An Introduction to Genetic Analysis
Chapter 25
Quantitative Genetics
Chapter 25
Quantitative Genetics
Key Concepts
In natural populations, variation in most characters takes the form of a continuous phenotypic
range rather than discrete phenotypic classes. In other words, the variation is quantitative, not
qualitative.
Mendelian genetic analysis is extremely difficult to apply to such continuous phenotypic
distributions, so statistical techniques are employed instead.
A major task of quantitative genetics is to determine the ways in which genes interact with the
environment to contribute to the formation of a given quantitative trait distribution.
The genetic variation underlying a continuous character distribution can be the result of
segregation at a single genetic locus or at numerous interacting loci that produce cumulative
effects on the phenotype.
The estimated ratio of genetic to environmental variation is not a measure of the relative
contribution of genes and environment to phenotype.
Estimates of genetic and environmental variance are specific to the single population and the
particular set of environments in which the estimates are made.
Introduction
Ultimately, the goal of genetics is the analysis of the genotype of organisms. But the genotype
can be identified — and therefore studied—only through its phenotypic effect. We recognize
two genotypes as different from each other because the phenotypes of their carriers are
different. Basic genetic experiments, then, depend on the existence of a simple relation
between genotype and phenotype. That is why studies of DNA sequences are so important,
because we can read off the genotype directly from this most basic of all phenotypes. In
general, we hope to find a uniquely distinguishable phenotype for each genotype and only a
simple genotype for each phenotype. At worst, when one allele is completely dominant, it
may be necessary to perform a simple genetic cross to distinguish the heterozygote from the
homozygote. Where possible, geneticists avoid studying genes that have only partial
penetrance and incomplete expressivity (see Chapter 4) because of the difficulty of making
genetic inferences from such traits. Imagine how difficult (if not impossible) it would have
been for Benzer to study the fine structure of the gene in phage, if the only effect of the rII
mutants was a 5 percent reduction from wild type in their ability to grow on E. coli K. For the
most part, then, the study of genetics presented in the preceding chapters has been the study of
allelic substitutions that cause qualitative differences in phenotype.
However, the actual variation between organisms is usually quantitative, not qualitative.
Wheat plants in a cultivated field or wild asters at the side of the road are not neatly sorted
into categories of “tall” and “short,” any more than humans are neatly sorted into categories
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An Introduction to Genetic Analysis
Chapter 25
Quantitative Genetics
of “black” and “white.” Height, weight, shape, color, metabolic activity, reproductive rate,
and behavior are characteristics that vary more or less continuously over a range (Figure 25-1).
Even when the character is intrinsically countable (such as eye facet or bristle number in
Drosophila), the number of distinguishable classes may be so large that the variation is nearly
continuous. If we consider extreme individuals—say, a corn plant 8 feet tall and another one 3
feet tall—a cross between them will not produce a Mendelian result. Such a corn cross will
produce plants about 6 feet tall, with some clear variation among siblings. The F2 from selfing
the F1 will not fall into two or three discrete height classes in ratios of 3:1 or 1:2:1. Instead,
the F2 will be continuously distributed in height from one parental extreme to the other. This
behavior of crosses is not an exception; it is the rule for most characters in most species.
Mendel obtained his simple results because he worked with horticultural varieties of the
garden pea that differed from one another by single allelic differences that had drastic
phenotypic effects. Had Mendel conducted his experiments on the natural variation of the
weeds in his garden, instead of abnormal pea varieties, he would never have discovered
Mendel's laws. In general, size, shape, color, physiological activity, and behavior do not
assort in a simple way in crosses.
The fact that most phenotypic characters vary continuously does not mean that their variation
is the result of some genetic mechanisms different from the Mendelian genes with which we
have been dealing. The continuity of phenotype is a result of two phenomena. First, each
genotype does not have a single phenotypic expression but a norm of reaction (see Chapter 1)
that covers a wide phenotypic range. As a result, the phenotypic differences between
genotypic classes become blurred, and we are not able to assign a particular phenotype
unambiguously to a particular genotype. Second, many segregating loci may have alleles that
make a difference in the phenotype under observation. Suppose, for example, that five equally
important loci affect the number of flowers that will develop in an annual plant and that each
locus has two alleles (call them + and −). For simplicity, also suppose that there is no
dominance and that a + allele adds one flower, whereas a − allele adds nothing. Thus, there
are 35 = 243 different possible genotypes [three possible genotypes (+/+, +/−, and −/−) at
each of five loci], ranging from
but there are only 11 phenotypic classes (10, 9, 8, . . . , 0) because many of the
genotypes will have the same numbers of + and − alleles. For example, although there is only
one genotype with 10 + alleles and therefore an average phenotypic value of 10, there are 51
different genotypes with 5 + alleles and 5 − alleles; for example,
Thus, many different genotypes may have the same average phenotype. At the same time,
because of environmental variation, two individuals of the same genotype may not have the
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An Introduction to Genetic Analysis
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Quantitative Genetics
same phenotype. This lack of a one-to-one correspondence between genotype and phenotype
obscures the underlying Mendelian mechanism. If we cannot study the behavior of the
Mendelian factors controlling such traits directly, then what can we learn about their genetics?
Using current experimental techniques, geneticists can answer the following questions about
the genetics of a continuously varying character in a population (say, height in a human
population). These questions constitute the study of quantitative genetics—the study of the
genetics of continuously varying characters:
1. Is the observed variation in the character influenced at all by genetic variation? Are there
alleles segregating in the population that produce some differential effect on the character or
is all the variation simply the result of environmental variation and developmental noise (see
Chapter 1)?
2. If there is genetic variation, what are the norms of reaction of the various genotypes?
3. How important is genetic variation as a source of total phenotypic variation? Are the norms
of reaction and the environments such that nearly all the variation is a consequence of
environmental difference and developmental instabilities or does genetic variation
predominate?
4. Do many loci (or only a few) vary with respect to the character? How are they distributed
over the genome?
5. How do the different loci interact with one another to influence the character? Is there
dominance, and is there any epistasis (interaction between genes at different loci)?
6. Is there any nonnuclear inheritance (for example, any maternal effect)?
In the end, the purpose of answering these questions is to be able to predict what kinds of
offspring will be produced by crosses of different phenotypes.
The precision with which these questions can be framed and answered varies greatly. In
experimental organisms, on the one hand, it is relatively simple to determine whether there is
any genetic influence at all, but extremely laborious experiments are required to localize the
genes (even approximately). In humans, on the other hand, it is extremely difficult to answer
even the question of the presence of genetic influence for most traits, because it is almost
impossible to separate environmental from genetic effects in an organism that cannot be
manipulated experimentally. As a consequence, we know a relatively large amount about the
genetics of bristle number in Drosophila but virtually nothing about the genetics of complex
human traits; a few (such as skin color) clearly are influenced by genes, whereas others (such
as the specific language spoken) clearly are not. The purpose of this chapter is to develop the
basic statistical and genetic concepts needed to answer these questions and to provide some
examples of the applications of these concepts to particular characters in particular species.
Some basic statistical notions
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An Introduction to Genetic Analysis
Chapter 25
Quantitative Genetics
To consider the answers to these questions about the most common kinds of genetic variation,
we must first examine a number of statistical tools that are essential in the study of
quantitative genetics.
Distributions
The outcome of a cross for a Mendelian character can be described in terms of the proportions
of the offspring that fall into several distinct phenotypic classes or often simply in terms of the
presence or absence of a class. For example, a cross between a red-flowered plant and a
white-flowered plant might be expected to yield all red-flowered plants or, if it were a
backcross, 1/2 red-flowered plants and 1/2 white-flowered plants. However, we require a
different mode of description for quantitative characters. The basic concept is that of the
statistical distribution. If the heights of a large number of male undergraduates are measured
to the nearest 5 centimeters (cm), they will vary (say, between 145 and 195 cm), but many
more male undergraduates will be in the middle categories (say, between 170 and 180 cm)
than at the extremes.
Representing each measurement class as a bar, with its height proportional to the number of
individuals in each class, we can graph the result as shown in Figure 25-2a. Such a graph of
numbers of individuals observed against measurement class is a frequency histogram. Now
suppose that five times as many individuals are measured, each to the nearest centimeter. The
classes in Figure 25-2a are now subdivided to produce a histogram like the one shown in
Figure 25-2b. If we continue this process, refining the measurement but proportionately
increasing the number of individuals measured, then the histogram eventually takes on the
continuous appearance of Figure 25-2c, which is the distribution function of heights in the
population.
This continuous curve is an idealization, because no measurement can be taken with infinite
accuracy or on an unlimited number of individuals. Moreover, the measured variate itself may
be intrinsically discontinuous because it is the count of some number of discrete objects such
as eye facets or bristles. It is sometimes convenient, however, to develop concepts by using
this slightly idealized picture as a shorthand for the more cumbersome observed frequency
histogram (Figure 25-2a).
Statistical measures
Although a distribution contains all of the information about a set of measurements, we would
like to be able to distill this information into a few characteristic numbers that convey the
necessary information about the distribution without giving it in detail. The characteristics of
a distribution that we would like to specify are as follows:
1. Where is the distribution located along the range of possible values? Are the observed
values near 100 or near 1000? Therefore we need a measure of central tendency.
2. How much variation is there among individual measurements? Are they all concentrated
around the central measurement or do they vary widely across a large range? That is, we need
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Chapter 25
Quantitative Genetics
a measure of dispersion.
3. If we are considering more than one characteristic, how are the values of these different
characteristics related? Is there any relation between body size and fertility? Is it positive or
negative? Do larger parents have larger offspring? If so, we would regard this as evidence that
genes influence body size. Thus, we need measures of relation between measurements.
Among the most commonly used measures of central tendency are the most frequent
observation, the mode, and the arithmetic average of the observations, the mean. The
dispersion of a distribution is almost always measured by the variance, which is the average
squared distance of the observations from their mean. The relation between different variables
is measured by their correlation, which is the average product of the deviation of one
variable from its own mean times the deviation of the other variable from its own mean.
These most commonly used measures of central tendency, dispersion, and relation are
considered in detail in the Statistical Appendix to this chapter. The detailed discussion is in a
separate section in order not to interrupt the flow of logic of the consideration of quantitative
genetics. It should not be assumed, however, that an understanding of those statistical
concepts is somehow secondary. A proper understanding of quantitative genetics requires a
grasp of that material.
