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Mendelโs Principles of Heredity
Synopsis
Chapter 2 covers the basic principles of inheritance that can be summarized as Mendelโs Laws of
Segregation (for one gene) and Independent Assortment (for more than one gene).
Key terms
genes and alleles of genes โ A gene determines a trait; and there are different alleles or
forms of a gene. The color gene in peas has two alleles: the yellow allele and the
green allele.
genotype and phenotype โ Genotype is the genetic makeup of an organism (written as
alleles of specific genes), while phenotype is what the organism looks like.
homozygous and heterozygous โ When both alleles of a gene are the same, the
individual is homozygous for that gene (or pure-breeding). If the two alleles are
different, the organism is heterozygous (also called a hybrid).
dominant and recessive โ The dominant allele is the one that controls the phenotype
in the heterozygous genotype; the recessive allele controls the phenotype only in a
homozygote.
monohybrid or dihybrid cross โ a cross between individuals who are both
heterozygotes for one gene (monohybrid) or for two genes (dihybrid)
testcross โ performed to determine whether or not an individual with the dominant
trait is homozygous or heterozygous; an individual with the dominant phenotype
but unknown genotype is crossed with an individual with the recessive phenotype
Key ratios
3:1 โ Ratio of progeny phenotypes in a cross between monohybrids
[Aa ร Aa โ 3 Aโ (dominant phenotype) : 1 aa (recessive phenotype)]
1:2:1 โ Ratio of progeny genotypes in a cross between monohybrids
(Aa ร Aa โ 1 AA : 2 Aa : 1aa)
1:1 โ Ratio of progeny genotypes in a cross between a heterozygote and a recessive homozygote
(Aa ร aa โ 1 Aa : 1aa : 1aa)
1:0 โ All progeny have the same phenotype. Can result from several cases:
[AA ร โ โ โ Aโ (all dominant phenotype)]
[aa ร aa โ aa (all recessive phenotype)]
9:3:3:1 โ Ratio of progeny phenotypes in a dihybrid cross
(Aa Bb ร Aa Bb โ 9 Aโ Bโ : 3 Aโ bb : 3 aa Bโ : 1 aa)
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Problem Solving
The essential component of solving most genetics problems is to DIAGRAM THE CROSS in a
consistent manner. In most cases you will be given information about phenotypes, so the
diagram would be:
Phenotype of one parent ร phenotype of the other parent โ phenotype(s) of progeny
The goal is to assign genotypes to the parents and then use these predicted genotypes to
generate the genotypes, phenotypes, and ratios of progeny. If the predicted progeny match the
observed data you were provided, then your genetic explanation is correct.
The points listed below will be particularly helpful in guiding your problem solving:
โข Remember that there are two alleles of each gene when describing the genotypes
of individuals. But if you are describing gametes, remember that there is only one
allele of each gene per gamete.
โข You will need to determine whether a trait is dominant or recessive. Two main
clues will help you answer this question.
o First, if the parents of a cross are true-breeding for the alternative forms of the
trait, look at the phenotype of the F1 progeny. Their genotype must be
heterozygous, and their phenotype is thus controlled by the dominant allele of
the gene.
o Second, look at the F2 progeny (that is, the progeny of the F1 hybrids). The 3/4
portion of the 3:1 phenotypic ratio indicates the dominant phenotype.
โข You should recognize the need to set up a testcross (to establish the genotype of an
individual showing the dominant phenotype by crossing this individual to a recessive
homozygote).
โข You must keep in mind the basic rules of probability:
o Product rule: If two outcomes must occur together as the result of independent
events, the probability of one outcome AND the other outcome is the product
of the two individual probabilities.
o Sum rule: If there is more than one way in which an outcome can be produced,
the probability of one OR the other occurring is the sum of the two mutually
exclusive individual probabilities.
โข Remember that Punnett squares are not the only means of analyzing a cross;
branched-line diagrams and calculations of probabilities according to the
product and sum rules are more efficient ways of looking at complicated crosses
involving more than one or two genes.
โข You should be able to draw and interpret pedigrees. When the trait is rare, look in
particular for vertical patterns of inheritance characteristic of dominant traits, and
horizontal patterns that typify recessive traits. Check your work by assigning genotypes
to all individuals in the pedigree and verifying that these make sense.
โข The vocabulary problem (the first problem in the set) is a useful gauge of how well you
know the terms most critical for you understanding of the chapter.
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Vocabulary
1.
a. phenotype
4. observable characteristic
b. alleles
3. alternate forms of a gene
c. independent
assortment
6. alleles of one gene separate into gametes randomly
with respect to alleles of other genes
d. gametes
7. reproductive cells containing only one copy of each
gene
e. gene
11. the heritable entity that determines a characteristic
f. segregation
13. the separation of the two alleles of a gene into
different gametes
g. heterozygote
10. an individual with two different alleles of a gene
h. dominant
2. the allele expressed in the phenotype of the
heterozygote
i.
F1
14. offspring of the P generation
j.
testcross
9. the cross of an individual of ambiguous genotype
with a homozygous recessive individual
k. genotype
12. the alleles an individual has
l.
recessive
8. the allele that does not contribute to the phenotype
of the heterozygote
m. dihybrid cross
5. a cross between individuals both heterozygous for
two genes
n. homozygote
1. having two identical alleles of a given gene
Section 2.1
2.
Prior to Mendel, people held two basic misconceptions about inheritance. First
was the common idea of blended inheritance: that the parental traits become mixed in
the offspring and forever changed. Second, many thought that one parent contributes
the most to an offspringโs inherited features. (For example, some people thought they
saw a fully formed child in a human sperm.)
In addition, people who studied inheritance did not approach the problem in
an organized way. They did not always control their crosses. They did not look at
traits with clear-cut alternative phenotypes. They did not start with pure-breeding lines.
They did not count the progeny types in their crosses. For these reasons, they could not
develop the same insights as did Mendel.
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3.
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Several advantages exist to using peas for the study of inheritance:
(1) Peas have a fairly rapid generation time (at least two generations per year if
grown in the field, three or four generations per year if grown in greenhouses.
(2) Peas can either self-fertilize or be artificially crossed by an experimenter.
(3) Peas produce large numbers of offspring (hundreds per parent).
(4) Peas can be maintained as pure-breeding lines, simplifying the ability to perform
subsequent crosses.
(5) Because peas have been maintained as inbred stocks, two easily distinguished
and discrete forms of many traits are known.
(6) Peas are easy and inexpensive to grow.
In contrast, studying genetics in humans has several disadvantages:
(1) The generation time of humans is very long (roughly 20 years).
(2) There is no self-fertilization in humans, and it is not ethical to manipulate
crosses.
(3) Humans produce only a small number of offspring per mating (usually only
one) or per parent (almost always fewer than 20).
(4) Although people who are homozygous for a trait do exist (analogous to purebreeding stocks), homozygosity cannot be maintained because mating with
another individual is needed to produce the next generation.
(5) Because human populations are not inbred, most human traits show a
continuum of phenotypes; only a few traits have two very distinct forms.
(6) People require a lot of expensive care to โgrowโ.
There is nonetheless one major advantage to the study of genetics in humans:
Because many inherited traits result in disease syndromes, and because the worldโs
population now exceeds 6 billion people, a very large number of people with diverse,
variant phenotypes can be recognized. These variations are the raw material of genetic
analysis.
Section 2.2
4.
a. Two phenotypes are seen in the second generation of this cross: normal and
albino. Thus, only one gene is required to control the phenotypes observed.
b. Note that the phenotype of the first generation progeny is normal color, and that in
the second generation, there is a ratio of 3 normal : 1 albino. Both of these
observations show that the allele controlling the normal phenotype (A) is
dominant to the allele controlling the albino phenotype (a).
c. In a test cross, an individual showing the dominant phenotype but that has an
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unknown genotype is mated with an individual that shows the recessive phenotype
and is therefore homozygous for the recessive allele. The male parent is albino, so
the male parentโs genotype is aa. The normally colored offspring must receive an
A allele from the mother, so the genotype of the normal offspring is Aa. The
albino offspring must receive an a allele from the mother, so the genotype of the
albino offspring is aa. Thus, the female parent must be heterozygous Aa.
