Download An Introduction to Genetic Analysis Chapter 24 Population Genetics

An Introduction to Genetic Analysis
Chapter 24
Population Genetics
Chapter 24
Population Genetics
Key Concepts
The goal of population genetics is to understand the genetic composition of a population and
the forces that determine and change that composition.
In any species, a great deal of genetic variation within and between populations arises from
the existence of various alleles at different gene loci.
A fundamental measurement in population genetics is the frequency at which the alleles are
found at any gene locus of interest.
The frequency of a given allele in a population can be changed by recurrent mutation,
selection, or migration or by random sampling effects.
In an idealized population, in which no forces of change are acting, a randomly interbreeding
population would show constant genotypic frequencies for a given locus.
Introduction
So far in our investigation of genetics, we have been concerned with processes that take place
in individual organisms and cells. How does the cell copy DNA and what causes mutations?
How do the mechanisms of segregation and recombination affect the kinds and proportions of
gametes produced by an individual organism? How is the development of an organism
affected by the interactions between its DNA, the cell machinery of protein synthesis, cellular
metabolic processes, and the external environment? But organisms do not live only as isolated
individuals. They interact with each other in groups, populations, and there are questions
about the genetic composition of those populations that cannot be answered only from a
knowledge of the basic individual-level genetic processes. Why are the alleles of the protein
Factor VIII and Factor IX genes that cause hemophilia so rare in all human populations,
whereas sickle-cell anemia is very common in some parts of Africa? What changes in the
frequency of sickle-cell anemia are to be expected in the descendants of Africans in North
America as a consequence of the change in environment and of the interbreeding between
Africans and Europeans and Native Americans? What genetic changes occur in a population
of insects subject to insecticides generation after generation? What is the consequence of an
increase or decrease in the rate of mating between close relatives? All are questions of what
determines the genetic composition of populations and how that composition may be expected
to change in time. These questions are the domain of population genetics.
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An Introduction to Genetic Analysis
Chapter 24
Population Genetics
MESSAGE
Population genetics relates the processes of individual heredity and development to
the genetic composition of populations and to changes in that composition in time.
To relate the basic individual-level processes to population genetic composition, we must
investigate the following phenomena:
1. The effect of the mating pattern on different genotypes in the population. Individuals may
mate at random or they may mate preferentially with close relatives (inbreeding) or
preferentially on the basis of their genotypic or phenotypic similarity (assortative mating).
2. The changes in population composition due to immigration of individuals from other
populations.
3. The rate of introduction of genetic variation into the population by mutation and
recombination.
4. The effect of the differential rate of reproduction by different genotypes and the differential
chance of survival of genetically different offspring of these matings. These differential rates
are the result of natural selection.
5. The consequences of random fluctuations in the actual reproductive rates of different
genotypes because any given individual has only a few offspring and the total population size
is limited.
Population genetics is both an experimental and a theoretical science. On the experimental
side, it provides descriptions of the actual patterns of genetic variation in populations and
estimates the parameters of the processes of mating, mutation, natural selection, and random
variation in reproductive rates. On the theoretical side, it makes predictions of what the
genetic composition of populations should be and how they can be expected to change as a
consequence of the various forces operating on them.
MESSAGE
Population genetics is the experimental and theoretical study of the pattern of
inherited variation in populations and its modulation in time and space.
Observations of variation
Population genetics necessarily deals with genotypic variation, but, by definition, only
phenotypic variation can be observed. The relation between phenotype and genotype varies in
simplicity from character to character. At one extreme, the phenotype may be the observed
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An Introduction to Genetic Analysis
Chapter 24
Population Genetics
DNA sequence of a stretch of the genome. In this case, the distinction between genotype and
phenotype disappears, and we can say that we are, in fact, directly observing the genotype. At
the other extreme lie the bulk of characters of interest to plant and animal breeders and to
most evolutionists—the variations in yield, growth rate, body shape, metabolic ratio, and
behavior that constitute the obvious differences between varieties and species. These
characters have a very complex relation to genotype, and we must use the methods introduced
in Chapter 25 to say anything at all about the genotypes. But, as we shall see in Chapter 25, it
is not possible to make very precise statements about the genotypic variation underlying
quantitative characters. For that reason, most of the study of experimental population genetics
has concentrated on characters with simple relations to the genotype, much like the characters
studied by Mendel. A favorite object of study for human population geneticists, for example,
has been the various human blood groups. The qualitatively distinct pheno-types of a given
blood group—say, the MN group—are encoded by alternative alleles at a single locus, and the
phenotypes are insensitive to environmental variations.
The study of variation, then, consists of two stages. The first is a description of the phenotypic
variation. The second is a translation of these phenotypes into genetic terms and the
redescription of the variation genetically. If there is a perfect one-to-one correspondence
between genotype and phenotype, then these two steps merge into one, as in the MN blood
group. If the relation is more complex—for example, as the result of dominance,
heterozygotes resemble homozygotes—it may be necessary to carry out experimental crosses
or to observe pedigrees to translate phenotypes into genotypes. This is the case for the human
ABO blood group (see page 110).
The simplest description of Mendelian variation is the frequency distribution of genotypes in
a population. Table 24-1 shows the frequency distribution of the three genotypes at the MN
blood group locus in several human populations. Note that there is variation between
individuals in each population, because there are different genotypes pres-ent, and there is
variation in the frequencies of these genotypes from population to population. More typically,
instead of the frequencies of the diploid genotypes, the frequencies of the alternative alleles
are used. The frequency of an allele is simply the proportion of that allelic form of the gene
among all copies of the gene in the population. There are twice as many gene copies in the
population as there are individuals, because every individual is diploid and homozygotes for
an allele have two copies of that allele, whereas heterozygotes have only one copy. So we
calculate the frequency of an allele by counting homozygotes and adding half the
heterozygotes. Thus, if the frequency of A/A individuals were, say, 0.36 and the frequency of
A/a individuals were 0.48, the allele frequency of A would be 0.36 + 0.48/2 = 0.60. Box 24-1
gives the general form of this calculation. Table 24-1 shows the values of p and q, the gene
frequency or allele frequency of the two alleles in the different populations.
A measure of genetic variation (in contrast with its description by gene frequencies) is the
amount of heterozygosity at a locus in a population, which is given by the total frequency of
heterozygotes at a locus. If one allele is in very high frequency and all others are near zero,
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An Introduction to Genetic Analysis
Chapter 24
Population Genetics
then there will be very little heterozygosity because, by necessity, most individuals will be
homozygous for the common allele. We expect heterozygosity to be greatest when there are
many alleles at a locus, all at equal frequency. In Table 24-1, the heterozygosity is simply
equal to the frequency of the M/N genotype in each population. When more than one locus is
considered, there are two possible ways of calculating heterozygosity. Locus S (the secretor
factor, determining whether the M and N proteins are also contained in the saliva) is closely
linked to the M N locus in humans. Table 24-2 shows the frequencies, commonly symbolized
by g's, of the four haplotypes (M S, M s, N S, and N s) in various populations. First, we can
calculate the frequency of heterozygotes at each locus separately. Alternatively, we can
consider each haplotype as a unit, as in Table 24-2, and calculate the proportion of all
individuals who carry two different haplotypic or gametic forms. This form of heterozygosity
is also referred to as haplotype diversity or gametic diversity. The results of both calculations
are given in Table 24-2. Note that the haplotype diversity is always greater than the average
heterozygosity of the separate loci, because an individual is a haplotypic heterozygote if
either of its loci is heterozygous. (See the discussion of the Hardy-Weinberg equilibrium in
Box 24-2 on page 722 for the calculation of heterozygosity.)
Simple Mendelian variation can be observed within and between populations of any species at
various levels of phenotype, from external morphology down to the amino acid sequence of
enzymes and other proteins. Indeed, with the new methods of DNA sequencing, variations in
DNA sequence (such as third-position variants that are not differentially coded in amino acid
sequences and even variations in nontranslated intervening sequences) have been observed.
Every species of organism ever examined has revealed considerable genetic variation, or
polymorphism, manifested at one or more levels of phenotype, within populations, between
populations, or both. A gene or a phenotypic trait is said to be polymorphic if there is more
than one form of the gene or trait in a population. Genetic variation that might be the basis for
evolutionary change is ubiquitous. The tasks for population geneticists are to describe that
ubiquitous variation quantitatively and to build a theory of evolutionary change that can use
these observations in prediction.
It is impossible in this text to provide an adequate picture of the immense richness of genetic
variation that exists in species. We can consider only a few examples of the different kinds of
Mendelian variation to gain a sense of the genetic diversity within species. Each of these
examples can be multiplied many times over in other species and with other traits.
Morphologic variation.
The shell of the land snail Cepaea nemoralis may be pink or yellow, depending on two alleles
at a single locus, with pink dominant to yellow. In addition, the shell may be banded or
unbanded (Figure 24-1) as a result of segregation at a second linked locus, with unbanded
dominant to banded. Table 24-3 shows the variation of these two loci in several European
colonies of the snail. The populations also show polymorphism for the number of bands and
the height of the shells, but these characters have complex genetic bases.
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An Introduction to Genetic Analysis
Chapter 24
Population Genetics
Examples of naturally occurring morphologic variation within plant species are Plectritis (see
Figure 1-14), Collinsia (blue-eyed Mary, page 58), and clover (see Figure 4-5).
Chromosomal polymorphism.
Although the karyotype is often regarded as a distinctive characteristic of a species, in fact,
numerous species are polymorphic for chromosome number and morphology. Extra
chromosomes (supernumeraries), reciprocal translocations, and inversions segregate in many
populations of plants, insects, and even mammals.
Table 24-4 gives the frequencies of supernumerary chromosomes and translocation
heterozygotes in a population of the plant Clarkia elegans from California. The “typical”
species karyotype would be hard to identify.
Immunologic polymorphism.
A number of loci in vertebrates encode antigenic specificities such as the ABO blood types.
More than 40 different specificities on human red cells are known, and several hundred are
known in cattle. Another major polymorphism in humans is the HLA system of cellular
antigens, which are implicated in tissue graft compatibility. Table 24-5 gives the allelic
frequencies for the ABO blood group locus in some very different human populations. The
polymorphism for the HLA system is vastly greater. There appear to be two main loci, each
with five distinguishable alleles. Thus, there are 52 = 25 different possible gametic types,
making 25 different homozygous forms and (25)(24)/2 = 300 different heterozygotes. All
genotypes are not phenotypically distinguishable, however; so only 121 phenotypic classes
can be seen. L. L. Cavalli-Sforza and W. F. Bodmer report that, in a sample of only 100
Europeans, 53 of the 121 possible phenotypes were actually observed!
Protein polymorphism.
