You’ll notice that a + d sum to g. The additive (a) and dominance (d) deviations are dependent on the allele frequencies because the phenotype of an offspring receiving a B₁ allele from one parent will depend on whether that allele combines with a B₁ or B₂ allele from the other parent, and that outcome depends on the frequencies of the alleles in the population.

The additive deviation (a) has an important meaning in plant and animal breeding. It is the breeding value, or the part of an individual’s deviation from the population mean that is due to additive effects. This is the part that is transmitted to its progeny. Thus, if we wanted to increase the number of flowers per plant in the population, the B₂/B₂ individuals have the highest breeding value. Breeding values can also be calculated for the genome overall for an individual. Animal breeders estimate the genomic breeding values of individual animals, and these estimates can determine the economic value of the animal.

We have partitioned the genetic deviation (g) into the additive (a) and dominance (d) deviations. Using algebra similar to that described in Box 19-2, we can also partition the genetic variance into the additive and dominance variances as follows:

V_g = V_a + V_d

where V_a is the additive genetic variance and V_d is the dominance variance. V_a is the variance of the additive deviations or the variance of the breeding values. It is the part of the genetic variation that is transmitted from parents to their offspring. V_d is the variance of the dominance deviations. Finally, we can substitute these terms in the equation for the phenotypic variance presented earlier in the chapter:

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where V_e is the environmental variance. This equation assumes that the additive and dominance components are not correlated with the environmental effects. This assumption will be true in experiments in which individuals are randomly assigned to environments.

Thus far, we have described models with genetic, environmental, additive, and dominance deviations and variances. In quantitative genetics, the models can get even more complex. In particular, the models can be expanded to include interaction between factors. If one factor alters the effect of another factor, then there is an interaction. Box 19-4 briefly reviews how interactions are factored into quantitative genetic models.

We can now define narrow-sense heritability, which is symbolized by a lowercase h squared (h²), as the ratio of the additive variance to the total phenotypic variance:

This form of heritability measures the extent to which variation among individuals in a population is predictably transmitted to their offspring. Narrow-sense heritability is the form of heritability of interest to plant and animal breeders because it provides a measure of how well a trait will respond to selective breeding.

To estimate h², we need to measure V_a, but how can this be accomplished? Using algebra and logic similar to that we used to show that V_g can be estimated using the covariance between monozygotic twins reared separately (see Box 19-3), it can also be shown that the covariance between a parent and its offspring is equal to one-half the additive variance:

The parent-offspring covariance is one-half of V_a because the offspring inherits only one-half of its genes from the parent. Combining this formula with the one for h², we get

To estimate V_a using the covariance between parents and offspring requires controlling environmental factors in experiments. This can be a challenge because parents and offspring are necessarily reared at different times. V_a can also be estimated using the covariance between half-sibs, in which case all individuals in the experiment can be reared at the same time in the same environment. Half-sibs share one-fourth of their genes, so V_a equals 4 × the covariance between half-sibs.

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If you compare the equation for h² to the one for H² (see Box 19-3), you will see that both involve the ratio of a covariance to a variance. The correlation coefficient introduced earlier in the chapter is also the ratio of a covariance to a variance. We are using the degree of correlation among relatives to infer the extent to which traits are heritable.

Here is an exercise that your class can try. Have each student submit his or her height and the height of their same-sex parent. Using these data and spreadsheet computer software, calculate the covariance between parents and their offspring (the students). Then estimate h² as two times the covariance divided by the phenotypic variance. For the total phenotypic variance (V_X) in the denominator of the equation, you can use the variance among the parents. Data for male and female students should be analyzed separately.

Typically, values for narrow-sense heritability of height in humans are about 0.8, meaning that about 80 percent of the variance is additive, or transmissible, from parent to offspring. The results for your class could deviate from this value for several reasons. First, if your class is small, sampling error can affect the accuracy of your estimate of h². Second, you will not be conducting a randomized experiment. If parents re-create in their households the growth-promoting (or growth-limiting) environments that they experienced as children, then there will be a correlation between the environments of the parents and their offspring. This correlation of environments violates an assumption of the analysis. Third, the population of students in your class may not be representative of the population in which the 0.8 value was obtained.

Figure 19-8 is a scatter plot with the height data for male and female students and their parents. There is a clear correlation between the heights of the students and their same-sex parent. These data give estimates of narrow-sense heritability of 0.86 for mother-daughter and 0.82 for father–son. The results are close to the value of h² equals 0.8 obtained from studies in which the children were separated at birth from their parents and reared in adoptive households.

Here are a few more points about narrow-sense heritability. First, when h² = 1.0 (V_a = V_X), the expected value for an offspring’s phenotype will equal the mid-parent value. All the variation in the population is additive and heritable in the narrow sense. Second, when h² = 0.0 (V_a = 0), the expected value of any offspring’s phenotype will be the population mean. All the variation in the population is due either to dominance or to environmental factors, and thus it is not transmissible to offspring. Finally, as with broad-sense heritability (H²), narrowsense heritability is the property of the specific environment and population in which it was measured. An estimate from one population and environment may not be meaningful for another population or environment.

Narrow-sense heritability is an important concept both in plant and animal breeding and in evolution. For a breeder, h² indicates which traits can be improved by artificial selection. For an evolutionary biologist, h² is critical to understanding how populations will change in response to natural selection imposed by a changing environment. Table 19-5 lists estimates of narrow-sense heritability for some traits and organisms.