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5.
Because two different phenotypes result from the mating of two cats of the same
phenotype, the short-haired parent cats must have been heterozygous. The phenotype
expressed in the heterozygotes (the parent cats) is the dominant phenotype. Therefore,
short hair is dominant to long hair.
6.
a. Two affected individuals have an affected child and a normal child. This outcome is
not possible if the affected individuals were homozygous for a recessive allele
conferring piebald spotting, and if the trait is controlled by a single gene. Therefore,
the piebald trait must be the dominant phenotype.
b. If the trait is dominant, the piebald parents could be either homozygous (PP) or
heterozygous (Pp). However, because the two affected individuals have an
unaffected child (pp), they both must be heterozygous (Pp). A diagram of the
cross follows:
piebald ร piebald โ 1 piebald : 1 normal
Pp
Pp
Pp
pp
Note that although the apparent ratio is 1:1, this is not a testcross but is instead a
cross between two monohybrids. The reason for this discrepancy is that only two
progeny were obtained, so this number is insufficient to establish what the true ratio
would be (it should be 3:1) if many progeny resulted from the mating.
7.
You would conduct a testcross between your normal-winged fly (Wโ) and a
short-winged fly that must be homozygous recessive (ww). The possible results are
diagrammed here; the first genotype in each cross is that of the normal-winged fly
whose genotype was originally unknown.
WW ร ww โ all Ww (normal wings)
Ww ร ww โ ยฝ Ww (normal wings) : ยฝ ww (short wings)
8.
First diagram the crosses:
closed ร open โ F1 all open โ F2 145 open : 59 closed
F1 open ร closed โ 81 open : 77 closed
The results of the crosses fit the pattern of inheritance of a single gene, with the
open trait being dominant and the closed trait recessive. The first cross is similar
to those Mendel did with pure-breeding parents, although you were not provided with
the information that the starting plants were true-breeding. The phenotype of the F1
plants is open, indicating that open is dominant. The closed parent must be
homozygous for the recessive allele. Because only one phenotype is seen among
the F1 plants, the open parent must be homozygous for the dominant allele.
Thus, the parental cucumber plants were indeed true-breeding homozygotes.
The result of the self-fertilization of the F1 plants shows a 3:1 ratio of the open :
closed phenotypes among the F2 progeny. The 3:1 ratio in the F2 shows that a single
gene controls the phenotypes and that the F1 plants are all hybrids (that is, they
are heterozygotes).
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The final cross verifies the F1 plants from the first cross are heterozygous hybrids
because this testcross yields a 1:1 ratio of open: closed progeny. In summary, all the
data are consistent with the trait being determined by one gene with two alleles, and
open being the dominant trait.
9.
The dominant trait (short tail) is easier to eliminate from the population by
selective breeding. The reason is you can recognize every animal that has inherited the
short tail allele, because only one such dominant allele is needed to see the phenotype.
If you prevent all the short-tailed animals from mating, then the allele would become
extinct.
On the other hand, the recessive dilute coat color allele can be passed unrecognized
from generation to generation in heterozygous mice (who are carriers). The heterozygous
mice do not express the phenotype, so they cannot be distinguished from homozygous
dominant mice with normal coat color. You could prevent the homozygous recessive
mice with the dilute phenotype from mating, but the allele for the dilute phenotype
would remain among the carriers, which you could not recognize.
10. The problem already states that only one gene is involved in this trait, and that the
dominant allele is dimple (D) while the recessive allele is nondimple (d).
a. Diagram the cross described in this part of the problem:
nondimple โ ร dimpled โ โ proportion of F1 with dimple?
Note that the dimpled woman in this cross had a dd (nondimpled) mother, so the
dimpled woman MUST be heterozygous. We can thus rediagram this cross with
genotypes:
dd (nondimple) โ ร Dd (dimple) โ โ ยฝ Dd (dimpled) : ยฝ dd (nondimpled)
One half of the children produced by this couple would be dimpled.
b. Diagram the cross:
dimple (D?) โ ร nondimpled (dd) โ โ nondimple F1 (dd)
Because they have a nondimple child (dd), the husband must have a d allele to
contribute to the offspring. The husband is thus of genotype Dd.
c. Diagram the cross:
dimple (D?) โ ร nondimpled (dd) โ โ eight F1, all dimpled (Dโ)
The D allele in the children must come from their father. The father could be either
DD or Dd, but it is most probable that the fatherโs genotype is DD. We cannot
rule out completely that the father is a Dd heterozygote. However, if this was the
case, the probability that all 8 children would inherit the D allele from a Dd parent is
only (1/2)8 = 1/256.
11. a. The only unambiguous cross is:
homozygous recessive ร homozygous recessive โ all homozygous recessive
The only cross that fits this criteria is: dry ร dry โ all dry. Therefore, dry is the
recessive phenotype (ss) and sticky is the dominant phenotype (Sโ).
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b. A 1:1 ratio comes from a testcross of heterozygous sticky (Ss) ร dry (ss). However,
the sticky x dry matings here include both the Ss ร ss AND the homozygous
sticky (SS) ร dry (ss).
A 3:1 ratio comes from crosses between two heterozygotes, Ss ร Ss, but the
sticky individuals are not only Ss heterozygotes but also SS homozygotes. Thus the
sticky x sticky matings in this human population are a mix of matings between two
heterozygotes (Ss ร Ss), between two homozygotes (SS ร SS) and between a
homozygote and heterozygote (SS ร Ss). The 3:1 ratio of the heterozygote cross
is therefore obscured by being combined with results of the two other
crosses.
12. Diagram the cross:
black ร red โ 1 black : 1 red
No, you cannot tell how coat color is inherited from the results of this one
mating. In effect, this was a test cross โ a cross between animals of different
phenotypes resulting in offspring of two phenotypes. This does not indicate whether
red or black is the dominant phenotype. To determine which phenotype is dominant,
remember that an animal with a recessive phenotype must be homozygous. Thus, if
you mate several red horses to each other and also mate several black horses to
each other, the crosses that always yield only offspring with the parental
phenotype must have been between homozygous recessives. For example, if all
the black ร black matings result in only black offspring, black is recessive. Some of the
red ร red crosses (that is, crosses between heterozygotes) would then result in both red
and black offspring in a ratio of 3:1. To establish this point, you might have to do
several red ร red crosses, because some of these crosses could be between red horses
homozygous for the dominant allele. You could of course ensure that you were
sampling heterozygotes by using the progeny of black ร red crosses (such as that
described in the problem) for subsequent black ร black or red ร red crosses.
13. a. 1/6 because a die has 6 different sides.
b. There are three possible even numbers (2, 4, and 6). The probability of obtaining
any one of these is 1/6. Because the 3 events are mutually exclusive, use the sum
rule: 1/6 + 1/6 + 1/6 = 3/6 = 1/2.
c. You must roll either a 3 or a 6, so 1/6 + 1/6 = 2/6 = 1/3.
d. Each die is independent of the other, thus the product rule is used: 1/6 ร 1/6 =
1/36.
e. The probability of getting an even number on one die is 3/6 = 1/2 (see part [b]).
