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A Brief Introduction to the Hardy-Weinberg Formulas and Population Genetics
In population genetics, we study the frequency of particular alleles in the population. In our
previous genetics studies we were interested in what happens if two particular individuals mate or
are crossed. For example, in snapdragons, where red flower color is incompletely dominant to
white flower color, what happens if we cross a red and pink flower? In population genetics we
don’t care about what happens when particular individuals mate. Instead we track what happens
to particular alleles in a whole population.
Hardy and Weinberg were pioneers in the field of population genetics. They came up with 2
2
2
formulas, p + q = 1 and p + 2pq + q =1.
In order for these 2 formulas to be accurate it is necessary to make 5 assumptions about
conditions. Use your text to write out those conditions here:
1)
2)
3)
4)
5)
If you look at the Hardy-Weinberg conditions you will see that these conditions can never all exist.
For example, one condition is that the population is infinitely large. Another is that all alleles are
equally fit for the environment. Yet another is that individuals mate randomly. These and the other
two conditions are all contrary to what we have just learned about evolution. So why do we pay
any attention to the Hardy-Weinberg equations? There are two answers:
A. First, notice that if all Hardy-Weinberg conditions are met, you cannot even have
microevolution. There cannot be a change in gene or allele frequency in a population
from one generation to the next if all 5 Hardy-Weinberg conditions are met. Some people
feel that understanding this helps you to understand evolution better.
B. Second, and more interestingly, even though the Hardy-Weinberg conditions never
completely exist, if we assume that they are true, we can make some interesting
predictions that are approximately true.
But first, let’s understand how we do Hardy-Weinberg calculations. To keep the math simple we
can only deal with situations in which there are two alleles. So we can handle the case of purple
and white pea flowers where there is just a purple and a white allele. We can not handle the
ABO blood systems with its 3 alleles (unless we do more math than most of you want to do).
If one of the 2 alleles is dominant we let p=the fraction of the alleles in the whole population that
are of the dominant type and q=the fraction of the alleles in the whole population that are of the
recessive type.
For example, in pea plants, if 60% of the flower color alleles were purple, p would = 0.6 =
60% = (60/100) = (6/10). (You can work in either fractions or decimals. You may think in
terms of percentages, but you will probably want to convert the percentages to fractions
or decimals when doing calculations.) In this example, since there are only 2 alleles, if
we know the frequency of one allele, we easily figure the frequency of the second allele.
Thus q, the frequency of the white allele in this example, will be = 0.4 = 40% = (40/100) =
(4/10).
(If neither allele is dominant we can let p = the frequency of the more common of the two
alleles.)
Without getting into a religious discussion, it is helpful to explain Hardy-Weinberg by assuming
that when a diploid organism is made, The Creator, wearing a blindfold, reaches into a sack of
alleles and picks first one allele, and then reaches in a second time and picks the second allele.
So, with the above figures in mind, let’s assume The Creator is making pea plants. What is the
chance of a pea plant with white flowers? (This is the same question as, what fraction of pea
plants will have white flowers?)
Since the white allele is recessive, the only way to get a pea plant with white flowers is if
The Creator randomly selects a white allele on the first pick and also randomly selects a
white allele on the second pick. Since 40% of the alleles are white, the chance that The
Creator will pick a white allele on the first pick = 40% = 0.4 = (4/10). Since the population
is infinitely large, withdrawing the single white allele on the first pick will have no effect on
the second pick and the chance that The Creator will randomly pick a white allele on the
second pick will also = 40% = 0.4 = (4/10). Thus the chance of getting two white alleles,
the requirement for a pea plant with white flowers, = (4/10)(4/10) = 16/100 = 0.16 (This is
like what’s the chance of rolling snake eyes, two ones, when you roll two dice. There is a
1/6 chance that the first die will come up one, and a 1/6 chance that the second die will
come up one. Since the two events are independent, the chance of both happening is
their product, (1/6) (1/6) = 1/36.)
Notice that the chance of getting the recessive genotype (which has to be homozygous
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recessive) = q .
All pea plants which are not white-flowered (homozygous recessive) must be of the dominant
2
2
2
phenotype. So, since p + 2pq + q =1, if you subtract q from 1 you get the frequency of the
2
dominant phenotype which can also be expressed as p + 2pq. (You can always perform the
2
2
same operation on both side of an = sign. Hardy and Weinberg got p + 2pq + q =1 by squaring
2
both sides of the equation, p + q = 1. If you want to understand why p + 2pq = the frequency of
the dominant phenotype, here is the explanation:
There are 3 ways to get a purple flowered pea plant. First, The Creator can select two
2
purple alleles. The chance of this happening is p . Second, The Creator could pick a
purple allele the first time and a white allele the second time. The chance of this
happening is pq. Finally, The Creator could pick a white allele the first time and a purple
allele the second time. The chance of this happening is also pq. Thus the chance of a
2
2
purple flowered pea plant = p + pq + pq = p + 2pq.)