Studies of genetic polymorphism have been carried down to the level of the polypeptides
encoded by the structural genes themselves. If there is a nonredundant codon change in a
structural gene (say, GGU to GAU), the result is an amino acid substitution in the polypeptide
produced at translation (in this case, aspartic acid is substituted for glycine). If a specific
protein could be purified and sequenced from separate individuals, then it would be possible
to detect genetic variation in a population at this level. In practice, such detection is tedious
for large organisms and impossible for small ones unless a large mass of protein can be
produced from a homozygous line.
There is, however, a practical substitute for sequencing that makes use of the change in the
physical properties of a protein when an amino acid is substituted. Five amino acids (glutamic
acid, aspartic acid, arginine, lysine, and histidine) have ionizable side chains that give a
protein a characteristic net charge, depending on the pH of the surrounding medium. Amino
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An Introduction to Genetic Analysis
Chapter 24
Population Genetics
acid substitutions may directly replace one of these charged amino acids or a noncharged
substitution near one of them in the polypeptide chain may affect the degree of ionization of
the charged amino acid or a substitution at the joining between two α helices may cause a
slight shift in the three-dimensional packing of the folded polypeptide. In all these cases, the
net charge on the polypeptide will be altered because the net charge on a protein is not simply
the sum of all the individual charges on its amino acids but depends on their exposure to the
liquid medium surrounding them.
To detect the change in net charge, protein can be subjected to gel electrophoresis. Figure
24-2 shows the outcome of such an electrophoretic separation of variants of an esterase
enzyme in Drosophila pseudoobscura, where each track is the protein from a different
individual fly. Figure 24-3 shows a similar gel for different variant human hemoglobins. In
this case, most individuals are heterozygous for the variant and normal hemoglobin A. Table
24-6 shows the frequencies of different alleles for three enzyme-encoding loci in D.
pseudoobscura in several populations: a nearly monomorphic locus (malic dehydrogenase), a
moderately polymorphic locus (α-amylase), and a highly polymorphic locus (xanthine
dehydrogenase).
The technique of gel electrophoresis (or sequencing) differs fundamentally from other
methods of genetic analysis in allowing the study of loci that are not segregating, because the
presence of a polypeptide is prima facie evidence of a structural gene—that is, a DNA
sequence encoding a protein. Thus, it has been possible to ask what proportion of all
structural genes in the genome of a species is polymorphic and what the average
heterozygosity is in a population. Very large numbers of species have been sampled by this
method, including bacteria, fungi, higher plants, vertebrates, and invertebrates. The results are
remarkably consistent over species. About one-third of structural-gene loci are polymorphic,
and the average heterozygosity in a population over all loci sampled is about 10 percent. This
means that scanning the genome in virtually any species would show that about 1 in every 10
loci is in heterozygous condition and that about one-third of all loci have two or more alleles
segregating in any population. Thus the potential of variation for evolution is immense. The
disadvantage of the electrophoretic technique is that it detects variation only in structural
genes. If most of the evolution of shape, physiology, and behavior rests on changes in
regulatory genetic elements, then the observed variation in structural genes is beside the
point.
DNA sequence polymorphism
DNA analysis makes it possible to examine variation among individuals and between species
in their DNA sequences. There are two levels at which such studies can be done. Studying
variation in the sites recognized by restriction enzymes provides a coarse view of base-pair
variation. At a finer level, methods of DNA sequencing allow variation to be observed base
pair by base pair.
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Population Genetics
Restriction-site variation.
A restriction enzyme that recognizes six-base sequences (a “six cutter”) will recognize an
appropriate sequence approximately once every 46 = 4096 base pairs along a DNA molecule
[determined from the probability that a specific base (of which there are four) will be found at
each of the six positions]. If there is polymorphism in the population for one of the six bases
at the recognition site, then there will be a restriction fragment length polymorphism (RFLP)
in the population, because in one variant the enzyme will recognize and cut the DNA, whereas
in the other variant it will not (see pages 398–399). A panel of, say, eight enzymes will then
sample every 4096/8 500 base pairs for such polymorphisms. However, when one is found,
we do not know which of the six base pairs at the recognition site is polymorphic.
If we use enzymes that recognize four-base sequences (“four cutters”), there is a recognition
site every 44 = 256 base pairs; so a panel of eight different enzymes can sample about once
every 32 base pairs along the enzyme. In addition to single base-pair changes that destroy
restrictionenzyme recognition sites, there are insertions and deletions of stretches of DNA that
also cause restriction fragment lengths to vary.
Extensive samples have been made for several regions of the genome in a number of species
of Drosophila with the use of both four-cutting and six-cutting enzymes. The result of one
such study of the xanthine dehydrogenase gene in Drosophila pseudoobscura is shown in
Figure 24-4. The figure shows, symbolically, the restriction pattern of 53 chromosomes
(haplotypes) sampled from nature, polymorphic for 78 restriction sites along a sequence 4.5
kb in length. Among the 53 haplotypes, there are 48 different ones. (Try to find the identical
pairs.) Clearly there is an immense amount of nucleotide variation at the xanthine
dehydrogenase locus in nature.
Twenty restriction-enzyme studies of different regions of the X chromosome and the two
large autosomes of Drosophila melanogaster have found between 0.1 and 1.0 percent
heterozygosity per nucleotide site, with an average of 0.4 percent. A study of the very small
fourth chromosome, however, found no polymorphism at all.
Tandem repeats.
Another form of DNA sequence variation that can be revealed by restriction fragment surveys
arises from the occurrence of multiply repeated DNA sequences. In the human genome, there
are a variety of different short DNA sequences dispersed throughout the genome, each one of
which is multiply repeated in a tandem row. The number of repeats may vary from a dozen to
more than 100 in different individual genomes. Such sequences are known as variable
number tandem repeats (VNTRs). If the restriction enzymes cut sequences that flank either
side of such a tandem array, a fragment will be produced whose size is proportional to the
number of repeated elements. The different-sized fragments will migrate at different rates in
an electrophoretic gel. Unfortunately, the individual elements are too short to allow
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An Introduction to Genetic Analysis
Chapter 24
Population Genetics
distinguishing between, say, 64 and 68 repeats, but size classes (bins) can be established, and
a population can be assayed for the frequencies of the different classes. Table 24-7 shows the
data for two different VNTRs sampled in two American Indian groups from Brazil. In one
case, D14S1, the Karitiana are nearly homozygous, whereas the Surui are very variable; in the
other case, D14S13, both populations are variable but with different frequency patterns.
Complete sequence variation.
Studies of variation at the level of single base pairs by DNA sequencing can provide
information of two kinds. First, translating the sequences of coding regions obtained from
different individuals in a population or from different species allows the exact amino acid
sequence differences to be determined. Electrophoretic studies can show that there is variation
in amino acid sequences but cannot identify how many or which amino acids differ between
individuals. So, when DNA sequences were obtained for the various electrophoretic variants
of esterase-5 in Drosophila pseudoobscura (see Figure 24-2), electrophoretic classes were
found to differ from each other by an average of 8 amino acids, and the 20 different kinds of
amino acids were involved in polymorphisms at about the frequency that they were present in
the protein. Such studies also show that different regions of the same protein have different
amounts of polymorphism. For the esterase-5 protein, consisting of 545 amino acids, 7
percent of amino acid positions are polymorphic, but the last amino acids at the carboxyl
terminus of the protein are totally invariant between individuals, probably because of
functional constraints on these amino acids.
Second, DNA base-pair variation can also be studied for those base pairs that do not
determine or change the protein sequence. Such base-pair variation can be found in DNA in
introns, in 5′-flanking sequences that may be regulatory, in nontranscribed DNA 3′ to the gene,
and in those nucleotide positions within codons (usually third positions) whose variation does
not result in amino acid substitutions. Within coding sequences, these so-called silent or
synonymous base-pair polymorphisms are much more common than are changes that result in
amino acid polymorphism, presumably because many amino acid changes interfere with
normal function of the protein and are eliminated by natural selection. An examination of the
codon translation table (see Figure 10-27) shows that approximately 25 percent of all random
base-pair changes would be synonymous, giving an alternative codon for the same amino acid,
whereas 75 percent of random changes would change the amino acid coded. For example, a
change from AAT to AAC still encodes asparagine, but a change to ATT, ACT, AAA, AAG,
AGT, TAT, CAT, or GAT, all single-base-pair changes from AAT, changes the amino acid
encoded. So, if mutations of base pairs are at random and if the substitution of an amino acid
made no difference to function, we would expect a 3:1 ratio of amino acid replacement to
silent polymorphisms. The actual ratios found in Drosophila vary from 2:1 to 1:10. Clearly,
there is a great excess of synonymous polymorphism, showing that most amino acid changes
are subject to natural selection. It should not be assumed, however, that silent sites in coding
sequences are entirely free from constraints. Different alternative triplet codings for the same
amino acid may differ in speed and accuracy of transcription, and the mRNA corresponding
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An Introduction to Genetic Analysis
Chapter 24
Population Genetics
to different alternative triplets may have different accuracy and speed of translation because
of limitations on the pool of tRNAs available. Evidence for the latter effect is that alternative
synonymous triplets for an amino acid are not used equally, and the inequality of use is much
more pronounced for genes that are transcribed at a very high rate.
There are also constraints on 5′ and 3′ noncoding sequences and on intron sequences. Both 5′
and 3′ noncoding DNA sequences contain signals for transcription, and introns may contain
enhancers of transcription (see Chapter 11).
MESSAGE
Within species, there is great genetic variation. This variation is manifest at the
morphologic level of chromosome form and number and at the level of DNA
segments that may have no observable developmental effects.
Quantitative variation
Not all variation in traits can be described in terms of allelic frequencies, because many
characteristics, such as height, vary continuously over a range rather than falling into a few
qualitatively distinct classes. There is no allele for being 5′8″ or 5′4″ tall. Such characters, if
they are varying as a consequence of genetic variation, will be affected by several or many
genes and by environmental variation as well. Special techniques are needed for the study of
such quantitative traits, and these techniques are presented in Chapter 25. For the moment,
we confine ourselves to the question of whether genetic differences between individuals affect
the trait at all. In experimental organisms, a simple way to answer this question is to choose
two groups of parents that differ markedly in the trait and to raise offspring from both groups
in the same environment. If the offspring of the two groups are different, then the trait is said
to be heritable (see Chapter 25 for a more detailed discussion of the concept and estimation of
heritability). A simple measure of the degree of heritability of the variation is the ratio of the
difference between the offspring groups to the difference between the parental groups. So, if
two groups of Drosophila parents differed by, say, 0.1 mg in weight, whereas the offspring
groups, raised in identical environments, differed by 0.03 mg, the heritability of weight
difference would be estimated as 30 percent. When this technique is applied to morphologic
variation in Drosophila, virtually every variable trait is found to have some heritability. It is
important to note that this method cannot be applied to organisms for which no rigorous
control over developmental environment is possible. In humans, for example, children of
different parental groups differ from one another not only because their genetic makeup is
different, but also because the environments of different families, social classes, and nations
are different. Japanese are, on the average, shorter than Europeans, but the difference between
children of Japanese ancestry and children of European ancestry, both born in North America,
is less and becomes even less in the second generation, presumably because of diet. It is not
clear whether all the differences in height would disappear or even be reversed if the family
environments were identical.
