Download Mendelian Genetics

Mendelian Genetics
Objectives
1. To understand the Principles of Segregation and Independent Assortment.
2. To understand how Mendel’s principles can explain transmission of characters from
one generation to the next.
3. To understand and use the χ2 goodness-of-fit statistical test to evaluate hypotheses
about mechanisms of inheritance.
I. Background
Much of our understanding of genetics today is taken for granted, as the use of DNA has
become common in forensics and medicine. However, compared to many fields in
biology, genetics is relatively young. Mendel’s “Laws” of inheritance were rediscovered
in the early 1900s, and Watson and Crick described the structure of DNA only about fifty
years ago (if you are interested in exploring the history of genetics, you can find a list of
good web sites at http://dmoz.org/Science/Biology/Genetics/History/)
People have always been interested in explaining how traits were passed from one
generation to the next. In the 19th century, the leading hypothesis of heritability was that
offspring resembled their parents because parental traits blended. A prediction of this
hypothesis is that offspring of a cross of purple-flowered and white-flowered peas will
have pale purple flowers. We can sum up their position and the outcome of Mendel’s
experiments in the following table:
OBSERVATION
QUESTION
HYPOTHESIS /
PREDICTION
EXPERIMENT(S)
EVALUATION/ DRAW
INFERENCES
The characters of the offspring appear to be a blend of the
parental traits.
How are traits inherited?
Blending of heritable information from both parents
produces offspring with intermediate traits.
Mendel’s pea experiments.
Data: The offspring from certain crosses, such as purple x
white-flowered peas, resemble only one parent, not a blend.
The data do not support the hypothesis. Mendel proposed
alternative models.
Mendel conducted these breeding experiments over and over again, with different traits;
all of them with two phenotypes, and from his results developed his “laws” of
segregation and independent assortment. We place Mendel’s experiments into our
structure for the scientific method in the following table:
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OBSERVATION
Pea flower colour is either white or purple, but never a blend.
QUESTION
If blending does not explain how traits are inherited, then how
are traits inherited?
Some “particle” determining flower color is passed intact
from parent to offspring (Mendel did not coin the term
“gene”). Prediction: the “particle” determining white color is
still present in the offspring, but its expression is masked.
Cross the offspring resulting from purple x white crosses (F1),
and see if some “grand-offspring” (F2) are white .
Data: about 25% of the F2 are white, the remainder are purple
The data support the hypothesis. Mendel predicted (and then
experimentally confirmed) that about 2/3 of the purple F2
would carry the “white” particle (i.e. had both a purple and a
white particle), and that other characteristics such as pea color
and plant height would show similar patterns of inheritance.
HYPOTHESIS /
PREDICTION
EXPERIMENT
EVALUATE/DRAW
INFERENCES
We can put this experiment into the context of modern genetics by using modern
terminology. In the parental generation (P), plants are true-breeding: for the trait of
interest such as flower color, crosses within the true-breeding group always produce
offspring that resemble the parents. This is because the parents are homozygous
(identical alleles) for either the purple or white color. When Mendel crossed the truebreeding purple and white flowers, he always obtained plants, the F1 generation, with
purple flowers, but the allele for “white” is not lost: we refer to these individuals as
heterozygous. When he then crossed two F1 heterozygotes and determined the flower
color of the F2, he found 705 individuals with purple flowers and 224 individuals with
white flowers. He referred to this as “Segregation” because he stated that the particle for
white color segregated (separated) from the particle for purple color. He further proposed
that these two “particles” are alternative states (alleles) for the same character (“gene”).
In modern terms; these two states are two different alleles for the same gene, where we
define a gene (or genetic locus) as a specific section of a chromosome coding for a
particular protein. Different alleles of one genetic locus, code for different versions of
the same protein.
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II. The monohybrid cross
The experiment described above is a monohybrid cross: the true-breeding parents are
homozygous for different flower colors, and the F1 are hybrids for that one trait. Our
understanding of meiosis provides a mechanistic explanation for the pattern of
segregation in monohybrid crosses. In meiosis, when the gametes are being formed, each
gamete only gets one copy of each chromosome (one member of each chromosome
“pair”). In the F1 plants, each gamete will either get the allele for purple or the allele for
white flower color: no gametes can have both alleles because these alleles are on each of
the homologous (paired) chromosomes.