Genotypes and phenotypic distribution
Using the concepts of distribution, mean, and variance, we can understand the difference
between quantitative and Mendelian genetic traits. Suppose that a population of plants
contains three genotypes, each of which has some differential effect on growth rate.
Furthermore, assume that there is some environmental variation from plant to plant because of
inhomogeneity in the soil in which the population is growing and that there is some
developmental noise (see Chapter 1). For each genotype, there will be a separate distribution
of phenotypes with a mean and a standard deviation that depend on the genotype and the set
of environments. Suppose that these distributions look like the three height distributions in
Figure 25-3a. Finally, assume that the population consists of a mixture of the three genotypes
but in the unequal proportions 1:2:3 (a/a:A/a:A/A). Then the phenotypic distribution of
individuals in the population as a whole will look like the black line in Figure 25-3b, which is
the result of summing the three underlying separate genotypic distributions, weighted by their
frequencies in the population. This weighting by frequency is indicated in Figure 25-3b by the
different heights of the component distributions that add up to the total distribution. The mean
of this total distribution is the average of the three genotypic means, again weighted by the
frequencies of the genotypes in the population. The variance of the total distribution is
produced partly by the environmental variation within each genotype and partly by the
slightly different means of the three genotypes.
Two features of the total distribution are noteworthy. First, there is only a single mode.
Despite the existence of three separate genotypic distributions underlying it, the population
distribution as a whole does not reveal the separate modes. Second, any individual whose
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Quantitative Genetics
height lies between the two arrows could have come from any one of the three genotypes,
because they overlap so much. The result is that we cannot carry out any simple Mendelian
analysis to determine the genotype of an individual organism. For example, suppose that the
three genotypes are the two homozygotes and the heterozygote for a pair of alleles at a locus.
Let a/a be the short homozygote and A/A be the tall one, with the heterozygote being of
intermediate height. Because the phenotypic distributions overlap so much, we cannot know
to which genotype a given individual belongs. Conversely, if we cross a homozygote a/a and
a heterozygote A/a, the offspring will not fall into two discrete classes in a 1:1 ratio but will
cover almost the entire range of phenotypes smoothly. Thus, we cannot know that the cross is
in fact a/a × A/a and not a/a × A/A or A/a × A/a.
Suppose we grew the hypothetical plants in Figure 25-3 in an environment that exaggerated
the differences between genotypes—for example, by doubling the growth rate of all
genotypes. At the same time, we were very careful to provide all plants with exactly the same
environment. Then, the phenotypic variance of each separate genotype would be reduced
because all the plants were grown under identical conditions; at the same time, the differences
between genotypes would be exaggerated by the more rapid growth. The result (Figure 25-4b)
would be a separation of the population as a whole into three nonoverlapping phenotypic
distributions, each characteristic of one genotype. We could now carry out a perfectly
conventional Mendelian analysis of plant height. A “quantitative” character has been
converted into a “qualitative” one. This conversion has been accomplished by finding a way
to make the differences between the means of the genotypes large compared with the
variation within genotypes.
MESSAGE
A quantitative character is one for which the average phenotypic differences between
genotypes are small compared with the variation between individuals within
genotypes.
It is sometimes assumed that continuous variation in a character is necessarily caused by a
large number of segregating genes, so continuous variation is taken as evidence for control of
the character by many genes. But, as we have just shown, this is not necessarily true. If the
difference between genotypic means is small compared with the environmental variance, then
even a simple one-gene–two-allele case can result in continuous phenotypic variation.
If the range of a character is limited and if many segregating loci influence it, then we expect
the character to show continuous variation, because each allelic substitution must account for
only a small difference in the trait. This multiple-factor hypothesis (that large numbers of
genes, each with a small effect, are segregating to produce quantitative variation) has long
been the basic model of quantitative genetics, although there is no convincing evidence that
such groups of genes really exist. A special name, polygenes, has been coined for these
hypothetical factors of small-but-equal effect, in contrast to the genes of simple Mendelian
analysis.
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Quantitative Genetics
It is important to remember, however, that the number of segregating loci that influence a trait
is not what separates quantitative and qualitative characters. Even in the absence of large
environmental variation, it takes only a few genetically varying loci to produce variation that
is indistinguishable from the effect of many loci of small effect. As an example, we can
consider one of the earliest experiments in quantitative genetics, that of Wilhelm Johannsen
on pure lines. By inbreeding (mating close relatives), Johannsen produced 19 homozygous
lines of bean plants from an originally genetically heterogeneous population. Each line had a
characteristic average seed weight ranging from 0.64 g per seed for the heaviest line to 0.35 g
per seed for the lightest line. It is by no means clear that all these lines were genetically
different (for example, five of the lines had seed weights of 0.450, 0.453, 0.454, 0.454, and
0.455 g), but let's take the most extreme position—that the lines were all different. These
observations would be incompatible with a simple one-locus–two-allele model of gene action.
In that case, if the original population were segregating for the two alleles A and a, all inbred
lines derived from that population would have to fall into one of two classes: A/A or a/a. If, in
contrast, there were, say, 100 loci, each of small effect, segregating in the original population,
then a vast number of different inbred lines could be produced, each with a different
combination of homozygotes at different loci.
However, we do not need such a large number of loci to obtain the result observed by
Johannsen. If there were only five loci, each with three alleles, then 35 = 243 different kinds
of homozygotes could be produced from the inbreeding process. If we make 19 inbred lines at
random, there is a good chance (about 50 percent) that each of the 19 lines will belong to a
different one of the 243 classes. So Johannsen's experimental results can be easily explained
by a relatively small number of genes. Thus, there is no real dividing line between polygenic
traits and other traits. It is safe to say that no phenotypic trait above the level of the amino
acid sequence in a polypeptide is influenced by only one gene. Moreover, traits influenced by
many genes are not equally influenced by all of them. Some genes will have major effects on
a trait; others, minor effects.
MESSAGE
The critical difference between Mendelian and quantitative traits is not the number of
segregating loci but the size of phenotypic differences between genotypes compared
with the individual variation within genotypic classes.
Norm of reaction and phenotypic distribution
The phenotypic distribution of a trait, as we have seen, is a function of the average differences
between genotypes and of the variation between genotypically identical individuals. But both
are in turn functions of the sequence of environments in which the organisms develop and live.
For a given genotype, each environment will result in a given phenotype (for the moment,
ignoring developmental noise). Then a distribution of environments will be reflected
biologically as a distribution of phenotypes. The way in which the environmental distribution
is transformed into the phenotypic distribution is determined by the norm of reaction, as
shown in Figure 25-5. The horizontal axis is environment (say, temperature) and the vertical
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Chapter 25
Quantitative Genetics
axis is phenotype (say, plant height). The norm of reaction curve for the genotype shows how
each particular temperature results in a particular plant height. So, the dashed lines from the
18°C point on the temperature axis is reflected off the norm of reaction curve to a
corresponding plant height on the vertical phenotype axis, and so forth for each temperature.
If a large number of individuals develop at, say, 20°C, then a large number of individuals will
have the phenotype that corresponds to 20°C, as shown by the dashed line; and, if only small
numbers develop at 18°C, few plants will have the corresponding plant height. Then the
frequency distribution of developmental environments will be reflected as a frequency
distribution of phenotypes as determined by the shape of the norm of reaction curve. It is as if
an observer, standing at the vertical phenotype axis, were seeing the environmental
distribution, not directly, but reflected in the curved mirror of the norm of reaction. The shape
of the curvature will determine how the environmental distribution is distorted on the
phenotype axis. So, the norm of reaction in Figure 25-5 falls very rapidly at lower
temperatures (the phenotype changes rapidly with small changes in temperature) but flattens
out at higher temperatures, so the plant height is much less sensitive to temperature
differences at the higher temperatures. The result is that the symmetric environmental
distribution is converted into an asymmetric phenotype distribution with a long tail at the
larger plant heights, corresponding to the lower temperatures.
By means of the same analysis, Figure 25-6 shows how a population consisting of two
genotypes with different norms of reaction has a phenotypic distribution that depends on the
distribution of environments. If the environments are distributed as shown by the black
distribution curve, then the resulting population of plants will have a unimodal distribution,
because the difference between genotypes is very small in this range of environments
compared with the sensitivity of the norms of reaction to small changes in temperature. If the
distribution of environments is shifted to the right, however, as shown by the gray distribution
curve, a bimodal distribution of phenotypes results, because the norms of reaction are nearly
flat in this environmental range but very different from each other.
MESSAGE
A distribution of environments is reflected biologically as a distribution of
phenotypes. The transformation of environmental distribution into phenotypic
distribution is determined by the norm of reaction.
Determining norms of reaction
Remarkably little is known about the norms of reaction for any quantitative traits in any
species—partly because it is difficult in most sexually reproducing species to replicate a
genotype so that it can be tested in different environments. For this reason, for example, we
do not have a norm of reaction for any genotype for any human quantitative trait.
In domesticated plants and animals
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Chapter 25
Quantitative Genetics
A few norm of reaction studies have been carried out with plants that can be clonally
propagated. The results of one of these experiments are presented in Chapter 1. It is possible
to replicate genotypes in sexually reproducing organisms by the technique of mating close
relatives, or inbreeding. By selfing (where possible) or by mating brother and sister repeatedly
generation after generation, a segregating line (one that contains both homozygotes and
heterozygotes at a locus) can be made homozygous.
The purpose of creating homozygous lines is to produce groups of organisms within which all
individuals are genetically identical. These genetically identical individuals can then be
allowed to develop in different environments to produce a norm of reaction. Alternatively,
two different homozygous lines can be crossed and the F1 offspring, all genetically identical
with one another, can be characterized in different environments.
Ideally for a norm of reaction study all the individuals should be absolutely identical
genetically, but the process of inbreeding increases the homozygosity of the group slowly,
generation after generation, depending on the closeness of the relatives that are mated. In corn,
for example, a single individual is chosen and self-pollinated. Then in the next generation, a
single one of its offspring is chosen and self-pollinated. In the third generation, a single one of
its offspring is chosen and self-pollinated, and so forth. Suppose that the original individual in
the first generation is already a homozygote at some locus. Then all of its offspring from
self-pollination will also be homozygous and identical at the locus. Future generations of
self-pollination will simply preserve the homozygosity. If, on the other hand, the original
individual is a heterozygote, then the selfing A/a × A/a will produce 1/4 A/A homozygotes and
1/4 a/a homozygotes. If a single offspring is chosen in this subsequent generation to
propagate the line, then there is a 50 percent chance that it is now a homozygote. If, by bad
luck, the chosen individual should still be a heterozygote, there is another 50 percent chance
that the selected individual in the third generation is homozygous, and so forth. Of the
ensemble of all heterozygous loci, then, after one generation of selfing, only 1/2 will still be
heterozygous; after two generations, 1/4; after three, 1/8. In the nth generation,
where Hetn is the proportion of heterozygous loci in the nth generation and Het0 is the
proportion in the 0 generation. When selfing is not possible, brother–sister mating will
accomplish the same end, although more slowly. Table 25-1 is a comparison of the amount of
heterozygosity left after n generations of selfing and brother–sister mating.