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An Introduction to Genetic Analysis
Chapter 24
Population Genetics
Effect of sexual reproduction on variation
Before Mendel, blending inheritance was the standard model. This concept has powerful
consequences for population variation.
Suppose that some trait (say, height) has a distribution in the population and that individuals
mate more or less at random. If intermediate individuals mated with each other, they would
produce only intermediate offspring, according to a blending model. The mating of a tall with
a short individual also would produce only intermediate offspring. Only the mating of tall
with tall individuals and short with short individuals would preserve extreme types. The net
result of all matings would be an increase in intermediate types and a decrease in extreme
types. The variance of the distribution would shrink, simply as a result of sexual reproduction.
In fact, it can be shown that the variance is cut in half in each generation, so that the
population would be essentially uniformly intermediate in height before very many
generations had passed.
The particulate nature of inheritance changes this picture completely. Because of the discrete
nature of the Mendelian genes and the segregation of alleles at meiosis, a cross of
intermediate with intermediate individuals does not result in all intermediate offspring. On the
contrary, extreme types (homozygotes) segregate out of the cross. To see the consequence of
Mendelian inheritance for genetic variation, consider a population in which males and females
mate with each other at random with respect to some gene locus A; that is, individuals do not
choose their mates preferentially with respect to the partial genotype at the locus. Such
random mating is equivalent to mixing all the sperm and all the eggs in the population
together and then matching randomly drawn sperm with randomly drawn eggs.
The outcome of such a random pairing of sperm and eggs is easy to calculate. If, in some
population, the allele frequency of A is 0.60 in sperm and eggs, then the chance that a
randomly chosen sperm and a randomly chosen egg are both A is 0.60 × 0.60 = 0.36. Thus, in
a randommating population with this allele frequency, offspring will be 36 percent A/A. In the
same way, the frequency of a/a offspring will be 0.40 × 0.40 = 0.16. Heterozygotes will be
produced by the fusion either of an A sperm with an a egg or of an a sperm with an A egg. If
gametes pair at random, then the chance of an A sperm and an a egg is 0.60 × 0.40, and the
reverse combination has the same probability, so the frequency of heterozygous offspring is 2
× 0.6 × 0.4 = 0.48. Moreover, the process of random mating has done nothing to change allele
frequencies, as can be easily checked by calculating the frequencies of the alleles A and a
among the offspring by using the method described on page 715. So the proportions of
homozygotes and heterozygotes in each successive generation will remain the same. Box 24-2
gives a general form of this equilibrium result.
MESSAGE
Mendelian segregation has the property that random mating results in an equilibrium
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An Introduction to Genetic Analysis
Chapter 24
Population Genetics
distribution of genotypes after only one generation, so genetic variation is maintained.
The equilibrium distribution
is called the Hardy-Weinberg equilibrium after those who independently discovered it. (A
third independent discovery was made by the Russian geneticist Sergei Chetverikov.)
The Hardy-Weinberg equilibrium means that sexual reproduction does not cause a constant
reduction in genetic variation in each generation; on the contrary, the amount of variation
remains constant generation after generation, in the absence of other disturbing forces. The
equilibrium is the direct consequence of the segregation of alleles at meiosis in heterozygotes.
Numerically, the equilibrium shows that, irrespective of the particular mixture of genotypes in
the parental generation, the genotypic distribution after one round of mating is completely
specified by the allelic frequency p. For example, consider three hypothetical populations:
The allele frequency p of A in the three populations is:
So, despite their very different genotypic compositions, they have the same allele frequency.
After one generation of random mating, however, each of the three populations will have the
same genotypic frequencies:
and they will remain so indefinitely.
One consequence of the Hardy-Weinberg proportions is that rare alleles are virtually never in
homozygous condition. An allele with a frequency of 0.001 is present in homo-zygotes at a
frequency of only 1 in a million; most copies of such rare alleles are found in heterozygotes.
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An Introduction to Genetic Analysis
Chapter 24
Population Genetics
In general, because two copies of an allele are in homozygotes but only one copy of that allele
is in each heterozygote, the relative frequency of the allele in heterozygotes (in contrast with
homozygotes) is, from the Hardy-Weinberg equilibrium frequencies,
which for q = 0.001 is a ratio of 999:1. The general relation between homozygote and
heterozygote frequencies as a function of allele frequencies is shown in Figure 24-5.
In our derivation of the equilibrium, we assumed that the allelic frequency p is the same in
sperm and eggs. The Hardy-Weinberg equilibrium theorem does not apply to sex-linked genes
if males and females start with unequal gene frequencies.
The Hardy-Weinberg equilibrium was derived on the assumption of “random mating,” but we
must carefully distinguish two meanings of that process. First, we may mean that individuals
do not choose their mates on the basis of some heritable character. Human beings are random
mating with respect to blood groups in this first sense, because they generally do not know the
blood type of their prospective mates, and, even if they did, it is unlikely that blood type
would be used as a criterion for choice. In the first sense, random mating will occur with
respect to genes that have no effect on appearance, behavior, smell, or other characteristics
that directly influence mate choice.
The second sense of random mating is relevant when there is any division of a species into
subgroups. If there is genetic differentiation between subgroups so that the frequencies of
alleles differ from group to group and if individuals tend to mate within their own subgroup
(endogamy), then, with respect to the species as a whole, mating is not at random and
frequencies of genotypes will depart more or less from Hardy-Weinberg frequencies. In this
sense, human beings are not random mating, because ethnic and racial groups differ from one
another in gene frequencies and people show high rates of endogamy not only within major
races, but also within local ethnic groups. Spaniards and Russians differ in their ABO blood
group frequencies, Spaniards marry Spaniards and Russians marry Russians, so there is
unintentional nonrandom mating with respect to ABO blood groups. Table 24-8 shows
random mating in the first sense and nonrandom mating in the second sense for the MN blood
group. Within Eskimo, Egyptian, Chinese, and Australian subpopulations, females do not
choose their mates by MN type, and, thus, Hardy-Weinberg equilibrium exists within the
subpopulations. But Egyptians do not mate with Eskimos or Australian aborigines, so the
nonrandom associations in the human species as a whole result in large differences in
genotype frequencies and departure from Hardy-Weinberg equilibrium.
Sources of variation
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The variational theory of evolution has a peculiar selfdefeating property. If evolution occurs
by the differential reproduction of different variants, we expect the variant with the highest
rate of reproduction eventually to take over the population and all other genotypes to
disappear. But then there is no longer any variation for further evolution. The possibility of
continued evolution therefore is critically dependent on renewed variation.
For a given population, there are three sources of variation: mutation, recombination, and
immigration of genes. However, recombination by itself does not produce variation unless
alleles are segregating already at different loci; otherwise there is nothing to recombine.
Similarly, immigration cannot provide variation if the entire species is homo-zygous for the
same allele. Ultimately, the source of all variation must be mutation.
Variation from mutations
Mutations are the source of variation, but the process of mutation does not itself drive
evolution. The rate of change in gene frequency from the mutation process is very low
because spontaneous mutation rates are low (Table 24-9). The mutation rate is defined as the
probability that a copy of an allele changes to some other allelic form in one generation.
Suppose that a population were completely homozygous A and mutations to a occurred at the
rate of 1/100,000 Then, in the next generation, the frequency of a alleles would be only 1.0
× 1/100,000 = 0.00001 and the frequency of A alleles would be 0.99999. After yet
another generation of mutation, the frequency of a would be increased by 0.99999 ×
1/100,000 = 0.00009 to a new frequency of 0.000019, whereas the original allele would be
reduced in frequency to 0.999981. It is obvious that the rate of increase of the new allele is
extremely slow and that it gets slower every generation because there are fewer copies of the
old allele still left to mutate. A general formula for the change in allele frequency under
mutation is given in Box 24-3.
MESSAGE
Mutation rates are so low that mutation alone cannot account for the rapid evolution
of populations and species.
If we look at the mutation process from the standpoint of the increase of a particular new
allele rather than the decrease of the old form, the process is even slower. Most mutation rates
that have been determined are the sum of all mutations of A to any mutant form with a
detectable effect. Any specific base substitution is likely to be at least two orders of
magnitude lower in frequency than the sum of all changes. So, precise reverse mutations
(“back mutations”) to the original allele A are unlikely, although many mutations may
produce alleles that are phenotypically similar to the original.
It is not possible to measure locus-specific mutation rates for continuously varying characters,
but the rate of accumulation of genetic variance can be determined. Beginning with a
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completely homozygous line of Drosophila derived from a natural population, 1/1000 to
1/500 of the genetic variance in bristle number in the original population is restored each
generation by spontaneous mutation.
Variation from recombination
The creation of genetic variation by recombination can be a much faster process than its
creation by mutation. When just two chromosomes with “normal” survival, taken from a
natural population of Drosophila, are allowed to recombine for a single generation, they
produce an array of chromosomes with 25 to 75 percent as much genetic variation in survival
as was present in the entire natural population from which the parent chromosomes were
sampled. This outcome is simply a consequence of the very large number of different
recombinant chromosomes that can be produced even if we take into account only single
crossovers. If a pair of homologous chromosomes is heterozygous at n loci, then a crossover
can take place in any one of the n − 1 intervals between them, and, because each
recombination produces two recombinant products, there are 2(n − 1) new unique gametic
types from a single generation of crossing-over, even considering only single crossovers. If
the heterozygous loci are well spread out on the chromosomes, these new gametic types will
be frequent and considerable variation will be generated. Asexual organisms or organisms,
such as bacteria, that very seldom undergo sexual recombination do not have this source of
variation, so new mutations are the only way in which a change in gene combinations can be
achieved. As a result, asexual organisms may evolve more slowly under natural selection than
sexual organisms.
Variation from migration
A further source of variation is migration into a population from other populations with
different gene frequencies. The resulting mixed population will have an allele frequency that
is somewhere intermediate between its original value and the frequency in the donor
population. Suppose a population receives a group of migrants whose number is equal to, say,
10 percent of its native population size. Then the newly formed mixed population will have an
allele frequency that is a 0.90:0.10 mixture between its original allele frequency and the allele
frequency of the donor population. If its original allele frequency of A were, say, 0.70,
whereas the donor population had an allele frequency of only, say, 0.40, the new mixed
population would have a frequency of 0.70 × 0.90 + 0.40 × 0.10 = 0.67. Box 24-4 derives the
general result. The change in gene frequency is proportional to the difference in frequency
between the recipient population and the average of the donor populations. Unlike the
mutation rate, the migration rate (m) can be large, so the change in frequency may be
substantial.