This is also the probability of getting an odd number on the second die. This result
could happen either of 2 ways โ you could get the odd number first and the even
number second, or vice versa. Thus the probability of both occurring is 1/2 ร 1/2 ร
2 = 1/2.
f. The probability of any specific number on a die = 1/6. The probability of the same
number on the other die =1/6. The probability of both occurring at same time is
1/6 x 1/6 = 1/36. The same probability is true for the other 5 possible numbers on
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the dice. Thus the probability of any of these mutually exclusive situations occurring
is 1/36 + 1/36 + 1/36 + 1/36 + 1/36 + 1/36 = 6/36 = 1/6.
g. The probability of getting two numbers both over four is the probability of getting a
5 or 6 on one die (1/6 + 1/6 = 1/3) and 5 or 6 on the other die (1/3). The results
for the two dice are independent events, so 1/3 ร 1/3 = 1/9.
14. The probability of drawing a face card = 0.231 (= 12 face cards / 52 cards). The
probability of drawing a red card = 0.5 (= 26 red cards / 52 cards). The probability
of drawing a red face card = probability of a red card ร probability of a face card
= 0.231 ร 0.5 = 0.116.
15. a. The Aa bb CC DD woman can produce 2 genetically different eggs that vary in their
allele of the first gene (A or a). She is homozygous for the other 3 genes and can
only make eggs with the b C D alleles for these genes. Thus, using the product rule
(because the inheritance of each gene is independent), she can make 2 ร 1 ร 1 ร 1
= 2 different types of gametes: (A b C D and a b C D).
b. Using the same logic, an AA Bb Cc dd woman can produce 1 ร 2 ร 2 ร 1 = 4
different types of gametes: A (B or b) (C or c) d.
c. A woman of genotype Aa Bb cc Dd can make 2 ร 2 ร 1 ร 2 = 8 different types of
gametes: (A or a) (B or b) c (D or d).
d. A woman who is a quadruple heterozygote can make 2 ร 2 ร 2 ร 2 = 16 different
types of gametes: (A or a) (B or b) (C or c) (D or d). This problem (like those in
parts (a-c) above) can also be visualized with a branched-line diagram.
C
B
c
A
C
b
c
D
d
D
d
D
d
D
d
ABCD
ABCd
ABcD
ABcd
AbCD
AbCd
AbcD
Abcd
C
B
c
a
C
b
c
D
d
D
d
D
d
D
d
aBCD
aBCd
aBcD
aBcd
abCD
abCd
abcD
abcd
16. a. The probability of any phenotype in this cross depends only on the gamete from
the heterozygous parent. The probability that a child will resemble the quadruply
heterozygous parent is thus 1/2A ร 1/2B ร 1/2C ร 1/2D = 1/16. The probability
that a child will resemble the quadruply homozygous recessive parent is 1/2a ร
1/2b ร 1/2c ร 1/2d = 1/16. The probability that a child will resemble either
parent is then 1/16 + 1/16 = 1/8. This cross will produce 2 different phenotypes
for each gene or 2 ร 2 ร 2ร2 = 16 potential phenotypes.
b. The probability of a child resembling the recessive parent is 0; the probability of a
child resembling the dominant parent is 1 ร 1 ร 1 ร 1 = 1. The probability that a
child will resemble one of the two parents is 0 + 1 = 1. Only 1 phenotype is
possible in the progeny (dominant for all 4 genes), as (1)4 = 1.
c. The probability that a child would show the dominant phenotype for any one gene
is 3/4 in this sort of cross (remember the 3/4 : 1/4 monohybrid ratio of
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phenotypes), so the probability of resembling the parent for all four genes is
(3/4)4 = 81/256. There are 2 phenotypes possible for each gene, so (2)4 = 16
different kinds of progeny.
d. All progeny will resemble their parents because all of the alleles from both
parents are identical, so the probability = 1. There is only 1 phenotype possible for
each gene in this cross; because (1)4 = 1, the child can have only one possible
phenotype when considering all four genes.
17. a. The combination of alleles in the egg and sperm allows only one genotype for the
zygote: aa Bb Cc DD Ee.
b. Because the inheritance of each gene is independent, you can use the product rule
to determine the number of different types of gametes that are possible: 1 x 2 x 2 x
1 x 2 = 8 types of gametes. To figure out the types of gametes, consider the
possibilities for each gene separately and then the possible combinations of genes in
a consistent order. For each gene the possibilities are: a, (B : b), (C : c), D, and (E : e).
The possibilities can be determined using the product rule. Thus for the first 2
genes [a] ร [B : b] gives [a B : a b] ร [C : c] gives [a B C : a B c : a b C : a b c] ร [D]
gives [a B C D : a B c D : a b C D : a b c D] ร [E : e] gives [a B C D E : a B C D e
: a B c D E : a B c D e : a b C D E : a b C D e : a b c D E : a b c D e].
This problem can also be visualized with a branched-line diagram:
C
D
c
D
C
D
c
D
B
a
b
E
aBCDE
e
aBCDe
E
aBcDE
e
aBcDe
E
abCDE
e
abCDe
E
abcDE
e
abcDe
18. The first two parts of this problem involve the probability of occurrence of two
independent traits: the sex of a child and galactosemia. The parents are heterozygous
for galactosemia, so there is a 1/4 chance that a child will be affected (that is,
homozygous recessive). The probability that a child is a girl is 1/2. The probability of an
affected girl is therefore 1/2 ร 1/4 = 1/8.
a. Fraternal (non-identical) twins result from two independent fertilization events and
therefore the probability that both will be girls with galactosemia is the product of
their individual probabilities (see above); 1/8 ร 1/8 = 1/64.
b. For identical twins, one fertilization event gave rise to two individuals. The
probability that both are girls with galactosemia is 1/8.
For parts c-g, remember that each child is an independent genetic event. The sex of the
children is not at issue in these parts of the problem.
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c. Both parents are carriers (heterozygous), so the probability of having an unaffected
child is 3/4. The probability of 4 unaffected children is 3/4 x 3/4 x 3/4 x 3/4 =
81/256.
d. The probability that at least one child is affected is all outcomes except the one
mentioned in part (c). Thus, the probability is 1 – 81/256 = 175/256. Note that this
general strategy for solving problems, where you first calculate the probability of all
events except the one of interest, and then subtract that number from 1, is often
useful for problems where direct calculations of the probability of interest appear to
be very difficult.
e. The probability of an affected child is 1/4 while the probability of an unaffected
child is 3/4. Therefore 1/4 ร1/4 ร 3/4 ร 3/4 = 9/256.
f. The probability of 2 affected and 1 unaffected in any one particular birth order is
1/4 ร 1/4 ร 3/4 = 3/64. There are 3 mutually exclusive birth orders that could
produce 2 affecteds and 1 unaffected โ unaffected child first born, unaffected child
second born, and unaffected child third born. Thus, there is a 3/64 + 3/64 + 3/64
= 9/64 chance that 2 out of 3 children will be affected.
g. The phenotype of any particular child is independent of all others, so the probability
of an affected child is 1/4.
19. Diagram the cross, where P is the normal pigmentation allele and p is the albino allele:
normal (P?) ร normal (P?) โ albino (pp)
An albino must be homozygous recessive pp. The parents are normal in pigmentation
and therefore could be PP or Pp. Because they have an albino child, both parents must
be carriers (Pp). The probability that their next child will have the pp genotype is
1/4.
20. Diagram the cross:
yellow round ร yellow round โ 156 yellow round : 54 yellow wrinkled
The monohybrid ratio for seed shape is 156 round : 54 wrinkled = 3 round : 1 wrinkled.
The parents must therefore have been heterozygous (Rr) for the pea shape gene. All the
offspring are yellow and therefore have the Yy or YY genotype. The parent plants
were Yโ Rr ร YY Rr (that is, you know at least one of the parents must have been
YY).
21. Diagram the cross:
smooth black โ ร rough white โ โ F1 rough black
โ F2 8 smooth white : 25 smooth black : 23 rough white : 69 rough black
a. Since only one phenotype was seen in the first generation of the cross, we can
assume that the parents were true breeding, and that the F1 generation consists of
heterozygous animals. The phenotype of the F1 progeny indicates that rough
and black are the dominant phenotypes. Four phenotypes are seen in the F2
generation so there are two genes controlling the phenotypes in this cross.