In natural populations
To carry out a norm of reaction study of a natural population, a large number of lines are
sampled from the population and inbred for a sufficient number of generations to guarantee
that each line is virtually homozygous at all its loci. Each line is then homozygous at each
locus for a randomly selected allele present in the original population. The inbred lines
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Quantitative Genetics
themselves cannot be used to characterize norms of reaction in the natural population, because
such totally homozygous genotypes do not exist in the original population. Each inbred line
can be crossed to every other inbred line to produce heterozygotes that reconstitute the
original population, and an arbitrary number of individuals from each cross can be produced.
If inbred line 1 has the genetic constitution A/A·B/B·c/c·d/d·E/E . . . and inbred line 2
is a/a·B/B·C/C·d/d·e/e . . . , then a cross between them will produce a large number of
offspring, all of whom are identically A/a·B/B·C/c·d/d·E/e . . . and can be raised in
different environments.
Inbreeding by mating of close relatives for many generations results in total homozygosity for
the entire genome. In species such as Drosophila in which the necessary dominant markers
and crossover suppressors are available, it is possible to produce lines that are homozygous
for only a single chromosome, rather than for the whole set, as shown for an autosome in
Figure 25-7. A single male from the population to be sampled is crossed to a female carrying
a chromosome with a crossover suppressor C (usually a complex inversion), a recessive lethal
l, and a dominant visible marker M1 heterozygous with a second dominant visible M2. In the
F1, a single male carrying the ClM1 chromosome is chosen. This male, which is also carrying
a wild-type chromosome from the population, is again crossed to the marker stock. In the F2,
all flies showing the M1 trait but not M2 are necessarily all heterozygotes for copies of the
original wild-type chromosome because ClM1/ClM1 is lethal, and no crossovers have taken
place. In the F3, all wild-type flies are identically homozygous for the wild-type chromosome
and are now available to make a stock for norm of reaction studies for crosses. (See Chapter
15 for another use of this technique.)
Results of norm of reaction studies
Very few norm of reaction studies have been carried out for quantitative characters found in
natural populations, but many more have been carried out for domesticated species such as
corn, which can be self-pollinated, or strawberries, which can be clonally propagated. The
outcomes of such studies resemble those given in Figure 25-8, which shows the norms of
reaction for abdominal bristle number as a function of temperature for second chromosome
homozygotes of D. pseudoobscura. No genotype is consistently above or below the others.
Instead, there are small differences between genotypes, and the direction of these differences
is not consistent over a wide range of environments.
These factors have two important consequences. First, the selection of “superior” genotypes
in domesticated animals and cultivated plants will result in very specifically adapted varieties
that may not show their superior proper-ties in other environments. To some extent, this
problem is overcome by deliberately testing genotypes in a range of environments (for
example, over several years and in several locations). It would be even better, however, if
plant breeders could test their selections in a variety of controlled environments in which
different environmental factors could be separately manipulated. The consequences of actual
plant-breeding practices can be seen in Figure 25-9, in which the yields of two varieties of
corn are shown as a function of different farm environments. Variety 1 is an older variety of
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Quantitative Genetics
hybrid corn; variety 2 is a later “improved” hybrid. These performances are compared at a
low planting density, which prevailed when variety 1 was developed, and at a high planting
density characteristic of farming practice when hybrid 2 was selected. At the high density, the
new variety is clearly superior to the old variety in all environments (Figure 25-9a). At the
low density, however, the situation is quite different. First, note that the new variety is less
sensitive to environment than is the older hybrid, as evidenced by its flatter norm of reaction.
Second, the new “improved” variety is actually poorer under the best farm conditions. Third,
the yield improvement of the new variety is not apparent under the low densities characteristic
of earlier agricultural practice.
The second consequence of the nature of reaction norms is that, even if it should turn out that
there is genetic variation for various mental and emotional traits in the human species, which
is by no means clear, this variation is unlikely to favor one genotype over another across a
range of environments. We must beware of hypothetical norms of reaction for human
cognitive traits that show one genotype unconditionally superior to another. Even putting
aside all questions of moral and political judgment, there is simply no basis for describing
different human genotypes as “better” or “worse” on any scale, unless the investigator is able
to make a very exact specification of environment.
MESSAGE
Norm of reaction studies show only small differences between natural genotypes, and
these differences are not consistent over a wide range of environments. Thus,
“superior” genotypes in domesticated animals and cultivated plants may be superior
only in certain environments. If it should turn out that humans exhibit genetic
variation for various mental and emotional traits, this variation is unlikely to favor
one genotype over another across a range of environments.
The most basic question to be asked about a quantitative trait is whether the observed
variation in the character is influenced by genes at all. It is important to note that this is not
the same as asking whether genes play any role in the character's development.
Gene-mediated developmental processes lie at the base of every character, but variation from
individual to individual is not necessarily the result of genetic variation. Thus, the possibility
of speaking any language at all depends critically on the structures of the central nervous
system as well as of the vocal cords, tongue, mouth, and ears, which depend in turn on the
nature of the human genome. There is no environment in which cows will speak. But,
although the particular language that is spoken by humans varies from nation to nation, that
variation is totally nongenetic.
MESSAGE
The question of whether a trait is heritable is a question about the role that differences
in genes play in the phenotypic differences between individuals or groups.
Familiality and heritability
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Quantitative Genetics
In principle, it is easy to determine whether any genetic variation influences the phenotypic
variation among organisms for a particular trait. If genes are involved, then (on average)
biological relatives should resemble one another more than unrelated individuals do. This
resemblance would be seen as a positive correlation between parents and offspring or between
siblings (offspring of the same parents). Parents who are larger than the average would have
offspring who are larger than the average; the more seeds that a plant produces, the more
seeds that its siblings would produce. Such correlations between relatives, however, are
evidence for genetic variation only if the relatives do not share common environments more
than nonrelatives do. It is absolutely fundamental to distinguish familiality from heritability.
Traits are familial if members of the same family share them, for whatever reason. Traits are
heritable only if the similarity arises from shared genotypes.
There are two general methods for establishing the heritability of a trait as distinct from its
familial occurrence. The first depends on phenotypic similarity between relatives. For most of
the history of genetics, this method has been the only one available; so nearly all the evidence
about heritability for most traits in experimental organisms and in humans has been
established by using this approach. The second method, using marker-gene segregation,
depends on showing that genotypes carrying different alleles of marker genes also differ in
their average phenotype for the quantitative character. If the marker genes (which have
nothing to do with the character under study) are seen to vary in relation to the character,
presumably they are linked to genes that do influence the character and its variation. Thus,
heritability is demonstrated even if the actual genes causing the variation are not known. This
method requires that the genome of the organism being studied have large numbers of
detectable genetically variable marker loci spread throughout the genome. Such marker loci
can be observed from electrophoretic studies of protein variation or, in vertebrates, from
immunological studies of blood group genes. For example, within flocks, chickens of
different blood groups show some difference in egg weight.
Since the introduction of molecular methods for the study of DNA sequence variation, very
large numbers of variable nucleotide positions have been discovered in a great variety of
organisms. This molecular variation includes both single nucleotide replacements and
insertions and deletions of longer nucleotide sequences. These variations are usually detected
by the gain or loss of sites of cleavage of restriction enzymes or by length variation of DNA
sequences between two fixed restriction sites, both of which are a form of restriction fragment
length polymorphisms (RFLPs). In tomatoes, for example, strains carrying different RFLP
variants differ in fruit characteristics.
However, because so much of what is known or claimed about heritability still depends on
phenotypic similarity between relatives, especially in human genetics, we will begin the
examination of the problem of heritability by analyzing phenotypic similarity.
Phenotypic similarity between relatives
In experimental organisms, there is no problem in separating environmental from genetic
similarities. The offspring of a cow producing milk at a high rate and the offspring of a cow
12
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An Introduction to Genetic Analysis
Chapter 25
Quantitative Genetics
producing milk at a low rate can be raised together in the same environment to see whether,
despite the environmental similarity, each resembles its own parent. In natural populations,
and especially in humans, this is difficult to do. Because of the nature of human societies,
members of the same family not only share genes, but also have similar environments. Thus,
the observation of simple familiality of a trait is genetically uninterpretable. In general, people
who speak Hungarian have Hungarian-speaking parents and people who speak Japanese have
Japanese-speaking parents. Yet the massive experience of immigration to North America has
demonstrated that these linguistic differences, although familial, are nongenetic. The highest
correlations between parents and offspring for any social traits in the United States are those
for political party and religious sect, but they are not heritable. The distinction between
familiality and heredity is not always so obvious. The Public Health Commission, which
originally studied the vitamindeficiency disease pellegra in the southern United States in 1910,
came to the conclusion that it was genetic because it ran in families.
To determine whether a trait is heritable in human populations, we must use adoption studies
to avoid the usual environmental similarity between biological relatives. The ideal
experimental subjects are identical twins reared apart, because they are genetically identical
but environmentally different. Such adoption studies must be so contrived that there is no
correlation between the social environment of the adopting family and that of the biological
family. These requirements are exceedingly difficult to meet; so, in practice, we know very
little about whether human quantitative traits that are familial are also heritable. Skin color is
clearly heritable, as is adult height—but even for these traits we must be very careful. We
know that skin color is affected by genes from studies of cross-racial adoptions and
observations that the offspring of black African slaves were black even when they were born
and reared in Canada. But are the differences in height between Japanese and Europeans
affected by genes? The children of Japanese immigrants who are born and reared in North
America are taller than their parents but shorter than the North American average, so we
might conclude that there is some influence of genetic difference. However,
second-generation Japanese Americans are even taller than their American-born parents. It
appears that some environmental–cultural influence or perhaps a maternal effect is still felt in
the first generation of births in North America. We cannot yet say whether genetic differences
in height distinguish North Americans of, say, Japanese and Swedish ancestry.
