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C
H A P T E R
MENDELIAN GENETICS
CHAPTER2
2.1 Introduction
Simple inheritance follows very basic laws, called Mendelian
Inheritance. These laws are consistent with the biochemistry of
DNA, the biology of the cell, and with the laws of probability.
While they are quite useful in helping us to organize our
understanding of Genetics, each of the laws is violated for some
genes in some organisms.
2.2 Simple Mendelian Genetics
Simple Mendelian Genetics follows from Mendel’s Postulates:
• Genetics Factors come in discrete units (factors).
• Every individual has two factors, one from each of its parents.
• When an individual has two different factors, one of them will be
dominant to the other (recessive).
• There is random segregation of factors in creating gametes.
• There is independent assortment of pairs of factors.
2.2.1 Gamete and Genotype Formation
Gametes are created separately for each locus (although not
independently in a statistical sense if the loci are linked). A gamete
will have a single allele for each locus.
EXAMPLE: We are working with three loci (A, B, and C). A
gamete must specify the allele for each of the three loci. If an
individual has genotype AaBbCC, then that individual can produce
four gamete types, ABC, AbC, aBC, and abC.
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A genotype will have a pair of alleles for each locus. The genotype
for an offspring will contain all the loci, with a single allele from
each parent.
2.2.2 Mendelian Crosses
Terms to Remember
Dominant, Recessive
Pure Breeding, True Breeding, Homozygous, Heterozygous
Gene, Locus, Allele, P0, F1, F2, Backcross, Testcross. Be able to
distinguish between parent and parental type.
Simple Mendelian Crosses
For the lesser desert bluebell, flower color is controlled by two
Mendelian Loci. The first locus determines the basic flower color –
Blue is dominant to Purple. We use the symbols B for the blue
allele, and b for the purple allele. Since Blue is dominant, BB and
Bb are both blue, while bb is purple. The second locus determines
the color of the star shape in the middle – White is dominant to
Ivory. For this, the white allele is designated I and the ivory allele is
designated i.
We can keep track of the crosses by generation designations.
The first generation is called the parental generation.
Often, we will start with the individuals in the parental
generation chosen from pure breeding strains.
• P0
The offspring from the first generation are called the
first filial generation. Often, these individuals will be
heterozygous for all the traits of interest.
• F1
If we interbreed the offspring of a filial
generation, their offspring will be the next filial generation.
F2 is the offspring of the mating, F1 × F1.
• F2, F3, etc.
What are the expected frequencies of the offspring of the given
crosses? In all cases, this can be done with Punnett Squares.
Pollen (Male)
Seed (Female)
Crosses
Blue
Purple
F1 × F1
Blue
Purple
F1 Backcross to Pollen
Parent
White
Ivory
F1 Testcrossed
White
Ivory
F2 = F1 × F1
Blue Ivory
Purple Ivory
F1 × F1
Blue Ivory
Purple Ivory
F1 × F1
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Pollen (Male)
Seed (Female)
Crosses
Blue White
Purple Ivory
F1 × F1
Blue Ivory
Purple White
F1 Testcrossed
For Punnett Squares, you must first decide the possible gametes
from each of the parents, and the probability for each gamete. For
these simple crosses, the probabilities for the gametes will be equal.
Consider the cross BB × bb. Since each individual is homozygous,
they can only produce gametes of a single type. The Punnett square
looks like:
b
B
Bb
If we cross two heterozygous individuals (Bb × Bb, called a
monohybrid cross), then each can produce gametes B and b in equal
frequencies. The Punnett Square can look like:
B
b
B
BB
Bb
b
Bb
bb
-1 B
2
-1 b
2
-1 B
2
-1 BB
4
-1 Bb
4
-1 b
2
-1 Bb
4
-1 bb
4
When the frequencies are equal, the Punnett Square on the left is
faster. Many times, we do not have equal frequencies. In that case,
we would use Punnett Squares similar to the one on the right. The
numbers given are the frequencies of the gametes or genotypes.