Therefore, R = rough, r = smooth; B = black, b = white. In the F2 generation,
consider each gene separately. For the coat texture, there were 8 + 25 = 33 smooth
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: 23 + 69 = 92 round, or a ratio of ~1 smooth : ~3 round. For the coat color, there
were 8 + 23 = 31 white : 25 + 69 = 94 black, or about ~1 white : ~3 black, so the
F2 progeny support the conclusion that the F1 animals were heterozygous for both
genes.
b. An F1 male is heterozygous for both genes, or Rr Bb. The smooth white female
must be homozygous recessive; that is, rr bb. Thus, Rr Bb ร rr bb โ 1/2 Rr
(rough) : 1/2 rr (smooth) and 1/2 Bb (black) : 1/2 bb (white). The inheritance of
these genes is independent, so apply the product rule to find the expected
phenotypic ratios among the progeny, or 1/4 rough black : 1/4 rough white : 1/4
smooth black : 1/4 smooth white.
22. Diagram the cross:
YY rr ร yy RR โ all Yy Rr โ 9/16 Yโ Rโ (yellow round) : 3/16 Yโ rr
(yellow wrinkled) : 3/16 yy Rโ (green round) : 1/16 yy rr (green wrinkled).
Each F2 pea results from a separate fertilization event. The probability of 7 yellow
round F2 peas is (9/16)7 = 4,782,969/268,435,456 = 0.018.
23. a. First diagram the cross, and then figure out the monohybrid ratios for each gene:
Aa Tt ร Aa Tt โ 3/4 Aโ (achoo) : 1/4 aa (non-achoo) and 3/4 Tโ
(trembling) : 1/4 tt (non-trembling).
The probability that a child will be Aโ (and have achoo syndrome) is independent
of the probability that it will lack a trembling chin, so the probability of a child with
achoo syndrome but without trembling chin is 3/4 Aโ ร 1/4 tt = 3/16.
b. The probability that a child would have neither dominant trait is 1/4 aa ร 1/4 tt =
1/16.
24. The F1 must be heterozygous for all the genes because the parents were pure-breeding
(homozygous). The appearance of the F1 establishes that the dominant phenotypes for
the four traits are tall, purple flowers, axial flowers and green pods.
a. From a heterozygous F1 ร F1, both dominant and recessive phenotypes can be seen
ย
for each gene. Thus, you expect 2 ร 2 ร 2 ร 2 = 16 different phenotypes when
considering the four traits together. The possibilities can be determined using the
product rule with the pairs of phenotypes for each gene, because the traits are
inherited independently. Thus: [tall : dwarf] ร [green : yellow] gives [tall green : tall
yellow : dwarf green : dwarf yellow] ร [purple : white] gives [tall green purple : tall
yellow purple : dwarf green purple : dwarf yellow purple : tall green white : tall
yellow white : dwarf green white : dwarf yellow white] ร [terminal : axial] which
gives tall green purple terminal : tall yellow purple terminal : dwarf green
purple terminal : dwarf yellow purple terminal : tall green white terminal : tall
yellow white terminal : dwarf green white terminal : dwarf yellow white
terminal : tall green purple axial : tall yellow purple axial : dwarf green purple
axial : dwarf yellow purple axial : tall green white axial : tall yellow white
axial : dwarf green white axial : dwarf yellow white axial. The possibilities can
also be determined using the branch method shown on the next page, which might
in this complicated problem be easier to track
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purple
green
white
tall
purple
yellow
white
purple
green
white
dwarf
purple
yellow
white
axial
terminal
tall, green, purple, axial
tall, green, purple, terminal
axial
tall, green, white, axial
terminal
tall, green, white, purple
axial
tall, yellow, purple, axial
terminal
axial
terminal
axial
terminal
tall, yellow, purple, terminal
tall, yellow, white, axial
tall, yellow, white, terminal
dwarf, green, purple, axial
dwarf, green, purple, terminal
axial
dwarf, green, white, axial
terminal
dwarf, green, white, purple
axial
dwarf, yellow, purple, axial
terminal
axial
terminal
ย
dwarf, yellow, purple, terminal
dwarf, yellow, white, axial
dwarf, yellow, white, terminal
b. Designate the alleles: T = tall, t = dwarf; G = green; g = yellow; P = purple, p =
white; A = axial, a = terminal. The cross Tt Gg Pp Aa (an F1 plant) ร tt gg pp AA
(the dwarf parent) will produce 2 phenotypes for the tall, green and purple genes,
but only 1 phenotype (axial) for the fourth gene or 2 ร 2 ร 2 ร 1 = 8 different
phenotypes. The first 3 genes will give a 1/2 dominant : 1/2 recessive ratio of the
phenotypes (for example 1/2 T : 1/2 t) as this is in effect a test cross for each gene.
Thus, the proportion of each phenotype in the progeny will be 1/2 ร 1/2 ร
1/2 ร 1 = 1/8.
Using either of the methods described in part (a), the progeny will be 1/8 tall
green purple axial : 1/8 tall yellow purple axial : 1/8 dwarf green purple axial
: 1/8 dwarf yellow purple axial : 1/8 tall green white axial : 1/8 tall yellow
white axial : 1/8 dwarf green white axial : 1/8 dwarf yellow white axial.
25. For each separate cross, determine the number of genes involved. Remember that 4
phenotypic classes in the progeny means that 2 genes control the phenotypes. Next,
determine the phenotypic ratio for each gene separately. A 3:1 monohybrid ratio tells
you which phenotype is dominant and that both parents were heterozygous for the trait;
in contrast, a 1:1 ratio results from a testcross where the dominant parent was
heterozygous.
a. There are 2 genes in this cross (4 phenotypes). One gene controls purple : white
with a monohybrid ratio of 94 + 28 = 122 purple : 32 + 11 = 43 white or ~3 purple
: ~1 white. The second gene controls spiny : smooth with a monohybrid ratio of 94
+ 32 =126 spiny : 28 + 11 = 39 smooth or ~3 spiny : ~1 smooth. Thus, designate
the alleles P = purple, p = white; S = spiny, s = smooth. This is a
straightforward dihybrid cross: Pp Ss ร Pp Ss โ 9 Pโ Sโ : 3 Pโ ss : 3 pp Sโ : 1
pp ss.
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b. The 1 spiny : 1 smooth ratio indicates a test cross for the pod shape gene. Because
all progeny were purple, at least one parent plant must have been homozygous for
the P allele of the flower color gene. The cross was either PP Ss ร Pโ ss or Pโ
Ss ร PP ss.
c. This is similar to part (b), but here all the progeny were spiny so at least one parent
must have been homozygous for the S allele. The 1 purple : 1 white test cross ratio
indicates that the parents were either Pp Sโ ร pp SS or Pp SS ร pp Sโ.
d. Looking at each trait individually, there are 89 + 31 = 120 purple : 92 + 27 = 119
white. A 1 purple : 1 white monohybrid ratio denotes a test cross. For the other
gene, there are 89 + 92 = 181 spiny : 31 + 27 =58 smooth, or a 3 spiny : 1 smooth
ratio indicating that the parents were both heterozygous for the S gene. The
genotypes of the parents were pp Ss ร Pp Ss.
e. There is a 3 purple : 1 white ratio among the progeny, so the parents were both
heterozygous for the P gene. All progeny have smooth pods so the parents were
both homozygous recessive ss. The genotypes of the parents are Pp ss ร Pp ss.
f. There is a 3 spiny : 1 smooth ratio, indicative of a cross between heterozygotes (Ss
ร Ss). All progeny were white so the parents must have been homozygous recessive
pp. The genotypes of the parents are pp Ss ร pp Ss.