We must understand migration as meaning any form of the introduction of genes from one
population into another. So, for example, genes from Europeans have “migrated” into the
population of African origin in North America steadily since the Africans were introduced as
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slaves. We can determine the amount of this migration by looking at the frequency of an allele
that is found only in Europeans and not in Africans and comparing its frequency among
blacks in North America.
We can use the formula for the change in gene frequency from migration if we modify it
slightly to account for the fact that several generations of admixture have taken place. If the
rate of admixture has not been too great, then (to a close order of approximation) the sum of
the single-generation migration rates over several generations (let's call this M) will be related
to the total change in the recipient population after these several generations by the same
expression as the one used for changes due to migration. If, as before, P is the allelic
frequency in the donor population and p0 is the original frequency among the recipients, then
For example, the Duffy blood group allele Fya is absent in Africa but has a frequency of 0.42
in whites from the state of Georgia. Among blacks from Georgia, the Fya frequency is 0.046.
Therefore, the total migration of genes from whites into the black population since the
introduction of slaves in the eighteenth century is
When the same analysis is carried out on American blacks from Oakland (California) and
Detroit, M is 0.22 and 0.26, respectively, showing either greater admixture rates in these cities
than in Georgia or differential movement into these cities by American blacks who have more
European ancestry. In any case, the genetic variation at the Fy locus has been increased by
this admixture.
Inbreeding and assortative mating
Random mating with respect to a locus is common, but it is not universal. Two kinds of
deviation from random mating must be distinguished. First, individuals may mate with each
other nonrandomly because of their degree of common ancestry; that is, their degree of
genetic relationship. If mating between relatives occurs more commonly than would occur by
pure chance, then the population is inbreeding. If mating between relatives is less common
than would occur by chance, then the population is said to be undergoing enforced
outbreeding, or negative inbreeding.
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Second, individuals may tend to choose each other as mates, not because of their degree of
genetic relationship but because of their degree of resemblance to each other at some locus.
Bias toward mating of like with like is called positive assortative mating. Mating with unlike
partners is called negative assortative mating. Assortative mating is never complete.
Inbreeding and assortative mating are not the same. Close relatives resemble each other more
than unrelated individuals on the average but not necessarily for any particular trait in
particular individuals. So inbreeding can result in the mating of quite dissimilar individuals.
On the other hand, individuals who resemble each other for some trait may do so because they
are relatives, but unrelated individuals also may have specific resemblances. Brothers and
sisters do not all have the same eye color, and blue-eyed people are not all related to one
another.
Assortative mating for some traits is common. In humans, there is a positive assortative
mating bias for skin color and height, for example. An important difference between
assortative mating and inbreeding is that the former is specific to a trait, whereas the latter
applies to the entire genome. Individuals may mate assortatively with respect to height but at
random with respect to blood group. Cousins, on the other hand, resemble each other
genetically on the average to the same degree at all loci.
For both positive assortative mating and inbreeding, the consequence to population structure
is the same: there is an increase in homozygosity above the level predicted by the
Hardy-Weinberg equilibrium. If two individuals are related, they have at least one common
ancestor. Thus, there is some chance that an allele carried by one of them and an allele carried
by the other are both descended from the identical DNA molecule. The result is that there is
an extra chance of homozygosity by descent, to be added to the chance of homozygosity (p2
+ q2) that arises from the random mating of unrelated individuals. The probability of
homozygosity by descent is called the inbreeding coefficient (F). Figure 24-6 and Box 24-5
illustrate the calculation of the probability of homozygosity by descent. Individuals I and II
are full sibs because they share both parents. We label each allele in the parents uniquely to
keep track of them. Individuals I and II mate to produce individual III. If individual I is A1/A3
and the gamete that it contributes to III contains the allele A1, then we would like to calculate
the probability that the gamete produced by II is also A1. The chance is 1/2 that II will receive
A1 from its father, and, if it does, the chance is 1/2 that II will pass A1 on to the gamete in
question. Thus, the probability that III will receive an A1 from II is 1/2 × 1/2 = 1/4 and this is
the chance that III—the product of a full-sib mating—will be homozygous by descent.
Such close inbreeding can have deleterious consequences. Let's consider a rare deleterious
allele a that, when homozygous, causes a metabolic disorder. If the frequency of the allele in
the population is p, then the probability that a random couple will produce a homozygous
offspring is only p2 (from the Hardy-Weinberg equilibrium). Thus, if p is, say, 1/1000, the
frequency of homozygotes will be 1 in 1,000,000. Now suppose that the couple are brother
and sister. If one of their common parents is a heterozygote for the disease, they may both
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Population Genetics
receive it and may both pass it on to their offspring. As the calculation shows, the rarer the
gene, the worse the relative risk of a defective offspring from inbreeding. For more-distant
relatives, the chance of homozygosity by descent is less but still substantial. For first cousins,
for example, the relative risk is 1/16p compared with random mating.
Systematic inbreeding between close relatives eventually leads to complete homozygosity of
the population but at different rates, depending on the degree of relationship. Which allele is
fixed within a line is a matter of chance. If, in the original population from which the inbred
lines are taken, allele A has frequency p and allele a has frequency q = 1 − p, then a
proportion p of the homozygous lines established by inbreeding will be homozygous A/A and
a proportion q of the lines will be a/a. Inbreeding takes the genetic variation present within the
original population and converts it into variation between homozygous inbred lines sampled
from the population (Figure 24-7).
Suppose that a population is founded by some small number of individuals who mate at
random to produce the next generation. Assume that no further immigration into the
population ever occurs again. (For example, the rabbits now in Australia probably have
descended from a single introduction of a few animals in the nineteenth century.) In later
generations, then, everyone is related to everyone else, because their family trees have
common ancestors here and there in their pedigrees. Such a population is then inbred, in the
sense that there is some probability of a gene's being homozygous by descent. Because the
population is, of necessity, finite in size, some of the originally introduced family lines will
become extinct in every generation, just as family names disappear in a closed human
population because, by chance, no male offspring are left. As original family lines disappear,
the population comes to be made up of descendants of fewer and fewer of the original founder
individuals, and all the members of the population become more and more likely to carry the
same alleles by descent. In other words, the inbreeding coefficient F increases, and the
heterozygosity decreases over time until finally F reaches 1.00 and heterozygosity reaches 0.
The rate of loss of heterozygosity per generation in such a closed, finite, randomly breeding
population is inversely proportional to the total number (2N) of haploid genomes, where N is
the number of diploid individuals in the population. In each generation, 1/2N of the remaining
heterozygosity is lost, so
where Ht and H0 are the proportions of heterozygotes in the tth and original generations,
respectively. As the number t of generations becomes very large, Ht approaches zero.
Balance between inbreeding and new variation
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Population Genetics
Any population of any species is finite in size, so all populations should eventually become
homozygous and differentiated from one another as a result of inbreeding. In nature, however,
new variation is always being introduced into populations by mutation and by some migration
between localities. Thus, the actual variation available for natural selection is a balance
between the introduction of new variation and its loss through local inbreeding. The rate of
loss of heterozygosity in a closed population is 1/2N, so any effective differentiation between
populations will be negated if new variation is introduced at this rate or higher.
Selection
So far in this chapter, we have considered changes in a population arising from forces of
mutation, migration, recombination, and breeding structure. But these changes are random
with respect to the way in which organisms make a living in the environments in which they
live. Changes in a species in response to a changing environment occur because the different
genotypes produced by mutation and recombination have different abilities to survive and
reproduce. The differential rates of survival and reproduction are what is meant by selection,
and the process of selection results in changes in the frequencies of the various genotypes in
the population. Darwin called the process of differential survival and reproduction of different
types natural selection by analogy with the artificial selection carried out by animal and
plant breeders when they deliberately select some individuals of a preferred type.
The relative probability of survival and rate of reproduction of a phenotype or genotype is
now called its Darwinian fitness. Although geneticists sometimes speak loosely of the fitness
of an individual, the concept of fitness really applies to the average survival and reproduction
of individuals in a phenotypic or genotypic class. Because of chance events in the life
histories of individuals, even two organisms with identical genotypes and identical
environments will differ in their survival and reproduction rates. It is the fitness of a genotype
on average over all its possessors that matters.
Fitness is a consequence of the relation between the phenotype of the organism and the
environment in which the organism lives, so the same genotype will have different fitnesses
in different environments. In part, this difference is because exposure to different
environments during development will result in different phenotypes for the same genotypes.
But, even if the phenotype is the same, the success of the organism depends on the
environment. Having webbed feet is fine for paddling in water but a positive disadvantage for
walking on land, as a few moments spent observing a duck walk will reveal. No genotype is
unconditionally superior in fitness to all others in all environments.
Furthermore, the environment is not a fixed situation that is experienced passively by an
organism. The environment of an organism is defined by the activities of the organism itself.
For example, dry grass is part of the environment of a junco, so juncos that are most efficient
at gathering it may waste less energy in nest building and thus have a higher reproductive
fitness. But dry grass is part of a junco's environment because juncos gather it to make nests.
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The rocks among which the grass grows are not part of the junco's environment, although the
rocks are physically present there. But the rocks are part of the environment of thrushes; these
birds use the rocks to break open snails. Moreover, the environment that is defined by the life
activities of an organism evolves as a result of those activities. The structure of the soil that is
in part determinative of the kinds of plants that will grow is altered by the growth of those
very plants. Environment is both the cause and the result of the evolution of organisms. As
primitive plants evolved photosynthesis, they changed the earth's atmosphere from one that
had had essentially no free oxygen and a high concentration of carbon dioxide to the
atmosphere that we know today, which contains 21 percent oxygen and only 0.03 percent
carbon dioxide. Plants that evolve today must do so in an environment created by the
evolution of their own ancestors.
Darwinian, or reproductive, fitness is not to be confused with “physical fitness” in the
everyday sense of the term, although they may be related. No matter how strong, healthy, and
mentally alert the possessor of a genotype may be, that genotype has a fitness of zero if for
some reason its possessors leave no offspring. Thus such statements as the “unfit are
outreproducing the fit, so the species may become extinct” are meaningless. The fitness of a
genotype is a consequence of all the phenotypic effects of the genes involved. Thus, an allele
that doubles the fecundity of its carriers but at the same time reduces the average lifetime of
its possessors by 10 percent will be more fit than its alternatives, despite its life-shortening
property. The most common example is parental care. An adult bird that expends a great deal
of its energy gathering food for its young will have a lower probability of survival than one
that keeps all the food for itself. But a totally selfish bird will leave no offspring, because its
young cannot fend for themselves. As a consequence, parental care is favored by natural
selection.