Male
(genotype Ww)
One gamete from one parent combines with one gamete from the other parent, but which
gamete combines with which (i.e., purple with purple or with white) is random. For each
gamete type (purple or white) from one parent, there is a 50:50 chance it will combine
with either white or purple from the other parent. This will produce, in the F2 generation,
purple/purple: purple/white (or white/purple): white/white genotypes in a 1:2:1 ratio.
Since the purple allele “masks” or is dominant to the white allele, the phenotypes of the
F2 generation will be 3 purple: 1 white.
Female (genotype Ww)
gamete
genotypes P
p
P
PP
Pp
p
Pp
pp
This is represented here in a Punnett
Square. P represents the allele for the
dominant purple flower trait and p
represents the allele for the recessive
white flower trait.
III.
Dihybrid cross.
Mendel also conducted experiments where he crossed true-breed plants that differed in
two different traits, producing F1 offspring that were hybrids at two genetic loci, or
dihybrids. Now it can be imagined that two traits affecting the same structure, pea color
and shape for instance, might move jointly to the next generation: if the original
homozygous parents have smooth, yellow peas then all descendents with smooth peas
will have yellow peas. The alternative hypothesis is that the traits are separate, and that
the inheritance of pea color will be independent of the inheritance of pea shape. These
two alternative hypotheses can be put into our table of the scientific method, and a single
experiment can test both of them:
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Observation
Question
Hypothesis 1,
Prediction 1
Hypothesis 2,
Prediction 2
Experiment
Evaluation,
Inferences
Peas display four different phenotypes with respect to their
seed characteristics: for color yellow or green, for shape
smooth or wrinkled
Does the inheritance of one trait (shape) influence the
inheritance of another trait (color)?
Yes: if the original pea parents have either smooth and
yellow peas or green and wrinkled peas, all the F2
offspring will have one of these combinations.
No: no matter what combination of traits the original
parents have, all possible combinations will be found in the
F2 generation.
Cross true-breeding plants then cross the F1 offspring.
Rear the resultant F2 plants and determine the frequencies
of the different combinations of traits.
Data: in the F2, all possible combinations of traits are
found.
Hypothesis 1 is rejected; there is no “linkage” of the two
different traits as they are passed from generation to
generation.
Male:
WwYy
In modern terminology, the inference drawn from this result is that the two genetic loci,
pea color and pea shape, are not “linked”. The predicted genotype and phenotype
distribution in offspring of a dihybrid cross using unlinked genetic loci can be calculated
in two fashions. First, you can do a 4 x 4 Punnett square (you should finish filling in all
possible genotypes):
Female: WwYy
gametes:
WY
WY
WWYY
Wy
WWYy
wY
WwYY
wy
WwYy
Wy
WWYy
wY
WwYY
wy
WwYy
Since W- is always round and Y- is always green, we can determine the phenotypes of
these offspring, and find a 9:3:3:1 ratio for W-Y- : wwY- : W-yy : wwyy.
A second, and much quicker, way to predict phenotype frequencies is to recall that in a
monohybrid cross, 3/4 of the offspring will have the dominant phenotype and 1/4 will
have the recessive phenotype. To estimate how many offspring from a dihybrid cross
will have, for instance, the phenotype “round green” (W-Y-) you simply multiply the
proportion of W- by the proportion of Y-, or 3/4 x 3/4 = 9/16. Similarly, 1/4 x 3/4 = 3/16
will have the phenotype “wrinkled green” (wwY-). This approach has the benefit that it
is easily expanded to many more genetic loci, when Punnett’s squares become awkward
or impossible to draw (one needs an additional dimension for every additional genetic
locus!)
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It is important to note that not all genetic loci assort independently. If two genetic loci
are relatively close together on the same chromosome, they will not move independently
into the gametes and hence will appear “linked” in experimental crosses.
Again, we can look to our understanding of meiosis for a
mechanistic explanation of independent assortment. Remember
that, during meiosis, homologous chromosomes pair along the
center of the cell during metaphase I. When they are lined up,
however, the orientation of each pair of chromosomes is
independent of the orientation of every other pair. In other
words, there is equal chance that the w-carrying chromosome
will be aligned with the y-carrying chromosome or with the Ycarrying chromosome.