Personality traits, temperament, and cognitive performance (including IQ scores), as well as a
whole variety of behaviors such as alcoholism and of mental disorders such as schizophrenia,
have been the subject of heritability studies in human populations. Many show familiality.
There is indeed a positive correlation between the IQ scores of parents and the scores of their
children (the correlation is about 0.5 in white American families), but the correlation does not
distinguish familiality from heritability. To make that distinction requires that the
environmental correlation between parents and children be broken, so adoption studies are
common. Because it is difficult to randomize the environments, even in cases of adoption,
evidence of heritability for human personality and behavior traits remains equivocal despite
the very large number of studies that exist. Prejudices about the causes of human differences
are widespread and deep, and, as a result, the canons of evidence adhered to in studies of the
13
勇者并非无所畏惧,而是判断出有比恐惧更重要的东西.
An Introduction to Genetic Analysis
Chapter 25
Quantitative Genetics
heritability of IQ, for example, have been much more lax than in studies of milk yield in
cows.
Figure 25-10 summarizes the usual method for testing heritability in experimental organisms.
Individuals from both extremes of the distribution are mated with their own kind, and the
offspring are raised in a common controlled environment. If there is an average difference
between the two offspring groups, the trait is heritable. Most morphological traits in
Drosophila, for example, turn out to be heritable—but not all of them. If flies with right
wings that are slightly longer than their left wings are mated together, their offspring have no
greater tendency to be “right winged” than do the offspring of “left winged” flies. As we shall
see later, this method can also be used to obtain quantitative information about heritability.
MESSAGE
In experimental organisms, environmental similarity can often be readily
distinguished from genetic similarity (heritability). In humans, however, it is very
difficult to determine whether a particular trait is heritable.
Heritability of a trait
The most basic question to be asked about a quantitative trait is whether the observed
variation in the character is influenced by genes at all. It is important to note that this is not
the same as asking whether genes play any role in the character's development.
Gene-mediated developmental processes lie at the base of every character, but variation from
individual to individual is not necessarily the result of genetic variation. Thus, the possibility
of speaking any language at all depends critically on the structures of the central nervous
system as well as of the vocal cords, tongue, mouth, and ears, which depend in turn on the
nature of the human genome. There is no environment in which cows will speak. But,
although the particular language that is spoken by humans varies from nation to nation, that
variation is totally nongenetic.
MESSAGE
The question of whether a trait is heritable is a question about the role that differences
in genes play in the phenotypic differences between individuals or groups.
Familiality and heritability
In principle, it is easy to determine whether any genetic variation influences the phenotypic
variation among organisms for a particular trait. If genes are involved, then (on average)
biological relatives should resemble one another more than unrelated individuals do. This
resemblance would be seen as a positive correlation between parents and offspring or between
siblings (offspring of the same parents). Parents who are larger than the average would have
offspring who are larger than the average; the more seeds that a plant produces, the more
seeds that its siblings would produce. Such correlations between relatives, however, are
evidence for genetic variation only if the relatives do not share common environments more
14
勇者并非无所畏惧,而是判断出有比恐惧更重要的东西.
An Introduction to Genetic Analysis
Chapter 25
Quantitative Genetics
than nonrelatives do. It is absolutely fundamental to distinguish familiality from heritability.
Traits are familial if members of the same family share them, for whatever reason. Traits are
heritable only if the similarity arises from shared genotypes.
There are two general methods for establishing the heritability of a trait as distinct from its
familial occurrence. The first depends on phenotypic similarity between relatives. For most of
the history of genetics, this method has been the only one available; so nearly all the evidence
about heritability for most traits in experimental organisms and in humans has been
established by using this approach. The second method, using marker-gene segregation,
depends on showing that genotypes carrying different alleles of marker genes also differ in
their average phenotype for the quantitative character. If the marker genes (which have
nothing to do with the character under study) are seen to vary in relation to the character,
presumably they are linked to genes that do influence the character and its variation. Thus,
heritability is demonstrated even if the actual genes causing the variation are not known. This
method requires that the genome of the organism being studied have large numbers of
detectable genetically variable marker loci spread throughout the genome. Such marker loci
can be observed from electrophoretic studies of protein variation or, in vertebrates, from
immunological studies of blood group genes. For example, within flocks, chickens of
different blood groups show some difference in egg weight.
Since the introduction of molecular methods for the study of DNA sequence variation, very
large numbers of variable nucleotide positions have been discovered in a great variety of
organisms. This molecular variation includes both single nucleotide replacements and
insertions and deletions of longer nucleotide sequences. These variations are usually detected
by the gain or loss of sites of cleavage of restriction enzymes or by length variation of DNA
sequences between two fixed restriction sites, both of which are a form of restriction fragment
length polymorphisms (RFLPs). In tomatoes, for example, strains carrying different RFLP
variants differ in fruit characteristics.
However, because so much of what is known or claimed about heritability still depends on
phenotypic similarity between relatives, especially in human genetics, we will begin the
examination of the problem of heritability by analyzing phenotypic similarity.
Phenotypic similarity between relatives
In experimental organisms, there is no problem in separating environmental from genetic
similarities. The offspring of a cow producing milk at a high rate and the offspring of a cow
producing milk at a low rate can be raised together in the same environment to see whether,
despite the environmental similarity, each resembles its own parent. In natural populations,
and especially in humans, this is difficult to do. Because of the nature of human societies,
members of the same family not only share genes, but also have similar environments. Thus,
the observation of simple familiality of a trait is genetically uninterpretable. In general, people
who speak Hungarian have Hungarian-speaking parents and people who speak Japanese have
Japanese-speaking parents. Yet the massive experience of immigration to North America has
demonstrated that these linguistic differences, although familial, are nongenetic. The highest
15
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An Introduction to Genetic Analysis
Chapter 25
Quantitative Genetics
correlations between parents and offspring for any social traits in the United States are those
for political party and religious sect, but they are not heritable. The distinction between
familiality and heredity is not always so obvious. The Public Health Commission, which
originally studied the vitamindeficiency disease pellegra in the southern United States in 1910,
came to the conclusion that it was genetic because it ran in families.
To determine whether a trait is heritable in human populations, we must use adoption studies
to avoid the usual environmental similarity between biological relatives. The ideal
experimental subjects are identical twins reared apart, because they are genetically identical
but environmentally different. Such adoption studies must be so contrived that there is no
correlation between the social environment of the adopting family and that of the biological
family. These requirements are exceedingly difficult to meet; so, in practice, we know very
little about whether human quantitative traits that are familial are also heritable. Skin color is
clearly heritable, as is adult height—but even for these traits we must be very careful. We
know that skin color is affected by genes from studies of cross-racial adoptions and
observations that the offspring of black African slaves were black even when they were born
and reared in Canada. But are the differences in height between Japanese and Europeans
affected by genes? The children of Japanese immigrants who are born and reared in North
America are taller than their parents but shorter than the North American average, so we
might conclude that there is some influence of genetic difference. However,
second-generation Japanese Americans are even taller than their American-born parents. It
appears that some environmental–cultural influence or perhaps a maternal effect is still felt in
the first generation of births in North America. We cannot yet say whether genetic differences
in height distinguish North Americans of, say, Japanese and Swedish ancestry.
Personality traits, temperament, and cognitive performance (including IQ scores), as well as a
whole variety of behaviors such as alcoholism and of mental disorders such as schizophrenia,
have been the subject of heritability studies in human populations. Many show familiality.
There is indeed a positive correlation between the IQ scores of parents and the scores of their
children (the correlation is about 0.5 in white American families), but the correlation does not
distinguish familiality from heritability. To make that distinction requires that the
environmental correlation between parents and children be broken, so adoption studies are
common. Because it is difficult to randomize the environments, even in cases of adoption,
evidence of heritability for human personality and behavior traits remains equivocal despite
the very large number of studies that exist. Prejudices about the causes of human differences
are widespread and deep, and, as a result, the canons of evidence adhered to in studies of the
heritability of IQ, for example, have been much more lax than in studies of milk yield in
cows.
Figure 25-10 summarizes the usual method for testing heritability in experimental organisms.
Individuals from both extremes of the distribution are mated with their own kind, and the
offspring are raised in a common controlled environment. If there is an average difference
between the two offspring groups, the trait is heritable. Most morphological traits in
Drosophila, for example, turn out to be heritable—but not all of them. If flies with right
16
勇者并非无所畏惧,而是判断出有比恐惧更重要的东西.
An Introduction to Genetic Analysis
Chapter 25
Quantitative Genetics
wings that are slightly longer than their left wings are mated together, their offspring have no
greater tendency to be “right winged” than do the offspring of “left winged” flies. As we shall
see later, this method can also be used to obtain quantitative information about heritability.
MESSAGE
In experimental organisms, environmental similarity can often be readily
distinguished from genetic similarity (heritability). In humans, however, it is very
difficult to determine whether a particular trait is heritable.
Quantifying heritability
If a trait is shown to have some heritability in a population, then it is possible to quantify the
degree of heritability. In Figure 25-3, we saw that the variation between phenotypes in a
population arises from two sources. First, there are average differences between the genotypes;
second, each genotype exhibits phenotypic variance because of environmental variation. The
total phenotypic variance of the population (S2p) can then be broken into two parts: the
variance between genotypic means (S2g) and the remaining variance (S2e) The former is called
the genetic variance, and the latter is called the environmental variance; however, as we
shall see, these names are quite misleading. Moreover, the breakdown of the phenotypic
variance into the sum of environmental and genetic variance leaves out the possibility of some
covariance between genotype and environment. For example, suppose it were true (we do not
know) that there are genes that influence musical ability. Parents with such genes might
themselves be musicians, who would create a more musical environment for their children,
who would then have both the genes and the environment promoting musical performance.
The result would be an increase in the phenotypic variances of musical ability and an
erroneous estimate of genetic and environmental variances. If the phenotype is the sum of a
genetic and an environmental effect, P = G + E, then, as explained on page 768 of the
Statistical Appendix, the variance of the phenotype is the sum of the genetic variance, the
environmental variance, and twice the covariance between the genotypic and environmental
effects.
If genotypes are not distributed randomly across environments, there will be some covariance
between genotype and environmental values, and the covariance will be hidden in the genetic
and environmental variances.
The degree of heritability can be defined as the part of the total variance that is due to genetic
variance:
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An Introduction to Genetic Analysis
Chapter 25
Quantitative Genetics
H2, so defined, is called the broad heritability of the character.