Two forms of selection
Because the differences in reproduction and survival between genotypes depend on the
environment in which the genotypes live and develop and because organisms may alter their
own environments, there are two fundamentally different forms of selection. In the simple
case, the fitness of an individual does not depend on the composition of the population; rather
it is a fixed property of the individual's phenotype and the external physical environment. For
example, the relative ability of two plants that live at the edge of the desert to get sufficient
water will depend on how deep their roots grow and how much water they lose through their
leaf surfaces. These characteristics are a consequence of their developmental patterns and are
not sensitive to the composition of the population in which they live. The fitness of a
genotype in such a case does not depend on how rare or how frequent it is in the population.
Fitness is then frequency independent.
In contrast, consider organisms that are competing to catch prey or to avoid being captured by
a predator. Then the relative abundances of two different genotypes will affect their relative
fitnesses. An example is Mullerian mimicry in butterflies. Some species of brightly colored
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butterflies (such as monarchs and viceroys) are distasteful to birds, which learn, after a few
trials, to avoid attacking butterflies with that pattern. If two species differ in pattern, there will
be selection to make them more similar because both will be protected and they share the
burden of the birds' initial learning period. The less frequent pattern will be at a disadvantage
with respect to the more frequent one, because birds will less often learn to avoid them.
Within a species, rarer patterns will be selected against for the same reason. The rarer the
pattern, the greater is the selective disadvantage, because birds will be unlikely to have had a
prior experience of a low-frequency pattern and therefore will not avoid it. This selection to
blend in with the crowd is an example of frequency-dependent fitness.
For reasons of mathematical convenience, most models of natural selection are based on
frequency-independent fitness. In fact, however, a very large number of selective processes
(perhaps most) are frequency dependent. The kinetics of the evolutionary process depend on
the exact form of frequency dependence, and, for that reason alone, it is difficult to make any
generalizations. The result of positive frequency dependence (such as competing predators,
where fitness increases with increasing frequency) is quite different from that of negative
frequency dependence (where fitness of a genotype declines with increasing frequency). For
the sake of simplicity and as an illustration of the main qualitative features of selection, we
deal only with models of frequency-independent selection in this chapter, but convenience
should not be confused with reality.
Measuring fitness differences
For the most part, the differential fitness of different genotypes can be most easily measured
when the genotypes differ at many loci. In very few cases (except for laboratory mutants,
horticultural varieties, and major metabolic disorders) does the effect of an allelic substitution
at a single locus make enough difference to the phenotype to be reflected in measurable
fitness differences. Figure 24-8 shows the probability of survival from egg to adult—that is,
the viability—of a number of second-chromosome homozygotes of D. pseudoobscura at three
different temperatures. As is generally the case, the fitness (in this case, a component of the
total fitness, viability) is different in different environments. A few homozygotes are lethal or
nearly so at all three temperatures, whereas a few have consistently high viability. Most
genotypes, however, are not consistent in viability between temperatures, and no genotype is
unconditionally the most fit at all temperatures. The fitness of these chromosomal
homozygotes was not measured in competition with each other; all are measured against a
common standard, so we do not know whether they are frequency dependent. An example of
frequency-dependent fitness is shown in the estimates for inversion homozygotes and
heterozygotes of D. pseudoobscura in Table 24-10.
Examples of clear-cut fitness differences associated with single-gene substitutions are the
many “inborn errors of metabolism,” where a recessive allele interferes with a metabolic
pathway and causes lethality of the homozygotes. An example in humans is phenylketonuria,
where tissue degeneration is the result of the accumulation of a toxic intermediate in the
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pathway of tyrosine metabolism. A case that illustrates the relation of fitness to environment
is sickle-cell anemia. An allelic substitution at the structural-gene locus for the β chain of
hemoglobin results in substitution of valine for the normal glutamic acid at chain position 6.
The abnormal hemoglobin crystallizes at low oxygen pressure, and the red cells deform and
hemolyze. Homozygotes HbS/HbS have a severe anemia, and survivorship is low.
Heterozygotes have a mild anemia and under ordinary circumstances exhibit the same or only
slightly lower fitness than normal homozygotes HbA/HbA. However, in regions of Africa with
a high incidence of falciparum malaria, heterozygotes (HbA/HbS) have a higher fitness than
normal homozygotes because the presence of some sickling hemoglobin apparently protects
them from the malaria. Where malaria is absent, as in North America, the fitness advantage of
heterozygosity is lost.
It has not been possible to measure fitness differences for most single-locus polymorphisms.
The evidence for differential net fitness for different ABO or MN blood types is shaky at best.
The extensive enzyme polymorphism present in all sexually reproducing species has for the
most part not been connected with measurable fitness differences, although, in Drosophila,
clear-cut differences in the fitness of different genotypes have been demonstrated in the
laboratory for a few loci such as those encoding α-amylase and alcohol dehydrogenase.
How selection works
The simplest way to see the effect of selection is to consider an allele, a, that is completely
lethal before reproductive age in homozygous condition, such as the allele that leads to
Tay-Sachs disease. Suppose that, in some generation, the allele frequency of this gene is 0.10.
Then, in a randommating population, the proportions of the three genotypes after fertilization
are
At reproductive age, however, the homozygotes a/a will have already died, leaving the
genotypes at this stage as
But these proportions add up to only 0.99 because only 99 percent of the population is still
surviving. Among the actual surviving reproducing population, the proportions must be
recalculated by dividing by 0.99 so that the total proportions add up to 1.00. After this
readjustment, we have
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Population Genetics
The frequency of the lethal a allele among the gametes produced by these survivors is then
and the change in allelic frequency in one generation, expressed as the new value minus the
old one, has been 0.091 − 0.100 = −0.019. We can repeat this calculation in each successive
generation to obtain the predicted frequencies of the lethal and normal alleles in a succession
of future generations.
The same kind of calculation can be carried out if genotypes are not simply lethal or normal,
but if each genotype has some relative probability of survival. This general calculation is
shown in Box 24-6. After one generation of selection, the new value of the frequency of A is
equal to the old value (p) multiplied by the ratio of the average fitness of A alleles to the
fitness of the whole population. If the fitness of A alleles is greater than the average fitness of
all alleles, then A / is greater than unity and p′ is larger than p. Thus, the allele A
increases in the population. Conversely, if A / is less than unity, A decreases. But the
mean fitness of the population ( ) is the average fitness of the A alleles and of the a alleles.
So if A is greater than the mean fitness of the population, it must be greater than a, the
mean fitness of a alleles.
MESSAGE
The allele with the higher average fitness increases in the population.
It should be noted that the fitnesses WA / A , WA / a , and Wa / a may be expressed as absolute
probabilities of survival and absolute reproduction rates or they may all be rescaled relative to
one of the fitnesses, which is given the standard value of 1.0. This rescaling has absolutely no
effect on the formula for p′, because it cancels out in the numerator and denominator.
MESSAGE
The course of selection depends only on relative fitnesses.
An increase in the allele with the higher fitness means that the average fitness of the
population as a whole increases, so selection can also be described as a process that increases
mean fitness. This rule is strictly true only for frequency-independent genotypic fitnesses, but
it is close enough to a general rule to be used as a fruitful generalization. This maximization
of fitness does not necessarily lead to any optimal property for the species as a whole, because
fitnesses are only defined relative to each other within a population. It is relative (not absolute)
fitness that is increased by selection. The population does not necessarily become larger or
grow faster, nor is it less likely to become extinct.
Rate of change in gene frequency
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The general expression for the change in gene frequency, derived in Box 24-6, is particularly
illuminating. It says that Δp will be positive (A will increase) if the mean fitness of A alleles is
greater than the mean fitness of a alleles, as we saw before. But it also shows that the speed of
the change depends not only on the difference in fitness between the alleles, but also on the
factor pq, which is proportional to the frequency of heterozygotes (2pq). For a given
difference in fitness of alleles, gene frequency will change most rapidly when the alleles A
and a are in intermediate frequency, so pq is large. If p is near 0 or 1 (that is, if A or a is
nearly fixed), then pq is nearly 0 and selection will proceed very slowly. Figure 24-9 shows
the S-shaped curve that represents the course of selection of a new favorable allele A that has
recently entered a population of homozygotes a/a. At first, the change in frequency is very
small because p is still close to 0. Then it accelerates as A becomes more frequent, but it slows
down again as A takes over and a becomes very rare. This is precisely what is expected from a
selection process. When most of the population is of one type, there is nothing to select. For
evolution by natural selection to occur, there must be genetic variance; the more variance, the
faster the process.
One consequence of the dynamics shown in Figure 24-9 is that it is extremely difficult to
significantly reduce the frequency of an allele that is already rare in a population. Thus,
eugenic programs designed to eliminate deleterious recessive genes from human populations
by preventing the reproduction of affected persons do not work. Of course, if all
heterozygotes could be prevented from reproducing, the gene could be eliminated (except for
new mutations) in a single generation. Because every human being is heterozygous for a
number of different deleterious genes, however, no one would be allowed to reproduce.
When alternative alleles are not rare, selection can cause quite rapid changes in allelic
frequency. Figure 24-10 shows the course of elimination of a malic dehydrogenase allele in a
laboratory population of D. melanogaster. The fitnesses in this case are:
The frequency of a is not reduced to 0, and further reduction in frequency will require longer
and longer times, as shown in the negative eugenics case.
MESSAGE
Unless alternative alleles are present in intermediate frequencies, selection (especially
against recessives) is quite slow. Selection is dependent on genetic variation.
Balanced polymorphism
Let's reexamine the general formula for allelic frequency change (see Box 24-6):
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Under what conditions will the process stop? When is Δp = 0? Two answers are: when p = 0
or when q = 0 (that is, when either allele A or allele a, respectively, has been eliminated from
the population). One of these events will eventually occur if A −
a is consistently
positive or negative, so that Δp is always positive or negative irrespective of the value of p.
The condition for such unidirectional selection is that the heterozygote fitness be somewhere
between the fitnesses of the two homozygotes: If A/A homozygotes are most fit, then A alleles
are more fit than a alleles in both the heterozygous and the homozygous condition. Then the
mean allelic fitness of A, A, is larger than the mean allelic fitness of a, a, no matter what
the frequencies of the genotypes may be. In this case, A −
a is positive, and A always
increases until it reaches p = 1. If, on the other hand, a/a are most fit, then A −
a is
negative, and a always increases until it reaches q = 1.
But there is another possibility for Δp = 0, even when p and q are not 0:
which can occur if the heterozygote is not intermediate between the homozygotes but has a
fitness that is more extreme than either homozygote. In this case, selection will lead to an
intermediate allele frequency, ^p (see Box 24-7).
There are, in fact, two qualitatively different possibilities for ^p. One possibility is that ^p is
an unstable equilibrium. There will be no change in frequency if the population has exactly
this value of p, but the frequency will move away from the equilibrium (toward p = 0 or p = 1)
if the slightest perturbation of frequency occurs. This unstable case will exist when the
heterozygote is lower in fitness than either homozygote; such a condition is an example of
underdominance. The alternative possibility is a stable equilibrium, or balanced
polymorphism, in which slight perturbations from the value of ^p will result in a return to ^p.