A dihybrid cross is when we cross two individuals who are
heterozygous for two loci. For our example, the individuals would
have the genotype BbIi. The possible gametes are BI, Bi, bI, bi,
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with equal frequencies. The Punnett Square can now be constructed,
with four gametes on each side, for a total of 16 possibilities.
-1 BI
4
-1 Bi
4
-1 bI
4
-1 bi
4
-1 BI
4
-1-- BBII
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-1- BBIi
16
-1- BbII
16
-1- BbIi
16
-1 Bi
4
-1-- BBIi
16
-1- BBii
16
-1- BbIi
16
-1- Bbii
16
-1 bI
4
-1-- BbII
16
-1-- BbIi
16
-1- bbII
16
-1- bbIi
16
-1 bi
4
-1- BbIi
16
-1- Bbii
16
-1- bbIi
16
-1- bbii
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Some of these combinations are duplicate genotypes. Recalling Blue
is dominant to Purple and White is dominant to Ivory, the expected
frequencies of the phenotypes among the offspring of the dihybrid
cross are easily calculated as given, where the dash (–) refers to
either of the two alleles.
Phenotype
Genotypes Expected Frequency
Blue White
B–I–
-9-16
Blue Ivory
B–ii
-3-16
Purple White
bbI–
-3-16
Purple Ivory
bbii
-1-16
Probability Facts
If Events A and B are independent, then
Probability (Event A and Event B) = Prob(Event A)×Prob(Event B)
If Events C and D are distinct, then
Probability(Event C or Event D) = Prob(Event C) + Prob(Event D)
2.3 Chi Square Goodness of Fit
The Chi Square Goodness of Fit Test is a way to compare the results
of a genetic test with a hypothesis about the form of inheritance. In
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this section, we will discuss the formal way to formulate and test
genetic hypotheses.
2.3.1 The Hypothesis
THE HYPOTHESIS is a simple statement of the Genetic Inheritance of
the trait(s) of interest.
EXAMPLE:
• The trait is Simple Mendelian.
• The traits have Duplicate Dominant Epistasis
• The trait is Simple Mendelian with a Penetrance of 80%
Ideally, THE HYPOTHESIS is created before the experiment. Often,
you will conduct a preliminary experiment to determine THE
HYPOTHESIS, then do a final experiment to test it.
2.3.2 Expression of the Hypothesis for the
Experiment
While we are interested in THE HYPOTHESIS, we will need to run an
experiment to look at the validity of it. A properly designed
experiment will be able to distinguish THE HYPOTHESIS from other
possibilities. The Experimental Hypothesis is a statement of what
we expect the results of the experiment to be if THE HYPOTHESIS is
correct. It will change, depending on the experiment used to test
THE HYPOTHESIS.
Example
• The dihybrid cross gives a 3:1 ratio of Blue to Red.
• The monohybrid cross gives a 2:1 ratio of Agouti to Yellow.
• The trihybrid cross gives a 28:16:15:6:2 ratio.
2.3.3 The Experiment
The experiment is designed to be able to distinguish THE
HYPOTHESIS from other possible alternatives. While the mono-, di-,
tri-hybrid crosses are used often, many crosses are possible.
Depending on the question, other experiments may be more
powerful, e.g., the testcross is the cross of choice when testing for
linkage.
2.3.4 The Data
For Chi Square Goodness of Fit Tests, the data consist of observed
numbers of individuals for each of the characteristics. Which
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characteristics are expected and their frequencies depend on the
Experimental Hypothesis.
2.4 An Example
We feel that brown vs. albino is a Simple Mendelian Characteristic.
To test this, we will conduct a monohybrid cross.
THE HYPOTHESIS
The Hypothesis is that Brown vs. Albino is Simple Mendelian with
Brown dominant to Albino.
The Experimental Hypothesis
For a Monohybrid cross, we expect a 3:1 ratio of Brown to Albino.
The Data
The experiment is run, and the data is tabulated as:
Phenotype Observed
Brown
82
Albino
23
Total
105
Chi-Square Goodness of Fit Test
Given the experimental hypothesis, we can calculate the expected
number of individuals from our experiment that have the various
phenotypes. Table 2-1shows how to compute the expected number
of individuals for each of the phenotypes, given the Hypothesis of
Simple Mendelian Inheritance.