It must be stressed that this measure of “genetic influence” tells us what part of the
population's variation in phenotype can be assigned to variation in genotype. It does not tell
us what parts of an individual's phenotype can be ascribed to its heredity and to its
environment. This latter distinction is not a reasonable one. An individual's phenotype is a
consequence of the interaction between its genes and its sequence of environments. It clearly
would be silly to say that you owe 60 inches of your height to genes and 10 inches to
environment. All measures of the “importance” of genes are framed in terms of the proportion
of variance ascribable to their variation. This approach is a special application of the more
general technique of the analysis of variance for apportioning relative weight to contributing
causes. The method was, in fact, invented originally to deal with experiments in which
different environmental and genetic factors were influencing the growth of plants. (For a
sophisticated but accessible treatment of the analysis of variance written for biologists, see R.
Sokal and J. Rohlf, Biometry, 3d ed. W. H. Freeman and Company, 1995.)
Methods of estimating H2
Genetic variance and heritability can be estimated in several ways. Most directly, we can
obtain an estimate of (S2e) by making a number of homozygous lines from the population,
crossing them in pairs to reconstitute individual heterozygotes, and measuring the phenotypic
variance within each heterozygous genotype. Because there is no genetic variance within a
genotypic class, these variances will (when averaged) provide an estimate of (S2e). This value
can then be subtracted from the value of (S2p) in the original population to give (S2g). With the
use of this method, any covariance between genotype and environment in the original
population will be hidden in the estimate of genetic variance and will inflate it.
Other estimates of genetic variance can be obtained by considering the genetic similarities
between relatives. Using simple Mendelian principles, we can see that half the genes of full
siblings will (on average) be identical. For identification purposes, we can label the alleles at a
locus carried by the parents differently, so that they are, say, A1/A2 and A3/A4. Now the older
sibling has a probability of 1/2 of getting A1 from its father, as does the younger sibling, so the
two siblings have a chance of 1/2 × 1/2 = 1/4 of both carrying A1. On the other hand,
they might both have received an A2 from their father; so, again, they have a probability of 1/4
of carrying a gene in common that they inherited from their father. Thus, the chance is 1/4
+ 1/4 = 1/2 that both siblings will carry an A1 or that both siblings will carry an A2. The
other half of the time, one sibling will inherit an A1 and the other will inherit an A2. So, as far
as paternally inherited genes are concerned, full siblings have a 50 percent chance of carrying
the same allele. But the same reasoning applies to their maternally inherited gene. Averaging
over their paternally and maternally inherited genes, half the genes of full siblings are
identical between them. Their genetic correlation, which is equal to the chance that they
carry the same allele, is 1/2.
If we apply this reasoning to half-siblings, say, with a common father but with different
mothers, we get a different result. Again, the two siblings have a 50 percent chance of
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An Introduction to Genetic Analysis
Chapter 25
Quantitative Genetics
inheriting an identical gene from their father, but this time they have no way of inheriting the
same gene from their mothers because they have two different mothers. Averaging the
maternally inherited and paternally inherited genes thus gives a probability of (1/2 +
0)/2 = 1/4 that these half-siblings will carry the same gene.
We might be tempted to use the theoretical correlation between, say, siblings to estimate H2.
If the observed phenotypic correlation were, for example, 0.4 and we expect on purely genetic
grounds a correlation of .05, then an estimate of heritability would be 0.4/0.5 = 0.8. But such
an estimate fails to take into account the fact that siblings may also be environmentally
correlated. Unless we are careful to raise the siblings in independent environments, the
estimate of H2 would be too large and could even exceed 1 if the observed phenotypic
correlation were greater than 0.5. To get around this problem, we use the differences between
phenotypic correlations of different relatives. For example, the difference in genetic
correlation between full and half-siblings is 1/2 − 1/4 = 1/4. Let's contrast this with
their phenotypic correlations. If the environmental similarity is the same for half- and full
siblings—a very important condition for estimating heritability—then environmental
similarities will cancel out if we take the difference in correlation between the two kinds of
siblings. This difference in phenotypic correlation will then be proportional to how much of
the variance is genetic. Thus:
but
so an estimate of H2 is:
where the correlation here is the phenotypic correlation.
We can use similar arguments about genetic similarities between parents and offspring and
between twins to obtain two other estimates of H2:
and
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勇者并非无所畏惧,而是判断出有比恐惧更重要的东西.
An Introduction to Genetic Analysis
Chapter 25
Quantitative Genetics
These formulas are derived from considering the genetic similarities between relatives. They
are only approximate and depend on assumptions about the ways in which genes act. The first
two formulas, for example, assume that genes at different loci add together in their effect on
the character. The last formula also assumes that the alleles at each locus show no dominance
(see the discussion of components of variance on pages 760–762).
All these estimates, as well as others based on correlations between relatives, depend
critically on the assumption that environmental correlations between individuals are the same
for all degrees of relationship. If closer relatives have more similar environments, as they do
in humans, the estimates of heritability are biased. It is reasonable to assume that most
environmental correlations between relatives are positive, in which case the heritabilities
would be overestimated. Negative environmental correlations also can exist. For example, if
the members of a litter must compete for food that is in short supply, there could be negative
correlations in growth rates among siblings.
The difference in correlation between monozygotic and dizygotic twins is commonly used in
human genetics to estimate H2 for cognitive or personality traits. Here the problem of degree
of environmental similarity is very severe. Identical (monozygotic) twins are generally treated
more similarly to each other than are fraternal (dizygotic) twins. People often give their
identical twins names that are similar, dress them alike, treat them identically, and, in general,
accent their similarities. As a result, heritability is overestimated.
Meaning of H2
Attention to the problems of estimating broad heritability distracts from the deeper questions
about the meaning of the ratio when it can be estimated. Despite its widespread use as a
measure of how “important” genes are in influencing a trait, H2 actually has a special and
limited meaning.
There are two conclusions that can be drawn from a properly designed heritability study. First,
if there is a nonzero heritability, then, in the population measured and in the environments in
which the organisms have developed, genetic differences have influenced the variation
between individuals, so genetic differences do matter to the trait. This finding is not trivial
and is a first step in a more detailed investigation of the role of genes. It is important to notice
that the reverse is not true. Finding no heritability for the trait is not a demonstration that
genes are irrelevant; rather, it demonstrates that, in the particular population studied, there is
no genetic variation at the relevant loci or that the environments in which the population
developed were such that different genotypes had the same phenotype. In other populations or
other environments, the trait might be heritable.
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An Introduction to Genetic Analysis
Chapter 25
Quantitative Genetics
MESSAGE
In general, the heritability of a trait is different in each population and in each set of
environnents; it cannot be extrapolated from one population and set of environments
to another.
Moreover, we must distinguish between genes being relevant to a trait and genetic differences
being relevant to differences in the trait. The experiment of immigration to North America has
proved that the ability to pronounce the sounds of North American English, rather than
French, Swedish, or Russian, is not a consequence of genetic differences between our
immigrant ancestors. But, without the appropriate genes, we could not speak any language at
all.
Second, the value of the H2 provides a limited prediction of the effect of environmental
modification under particular circumstances. If all the relevant environmental variation is
eliminated and the new constant environment is the same as the mean environment in the
original population, then H2 estimates how much phenotypic variation will still be present. So,
if the heritability of performance on an IQ test were, say, 0.4, then, if all children had the
same developmental and social environment as the “average child,” about 60 percent of the
variation in IQ test performance would disappear and 40 percent would remain.
The requirement that the new constant environment be at the mean of the old environmental
distribution is absolutely essential to this prediction. If the environment is shifted toward one
end or the other of the environmental distribution or a new environment is introduced, nothing
at all can be predicted. In the example of IQ performance, the heritability gives us no
information at all about how variable performance would be if children's developmental and
social environments were generally enriched. To understand why this is so, we must return to
the concept of the norm of reaction.
The separation of variance into genetic and environmental components S2g and S2e does not
really separate the genetic and environmental causes of variation. Consider Figure 25-9b.
When the environment is poor (50), corn variety 2 is much higher yielding than variety 1, so a
population made up of a mixture of the two varieties would have a lot of genetic variance for
yield. But, in an environment scoring 80, there is no difference in yield between genotypes 1
and 2, so a mixed population would have no genetic variance at all for yield in that
environment. Thus, genetic variance has been changed by changing the environment. On the
other hand, variety 2 is less sensitive to environment than variety 1, as shown by the slopes of
the lines. So a population made up mostly of genotype 2 would have a lower environmental
variance than one made up mostly of genotype 1. So, environmental variance in the
population is changed by changing the proportion of genotypes.
MESSAGE
Because genotype and environment interact to produce phenotype, no partition of
variation can actually separate causes of variation.
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An Introduction to Genetic Analysis
Chapter 25
Quantitative Genetics
As a consequence of the argument just given, knowledge of the heritability of a trait does not
permit us to predict how the distribution of that trait will change if either genotypic
frequencies or environmental factors change markedly.
MESSAGE
A high heritability does not mean that a trait is unaffected by its environment.
All that high heritability means is that, for the particular population developing in the
particular distribution of environments in which the heritability was measured, average
differences between genotypes are large compared with environmental variation within
genotypes. If the environment is changed, there may be large differences in phenotype.
Perhaps the most well known example of the erroneous use of heritability arguments to make
claims about the changeability of a trait is the case of human IQ performance and social
success. In 1969, an educational psychologist, A. R. Jensen, published a long paper in the
Harvard Educational Review, asking the question (in its title) “How much can we boost IQ
and scholastic achievement?” Jensen's conclusion was “not much.” As an explanation and
evidence of this unchangeability, he offered a claim of high heritability for IQ performance. A
great deal of criticism has been made of the evidence offered by Jensen for the high
heritability of IQ scores. But, irrespective of the correct value of H2 for IQ performance, the
real error of Jensen's argument lies in his equation of high heritability with unchangeability.
In fact, the heritability of IQ is irrelevant to the question raised in the title of his article.
To see why this is so, let us consider the results of adoption studies in which children are
separated from their biological parents in infancy and reared by adoptive parents. Although
results may vary quantitatively from study to study, there are three characteristics in common.
First, adopting parents generally have higher IQ scores than those of the biological parents.
Second, the adopted children have higher IQ scores than those of their biological parents.
Third, the adopted children show a higher correlation of IQ scores with their biological
parents than with their adoptive families. The following table is a hypothetical data set that
shows all these characteristics, in idealized form, to illustrate the concepts. The scores given
for parents are meant to be the average of mother and father.