The condition for this balance is that the heterozygote be greater in fitness than either
homozygote—a condition termed overdominance.
In nature, the chance that a gene frequency will remain balanced on the knife edge of an
unstable equilibrium is negligible, so we should not expect to find naturally occurring
polymorphisms in which heterozygotes are less fit than homozygotes. On the contrary, the
observation of a long-lasting polymorphism in nature might be taken as evidence of a superior
heterozygote.
Unfortunately, life confounds theory. The Rh locus (rhesus blood group) in humans has a
widespread polymorphism with Rh+ and Rh− alleles. In Europeans, the frequency of the Rh−
allele is about 0.4, whereas, in Africans, it is about 0.2. Thus, this human polymorphism must
be very old, antedating the origin of modern geographical races. But this polymorphism
causes a maternal–fetal incompatibility when an RH− mother (homozygous Rh−/Rh−) produces
an RH+ fetus (heterozygous Rh−/Rh+). This incompatibility results in hemolytic anemia (from
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Population Genetics
a destruction of red blood cells) and the death of the fetus in a moderate proportion of cases if
the mother has been previously sensitized by an earlier pregnancy with an incompatible fetus.
Thus, there is selection against heterozygotes, although it is frequency dependent, because it
occurs only when the mother is a homozygous recessive. This polymorphism is unstable and
should have disappeared from the species, yet it exists in most human populations. Many
hypotheses have been proposed to explain its apparent stability, but the mystery remains.
In contrast, no fitness difference at all can be demonstrated for many polymorphisms of blood
groups (and for the ubiquitous polymorphism of enzymes revealed by electrophoresis). It has
been suggested that such polymorphisms are not under selection at all but that
This situation of selective neutrality would also satisfy the requirement that A =
a, but,
instead of a stable equilibrium, it gives rise to a passive (neutral) equilibrium such that any
allele frequency p is as good as any other. This leaves unanswered the problem of how the
populations became highly polymorphic in the first place. The best case of overdominance for
fitness at a single locus remains that of sickle-cell anemia, where the two homozygotes are at
a disadvantage relative to the heterozygote for quite different reasons.
The best-studied cases of balanced polymorphism in nature and in the laboratory are the
inversion polymorphisms in several species of Drosophila.Figure 24-11 shows the course of
frequency change for the inversion ST (Standard) in competition with the alternative
chromosomal type CH (Chiricahua) in a laboratory population of D. pseudoobscura. The
inversions ST and CH are part of a chromosomal polymorphism in natural populations of this
species. The fitnesses estimated for the three genotypes in the laboratory are
Applying the formula for the equilibrium value ^p, we obtain ^p = 0.85, which agrees quite
well with the observations in Figure 24-11.
Another cause of genetic equilibrium in populations is the balance between the introduction of
new alleles by repeated mutation and their removal by natural selection. This balance is
probably the cause of many low-level polymorphisms for genetic diseases in human
populations. New deleterious mutations are constantly arising spontaneously or as the result
of the action of mutagens. These mutations may be completely recessive or partly dominant.
Selection removes them from the population, but there will be an equilibrium between their
appearance and removal.
The general expression for this equilibrium is that the frequency of the deleterious allele at
equilibrium depends on the ratio of the mutation rate, μ, to the intensity of selection, s, against
the deleterious genotype. For a completely recessive deleterious allele whose fitness in
homozygous state is 1 − s, the equilibrium frequency is
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These results are shown in detail in Box 24-8. So, for example, a recessive lethal (s = 1)
mutating at the rate of μ = 10−6 will have an equilibrium frequency of 10−3. Indeed, if we
knew that a gene was a recessive lethal and had no heterozygous effects, we could estimate its
mutation rate as the square of the allelic frequency. But the basis for such calculations must be
firm. Sickle-cell anemia was once thought to be a recessive lethal with no heterozygous
effects, which led to an estimated mutation rate in Africa of 0.1 for this locus.
A similar result can be obtained for a deleterious gene with some effect in heterozygotes. If
we let the fitnesses be WA / A = 1.0, WA / a = 1 − hs, and Wa / a = 1 − s for a partly dominant
gene, where h is the degree of dominance of the deleterious allele, then a similar calculation
gives us
Thus, if μ = 10−6 and the lethal is not totally recessive but has a 5 percent deleterious effect in
heterozygotes (s = 1.0, h = 0.05), then
which is smaller by two orders of magnitude than the equilibrium frequency for the purely
recessive case. In general, then, we can expect deleterious, completely recessive genes to have
frequencies much higher than those of partly dominant genes.
Artificial selection
In contrast with the difficulties of finding simple, wellbehaved cases in nature that exemplify
the simple formulas of natural selection, there is a vast record of the effectiveness of artificial
selection in changing populations phenotypically. These changes have been produced by
laboratory selection experiments and by selection of animals and plants in agriculture (as
examples, for increased milk production in cows and for rust resistance in wheat). No analysis
of these experiments in terms of allelic frequencies is possible, because individual loci have
not been identified and followed. Nevertheless, it is clear that genetic changes have occurred
in the populations and that some analysis of selected populations has been carried out
according to the methods described in Chapter 25. Figure 24-12 shows, as an example, the
large changes in average bristle number achieved in a selection experiment with D.
melanogaster.Figure 24-13 shows the changes in the number of eggs laid per chicken as a
consequence of 30 years of selection.
The usual method of selection is truncation selection. The individuals in a given generation
are pooled (irrespective of their families), a sample is measured, and only those individuals
above (or below) a given phenotypic value (the truncation point) are chosen as parents for the
next generation. This phenotypic value may be a fixed value over successive generations; then
selection is by constant truncation. More commonly, a fixed percentage of the population
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representing the highest (or lowest) value of the selected character is chosen; then selection is
by proportional truncation. With constant truncation, the intensity of selection decreases
with time, as more and more of the population exceeds the fixed truncation point. With
proportional truncation, the intensity of selection is constant, but the truncation point moves
upward as the population distribution moves. Figure 24-14 illustrates these two types of
truncation.
A common experience in artificial selection programs is that, as the population becomes more
and more extreme, its viability and fertility decrease. As a result, eventually no further
progress under selection is possible, despite the presence of genetic variance for the character,
because the selected individuals do not reproduce. The loss of fitness may be a direct
phenotypic effect of the genes for the selected character, in which case nothing much can be
done to improve the population further. Often, however, the loss of fitness is tied to linked
sterility genes that are carried along with the selected loci. In such cases, a number of
generations without selection allow recombinants to be formed, and selection can then be
continued, as in the upwardly selected line in Figure 24-12.
We must be very careful in our interpretation of long-term agricultural selection programs. In
the real world of agriculture, changes in cultivation methods, machinery, fertilizers,
insecticides, herbicides, and so forth are occurring along with the production of genetically
improved varieties. Increases in average yields are consequences of all of these changes. For
example, the average yield of corn in the United States increased from 40 bushels to 80
bushels per acre between 1940 and 1970. But experiments comparing old and new varieties of
corn in common environments show that only about half this increase is a direct result of new
corn varieties (the other half being a result of improved farming techniques). Furthermore, the
new varieties are superior to the old ones only at the high densities of modern planting for
which they were selected.
Random events
If a population is finite in size (as all populations are) and if a given pair of parents has only a
small number of offspring, then, even in the absence of all selective forces, the frequency of a
gene will not be exactly reproduced in the next generation, because of sampling error. If, in a
population of 1000 individuals, the frequency of a is 0.5 in one generation, then it may by
chance be 0.493 or 0.505 in the next generation because of the chance production of slightly
more or slightly fewer progeny of each genotype. In the second generation, there is another
sampling error based on the new gene frequency, so the frequency of a may go from 0.505 to
0.511 or back to 0.498. This process of random fluctuation continues generation after
generation, with no force pushing the frequency back to its initial state, because the
population has no “genetic memory” of its state many generations ago. Each generation is an
independent event. The final result of this random change in allelic frequency is that the
population eventually drifts to p = 1 or p = 0. After this point, no further change is possible;
the population has become homozygous. A different population, isolated from the first, also
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undergoes this random genetic drift, but it may become homozygous for allele A, whereas the
first population has become homozygous for allele a. As time goes on, isolated populations
diverge from each other, each losing heterozygosity. The variation originally present within
populations now appears as variation among populations.
One form of genetic drift occurs when a small group breaks off from a larger population to
found a new colony. This “acute drift,” called the founder effect, results from a single
generation of sampling, followed by several generations during which the population remains
small. The founder effect is probably responsible for the virtually complete lack of blood
group B in Native Americans, whose ancestors arrived in very small numbers across the
Bering Strait at the end of the last Ice Age, about 20,000 years ago.0
The process of genetic drift should sound familiar. It is, in fact, another way of looking at the
inbreeding effect in small populations discussed earlier. Whether regarded as inbreeding or as
random sampling of genes, the effect is the same. Populations do not exactly reproduce their
genetic constitutions; there is a random component of gene frequency change.
One result of random sampling is that most new mutations, even if they are not selected
against, never succeed in entering the population. Suppose that a single individual is
heterozygous for a new mutation. There is some chance that the individual in question will
have no offspring at all. Even if it has one offspring, there is a chance of 1/2 that the new
mutation will not be transmitted. If the individual has two offspring, the probability that
neither offspring will carry the new mutation is 1/4, and so forth. Suppose that the new
mutation is successfully transmitted to an offspring. Then the lottery is repeated in the next
generation, and again the allele may be lost. In fact, if a population is of size N, the chance
that a new mutation is eventually lost by chance is (2N − 1)/2N. (For a derivation of this result,
which is beyond the scope of this book, see Chapters 2 and 3 of Hartl and Clark, Principles of
Population Genetics.) But, if the new mutation is not lost, then the only thing that can happen
to it in a finite population is that eventually it will sweep through the population and become
fixed. This event has the probability of 1/2N In the absence of selection, then, the history of a
population looks like Figure 24-15. For some period of time, it is homozygous; then a new
mutation appears. In most cases, the new mutant allele will be lost immediately or very soon
after it appears. Occasionally, however, a new mutant allele drifts through the population, and
the population becomes homozygous for the new allele. The process then begins again.
Even a new mutation that is slightly favorable selectively will usually be lost in the first few
generations after it appears in the population, a victim of genetic drift. If a new mutation has a
selective advantage of s in the heterozygote in which it appears, then the chance is only 2s
that the mutation will ever succeed in taking over the population. So a mutation that is 1
percent better in fitness than the standard allele in the population will be lost 98 percent of the
time by genetic drift.
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MESSAGE
New mutations can become established in a population even though they are not
favored by natural selection simply by a process of random genetic drift. Even new
favorable mutations are often lost.