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In order to efficiently improve crops and livestock for traits of agronomic importance, the breeder must be able to predict an offspring’s phenotype from its parents’ phenotypes. Such predictions are made using the breeder’s knowledge of narrow-sense heritability. An individual’s phenotypic deviation (x) from the population mean is the sum of the additive, dominance, and environmental deviations:

x = a + d + e

The additive part is the heritable part that is transmitted to the offspring. Let’s look at a set of parents with phenotypic deviations x′ for the mother and x″ for the father. The parents’ dominance deviations (d′ and d″) are not transmitted to their offspring since new genotypes and new dominance interactions are created with each generation. Similarly, the parents do not transmit their environmental deviations (e′ and e″) to their offspring.

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Thus, the only factors that parents transmit to their offspring are their additive deviations (a′ and a″). Accordingly, we can estimate the offspring’s phenotypic deviation (x_o) as the mean of the additive deviations of its parents ( ).

So to predict the offspring’s phenotype, we need to know its parents’ additive deviations. We cannot directly observe the parents’ additive deviations, but we can estimate them. The additive deviation of an individual is the heritable part of its phenotypic deviation; that is,

where signifies an estimate of the additive deviation or breeding value. Thus, we can estimate the mean of the parents’ additive deviations as the product of h² times the mean of their phenotypic deviation and this product will be an estimate of the phenotypic deviation of the offspring ( ):

or

The offspring will have its own dominance and environmental deviations. However, these cannot be predicted. Since they are deviations, they will be zero on average over a large number of offspring.

Here is an example. Icelandic sheep are prized for the quality of their fleece. The average adult sheep in a particular population produces 6 lb of fleece per year. A sire that produces 6.5 lb per year is mated with a dam that produces 7.0 lb per year. The narrow-sense heritability of fleece production in this population is 0.4. What is the predicted fleece production for offspring of this mating? First, calculate the phenotypic deviations for the parents by subtracting the population mean from their phenotypic values:

Now multiply h² times to determine , the estimated phenotypic deviation of the offspring:

0.4 × 0.75 = 0.3

Finally, add the population mean (6.0) to the predicted phenotypic deviation of the offspring (0.3) and obtain the result that the predicted phenotype of the offspring is 6.3 lb of fleece per year.

It may seem surprising that the offspring are predicted to produce less fleece than either parent. However, this outcome is expected for a trait with a modest heritability of 0.4. Most (60 percent) of the superior performance of the parents is due to dominance and environmental factors that are not transmitted to the offspring. If the heritability were 1.0, then the predicted value for the offspring would be midway between the parents’. If the heritability were 0.0, then the predicted value for the offspring would be at the population mean since all the variation would be due to nonheritable factors.

Our final topic regarding narrow-sense heritability is the application of selection over the long term to improve the performance of a population for a complex trait. By applying selection, plant breeders over the past 10,000 years transformed a host of wild plant species into the remarkable array of fruit, vegetable, cereal, and spice crops that we enjoy today. Similarly, animal breeders applied selection to domesticate many wild species, transforming wolves into dogs, jungle fowl into chickens, and wild boar into pigs.

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Selection is a process by which only individuals with certain features contribute to the gene pool that forms the next generation (see Chapters 18 and 20). Selection applied by humans to improve a crop or livestock population is termed artificial selection to distinguish it from natural selection. Let’s look at an example of how artificial selection works.

Provitamin A is a precursor in the biosynthesis of vitamin A, an important nutrient for healthy eyes and a well-functioning immune system. Plant products are an important source of provitamin A for humans; however, people in many areas of the globe have too little provitamin A in their diets. To solve this problem, a plant breeder seeks to increase the provitamin A content of a maize population used in parts of Latin America where vitamin A deficiency is common. At present, this population produces 1.25 μg of provitamin A per gram of kernels. The variance for the population is 0.06 μg² (Figure 19-9). To improve the population, the breeder selects a group of plants that produce 1.5 μg or more of provitamin A per gram of kernels. The mean for the selected group is 1.63 μg. The breeder randomly intermates the selected plants and grows the offspring to produce the next generation, which has a mean of 1.44 μg per gram of kernels.

If the narrow-sense heritability of a trait is not known before performing an artificial selection experiment, one can use the results of such experiments to estimate it. Here’s an example using the case of provitamin A in maize. Let’s start with the equation from above:

and rewrite it as

is the mean deviation of the parents (the selected plants) from the population mean. This is known as the selection differential (S), the difference between the mean of the selected group and that of the base population. For our example,

is the mean deviation of the offspring from the population mean. This is known as the selection response (R), the difference between the mean of the offspring and that of the base population. For our example,

Now we can calculate the narrow-sense heritability for this trait in this population as

The underlying logic of this calculation is that the response represents the heritable or additive part of the selection differential.

Over the last century, quantitative geneticists have conducted a large number of selection experiments like this. Typically, these experiments are performed over many generations and are referred to as long-term selection studies. Each generation, the best individuals are selected to produce the subsequent generation. Such studies have been performed in economically important species such as crop plants and livestock and in many model organisms such as Drosophila, mice, and nematodes. This work has shown that virtually any species will respond to selection for virtually any trait. Populations contain deep pools of additive genetic variation.

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Here are two examples of long-term selection experiments. In the first experiment, fruit flies were selected for increased flight speed over a period of 100 generations (Figure 19-10a). Each generation, the speediest flies were selected and bred to form the next generation. Over the 100 generations, the average flight speed of the flies in the population increased from 2 to 170 cm/sec, and neither the flies nor the gains made by selection showed any signs of slowing down after 100 generations. In the second experiment, mice were selected over 10 generations for the amount of “wheel running” they did per day (Figure 19-10b). There was a 75 percent increase over just 10 generations. These studies and many more like them demonstrate the tremendous power of artificial selection and deep pools of additive genetic variation in species.