First, we can see that the children have a high correlation with their biological parents but a
low correlation with their adoptive parents. In fact, in our hypothetical example, the
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Chapter 25
Quantitative Genetics
correlation of children with biological parents is r = 1.00, but, with adoptive parents, it is r = 0.
(The correlation between two sets of numbers does not mean that the two sets are identical but
that, for each unit increase in one set, there is a constant proportion increase in the other set.
See page 768 of the Statistical Appendix at the end of this chapter.) This perfect correlation
with biological parents and zero correlation with adoptive parents means that H2 = 1, given
the arguments developed on page 755. All the variation in IQ score between the children is
explained by the variation between the biological parents.
Second, however, we notice that each of the IQ scores of the children is 20 points higher than
the IQ scores of their respective biological parents and that the mean IQ of the children is
equal to the mean IQ of the adoptive parents. Thus, adoption has raised the average IQ of the
children 20 points higher than the average IQ of their biological parents; so, as a group, the
children resemble their adoptive parents. So we have perfect heritability, yet high
environmental plasticity.
An investigator who is seriously interested in knowing how genes might constrain or
influence the course of development of any trait in any organism must study directly the
norms of reaction of the various genotypes in the population over the range of projected
environments. No less detailed information will do. Summary measures such as H2 are not
first steps toward a more complete analysis and therefore are not valuable in themselves.
MESSAGE
Heritability is not the opposite of phenotypic plasticity. A character may have perfect
heritability in a population and still be subject to great changes resulting from
environmental variation.
Locating the genes
It is not possible with purely genetic techniques to identify all the genes that influence the
development of a given trait. This is true even for simple qualitative traits—for example, the
genes that determine the total antigenic configuration of the membrane of the human red
blood cell. About 40 loci determining human blood groups are known at present; each has
been discovered by finding at least one person with an immunological specificity that differs
from the specificities of other people. Many other loci that determine red-cell membrane
structure may remain undiscovered because all the individuals studied are genetically
identical. Genetic analysis detects genes only when there is some allelic variation. In contrast,
molecular analysis, by dealing directly with DNA and its translated information, can identify
genes even when they do not vary—provided the gene products can be identified.
Even though a trait may show continuous phenotypic variation, the genetic basis for the
differences may be allelic variation at a single locus. Most of the classical mutations in
Drosophila are phenotypically variable in their expression, and in many cases the mutant
class differs little from wild type, so many individuals that carry the mutation are
indistinguishable from normal. Even the genes of the bithorax gene complex, which have
dramatic homeotic mutations that turn halteres into wings (see pages 698–699), also have
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weak alleles that increase the size of the haltere only slightly on the average, so individuals of
the mutant genotype may appear to be wild type.
It is sometimes possible to use prior knowledge of the biochemistry and development of an
organism to guess that variation at a known locus is responsible for at least some of the
variation in phenotype. This locus then is a candidate gene for investigation of continuous
phenotypic variation. An example is the variation in activity of the enzyme acid phosphatase
in human red blood cells. Because we are dealing with variation in enzyme activity, a good
hypothesis would be that there is allelic variation at the locus that encodes this enzyme. When
H. Harris and D. Hopkinson sampled an English population, they found that there were,
indeed, three allelic forms, A, B, and C, with different activities. Table 25-2 shows the mean
activity, the variance in activity, and the population frequency of the six genotypes. Figure
25-11 shows the distribution of activity in the entire population and how it is composed of the
distributions of the different genotypes. Table 25-2 shows that, of the variance in activity in
the total distribution (607.8), about half is explained by the average variance within genotypes
(310.7), so half (607.8 − 310.7 = 297.1) is accounted for by the variance between the means
of the six genotypes. Although much of the variation in activity is explained by the mean
differences between the genotypes, there remains variation within each genotype that may be
the result of environmental influences or of the segregation of other, as yet unidentified, genes.
This partial explanation of variation by alleles at a single identified locus is typical of what is
found by the candidate gene method, and the proportion of variance associated with the single
locus is usually less than what was found for acid phosphatase. For example, the three
common alleles for the gene apoE that encodes the protein apolipoprotein E account for only
about 16 percent of the variance in blood levels of low-density lipoproteins that carry
cholesterol and are implicated in excess cholesterol levels.
Marker-gene segregation
The genes segregating for a quantitative trait, so-called quantitative trait loci, or QTLs,
cannot be individually identified in most cases. It is possible, however, to localize those
regions of the genome in which the relevant loci lie and to estimate how much of the total
variation is accounted for by QTL variation in each region. This analysis is done in
experimental organisms by crossing two lines that differ markedly in the quantitative trait and
differ in alleles at well-known loci, marker genes, where the different genotypes can be
distinguished by criteria such as some visible phenotypic effect that is not confused with the
quantitative trait (say, eye color in Drosophila) or by the electrophoretic mobility of the
proteins that they encode or by the DNA sequence of the genes themselves. The F1 between
the two lines is then crossed with itself to make a segregating F2 or it may be backcrossed to
one of the parental lines. If there are QTLs closely linked to a marker gene, then the different
marker genotypes in the segregating generation will also carry the QTL alleles that were
linked to them in the original parental lines. Thus different marker genotypes in the F2 or
backcross will have different average phenotypes for the quantitative character.
Linkage analysis
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The localization of QTLs to small regions within a chromosome requires that there be closely
spaced marker loci along the chromosome. Moreover, it must be possible to have parental
lines that differ from each other in the alleles carried at these loci. For most of the history of
genetics, these requirements could not be met, even in a genetically well known species such
as Drosophila, because most marker loci were known from severe morphological mutants that
had deleterious effects on the viability and fecundity of their carriers. As a result, it was not
possible to create a line that carried large numbers of mutant alleles that would distinguish it
from an alternative line carrying the wild-type alleles. With the advent of molecular
techniques that can detect genetic polymorphism at the DNA level (see pages 718–721), very
high densities of variant loci have been discovered along the chromosomes of all species.
Especially useful are restriction-site polymorphisms and tandem repeats in DNA (see pages
718–719, and 720–721). Such polymorphisms are so common that any two lines selected for a
difference in quantitative traits are also sure to differ from each other at known molecular
marker loci spaced a few crossover units from each other along each chromosome.
An experimental protocol for localizing the genes uses groups of individuals that differ
markedly in the quantitative trait and differ at marker loci. These groups may be created by a
number of generations of divergent selection to create extreme lines or advantage may be
taken of already existing varieties of family groups that differ markedly in the trait. These
lines must then be surveyed for marker loci that differ between them. A cross is made
between the two lines, and the F1 is crossed with itself to produce a segregating F2 or is
crossed back to one of the parental lines to produce a segregating backcross. A large number
of offspring from the segregating generation are then measured for the quantitative trait and
characterized for their genotype at the marker loci. A marker locus that is unlinked or very
loosely linked to any QTLs will have the same average value of the quantitative trait for all its
genotypes, whereas one that is closely linked to some QTLs will differ in its mean phenotype
from one of its genotypes to another. How much difference there is in the mean phenotype
between the marker-locus genotypes depends both on the strength of the effect of the QTLs
and on the tightness of linkage between the QTLs and the marker locus. Suppose, for example,
that there are two selected lines that differ by a total of 100 units in some quantitative
character, that the high line is homozygous +/+ at a QTL, whereas the low line is
homozygous −/−, and that each + allele at this QTL accounts for 5 units of the total difference
between the lines. Further suppose that the high line is M/M and the low line is m/m at a
marker locus 10 crossover units away from the QTL. Then, as shown in Figure 25-12, there
are 4 units of difference between the average gamete carrying an M allele and an average
gamete carrying an m allele in the segregating F2, or 8 units of the difference between the two
original homozygous lines. Thus we have accounted for 8 percent of the average difference
between the lines, although the QTL actually accounts for 10 percent of the difference. The
discrepancy comes from the recombination between the marker gene and the QTL.
This technique has been used to locate chromosomal segments associated with traits such as
fruit weight in tomatoes, bristle number in Drosophila, and vegetative characters in maize. In
the maize case, 82 vegetative characters were examined in a cross between lines that differed
in 20 DNA markers. On the average, each trait was significantly associated with 14 different
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markers, but the proportion of the trait difference between the lines that was explained by any
given marker was usually very small. Figure 25-13 shows the proportion of the significant
marker–trait associations (on the y-axis) that accounted for different proportions of trait
difference between the lines. As Figure 25-13 shows, most associations accounted for less
than 1 percent of the trait difference between lines.
For many organisms (for example, humans), it is not possible to make homozygous lines
differing in some trait and then cross them to produce a segregating generation. For such
organisms, one can use the differences among sibs carrying different marker alleles from
heterozygous parents. This method has much less power to find QTLs especially when the
number of sibs in any family is small, as it is in human families. As a consequence, the
attempts to map QTLs for human traits have not been very successful, although the
segregating marker technique has been a success in finding loci whose mutations are
responsible for single-gene disorders.
More on analyzing variance
Knowledge of the broad heritability (H2) of a trait in a population is not very useful in itself,
but a finer subdivision of phenotypic variance can provide important information for plant and
animal breeders. The genetic variation and the environmental variation can themselves each
be further subdivided to provide information about gene action and the possibility of shaping
the genetic composition of a population.
Additive and dominance variance
Our previous consideration of gene action suggests that the phenotypes of homozygotes and
heterozygotes ought to have a simple relation. If one of the alleles encoded a less active gene
product or one with no activity at all and if one unit of gene product were sufficient to allow
full physiological activity of the organism, then we would expect complete dominance of one
allele over the other, as Mendel observed for flower color in peas. If, on the other hand,
physiological activity were proportional to the amount of active gene product, we would
expect the heterozygote phenotype to be exactly intermediate between the homozygotes
(show no dominance).
For many quantitative traits, however, neither of these simple cases is the rule. In general,
heterozygotes are not exactly intermediate between the two homozygotes but are closer to one
or the other (show partial dominance), even though there is an equal mixture of the primary
products of the two alleles in the heterozygote. Indeed, in some cases, the heterozygote
phenotype may lie outside the phenotypic range of the homozygotes altogether—a feature
termed overdominance. For example, newborn babies who are intermediate in size have a
higher chance of survival than very large or very small newborns. Thus, if survival were the
phenotype of interest, heterozygotes for genes influencing growth rate would show
overdominance for fitness although not for growth rate.