26. Three characters (genes) are analyzed in this cross. While we can usually tell which
alleles are dominant from the phenotype of the heterozygote, we are not told the
phenotype of the heterozygote (that is, the original pea plant that was selfed). Instead,
use the monohybrid phenotypic ratios to determine which allele is dominant and which
is recessive for each gene. Consider height first. There are 272 + 92 + 88 + 35 = 487
tall plants and 93 + 31 + 29 + 11 = 164 dwarf plants. This is a ratio of ~3 tall : ~1
dwarf, indicating that tall is dominant. Next consider pod shape, where there are 272
+ 92 + 93 + 31 = 488 inflated pods and 88 + 35 + 29 + 11 = 163 flat pods, or
approximately 3 inflated : 1 flat, so inflated is dominant. Finally, consider flower color.
There were 272 + 88 + 93 + 29 + 11 = 493 purple flowers and 92 + 35 + 31 + 11 =
169 white flowers, or ~3 purple : ~1 white. Thus, purple is dominant.
27. Diagram each of these crosses, remembering that you were told that tiny wings = t,
normal wings = T, narrow eye = n, and oval (normal) eye = N. You thus know that one
gene determines the wing trait and one gene determines the eye trait, and you further
know the dominance relationship between the alleles of each gene.
In cross 1, all of the parents and offspring show the tiny wing phenotype so there is
no variability in the gene controlling this trait, and all flies in this cross are tt. Note that
the eye phenotypes in the offspring are seen in a ratio of 3 oval : 1 narrow. This
phenotypic monohybrid ratio means that both parents are heterozygous for the gene
(Nn). Thus the genotypes for the parents in cross 1 are: tt Nn โ ร tt Nn โ.
In cross 2 consider the wing trait first. The female parent is tiny (tt) so this is a test
cross for the wings. The offspring show both tiny and normal in a ratio of 82 : 85 or a
ratio of 1 tiny : 1 normal. Therefore the normal male parent must be heterozygous for
this gene (Tt). For eyes the narrow parent is homozygous recessive (nn) so again this is a
test cross for this gene. Again both eye phenotypes are seen in the offspring in a ratio of
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1 oval : 1 narrow, so the oval female parent is a Nn heterozygote. Thus the genotypes
for the parents in cross 2 are: Tt nn โ ร tt Nn โ.
Consider the wing phenotype in the offspring of cross 3. Both wing phenotypes are
seen in a ratio of 64 normal flies : 21 tiny or a 3 normal : 1 tiny. Thus both parents are
Tt heterozygotes. The male parent is narrow (nn), so cross 3 is a test cross for eyes. Both
phenotypes are seen in the offspring in a 1 normal : 1 narrow ratio, so the female parent
is heterozygous for this gene. The genotypes of the parents in cross 3 are: Tt nn โ
ร Tt Nn โ.
When examining cross 4 you notice a monohybrid phenotypic ratio of 3 normal : 1
tiny for the wings in the offspring. Thus both parents are heterozygous for this gene
(Tt). Because the male parent has narrow eyes (nn), this cross is a test cross for eyes. All
of the progeny have oval eyes, so the female parent must be homozygous dominant for
this trait. Thus the genotypes of the parents in cross 4 are: Tt nn โ ร Tt NN โ.
28. a. Analyze each gene separately: Tt ร Tt will give 3/4 Tโ (normal wing) offspring.
The cross nn x Nn will give 1/2 Nโ (normal eye) offspring. To calculate the
probability of the normal offspring apply the product rule to the normal portions of
the monohybrid ratios by multiplying these two fractions: 3/4 Tโ ร 1/2 Nโ = 3/8
Tโ Nโ. Thus 3/8 of the offspring of this cross will have normal wings and oval
eyes.
b. Diagram the cross:
Tt nn โ ร Tt Nn โ โ ?
Find the phenotypic monohybrid ratio separately for each gene in the offspring.
Then multiply these monohybrid ratios to find the phenotypic dihybrid ratio. A
cross of Tt ร Tt โ 3/4 Tโ (normal wings) : 1/4 tt (tiny wings). For the eyes the
cross is nn ร Nn โ 1/2 Nโ (oval) : 1/2 nn (narrow). Applying the product rule
gives 3/8 Tโ Nโ (normal oval) : 3/8 Tโ nn (normal narrow) : 1/8 tt Nโ (tiny oval) :
1/8 tt nn (tiny narrow). When you multiply each fraction by 200 progeny you will
see 75 normal oval : 75 normal narrow : 25 tiny oval : 25 tiny narrow.
29. a. The protein specified by the pea color gene is an enzyme called Sgr, which is
required for the breakdown of the green pigment chlorophyll. (See Fig. 2.20b on p.
29.)
b. The y allele could be a null allele because it does not specify the production of
any of the Sgr enzyme. ย
c. The Y allele is dominant because in the heterozygote, the single Y allele will lead
to the production of some Sgr enzyme, even if the y allele cannot specify any Sgr.
The amount of the Sgr enzyme made in heterozygotes is sufficient for yellow
color.
d. In yy peas, the green chlorophyll cannot be broken down, so this pigment stays
in the peas, which remain green in color. ย
e. If the amount of Sgr protein is proportional to the number of functional copies of
the gene, then YY homozygotes should have twice the amount of Sgr protein as do
Yy heterozygotes. Yet both YY and Yy peas are yellow. These observations suggest
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that half the normal amount of Sgr enzyme is sufficient for the pea to break
down enough chlorophyll that the pea will still be yellow. ย
f. Just as was seen in part (e), for many genes (including that for pea color), half the
amount of the protein specified by the gene is sufficient for a normal phenotype.
Thus, in most cases, even if the gene is essential, heterozygotes for null alleles will
survive. The advantage of having two copies of essential genes is then that
even if one normal allele becomes mutated (changed) so that it becomes a
null allele, the organism can survive because half the normal amount of gene
product is usually sufficient for survival.
g. Yes, a single pea pod could contain peas with different phenotypes because a
pod is an ovary that contains several ovules (eggs), and each pea represents a single
fertilization event involving one egg and one sperm (from one pollen grain). If the
female plant was Yy, or yy, then it is possible that some peas in the same pod would
be yellow and others green. For example, fertilization of a y egg with Y pollen
would yield a yellow pea, but if the pollen grain was y, the pea would be green.
However, a pea pod could not contain peas with different phenotypes if the female
plant was YY, because all the peas produced by this plant would be yellow.
h. Yes, it is possible that a pea pod could be different in color from a pea
growing within it. One reason is that, as just seen in part (g), a single pod can
contain green and yellow peas. But a more fundamental reason is that one gene
controls the phenotype of pea color, while a different gene controls the separate
phenotype of pod color.
30. If the alleles of the pea color and pea shape genes inherited from a parent in the P
generation always stayed together and never separated, then the gametes produced by
the doubly heterozygous F1 individuals in Fig. 2.15 on p. 25 would be either Y R or y r.
(Note that only two possibilities would exist, and these would be in equal frequencies.)
On a Punnett square (male gametes shaded in blue, female gametes in red):
YR
yr
ยฝ
ยฝ
YR
YY RR
Yy Rr
yr
Yy Rr
yy rr
ยฝ
ยฝ
ยผ
ยผ
ยผ
ยผ
Thus the genotypic ratios of the F2 progeny would be ยผ YY RR, ยฝ Yy Rr, and
ยผ yy rr. The phenotypic ratios among the F2 progeny would be ยพ yellow round
and ยผ green wrinkled. These results make sense because if the alleles of the two
genes were always inherited as a unit, you would expect the same ratios as in a
monohybrid cross.