Another consequence of the interaction of random and selective forces is that the
effectiveness of the selective force in driving population composition depends on population
size. The magnitude of the random effect is proportional to the reciprocal of population size,
1/N, whereas the magnitude of a deterministic force depends on the migration rate, m, or
mutation rate, μ, or selection coefficient, s. Thus we can say, roughly, that migration and
mutation are effective if
The same is true of selection; selection is effective only if Ns ≥ 1. When Ns is small because
selection is weak or population size is small, then mutations are effectively neutral, even
though there is some selection of them. Small populations will be less affected by selection
than large populations even under otherwise identical conditions. For example, human
populations were very small for nearly all the history of our species, having grown large only
in the past few hundred generations. Thus, we may expect to find that many mutations that are
now under selection were effectively neutral for a long time and may have reached high
frequency by chance.
Summary
The study of changes within a population, or population genetics, relates the heritable changes
in populations or organisms to the underlying individual processes of inheritance and
development. Population genetics is the study of inherited variation and its modification in
time and space.
Identifiable inherited variation within a population can be studied by observing morphological
differences between individuals, examining the differences in specific amino acid sequences
of proteins, or even examining, most recently, the differences in nucleotide sequences within
the DNA. These kinds of observations have led to the conclusion that there is considerable
polymorphism at many loci within a population. A measure of this variation is the amount of
heterozygosity in a population. Population studies have shown that, in general, the genetic
differences between individuals within human races are much greater than the average
differences between races.
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The ultimate source of all variation is mutation. However, within a population, the
quantitative frequency of specific genotypes can be changed by recombination, immigration
of genes, continued mutational events, and chance.
One property of Mendelian segregation is that random mating results in an equilibrium
distribution of genotypes after one generation. However, inbreeding is one process that
converts genetic variation within a population into differences between populations by
making each separate population homozygous for a randomly chosen allele. On the other
hand, for most populations, a balance is reached for any given environment among inbreeding,
mutation from one allele to another, and immigration.
“Directed” changes of allelic frequencies within a population occur through the natural
selection of a favored genotype. In many cases, such changes lead to homozygosity at a
particular locus. On the other hand, the heterozygote may be more suited to a given
environment than either of the homozygotes, leading to a balanced polymorphism.
Environmental selection of specific genotypes is rarely this simple, however. More often than
not, phenotypes are determined by several interacting genes, and alleles at these different loci
will be selected for at different rates. Furthermore, closely linked loci, unrelated to the
phenotype in question, may have specific alleles carried along during the selection process. In
general, genetic variation is the result of the interaction of evolutionary forces. For instance, a
recessive, deleterious mutant may never be totally eliminated from a population, because
mutation will continue to resupply it to the population. Immigration also can reintroduce the
undesirable allele into the population. And, indeed, a deleterious allele may, under
environmental conditions of which we are unaware (including the remaining genetic makeup
of the individual), be selected for.
Unless alternative alleles are in intermediate frequencies, selection (especially against
recessives) is very slow, requiring many generations. In many populations, especially those of
small size, new mutations can become established even though they are not favored by natural
selection, simply by a process of random genetic drift.
Solved Problems
1. About 70 percent of all white North Americans can taste the chemical
phenylthiocarbamide, and the remainder cannot. The ability to taste is determined by the
dominant allele T, and the inability to taste is determined by the recessive allele t. If the
population is assumed to be in Hardy-Weinberg equilibrium, what are the genotypic and
allelic frequencies in this population?
Solution
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Because 70 percent are tasters (T/T), 30 percent must be nontasters (t/t). This homozygous
recessive frequency is equal to q2; so, to obtain q, we simply take the square root of 0.30:
2. In a large natural population of Mimulus guttatus, one leaf was sampled from each of a
large number of plants. The leaves were crushed and subjected to gel electrophoresis. The
gel was then stained for a specific enzyme X. Six different banding patterns were observed,
as shown in the diagram.
a. Assuming that these patterns are produced by a single locus, propose a genetic
explanation for the six types.
b. How can you test your idea?
c. What are the allelic frequencies in this population?
d. Is the population in Hardy-Weinberg equilibrium?
Solution
a. Inspection of the gel reveals that there are only three band positions: we will call them
slow, intermediate, and fast. Furthermore, any individual can show either one band or two.
The simplest explanation is that there are three al-leles of one locus (let's call them AS, AI,
and AF) and that the individuals with two bands are heterozygotes. Hence, 1 = S/S, 2 = I/I, 3 =
F/F, 4 = S/I, 5 = S/F, and 6 = I/F.
b. The hypothesis can be tested by making controlled crosses. For example, from a self of
type 5, we can predict 1/4 S/S, 1/2S/F, and 1/4 F/F.
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c. The frequencies can be calculated by a simple extension of the two-allele formulas. Hence:
d. The Hardy-Weinberg genotypic frequencies are:
which are precisely the observed frequencies. So it appears that the population is in
equilibrium.
3. In a large experimental Drosophila population, the fitness of a recessive phenotype is
calculated to be 0.90, and the mutation rate to the recessive allele is 5 × 10−5. If the
population is allowed to come to equilibrium, what allelic frequencies can be predicted?
Solution
Here mutation and selection are working in opposite directions, so an equilibrium is
predicted. Such an equilibrium is described by the formula
In the present question, μ = 5 × 10−5 and s = 1 − W = 1 − 0.9 = 0.1. Hence
Problems
1. What are the forces that can change the frequency of an allele in a population?See answer
2. In a population of mice, there are two alleles of the A locus (A1 and A2). Tests showed that
in this population there are 384 mice of genotype A1/A1, 210 of A1/A2, and 260 of A2/A2.
What are the frequencies of the two alleles in the population?See answer
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3. In a randomly mating laboratory population of Drosophila, 4 percent of the flies have black
bodies (black is the autosomal recessive b) and 96 percent have brown bodies (the normal
color B). If this population is assumed to be in Hardy-Weinberg equilibrium, what are the
allelic frequencies of B and b and the genotypic frequencies of B/B and B/b?
4. In a population, the D → d mutation rate is 4 × 10−6. If p = 0.8 today, what will p be after
50,000 generations?See answer
5. You are studying protein polymorphism in a natural population of a certain species of a
sexually reproducing haploid organism. You isolate many strains from various parts of the
test area and run extracts from each strain on electrophoretic gels. You stain the gels with a
reagent specific for enzyme X and find that in the population there is a total of, say, five
electrophoretic variants of enzyme X. You speculate that these variants represent various
alleles of the structural gene for enzyme X.
a. How could you demonstrate that the speculation is correct, both genetically and
biochemically? (You can make crosses, make diploids, run gels, test enzyme activities, test
amino acid sequences, and so forth.) Outline the steps and conclusions precisely.
b. Name at least one other possible way of generating the different electrophoretic variants,
and say how you would distinguish this possibility from the one described here.
6. A study made in 1958 in the mining town of Ashibetsu in the Hokkaido province of Japan
revealed the frequencies of MN blood type genotypes (for individuals and for married
couples) shown in the following table:
a. Show if the population is in Hardy-Weinberg equilibrium with respect to the MN blood
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Chapter 24
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types.
b. Show whether mating is random with respect to MN blood types. (Problem 6 is from J.
Kuspira and G. W. Walker, Genetics: Questions and Problems. Copyright © 1973 by
McGraw-Hill.)See answer
7. Consider the populations that have the genotypes shown in the following table:
a. Which of the populations are in Hardy-Weinberg equilibrium?
b. What are p and q in each population?
c. In population 10, it is discovered that the A → a mutation rate is 5 × 10−6 and that reverse
mutation is negligible. What must be the fitness of the a/a phenotype?
d. In population 6, the a allele is deleterious; furthermore, the A allele is incompletely
dominant, so A/A is perfectly fit, A/a has a fitness of 0.8, and a/a has a fitness of 0.6. If
there is no mutation, what will p and q be in the next generation? See answer
8. Colorblindness results from a sex-linked recessive allele. One in every 10 males is
colorblind.
a. What proportion of all women are colorblind?
b. By what factor is colorblindness more common in men (or, how many colorblind men
are there for each colorblind woman)?
c. In what proportion of marriages would colorblindness affect half the children of each
sex?
d. In what proportion of marriages would all children be normal?
e. In a population that is not in equilibrium, the frequency of the allele for colorblindness is
0.2 in women and 0.6 in men. After one generation of random mating, what proportion of
the female progeny will be colorblind? What proportion of the male progeny?
f. What will the allelic frequencies be in the male and in the female progeny in part e?
(Problem 8 courtesy of Clayton Person.)
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9. In a wild population of beetles of species X, you notice that there is a 3:1 ratio of shiny to
dull wing covers. Does this ratio prove that shiny is dominant? (Assume that the two states
are caused by the alleles of one gene.) If not, what does it prove? How would you elucidate
the situation?
See answer
10. It seems clear that most new mutations are deleterious. Why?
11. Most mutations are recessive to the wild type. Of those rare mutations that are dominant in
Drosophila, for example, the majority turn out either to be chromosomal aberrations or to
be inseparable from chromosomal aberrations. Explain why the wild type is usually
dominant.See answer
12. Ten percent of the males of a large and randomly mating population are colorblind. A
representative group of 1000 from this population migrates to a South Pacific island,
where there are already 1000 inhabitants and where 30 percent of the males are colorblind.
Assuming that Hardy-Weinberg equilibrium applies throughout (in the two original
populations before emigration and in the mixed population immediately after
immigration), what fraction of males and females can be expected to be colorblind in the
generation immediately after the arrival of the immigrants?
13. Using pedigree diagrams, find the probability of homozygosity by descent of the offspring
of (a) parent–offspring matings; (b) first-cousin matings; (c) aunt–nephew or uncle–niece
matings.
14. In a survey of Native American tribes in Arizona and New Mexico, albinos were
completely absent or very rare in most groups (there is one albino per 20,000 North
American Caucasians). However, in three Native American populations, albino
frequencies are exceptionally high: 1 per 277 Native Americans in Arizona; 1 per 140
Jemez in New Mexico; and 1 per 247 Zuni in New Mexico. All three of these populations
are culturally but not linguistically related. What possible factors might explain the high
incidence of albinos in these three tribes?
See answer
15. In an animal population, 20 percent of the individuals are A/A, 60 percent are A/a, and 20
percent are a/a. What are the allelic frequencies? In this population, mating is always with
like phenotype but is random within phenotype. What genotypic and allelic frequencies
will prevail in the next generation? Such assortative mating is common in animal
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populations. Another type of assortative mating is that which takes place only between
unlike phenotypes: answer the preceding question with this restriction imposed. What will
the end result be after many generations of mating of both types?
16. A Drosophila stock isolated from nature has an average of 36 abdominal bristles. By the
selective breeding of only those flies with more bristles, the mean is raised to 56 in 20
generations. What is the source of this genetic flexibility? The 56-bristle stock is infertile,
so selection is relaxed for several generations and the bristle number drops to about 45.