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Suppose that two alleles, a and A, segregate at a locus influencing height. In the environments
encountered by the population, the mean phenotypes (heights) and frequencies of the three
genotypes might be:
There is genetic variance in the population; the phenotypic means of the three genotypic
classes are different. Some of the variance arises because there is an average effect on
phenotype of substituting an allele A for an allele a; that is, the average height of all
individuals with A alleles is greater than that of all individuals with a alleles. By defining the
average effect of an allele as the average phenotype of all individuals that carry it, we
necessarily make the average effect of the allele depend on the frequencies of the genotypes.
The average effect is calculated by simply counting the a and A alleles and multiplying them
by the heights of the individuals in which they appear. Thus, 0.36 of all the individuals are
homozygous a/a, each a/a individual has two a alleles, and the average height of a/a
individuals is 10 cm. Heterozygotes make up 0.48 of the population, each has only one a
allele, and the average phenotypic measurement of A/a individuals is 18 cm. The total
“number” of a alleles is 2(0.36) + 1(0.48). Thus, the average effect of all the a alleles is:
and, by a similar argument,
This average difference in effect between A and a alleles of 5.60 cm accounts for some of the
variance in phenotype—but not for all of it. The heterozygote is not exactly intermediate
between the homozygotes; there is some dominance.
We would like to separate the so-called additive effect caused by substituting a alleles for A
alleles from the variation caused by dominance. The reason is that the effect of selective
breeding depends on the additive variation and not on the variation caused by dominance.
Thus, for purposes of plant and animal breeding or for making predictions about evolution by
natural selection, we must determine the additive variation. An extreme example will
illustrate the principle. Suppose that there is overdominance and that the phenotypic means
and frequencies of three genotypes are:
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It is apparent (and a calculation like the preceding one will confirm) that there is no average
difference between the a and A alleles, because each has an effect of 11 units. So there is no
additive variation, although there is obviously variation in phenotype between the genotypes.
The largest individuals are heterozygotes. If a breeder attempts to increase height in this
population by selective breeding, mating these heterozygotes together will simply reconstitute
the original population. Selection will be totally ineffective. This example illustrates the
general law that the effect of selection depends on the additive genetic variation and not on
genetic variation in general.
We partition the total genetic variance in a population into additive genetic variation(S2a, the
variance that arises because there is an average difference between the carriers of a alleles and
the carriers of A alleles, and a component called the dominance variance(S2d which results
from the fact that heterozygotes are not exactly intermediate between the monozygotes. Thus:
The components of variance in the first example, where a/a = 10, A/a = 18, and A/A = 20, can
be calculated by using the definitions of mean and variance developed earlier in this chapter.
Remembering that a mean is the sum of the values of a variable, each weighted by the
frequency with which that value occurs (see page 765), we can calculate the mean phenotype
to be:
The total genetic variance that arises from the variation among the mean phenotypes of the
three genotypes is:
The frequency of allele a is (by counting alleles):
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and the frequency of the A allele is:
The variance of allelic means is then
But we want the variance among diploid individuals that results from the allelic effects, and
every diploid individual carries two alleles; so:
and
The total phenotypic variance can now be written as
We define a new kind of heritability, the heritability in the narrow sense (h2), as
It is this heritability, not to be confused with H2, that is useful in determining whether a
program of selective breeding will succeed in changing the population. The greater the h2 is,
the greater the difference is between selected parents and the population as a whole that will
be preserved in the offspring of the selected parents.
MESSAGE
The effect of selection depends on the amount of additive genetic variance and not on
the genetic variance in general. Therefore, the narrow heritability, h2, not the broad
heritability, H2, is relevant for a prediction of response to selection.
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What has been described as the “dominance” variance is really more complicated. It is all the
genetic variation that cannot be explained by the average effect of substituting A for a. If more
than one locus affects the character, then any epistatic interactions between loci will appear as
variance not associated with the average effect of substituting alleles at the A locus. In
principle, we can separate this interaction variance(S2i from the dominance variance (S2d In
practice, however, this separation cannot be done with any semblance of accuracy, so all the
nonadditive variance appears as “dominance” variance.
Estimating genetic variance components
Genetic components of variance can be estimated from covariance between relatives, but the
derivation of these estimates is beyond the scope of an elementary text.
There is, however, another way to estimate h2 that reveals its real meaning. If we plot the
phenotypes of the offspring against the average phenotypes of their two parents (the
midparent value), we may observe a relation like the one illustrated in Figure 25-14. The
regression line will pass through the mean of all the parents and the mean of all the offspring,
which will be equal to each other because no change has occurred in the population between
generations. Moreover, taller parents have taller children and shorter parents have shorter
children, so the slope of the line is positive. But the slope is not unity; very short parents have
children who are somewhat taller and very tall parents have children who are somewhat
shorter than they themselves are. This slope of less than unity for the regression line arises
because heritability is less than perfect. If the phenotype were additively inherited with
complete fidelity, then the height of the offspring would be identical with the midparent value
and the slope of the line would be 1. On the other hand, if the offspring had no heritable
similarity to their parents, all parents would have offspring of the same average height and the
slope of the line would be 0. This suggests that the slope of the regression line of the offspring
value on the midparent value is an estimate of additive heritability. In fact, the relation is
precise.
The fact that the slope equals the additive heritability now allows us to use h2 to predict the
effects of artificial selection. Suppose that we select parents for the next generation who are
on the average 2 units above the general mean of the population from which they were chosen.
If h2 = 0.5, then the offspring who form the next, selected generation will lie 0.5(2.0) = 1.0
unit above the mean of the present population, because the regression coefficient predicts how
much increase in y will result from a unit increase in x. We can define the selection
differential as the difference between the selected parents and the unselected mean and the
selection response as the difference between their offspring and the preceding generation.
Then:
or
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The second expression provides us with yet another way to estimate h2: by selecting for one
generation and comparing the response with the selection differential. Usually this is carried
out for several generations, and the average response is used.
Remember that any estimate of h2, just as for H2, depends on the assumption of no greater
environmental correlation between closer relatives. Moreover, h2 in one population in one set
of environments will not be the same as h2 in a different population at a different time. Figure
25-15 shows the range of heritabilities reported in various studies for a number of traits in
chickens. The very small ranges are generally close to zero. For most traits for which a
substantial heritability has been reported in some population, there are big differences from
study to study.
Use of h2 in breeding
Even though h2 is a number that applies only to a particular population and a given set of
environments, it is still of great practical importance to breeders. A poultry geneticist
interested in increasing, say, growth rate is not concerned with the genetic variance over all
possible flocks and all environmental distributions. Given a particular flock (or a choice
between a few particular flocks) under the environmental conditions approximating present
husbandry practice, the question becomes: Can a selection scheme be devised to increase
growth rate and, if so, how fast? If one flock has a lot of genetic variance and another only a
little, the breeder will choose the former to carry out selection. If the heritability in the chosen
flock is very high, then the mean of the population will respond quickly to the selection
imposed, because most of the superiority of the selected parents will appear in the offspring.
The higher h2 is, the higher the parent–offspring correlation is. If, on the other hand, h2 is low,
then only a small fraction of the increased growth rate of the selected parents will appear in
the next generation.
If h2 is very low, some alternative scheme of selection or husbandry may be needed. In this
case, H2 together with h2 can be of use to the breeder. Suppose that h2 and H2 are both low,
which means that there is a lot of environmental variance compared with genetic variance.
Some scheme of reducing (S2a must be used. One method is to change the husbandry
conditions so that environmental variance is lowered. Another is to use family selection.
Rather than choosing the best individuals, the breeder allows pairs to produce several progeny,
and the mating is selected on the basis of the average performance of the progeny. Averaging
over progeny allows uncontrolled environmental and developmental noise variation to be
canceled out, and a better estimate of the genotypic difference between pairs can be made so
that the best pairs can be chosen as parents of the next generation.
If, on the other hand, h2 is low but H2 is high, then there is not much environmental variance.
The low h2 is the result of a small amount of additive genetic variance compared with
dominance and interaction variance. Such a situation calls for special breeding schemes that
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make use of nonadditive variance. One such scheme is the hybridinbred method, which is
used almost universally for corn. A large number of inbred lines are created by selfing. These
inbred lines are then crossed in many different combinations (all possible combinations, if this
is economically feasible), and the cross that gives the best hybrid is chosen. Then new inbred
lines are developed from this best hybrid, and again crosses are made to find the best
second-cycle hybrid. This scheme selects for dominance effects because it takes the best
heterozygotes; it has been the basis of major genetic advances in hybrid maize yield in North
America since 1930. Yield in corn does not appear to have large amounts of nonadditive
genetic variance, so it is debatable whether this technique ultimately produces higher-yielding
varieties than those that would have resulted from years of simple selection techniques based
on additive variance.
The hybrid method has been introduced into the breeding of all kinds of plants and animals.
Tomatoes and chickens, as examples, are now almost exclusively hybrids. Attempts also have
been made to breed hybrid wheat, but thus far the wheat hybrids obtained do not yield
consistently better than the nonhybrid varieties now used.
MESSAGE
The subdivision of genetic variation and environmental variation provides important
information about gene action that can be used in plant and animal breeding.
Summary
Many—perhaps most—of the phenotypic traits that we observe in organisms vary
continuously. In many cases, the variation of the trait is determined by more than a single
segregating locus. Each of these loci may contribute equally to a particular phenotype, but it is
more likely that they contribute unequally. The measurement of these phenotypes and the
determination of the contributions of specific alleles to the distribution must be made on a
statistical basis in these cases. Some of these variations of phenotype (such as height in some
plants) may show a normal distribution around a mean value; others (such as seed weight in
some plants) will illustrate a skewed distribution around a mean value.
In other characters, the variation in one phenotype may be correlated with the variation in
another. A correlation coefficient may be calculated for these two variables.
A quantitative character is one for which the average phenotypic differences between
genotypes are small compared with the variation between the individuals within the genotypes.
This situation may be true even for characters that are influenced by alleles at one locus. The
distribution of environments is reflected biologically as a distribution of phenotypes. The
transformation of environmental distribution into phenotypic distribution is determined by the
norm of reaction. Norms of reaction can be characterized in organisms in which large
numbers of genetically identical individuals can be produced.
With the use of genetically marked chromosomes, it is possible to determine the relative
contributions of different chromosomes to variation in a quantitative trait, to observe
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dominance and epistasis from whole chromosomes, and, in some cases, to map genes that are
segregating for a trait.
Traits are familial if they are common to members of the same family, for whatever reason.