31. Similar to what you saw in Fig. 2.20 on p. 29, the most likely biochemical
explanation is that the dominant allele L specifies functional G3ฮฒH enzyme,
while the recessive allele l is incapable of specifying any functional enzyme (in
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nomenclature you will see in later chapters, l is a null allele). The functional enzyme can
synthesize the growth hormone gibberellin, so plants/201/MH01954/har25316_disk1of1/0073525316/har25316_pagefiles
with the L allele are tall. Even
har25316_ch02_014-044.indd Page 42 14/05/14 9:26 PM user-f-w-198
half the normal amount of this enzyme is sufficient for the tall phenotype,
explaining why Ll heterozygotes are tall.
32. Note: Your copy of the text might be missing the key figure referred to in this Problem.
42
Chapter
2 Mendelโs
Principles
of Heredity
That
figure
follows
here:
bHLH
OH
HO
O
HO
DFR
OH
OH O
Colorless
a.
OH
OH
O
ANS
HO
O
OH
OH
OH OH
Colorless
OH
3GT
OH
Colorless
O
+
HO
O-Glc
OH
Anthocyanin
As in Problem 31 above, the dominant allele P most likely specifies a functional
32. The gene that likely controlled flower color (purple or
a. In what ways is the figure wrong or misleading, in
product
this genetic
case, nomenclature,
the protein bHLH),
recessive
p allele
cannot
white) in while
Mendelโs the
pea plants
has also been
identified.
terms
of either(in
standard
e flower
encodesisa purple
protein called
thespecify
expectations
for functional
a self-fertilization
of a hy- The Th
any
protein.
fact
thatcolor
thegene
hybrid
(as bHLH
shown on
cells require to make three different enzymes
brid, or the actual ways in which genes might
Fig. 2.8
p. 19)
indicates
thethat
normal
of active
protein is
(DFR,
ANS, amount
and 3GT) that
function bHLH
in the pathway
determine
theon
specific
phenotypes
seen?that
List half
as
shown above, leading to synthesis of the purple pigment
many
as you can.
(Assume
thatcolor.
the first column
sufficient
for
purple
anthocyanin.
[forming the โGโ in โGoogleโ] is the P generation;
a.be
What
is the most likelyby
explanation
the difference
theYes,
secondflower
column color
[the first
โoโ] ispotentially
the F1
could
controlled
genesforspecifying
the
generation, and the third column [โogleโ] is the F2
between the dominant allele (P) and the recessive alenzymes
DFR,
ANS,
or
3GT
in
addition
to
the
gene
specifying
the
bHLH
generation.)
lele (p) of the gene responsible for these flower colors?
Alleles
functional
enzymes
would
yieldpathway
purple
color,
b. Doprotein.
you think that
a singlespecifying
pea pod could
contain
b. Given the
biochemical
shown
above,while
could a those
peas
with
diff
erent
phenotypes?
Explain.
diff
erent
gene
have
been
the
one
governing
Mendelโs
that could not produce functional enzymesflower
would
cause white color. It is likely that
c. Do you think that a pea pod could be of one color
colors?
the
alleles
would
be dominant.
(say,
green)
whilefor
peaspurple
within the
pod could
be a
Section 2.3
different color (say, yellow)? Explain.
b.
30. What would have been the outcome (the genotypic and
phenotypic
Section
2.3ratios) in the F of Mendelโs dihybrid cross
shown in Fig. 2.15 on p. 25 if the alleles of the pea color
33. For each of the following human pedigrees, indicate
whether the inheritance pattern is recessive or dominant.
What feature(s) of the pedigree did you use to determine
the mode of inheritance? Give the genotypes of affected
gene (Y,y) and the pea shape gene (R,r) did not assort
Recessive
– two
unaffected
individuals
have anandaffected
child
(aa).
Therefore
the
individuals
of individuals
who carry
the disease
allele
independently
and instead
the alleles
inherited from
a
are not affected.
parentparents
always stayed
together as
unit? consanguineous but
involved
ina the
marriage
must both be carriers (Aa).
2
33. a.
31. Recall that Mendel obtained pure-breeding plants with
b. Dominant – the trait is seen in each generation(a)and
every affected person (Aโ) has
I
either long or short stems and that hybrids had long
parent.
Note
that an
III-3
even though both his parents
stemsan
(Fig.affected
2.8). Monohybrid
crosses
produced
F2 is unaffected (aa)
II
generation
with
a 3:1 ratiothis
of long
stems tonot
shortbe
stems,
are
affected;
would
possible
for
a
recessive
trait. The term โcarrierโ is
indicating that this difference in stem length is govIII
not
applicable,
because
everyone
with
a
single
A
allele
shows
the trait.
erned by a single gene. The gene that likely controlled
IV
this trait in Mendelโs plants has been discovered, and it
– two
unaffected,
carrier
parents (Aa) have an affected child (aa), as in
specifiRecessive
es an enzyme called
G3!H,
which catalyzes
the
V
reaction
shown
part
(a). below. The product of the reaction,
gibberellin, is a growth hormone that makes plants
(b) I
grow tall. What is the most likely hypothesis to explain
Cutis
laxa must
be a recessive
the diff
erence between
the dominant
allele (L) andtrait
the because affected child II-4 has normal parents.
recessive
allele (l)? II-4 is affected she must have receivedII a disease allele (CL) from both
Because
c.
34. a.
III
parents. The mother (I-3) and the father (I-4) are
both heterozygous (CL+ CL).
The
trait is thus recessive.
H
H
O
O
(c) I
G3ฮฒH
b. You are told
that this trait is rare, so unrelated people
in the pedigree, like I-2, are
II
CO
CO2H
HO
CO2H
H
almost certainly homozygous
normal (CL+ CL+). Diagram the cross that gives rise
III
Precursor
Gibberellin
to II-2: CL CL (I-1) ร CL+ CL+ (I-2) โ CL+ CL. Thus the probability that IIH
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2 is a carrier is very close to 100%. (In Chapter 21 you will find the definition of a
term called the allele frequency; if the value of the allele frequency in the population
under study is known, you can calculate the very low likelihood that II-2 is a
carrier.)
c. As described in part (a) both parents in this cross are carriers: CL+ CL ร CL+ CL.
II-3 is not affected so he cannot be the CL CL genotype. Therefore there is a 1/3
probability that he is the CL+ CL+ genotype and a 2/3 probability that he is a
carrier (CL + CL).
d. As shown in part (b), II-2 must be a carrier (CL+ CL). In order to have an affected
child II-3 must also be a carrier. The probability of this is 2/3 as shown in part (c).
The probability of two heterozygous parents having an affected child is 1/4. Apply
the product rule to these probabilities: 1 probability that II-2 is CL+ CL ร 2/3
probability that II-3 is CL+ CL ร 1/4 probability of an affected child from a mating
of two carriers = 2/12 = 1/6.
35. Diagram the cross! In humans this is usually done as a pedigree. Remember that the
affected siblings must be CF CF.
CF + CF
CF + CF
CF + CF CF + CF
I
II
III
sibling
CF CF
sibling
CF CF
?
a. The probability that II-2 is a carrier is 2/3. Both families have an affected
sibling, so both sets of parents (that is, all the people in generation I) must have
been carriers. Thus, the expected genotypic ratio in the children is 1/4 affected :
1/2 carrier : 1/4 homozygous normal. II-2 is NOT affected, so she cannot be CF
CF. Of the remaining possible genotypes, 2 are heterozygous. There is therefore a
2/3 chance that she is a carrier.
b. The probability that II-2 ร II-3 will have an affected child is 2/3 (the
probability that the mother is a carrier as seen in part [a]) ร 2/3 (the probability the
father is a carrier using the same reasoning) ร 1/4 (the probability that two carriers
can produce an affected child) = 1/9.
c. The probability that both parents are carriers and that their child will be a carrier is
2/3 ร 2/3 ร 1/2 = 2/9 (using the same reasoning as in part [b], except asking that
the child be a carrier instead of affected). However, it is also possible for CF+ CF+
ร CF+ CF parents to have children that are carriers. Remember that there are two
possible ways for this particular mating to occur: homozygous father ร
heterozygous mother or vice versa. Thus the probability of this sort of mating is 2 ร
1/3 (the probability that a particular parent is CF+ CF+) ร 2/3 (the probability that
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the other parent is CF+ CF ร 1/2 (the probability such a mating could produce a
carrier child) = 2/9. The probability that a child could be carrier from either of
these two scenarios (where both parents are carriers or where only one parent is a
carrier) is the sum of these mutually exclusive events, or 2/9 + 2/9 = 4/9.