Why doesn't it drop back to 36? When selection is reapplied, 56 bristles are soon attained,
but this time the stock is not sterile. How can this situation arise?
17. The fitnesses of three genotypes are WA / A = 0.9, WA / a = 1.0, and Wa / a = 0.7.
a. If the population starts at the allelic frequency p = 0.5, what is the value of p in the next
generation?
b. What is the predicted equilibrium allelic frequency? See answer
18. A/A and A/a individuals are equally fertile. If 0.1 percent of the population is a/a, what
selection pressure exists against a/a if the A → a mutation rate is 10−5?See answer
19. Gene B is a deleterious autosomal dominant. The frequency of affected individuals is 4.0
× 10−6. The reproductive capacity of these individuals is about 30 percent that of normal
individuals. Estimate μ, the rate at which b mutates to its deleterious allele B.
20. Of 31 children born of father–daughter matings, 6 died in infancy, 12 were very abnormal
and died in childhood, and 13 were normal. From this information, calculate roughly how
many recessive lethal genes we have, on average, in our human genomes. For example, if
the answer is 1, then a daughter would stand a 50 percent chance of carrying the gene, and
the probability of the union's producing a lethal combination would be 1/2 × 1/4 =
1/8. (So 1 is not the answer.) Consider also the possibility of undetected fatalities in utero
in such matings. How would they affect your result?See answer
21. If we define the total selection cost to a population of deleterious recessive genes as the
loss of fitness per individual affected (s) multiplied by the frequency of affected
individuals (q2), then
a. Suppose that a population is at equilibrium between mutation and selection for a
deleterious recessive gene, where s = 0.5 and μ = 10−5. What is the equilibrium frequency
of the gene? What is the genetic cost?
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Population Genetics
b. Suppose that we start irradiating individual members of the population, so the mutation
rate doubles. What is the new equilibrium frequency of the gene? What is the new genetic
cost?
c. If we do not change the mutation rate but we lower the selection intensity to s = 0.3
instead, what happens to the equilibrium frequency and the genetic cost?
Chapter 24*
1. Selection, mutation, migration, inbreeding, and random genetic drift
2. A1 = 0.57, A2 = 0.43
4. 0.65
6. a. p = [406 + 1/2(744)]/1482 = 0.52, q = [332 + 1/2(744)]/1482 = 0.48. If the population is
in Hardy-Weinberg equilibrium, the genotypes should be distributed as follows:
The population is in equilibrium.
b. If mating is random with respect to blood type, then the following frequency of matings
should occur:
The mating is random with respect to blood type.
7. a. and b.
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c. 0.898
d. p = 0.56, q = 0.44
9. The frequency of a phenotype in a population is a function of the frequency of alleles that
lead to that phenotype in the population. To determine dominance and recessiveness, do
standard Mendelian crosses.
11. Wild-type alleles are usually dominant because most mutations result in lowered or
eliminated function. To be dominant, the heterozygote has approximately the same
phenotype as the dominant homozygote. This will typically be true when the wild-type
allele produces a functional product and the mutant allele does not.
Chromosomal rearrangements are often dominant mutations because they can cause gross
changes in gene regulation or even cause fusions of several gene products. Novel
activities, overproduction of gene products, and so forth, are typical of dominant
mutations.
14. Albinos may have been considered lucky and encouraged to breed at very high levels in
comparison with nonalbinos. They may also have been encouraged to mate with each
other. Alternatively, in the tribes with a very low frequency, albinos may have been
considered very unlucky and destroyed at birth or prevented from marriage.
17. (a). 0.528;
(b). 0.75.
18. 0.01
20. 6.5
Boxes
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 24
Population Genetics
Calculation of Allele Frequency
Box 24-1
If fA / A , fA / a , and fa / a are the proportions of the three genotypes at a locus with two alleles,
then the frequency p(A) of the A allele and the frequency q(a) of the a allele are obtained by
counting alleles. Because each homozygote A/A consists only of A alleles and only half the
alleles of each heterozygote A/a are type A, the total frequency (p) of A alleles in the
population is:
Similarly, the frequency q of a alleles is given by:
So,
If there are multiple alleles, the frequency for each allele is simply the frequency of its
homozygote plus half the sum of the frequencies for all the heterozygotes in which it
appears.
Hardy-Weinberg Equilibrium
Box 24-2
If the frequency of allele A is p in both the sperm and the eggs and the frequency of allele a is
q = 1 − p, then the consequences of random unions of sperm and eggs are shown in the
adjoining Punnett square. The probability that both the sperm and the egg will carry A is p × p
= p2, so p2 will be the frequency of A/A homozygotes in the next generation. In like manner,
the chance of heterozygotes A/a will be (p × q) + (q × p) = 2pq, and the chance of
homozygotes a/a will be q × q = q2. The three genotypes, after a generation of random mating,
will be in the frequencies p2:2pq:q2. As the Punnett square shows, the allelic frequency of A
has not changed and is still p. Therefore, in the second generation, the frequencies of the three
genotypes will again be p2:2pq:q2, and so forth, forever.
39
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 24
Population Genetics
The Hardy-Weinberg equilibrium frequencies that result from random mating. The
frequencies of A and a among both eggs and sperm are p and q ( = 1 − p), respectively. The
total frequencies of the zygote genotypes are p2 for A/A, 2pq for A/a, and q2 for a/a. The
frequency of the allele A in the zygotes is the frequency of A/A plus half the frequency of A/a,
or p2 + pq = p(p + q) = p.
Effect of Mutation on Allele Frequency
Box 24-3
Let μ be the mutation rate from allele A to some other allele a (the probability that a gene
copy A will become a in the DNA replication preceding meiosis). If pt is the frequency of the
A allele in generation t, if qt = 1 − pt is the frequency of the a allele, and if there are no other
causes of gene frequency change (no natural selection, for example), then the change in allelic
frequency in one generation is:
where pt −1 is the frequency in the preceding generation. This tells us that the frequency of A
decreases (and the frequency of a increases) by an amount that is proportional to the mutation
rate μ and to the proportion p of all the genes that are still available to mutate. Thus Δp gets
smaller as the frequency of p itself decreases, because there are fewer and fewer A alleles to
mutate into a alleles. We can make the approximation that, after n generations of mutation,
40
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 24
Population Genetics
where e is the base of the natural logarithms. This relation of gene frequency to number of
generations is shown in the graph for μ = 10−5. After 10,000 generations of continued
mutation of A to a,
If the population started with only A alleles (p0 = 1.0), it would still have only 10 percent a
alleles after 10,000 generations at this rather high mutation rate and would require 60,000
additional generations to reduce p to 0.5. Even if mutation rates were doubled (say, by
environmental mutagens), the rate of evolution would be very slow. For example, radiation
levels of sufficient intensity to double the mutation rate over the reproductive lifetime of an
individual human are at the limit of occupational safety regulations, and a dose of radiation
sufficient to increase mutation rates by an order of magnitude would be lethal; so rapid
genetic change in the species would not be one of the effects of increased radiation. Although
we have many things to fear from environmental radiation pollution, turning into a species of
monsters is not one of them.
The change over generations in the frequency of a gene A due to mutation from A to a at a
constant mutation rate (μ) of 10−5.
Effect of Migration on Allele Frequency
Box 24-4
If pt is the frequency of an allele in the recipient population in generation t and P is the allelic
frequency in a donor population (or the average over several donor populations) and if m is
the proportion of the recipient population that is made up of new migrants from the donor
population, then the gene frequency in the recipient population in the next generation, pt +1, is
the result of mixing 1 − m genes from the recipient with m genes from the donor population.
Thus:
41
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 24
Population Genetics
and
Effect of the Mating of Close Relatives on Homozygosity
Box 24-5
The probability of a homozygous a/a offspring from a brother–sister mating is:
We assume that the chance that both grandparents are A/a is negligible. If p is very small, then
q is nearly 1.0 and the chance of an affected offspring is close to p/4. For p = 1/1000, there
is 1 chance in 4000 of an affected child, compared with the 1-in-a-million chance from a
random mating. In general, for full sibs, the ratio of risks will be:
Effect of Selection on Allele Frequency
Box 24-6
Suppose that a population is mating at random with respect to a given locus with two alleles
and that the population is so large that (for the moment) we can ignore inbreeding. Just after
the eggs have been fertilized, the zygotes will be in Hardy-Weinberg equilibrium:
42
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 24
Population Genetics
and p2 + 2pq + q2 = (p + q)2 = 1.0, where p is the frequency of A.
Further suppose that the three genotypes have probabilities of survival to adulthood
(viabilities) of WA / A :WA / a :Wa / a . For simplicity, let us also assume that all selective
differences are differences in survivorship between the fertilized egg and the adult stage.
Differences in fertility give rise to much more complex mathematical formulations. Then
among the progeny having reached adulthood, the frequencies will be:
These adjusted frequencies do not add up to unity, because the W's are all fractions smaller
than 1. However, we can readjust them so that they do, without changing their relation to each
other, by dividing each frequency by the sum of the frequencies after selection ( ):
So defined, is called the mean fitness of the population because it is, indeed, the mean of
the fitnesses of all individuals in the population. After this adjustment, we have
We can now determine the frequency p′ of the allele A in the next generation by summing up
genes:
Finally, we note that the expression pWA / A + qWA / a is the mean fitness of A alleles because,
from the Hardy-Weinberg frequencies, a proportion p of all A alleles are present in
homozygotes with another A and then they have a fitness of WA / A , whereas a proportion q of
all the A al-leles are present in heterozygotes with a and have a fitness of WA / a . Using A to
denote pWA / A + qWA / a yields the final new gene frequency:
43
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 24
Population Genetics
An alternative way to look at the process of selection is to solve for the change in allelic
frequency in one generation:
But
so
, the mean fitness of the population, is the average of the allelic fitnesses
A
and
a,
Substituting this expression for in the formula for Δp and remembering that q = 1 − p, we
obtain (after some algebraic manipulation)
Natural Selection Leading to Equilibrium of Allele Frequencies
Box 24-7
The average fitness of an allele is defined as the average of the fitnesses of the genotypes that
carry that allele:
Suppose that the fitness of heterozygotes, WA / a , is greater than the fitnesses of both
homozygotes, WA / A and Wa / a . Then, for part of the range of values of p, A <
a,
whereas A <
for
the
rest
of
the
range.
Just
between
those
ranges
is
a
value
of
p
a
^
(denoted by p) for which the mean fitnesses of the two alleles are equal. A little algebraic
manipulation of
gives us the solution for ^p:
44
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 24
Population Genetics
The equilibrium value is a simple ratio of the differences in fitness between the homozygotes
and the heterozygote. For example, suppose the fitnesses are:
The equilibrium will be ^p = 2/3.
45
勇者并非无所畏惧,而是判断出有比恐惧更重要的东西.