Traits are heritable, however, only if the similarity arises from common genotypes. In
experimental organisms, environmental similarities may be readily distinguished from genetic
similarities, or heritability. In humans, however, it is very difficult to determine whether a
particular trait is heritable. Norm of reaction studies show only small differences between
genotypes, and these differences are not consistent over a wide range of environments. Thus,
“superior” genotypes in domesticated animals and cultivated plants may be superior only in
certain environments. If it should turn out that humans exhibit genetic variation for various
mental and emotional traits, this variation is unlikely to favor one genotype over another
across a range of environments.
The attempt to quantify the influence of genes on a particular trait has led to the determination
of heritability in the broad sense (H2). In general, the heritability of a trait is different in each
population and each set of environments and cannot be extrapolated from one population and
set of environments to another. Because H2 characterizes present populations in present
environments only, it is fundamentally flawed as a predictive device. Heritability in the
narrow sense, h2, measures the proportion of phenotypic variation that results from
substituting one allele for another. This quantity, if large, predicts that selection for a trait will
succeed rapidly. If h2 is small, special forms of selection are required.
Solved Problems
1. Two inbred lines of beans are intercrossed. In the F1, the variance in bean weight is
measured at 1.5. The F1 is selfed; in the F2, the variance in bean weight is 6.1. Estimate
the broad heritability of bean weight in the F2 population of this experiment.
Solution
The key here is to recognize that all the variance in the F1 population must be environmental
because all individuals must be of identical genotype. Furthermore, the F2 variance must be a
combination of environmental and genetic components, because all the genes that are
heterozygous in the F1 will segregate in the F2 to give an array of different genotypes that
relate to bean weight. Hence, we can estimate
Therefore
and broad heritability is
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2. In an experimental population of Tribolium (flour beetles), the body length shows a
continuous distribution with a mean of 6 mm. A group of males and females with body
lengths of 9 mm are removed and interbred. The body lengths of their offspring average
7.2 mm. From these data, calculate the heritability in the narrow sense for body length in
this population.
Solution
The selection differential is 9 − 6 = 3 mm, and the selection response is 7.2 − 6 = 1.2 mm.
Therefore, the heritability in the narrow sense is:
Problems
1. Distinguish between continuous and discontinuous variation in a population, and give some
examples of each.
2. In a large herd of cattle, three different characters showing continuous distribution are
measured, and the variances in the following table are calculated:
a. Calculate the broad- and narrow-sense heritabilities for each character.
b. In the population of animals studied, which character would respond best to selection?
Why?
c. A project is undertaken to decrease mean fat content in the herd. The mean fat content is
currently 10.5 percent. Animals of 6.5 percent fat content are interbred as parents of the
next generation. What mean fat content can be expected in the descendants of these
animals?See answer
3. Suppose that two triple heterozygotes A/a ; B/b ; C/c are crossed. Assume that the three loci
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are in different chromosomes.
a. What proportions of the offspring are homozygous at one, two, and three loci,
respectively?
b. What proportions of the offspring carry 0, 1, 2, 3, 4, 5, and 6 alleles (represented by
capital letters), respectively?See answer
4. In Problem 3, suppose that the average phenotypic effect of the three genotypes at the A
locus is A/A = 4, A/a = 3, and a/a = 1 and that similar effects exist for the B and C loci.
Moreover, suppose that the effects of loci add to each other. Calculate and graph the
distribution of phenotypes in the population (assuming no environmental variance).
5. In Problem 4, suppose that there is a threshold in the phenotypic character so that, when the
phenotypic value is above 9, the individual Drosophila has three bristles; when it is
between 5 and 9, the individual has two bristles; and when the value is 4 or less, the
individual has one bristle. Describe the outcome of crosses within and between bristle
classes. Given the result, could you infer the underlying genetic situation?See answer
6. Suppose that the general form of a distribution of a trait for a given genotype is:
over the range of x where f is positive.
a. On the same scale, plot the distributions for three genotypes with the following means
and environmental variances:
b. Plot the phenotypic distribution that would result if the three genotypes were equally
frequent in a population. Can you see distinct modes? If so, what are they?
7. The following table shows a distribution of bristle number in Drosophila:
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Calculate the mean, variance, and standard deviation of this distribution.
See answer
8. The following sets of hypothetical data represent paired observations on two variables (x,
y). Plot each set of data pairs as a scatter diagram. Look at the plot of the points, and make
an intuitive guess about the correlation between x and y. Then calculate the correlation
coefficient for each set of data pairs, and compare this value with your estimate.
a. (1, 1); (2, 2); (3, 3); (4, 4); (5, 5); (6, 6).
b. (1, 2); (2, 1); (3, 4); (4, 3); (5, 6); (6, 5).
c. (1, 3); (2, 1); (3, 2); (4, 6); (5, 4); (6, 5).
d. (1, 5); (2, 3); (3, 1); (4, 6); (5, 4); (6, 2).
9. A book on the problem of heritability of IQ makes the following three statements. Discuss
the validity of each statement and its implications about the authors' understanding of h2
and H2.
a. “The interesting question then is . . . ‘How heritable?' The answer [0.01] has a
very different theoretical and practical application from the answer [0.99].” [The authors
are talking about H2.]
b. “As a rule of thumb, when education is at issue, H2 is usually the more relevant
coefficient, and when eugenics and dysgenics (reproduction of selected individuals) are
being discussed, h2 is ordinarily what is called for.”
c. “But whether the different ability patterns derive from differences in genes . . . is
not relevant to assessing discrimination in hiring. Where it could be relevant is in deciding
what, in the long run, might be done to change the situation.”
(From J. C. Loehlin, G. Lindzey, and J. N. Spuhler, Race Differences in Intelligence.
Copyright © 1975 by W. H. Freeman and Company.)See answer
10. Using the concepts of norms of reaction, environmental distribution, genotypic
distribution, and phenotypic distribution, try to restate the following statement in more
exact terms: “80 percent of the difference in IQ performance between the two groups is
genetic.” What would it mean to talk about the heritability of a difference between two
groups?
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勇者并非无所畏惧,而是判断出有比恐惧更重要的东西.
An Introduction to Genetic Analysis
Chapter 25
Quantitative Genetics
11. Describe an experimental protocol for studies of relatives that could estimate the broad
heritability of alcoholism. Remember that you must make an adequate observational
definition of the trait itself.See answer
12. A line selected for high bristle number in Drosophila has a mean of 25 sternopleural
bristles, whereas a low-selected line has a mean of only 2. Marker stocks involving the
two large autosomes II and III are used to create stocks with various mixtures of
chromosomes from the high (h) and low (l) lines. The mean number of bristles for each
chromosomal combination is as follows:
What conclusions can you reach about the distribution of genetic factors and their actions
from these data?
13. Suppose that number of eye facets is measured in a population of Drosophila under
various temperature conditions. Further suppose that it is possible to estimate total genetic
variance (S2g) as well as the phenotypic distribution. Finally, suppose that there are only
two genotypes in the population. Draw pairs of norms of reaction that would lead to the
following results:
a. An increase in mean temperature decreases the phenotypic variance.
b. An increase in mean temperature increases H2.
c. An increase in mean temperature increases S2g but decreases H2.
d. An increase in temperature variance changes a unimodal into a bimodal phenotypic
distribution (one norm of reaction is sufficient here).
14. Francis Galton compared the heights of male undergraduates with the heights of their
fathers, with the results shown in the graph below.
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勇者并非无所畏惧,而是判断出有比恐惧更重要的东西.
An Introduction to Genetic Analysis
Chapter 25
Quantitative Genetics
The average height of all fathers is the same as the average height of all sons, but the
individual height classes are not equal across generations. The very tallest fathers had
somewhat shorter sons, whereas the very short fathers had somewhat taller sons. As a
result, the best line that can be drawn through the points on the scatter diagram has a slope
of about 0.67 (solid line) rather than 1.00 (dashed line). Galton used the term regression to
describe this tendency for the phenotype of the sons to be closer than the phenotype of
their fathers to the population mean.
a. Propose an explanation for this regression.
b. How are regression and heritability related here? (Graph after W. F. Bodmer and L. L.
Cavalli-Sforza, Genetics, Evolution, and Man. Copyright © 1976 by W. H. Freeman and
Company.)
Chapter 25*
2. a. Shank length: H2 = 0.200
h2 = 0.150
Neck length: H2 = 0.600 h2 = 0.010
Fat content: H2 = 0.500 h2 = 0.400
b. Fat content would respond best to selection.
c. 8.9%
3. a. p(homozygous at 3 loci = 21/23 = 2/8 p(homozygotic at two loci) = 31/23 = 3/8
p(homozygotic at three loci) = 1/23 = 1/8
b. p(0 capital letters) = 1/64 p(4 capital letters) = 15/64
p(1 capital letter) = 3/32 p(5
capital letters) = 3/32
p(2 capital letters) = 15/64 p(6 capital letters) = 1/64
p(3
38
勇者并非无所畏惧,而是判断出有比恐惧更重要的东西.
An Introduction to Genetic Analysis
Chapter 25
Quantitative Genetics
capital letters) = 10/32
5. The population described would be distributed as follows:
Note that the three-bristle class contains 7 different genotypes, the two-bristle class
contains 19 different genotypes, and the one-bristle class contains only 1 genotype. It
would be very difficult to determine the underlying genetic situation by doing controlled
crosses and determining progeny frequencies.
7. Mean = 4.7;Variance = 0.2619;Standard deviation = 0.5117
9. a. H2 has meaning only with respect to the population that was studied in the environment
in which it was studied. Otherwise, it has no meaning.
b. Neither H2 nor h2 is a reliable measure that can be used to generalize from a particular
sample to a “universe” of the human population. They certainly should not be used in social
decision-making (as implied by the terms eugenics and dysgenics).
c. Again, H2 and h2 are not reliable measures, and they should not be used in any
decision-making with regard to social problems.
11. First, define alcoholism in behavioral terms. Next, realize that all observations must be
limited to the behavior that you used in the definition and that all conclusions from your
observations are applicable to only that behavior. To do your data gathering, you must
work with a population in which familiarity is distinguished from heritability. In practical
terms, this population must consist of persons who are genetically close but who are found
in all environments possible.
14. a. If you assume that individuals at the extreme of any spectrum are homozygous, then
their offspring are more likely to be heterozygous than the original individuals. That is,
they will be less extreme.
b. For Galton's data, regression is an estimate of heritability (h2), assuming that there were
few environmental differences between father and son.
39
勇者并非无所畏惧,而是判断出有比恐惧更重要的东西.