36. a. Because the disease is rare the affected father is most likely to be heterozygous (HD
HD+). There is a 1/2 chance that the son inherited the HD allele from his father
and will develop the disease.
b. The probability of an affected child is: 1/2 (the probability that Joe is HD HD+)
ร 1/2 (the probability that the child inherits the HD allele if Joe is HD HD+) =
1/4.
37. The trait is recessive because pairs of unaffected individuals (I-1 ร I-2 as well as II-3
ร II-4) had affected children (II-1, III-1, and III-2). There are also two cases in which
an unrelated individual must have been a carrier (II-4 and either I-1 or I-2), so the
disease allele appears to be common in the population.
38. a. The inheritance pattern seen in Fig. 2.22 on p. 32 could be caused by a rare
dominant mutation. In this case, the affected individuals would be heterozygous
(HD+ HD) and the normal individuals would be HD+ HD+. Any mating between
an affected individual and an unaffected individual would give 1/2 normal
(HD+HD+) : 1/2 affected (HD+ HD) children. However, the same pattern of
inheritance could be seen if the disease were caused by a common recessive
mutation. In the case of a common recessive mutation, all the affected individuals
would be HD HD. Because the mutant allele is common in the population, most or
even all of the unrelated individuals could be assumed to be carriers (HD+ HD).
Matings between affected and unaffected individuals would then also yield
phenotypic ratios of progeny of 1/2 normal (HD+ HD) : 1/2 affected (HD HD).
b. Determine the phenotype of the 14 children of III-6 and IV-6. If the disease is due
to a recessive allele, then III-6 and IV-6 must be homozygotes for this recessive
allele, and all their children must have the disease. If the disease is due to a
dominant allele, then III-6 and IV-6 must be heterozygotes (because they are
affected but they each had one unaffected parent), and 1/4 of their 14 children
would be expected to be unaffected.
Alternatively, you could look at the progeny of matings between unaffected
individuals in the pedigree such as III-1 and an unaffected spouse. If the disease
were due to a dominant allele, these matings would all be homozygous recessive ร
homozygous recessive and would never give affected children. If the disease is due
to a recessive mutation, then many of these individuals would be carriers, and if the
trait is common then at least some of the spouses would also be carriers, so such
matings could give affected children.
39. Diagram the cross by drawing a pedigree:
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a. Assuming the disease is very rare, the first generation is HH unaffected (I-1) ร hh
affected (I-2). Thus, both of the children (II-2 and II-3) must be carriers (Hh).
Again assuming this trait is rare in the population, those people marrying into the
family (II-1 and II-4) are homozygous normal (HH). Therefore, the probability that
III-1 is a carrier is 1/2; III-2 has the same chance of being a carrier. Thus the
probability that a child produced by these two first cousins would be affected
is 1/2 (the probability that III-1 is a carrier) ร 1/2 (the probability that III-2 is a
carrier) ร 1/4 (the probability the child of two carriers would have an hh genotype)
= 1/16 = 0.0625.
b. If 1/10 people in the population are carriers, then the probability that II-1 and II-4
are Hh is 0.1 for each. In this case an affected child in generation IV can only occur
if III-1 and III-2 are both carriers. III-1 can be a carrier as the result of 2 different
matings: (i) II-1 homozygous normal ร II-2 carrier or (ii) II-1 carrier ร II-2 carrier.
(Note that whether I-1 is HH or Hh, II-2 must be a carrier because of the normal
phenotype (II-2 cannot be hh) and the fact that one parent was affected.) The
probability of III-1 being a carrier is thus the probability of mating (i) ร the
probability of generating a Hh child from mating (i) + the probability of mating (ii)
ร the probability of generating an Hh child from mating (ii) = 0.9 (the probability
II-1 is HH, which is the probability for mating [i]) x 1/2 (the probability that III-1
will inherit h in mating [i]) + 0.1 (the probability II-1 is H, which is the probability
for mating [ii]) ร 2/3 (the probability that III-1 will inherit h in mating [ii];
remember that III-1 is known not to be hh) = 0.45 + 0.067 = 0.517. The chance
that III-2 will inherit h is exactly the same. Thus, the probability that IV-1 is hh =
0.517 (the probability III-1 is Hh) ร 0.517 (the probability that III-2 is Hh) ร 1/4
(the probability the child of two carriers will be hh) = 0.067. This number is
slightly higher than the answer to part (a), which was 0.0625, so the increased
likelihood that II-1 or II-4 is a carrier makes it only slightly more likely that
IV-1 will be affected.
40. a. Both diseases are known to be rare, so normal people marrying into the pedigree are
assumed to be homozygous normal. Nail-patella (N) syndrome is dominant
because all affected children have an affected parent. Alkaptonuria (a) is recessive
because the affected children are the result of a consanguineous mating between 2
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unaffected individuals (III-3 ร III-4). Because alkaptonuria is a rare disease, it
makes sense to assume that III-3 and III-4 inherited the same a allele from a
common ancestor. Genotypes: I-1 nn Aa; I-2 Nn AA (or I-1 nn AA and I-2 Nn
Aa); II-1 nn AA; II-2 nn Aa; II-3 Nn Aโ; II-4 nn Aโ; II-5 Nn Aa; II-6 nn
AA; III-1 nn AA; III-2 nn Aโ; III-3 nn Aa; III-4 Nn Aa; III-5 nn Aโ; III-6
nn Aโ; IV-1 nn Aโ; IV-2 nn Aโ; IV-3 Nn Aโ; IV-4 nn Aโ; IV-5 Nn aa; IV-6
nn aa; IV-7 nn Aโ.
b. The cross is nn Aโ (IV-2) ร Nn aa (IV-5). The ambiguity in the genotype of IV-2 is
due to the uncertainty of her father’s genotype (III-2). His parents’ genotypes are nn
AA (II-1) ร nn Aa (II-2) so there is a 1/2 chance III-2 is nn AA and a 1/2 chance
he is nn Aa. Thus, for each of the phenotypes below you must consider both
possible genotypes for IV-2. For each part below, calculate the probability of the
child inheriting the correct gametes from IV-2 ร the probability of obtaining the
correct gametes from IV-5 to give the desired phenotype. If both the possible IV-2
genotypes can produce the needed gametes, you will need to sum the two
probabilities.
For the child to have both syndromes (Nโ aa), IV-2 would have to contribute
an n a gamete. This could only occur if IV-2 were nn Aa. The probability IV-2 is nn
Aa is 1/4: For IV-2 to be nn Aa, III-2 would have had to be nn Aa and would also
have had to give an n a gamete to IV-2. The probability of each of those events is
1/2, so the chance of both of them occurring is 1/2 ร 1/2 = 1/4. (Note that we
can assume that II-2 is nn Aa because III-3 must have given two of her children an
a allele. Therefore, both II-2 and III-3 must be nn Aa.) If IV-2 is nn Aa, the chance
that he would give a child an n a gamete is 1/2. The probability that IV-5 would
supply an N a gamete is also 1/2. Thus, the probability that the child would
have both syndromes is 1/4 ร 1/2 ร 1/2 = 1/16. There is no need to sum
probabilities in this case because IV-2 cannot produce an n a gamete if his genotype
is nn AA.