Assigning letters to each allele (W:round, w:wrinkled, Y: green,
y: yellow), we can diagram these possibilities in a drawing of
metaphase I.
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IV.
Today’s Lab with Corn.
Corn Genetics: Kernel color and sugar content
Ears of corn are good study systems for learning about genetics, because each kernel
represents an independent union of gametes, and thus a cob has a population of
genetically unique kernels. The cobs you will be studying are variable for two traits of
interest, color and sugar content. You can detect sugar content differences because highsugar kernels are wrinkled and low-sugar kernels are smooth. Each of these
characteristics is determined by a single gene, with two alleles.
There are four kernel phenotypes in the above ear of corn: Purple & Smooth (A),
Purple & Wrinkled (B), Yellow & Smooth (C) and Yellow & Wrinkled (D).
For your experiments today, you will use what you have learned about the principles of
segregation and independent assortment to infer the mechanisms of inheritance for color
and sugar content in corn. You will collect data to test four hypotheses concerning these
traits by sampling kernels on the cob. Finally, you will evaluate your hypotheses by
comparing your data with your predicted results using the 2 statistical test (Chi Square).
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Evaluating Hypotheses by Predicting results:
Part of testing hypotheses is to describe what results you would expect to find if your
hypothesis is correct. The more exact your predictions are, the easier it is to use your
data to test the hypothesis.
Today’s Observation: Individual kernels of corn on a cob vary in color and shape.
Hypotheses and predictions: Below are four hypotheses regarding the inheritance of
corn kernel color and shape. Working with your partner and the teaching fellow, develop
predictions of what you would observe in the population of corn kernels if the hypothesis
is correct. These will be the “working hypotheses” guiding your experiments.
Hyp. 1. The genes for color and shape assort independently.
Prediction 1
Hyp.2. Purple is dominant to yellow. (or use the reverse)
Prediction 2
Hyp. 3. Sugar is dominant to starch (wrinkled dominant to smooth) (or the reverse)
Prediction 3
Hyp. 4. The corncob you are examining represents the result of a dihybrid cross.
Prediction 4
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Analysis of Experimental Results: You will use the χ2 goodness-of-fit test to determine
if your observed results are consistent with the predictions made based on Mendel’s
principles.
The 2 test allows you to determine whether your results could have happened by chance
alone. Typically, the null hypothesis being tested is that the observed results (O) and the
expected results (E) are not different (Ho: O = E). The alternative hypothesis is that the
observed counts and the expected counts are different (Ha: O ≠ E).
The general equation for the chi-square value for a particular experiment is:
Chi-Square = the sum over all categories for (observed cell counts – expected cell
2
O  E

2
2
counts) /expected cell counts or
 =
E
The last ingredient needed for the test is the “degrees of freedom” (df), which is
calculated as the number of classes, less one. Here, the “degrees of freedom” are related

to the number of categories contained
in the characteristic under study. For example if
the characteristic under study is the sex ratio of a trait there are two possible classes (male
or female) and one degree of freedom. This is the case since one class (e.g. female) is
“free” to vary in % of the total, but once its value is known the % of the other class
(male) is determined by default (no longer “free”). Likewise if there are three values that
make up100% of the characteristics (such as red, pink and white) there are two degrees of
freedom, two could vary but the last will be determined by default, and so on for
increasing numbers of classes.
The observed 2 value (calculated with the above equation) is compared to those in a
statistical table (see section below) containing values from a 2 distribution for each
number of degrees of freedom, and the associated probability “” that the observed data
match the null hypothesis (or conversely that any deviation from the expected was solely
the result of random chance).
A 2 value of 0 (zero) indicates no deviation from the expected values. Therefore, any 2
value greater than 0 represents some deviation from the expected values. The statistical
table provides a basis for evaluating the probability that the deviation from 0 has occurred
by chance. The results are expressed as a range of probability () values.