Table 2-1
Expected Values for the Phenotypes
Phenotype Expected Frequency Expected Number
Brown
3/4
3/4(105)=78.75
Albino
1/4
1/4(105)=26.25
Total
1
105
Even though we really don’t expect a fractional individual, it is
important to keep fractions in that column. Two decimal places is
usually sufficient.
We are now ready to compute a Chi-Square Goodness of Fit. The
Chi Square table is given in Table 2-2. Notice that in the last
column, the divisor is the Expected Number.
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Chi Square Table for Testing Brown vs. Albino
Table 2-2
Chi Square
Phenotype
Observed
Number
(Data)
Expected
Observed Number
Expected
(Hypothesis)
(O-E)
Brown
82
78.75
3.25
0.13
Albino
23
26.25
-3.25
0.40
Total
105
105
0
0.53
(-O
E--)-2
---–--E
2.4.1 Results of the Chi Square Test
Statistical Result:
A Chi Square Goodness-of-Fit Test is done to see if the hypothesis
is plausible. The Chi Square is a measure of the Distance of the
actual data to what you expect the data to be if the hypothesis is
correct.
The Chi Square Goodness-of-Fit has degrees of freedom (df)
associated with it. The degrees of freedom is a measure of the size of
the experiment. For these simple Genetic Tests, the df are equal to
the number of phenotypic classes (2 in this case) - 1. The minus 1
takes into account that since the sample size is fixed (105), once we
know the number in all but the last phenotypic class, we also know
the number in the last phenotypic class. The df for this Chi-Square
are 2-1 = 1.
In order to know whether or not the distance (Chi Square) is too
large, we compare it to a reference value. These references (called
the Table Value or Table Chi Square) can be found in most genetics
text book, and is also given here for 5% significance level in
Table 2-3.
Table 2-3
df
Chi Square Table for 5% Significance Level
1
2
3
4
5
6
7
8
9
10
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Chi-Square 3.84 5.99 7.82 9.49 11.07 12.59 14.07 15.51 16.92 18.31 19.68
There are two possible statistical conclusions, depending on how
close the data are to the hypothesis,
• If the Computed Chi Square is less than the Table Chi Square (as
we have with these data), we fail to reject the hypothesis. Given
our data, we see no reason to doubt our hypothesis (at the 5%
significance level). The hypothesis of Simple Mendelian
Characteristics is consistent with the data. It is usually written as,
“The data are consistent with the characteristics being inherited in
a Simple Mendelian manner at the 5% significance level.” This
statement is very weak, but it is as good as we can get.
For this experiment, the statistical result is “Fail to Reject.”
• If the Computed Chi Square had been greater than the Table Chi
Square, we reject the hypothesis. The hypothesis is not plausible
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(at the 5% significance level) given the data. It is usually written
as, “There is significant evidence at the 5% significance level to
conclude that the characteristics are not Simple Mendelian.” This
statement is quite strong — we can conclude that it is not Simple
Mendelian. Even so, we may be wrong. The hypothesis may be
correct, but we have concluded that it is not. In fact, the 5% that is
used as a significance level is the chance an experiment will
reject the hypothesis when it is, in fact, true. This rate is chosen
by the experimenter.
Genetic Conclusion
The Genetic Conclusion relates the statistical result back to THE
HYPOTHESIS.
If the Statistical Conclusion Is Reject the Hypothesis
If the statistical conclusion had been to reject the hypothesis, then
we state, “There is enough evidence from this experiment to
conclude that Brown vs. Albino is not Simple Mendelian.” (Note:
this is not the conclusion for this experiment).
If the Statistical Conclusion Is Fail to Reject the Hypothesis
If the statistical conclusion had been to fail to reject the hypothesis,
then we state, “Simple Mendelian Inheritance is consistent with this
experiment.” This is a very weak statement, particularly when the
experiment is not very large.
Genetic Conclusion for Brown vs. Albino
For the experiment given, the Genetic Conclusion is stated, “The
data are consistent with Brown vs. Albino as Simple Mendelian
Traits with Brown dominant to Albino.”
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