For the child to have only nail-patella syndrome (Nโ Aโ), IV-2 would have to
provide an n A gamete and IV-5 an N a gamete. This could occur if IV-2 were nn
Aa; the probability is 1/4 (the probability IV-2 is Aa) ร 1/2 (the probability of an
A gamete if IV-2 is Aa) ร 1/2 (the probability of an N a gamete from IV-5] =
1/16. This could also occur if IV-2 were nn AA. Here, the probability is 3/4 (the
probability IV-2 is nn AA) ร 1 (the probability of an n A gamete if IV-2 is nn AA)
ร 1/2 (the probability of an N a gamete from IV-5) = 3/8. Summing the
probabilities for the two mutually exclusive IV-2 genotypes, the probability that
the child of IV-2 and IV-5 would have only nail-patella syndrome is 1/16 +
3/8 = 7/16.
For the child to have just alkaptonuria (nn aa), IV-2 would have to contribute an
n a gamete. This could only occur if IV-2 were nn Aa. The probability IV-2 is nn Aa
is 1/4, and the probability of receiving an n a gamete from IV-2 if he is nn Aa is
1/2. The probability that IV-5 would supply an n a gamete is also 1/2. Thus, the
probability that the child of IV-2 and IV-5 would have only alkaptonuria is
1/4 x 1/2 x 1/2 = 1/16. There is no need to sum probabilities in this case because
IV-2 cannot produce an n a gamete if his genotype is nn AA.
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The probability of neither defect is 1 โ (sum of the first 3) = 1 – (1/16 +
7/16 + 1/16) = 1 – 9/16 = 7/16. You can make this calculation because there are
only the four possible outcomes and you have already calculated the probabilities of
three of them.
41. Diagram the cross(es):
midphalangeal ร midphalangeal โ 1853 midphalangeal : 209 normal
M?
ร
M?
โ
M? : mm
The following crosses are possible:
MM
ร
MM
โ
all MM
Mm
ร
MM
โ
all MMM
ร
Mm
โ
all MMm
ร
Mm
โ
3/4 M- : 1/4 mm
The 209 normal children must have arisen from the last cross, so approximately 3 x 209
= 630 children should be their M- siblings. Thus, about 840 of the children or ~40%
came from the last mating and the other 60% of the children were the result of one
or more of the other matings. This problem illustrates that much care in interpretation
is required when the results of many matings in mixed populations are reported (as
opposed to the results of matings where individuals have defined genotypes).
42. a. An equally likely possibility exists that any child produced by this couple will be
affected (A) or unaffected (U) . For two children, the possibilities are: AA, AU, UA,
UU. The case in which only the second child is affected is UA; this is one of the
four possibilities so the probability that only the second child is affected is 1/4.
b. From the list just presented in part (a), you can see that there are two possibilities in
which only one child is affected: AU and UA. The probability that either of these
two mutually exclusive possibilities will occur is the sum of their independent
probabilities: 1/4 + 1/4 = 1/2.
c. From the list just presented in part (a), you can see that there is only one possibility
in which no child is affected: UU. The probability of this event is 1/4.
d. If this family consisted of 10 children, the case in which only the second child
out of 10 is affected (that is, UAUUUUUUUU) has a probability of 1/210 =
1/1024 = ~0.00098. This probability is based on the facts that each birth is an
independent event, and that the chance of U and A are each 1/2. We thus use the
product rule to determine the chance that each of those 10 independent events will
occur in a particular way โ a particular birth order.
In a family of ten children, 10 different outcomes (birth orders) exist that satisfy
the criterion that only 1 child has the disease. Only the first child could have the
disease, only the second child, only the third child, etc. :
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1. AUUUUUUUUU
2. UAUUUUUUUU
3. UUAUUUUUUU
4. UUUAUUUUUU
5. UUUUAUUUUU
6. UUUUUAUUUU
7. UUUUUUAUUU
8. UUUUUUUAUU
9. UUUUUUUUAU
10. UUUUUUUUUA
We have already calculated that the chance of one of these outcomes in particular
(#2) is 1/1024. As each of the 10 possibilities has the same probability, the
probability that only one child is affected would be 10 x (1/1024) = 10/1024 =
~0.0098.
Only one possibility exists in which no child would be affected
(UUUUUUUUUU), and just like any other specific outcome, this one has a
probability of 1/1024 = ~0.00098.
e. One way to determine the probability that four children in a family of ten will have
the disease is to write down all possible outcomes for the criterion, as we did above
for the second answer in part (d). Then, also as we did above, sum their individual
probabilities, each of which is (1/2)10 just as before. If you start to do thisโฆโฆ
1. AAAAUUUUUU
2. AAAUAUUUUU
3. AAAUUAUUUU
4. AAAUUUAUUU
5. AAAUUUUAUU
6. AAAUUUUUAU
7. AAAUUUUUUA
8. AAUAAUUUUU
9. AAUAUAUUUU
10. AAUAUUAUUU
etc.
โฆ..you will realize fairly quickly that writing down every possible birth order in this
case is quite a difficult task and you are likely to miss some outcomes. In short โ
this is not a good way to find the answer! For questions like this, it is far preferable
to use a mathematical tool called the binomial theorem in order to determine the
number of possible outcomes that satisfy the criterion. The binomial theorem looks
like this:
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P (X will occur s times, and Y will occur t times, in n trials) =
n!
(ps ร qt)
s! ร t!
P = the probability of what is in parentheses
p = P(X)
q = P(Y)
X and Y are the only two possibilities, so p + q = 1.
Also, s + t = n.
Remember that ! means factorial: for example, 5! = 5 ร 4 ร 3 ร 2 ร 1.
To apply the binomial theorem to the question at hand (assuming you can still
remember what the question was!), weโll let X = a child has the disease (A), and Y =
a child does not have the disease (U). Then, s = 4, t = 6, n = 10, p = ยฝ, and q = ยฝ.
The answer to the question is then:
P (4 A and 6 U children out of 10) = (10! / 4! ร 6!) (1/24 ร 1/26).
Notice that (ps ร qt) = (1/24 ร 1/26) = 1/210. This factor of the binomial theorem
equation is the probability of each single birth order, as we saw previously in part
(d) above. To get the answer to our question, we need to multiply this factor (the
probability of each single birth order) by the number of different birth orders that
satisfy our criterion. From the equation in the box above, this second factor is
[n!/(s! ร t!)] = (10! / 4! ร 6!) = 210. Thus, the probability (P) of only 4 children
having the disease in a family of 10 children is 1/210 ร 210 โ 21%.
43. In the case of cystic fibrosis, the alleles causing the disease do not specify active
protein [in this case, the cystic fibrosis transmembrane receptor (CFTR)]. Some CF
disease alleles specify defective CFTR proteins that do not allow the passage of chloride
ions, while other CF disease alleles do not specify any CFTR protein at all. As you will
learn in a later chapter, such alleles are called loss-of-function alleles. In a heterozygote, the
normal CF+ allele still specifies active CFTR protein, which allows for the passage of
chloride ions. Because the phenotype of the heterozygote is unaffected, the amount of
active CFTR protein allows passage of enough chloride ions for the cells to function
normally. Again as you will see, most loss-of-function alleles are recessive to normal
alleles for similar reasons. (But it is important to realize that important exceptions are
known in which loss-of-function mutations are actually dominant to normal alleles.)
In the case of Huntington disease, the disease-causing allele is dominant. The
reason is that the huntingtin protein specified by this HD allele has, in addition
to its normal function (which is not entirely understood), a second function that is
toxic to nerve cells. This makes the HD disease allele a gain-of-function allele. The reason
HD is dominant to HD+ is that the protein specified by the disease allele will be toxic
to cells even if the cells have normal huntingtin specified by the normal allele. Most (but
again not all) gain-of-function mutations are dominant for similar reasons.
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