Chi-Square Probability Distribution

df
1
2
3
4
5
0.99 0.95
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.05
0.01
0.00
0.02
0.11
0.30
0.55
0.02
0.21
0.58
1.06
1.61
0.06
0.45
1.01
1.65
2.34
0.15
0.71
1.42
2.19
3.00
0.27
1.02
1.87
2.75
3.66
0.45
1.39
2.37
3.36
4.35
0.71
1.83
2.95
4.04
5.13
1.07
2.41
3.66
4.88
6.06
1.64
3.22
4.64
5.99
7.29
2.71
4.61
6.25
7.78
9.24
3.84
5.99
7.81
9.49
11.07
6.63
9.21
11.34
13.28
15.09
0.004
0.10
0.35
0.71
1.15
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The numbers in the table represent 2 values. This “2 statistic” reflects the magnitude of
the discrepancies between observed and expected counts. When the differences are
sizable, the value of 2 tends to be large. Therefore, large values of 2 suggest rejection
of the null hypothesis Ho.
A key result of an analysis of the behavior of 2, when the null hypothesis is correct and
the sample size is large enough (at least 5 per category), shows that its graphical
distribution (for each degree of freedom) is asymmetric with a longer tail on the right
(left resembles a “normal” curve).
In general, Ho will be rejected when 2 exceeds a specified critical value (the test is
upper-tailed). The critical value is usually chosen to control for the probability of a type I
error (rejecting Ho when Ho is true). Given that the distribution curve has an extended
right tail, and is otherwise normal, the critical value is often chosen to eliminate much of
the “right tail” of the curve. In practice this is usually taken as a 5% probability of match
(observed vs. expected) or less. In the table above the blue shaded region would be
considered to be a poor match and lead to the rejection of the null hypothesis.
The general method for calculating the 2 value for a goodness-of-fit test is to fill in a
table, shown as an example for a monohybrid cross:
class
A–
aa
total
observed
65
35
100
expected*
75
25
100
(O-E)
-10
+10
(O-E)2/E
1.333
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2 = 5.333
df=2-1 =1
In our example, 0.05>0> 0.01, or the probability of finding these (divergent) results
purely by chance is less than 5% but greater than 1%.
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Experimental Procedures
Methods: Obtain a cob of corn. These cobs are the F2 generation of unknown crosses.
Before you start, determine a protocol that you will follow to test each of the hypotheses
you made predictions for earlier. For each hypothesis, decide how many kernels you will
count (sample size – remember you will need a minimum of 5 counts for each “class” and
preferably many more!), where on the cob you will start (we recommend a new start
point for each sample), and how you will classify each kernel. Fill in your protocol for
each experiment in the section below.
Experiment 1
To support or reject the first hypothesis, no formal experimentation is required, only
observation of the cob.
Protocol: Study the specimen, and record the important observations.
Predicted results: if your hypothesis is correct, what results do you expect?
Results:
Evaluation:
Experiment to test hypothesis 2
Protocol: Number of kernels to be counted__________
Kernel characteristics (categories)______________
Predicted results: if your hypothesis is correct, what numerical results do you expect
(how many kernels in each category)? These will be your Expected values for your 2
test.
Results:
Evaluation: do the 2 test.
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Conclusion; do your results cause you to reject your hypothesis?
Experiment to test hypothesis 3
Protocol: Number of kernels to be counted__________
Kernel characteristics (categories)______________
Predicted results: if your hypothesis is correct, what numerical results do you expect
(how many kernels in each category)? These will be your Expected values for your 2
test.
Results:
Evaluation: do the 2 test.
Conclusion; do your results cause you to reject your hypothesis?
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Experiment to test hypothesis 4
Protocol: Number of kernels to be counted__________
Kernel characteristics (categories)______________
Predicted results: if your hypothesis is correct, what numerical results do you expect
(how many kernels in each category)? These will be your Expected values for your 2
test. (Hint: create a 4x4 Punnet Square for this cross as you did in the “background”
section for Mendel’s Peas. Then the “expected” can be calculated from the total
observations times the phenotype ratios -x/16- of the different classes.)
Results:
Evaluation: do the 2 test.
Conclusion; do your results cause you to reject your hypothesis?
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V. Questions and Interpretation of Data
1. Based on your results, what is the relationship between kernel color alleles? Support
your answer.
2. Based on your results, what is the relationship between kernel sugar content alleles?
Support your answer.
3. Do the genes for kernel color and sugar content assort independently among gametes?
Support your answer.
4. Does the kernel phenotype distribution support the idea that the cob is the result of a
dihybrid cross? What information supports your answer? If a dihybrid cross (i.e. F1 to
F2 of standard Mendelian crosses) is not indicated what conditions might contribute to
this finding.
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