From the above you can see that if p = 0.6, then q = 0.4 and 0.16 = 16/100 of the plants will be
white flowered and 84/100 of the plants will be purple flowered and 36/100 of the plants
(6/10)(6/10) will be homozygous purple flowered while 48/100 of the plants (2)(6/10)(4/10) will
be heterozygous purple flowered.
Now let’s do something interesting with Hardy-Weinberg.
Cystic fibrosis, CF, is caused by a recessive autosomal gene and that 1/2500 children born to
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people of Northern European descent have CF. The figure just cited, 1/2500 = q . Therefore q
has to be 1/50 = 0.02 and p = 49/50 = 0.98. The frequency of carriers in this population will be
about 2pq (frequency of heterozygotes) = (2)(.98)(.02)= 0.0392 = approximately 1/25. (I have to
say approximately because the 5 Hardy-Weinberg conditions are never met in a real population;
but the Hardy-Weinberg formulas can give good estimates even in a real population.)
Sickle cell anemia. You will recall that people heterozygous for the sickle cell allele have
increased resistance to malaria and are otherwise basically ok, but that people homozygous for
the sickle cell allele have the serious disease of sickle cell anemia. Since malaria is a tropical
disease, you would expect some African-Americans to suffer from it. In fact, your textbook says
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1/400 African-Americans has the disease. Thus q = 1/400 and q = 1/20 = 0.05 and p has to =
19/20 = 0.95 and 2pq = (2)(19/20)(1/20) = 0.095 which is approximately equal to 1/10. In other
words, about 1 out of 10 African-Americans is a carrier for sickle cell anemia.
Thalasemiais also a severe anemia caused by an autosomal recessive and that carriers of the
disease have increased malaria resistance. Luigi Cavalli-Sforza in Genes, Peoples and
Languages (University of California Press, 2000) does a very similar calculation for thalassemia
in regions of Italy where malaria was prevalent prior to World War II. One in 100 children born in
2
those regions suffers from thalassemia. Thus q = 1/100 so q = 1/10 and p = 9/10 and 2pq =
18/100. Thus, one in a hundred people get the disease, but 18 in a hundred people have
increased malarial resistance.
So let’s review. If two normal people of Northern European ancestry mate, they have to worry
about having a child with CF. If two normal African-Americans mate, they have to worry about
having a child with sickle cell anemia. Supposing, instead, a normal person of Northern European
ancestry mates with a normal African-American. There is a much reduced chance that this couple
will have a child with either disease. From a genetic standpoint it is healthier to mate outside your
ethnic group than within your ethnic group. This is part of the explanation for hybrid vigor. It also
helps explain why inbreeding is bad. Mating with your sibling or your first cousin increases the
chance that two people who are carriers for the same recessive disease will mate. Mating
outside your ethnic group decreases the chance that two people who are carriers for the same
recessive disease will mate.
Hardy-Weinberg approach also works with X-linked genes with this modification. For males only,
with an X-linked gene, p = the frequency of the dominant phenotype and q = the frequency of
the recessive phenotype since males only have one allele for an X-linked trait. Thus, if 0.01 =
1/100 is the frequency of the X-linked recessive gene for hemophilia, the frequency of hemophilia
in women will be approximately = q2 = (1/100)(1/100) = 1/10000 = 0.0001 and the frequency of
hemophilia in men will be approximately = q = (1/100) = 0.01
Biology 30
Hardy-Weinberg Problems
Instructions: Using a calculator, solve the following problems. Rounding off to the nearest whole
number or 1/10 when solving for p or q. If you do not have a calculator, set the problem up
without finding the square root. Use the following equations:
2
2
p + q = 1 or 100%.............. p + 2pq + q = 1 or 100%
1. Assume the Rh negative blood factor to be recessive. If 16% of the U.S. population is Rh
negative calculate the following:
a. What is the frequency of the q allele?
b. What is the frequency of the p allele?
2. Assume brown fur in guinea pigs to be dominant and white to be recessive. The allele for
brown fur = 0.8. Calculate the following:
a. The frequency of the white allele.
b. The expected frequency of homozygous dominants.
c. The expected frequency of heterozygotes.
d. The expected frequency of homozygous recessives.
3. People who can taste PTC are homozygous recessive. If 25% of the population can taste PTC
calculate the following:
a. The frequency of the p allele.
b. The frequency of the q allele.
c. The percent of heterozygotes.
d. The percentage of all tasters.
4. Answer the following:
a. If q = 0.2, calculate the allele frequency of p.
b. How many people out of 8000 will be homozygous dominant?
c. How many people out of 8000 will be heterozygous?
d. How many people out of 8000 will be homozygous recessive?
5. Assume resistance to be dominant. If 96% of a bacterial population is resistant to an antibiotic
calculate the following:
a. The frequency of the recessive allele.
b. The frequency of the dominant allele.
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c. The number of resistant bacterial in a population of 2 x 10 .
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d. The number of heterozygotes in the population of 2 x 10
6. 86 out of every 100 fruit flies have normal wings. Assume normal wings are dominant over
curly wings. Calculate the following:
a The frequency of the recessive allele.
b. The frequency of the dominant allele.
c. The number of offspring with normal wings in a population of 1000.
d. The number of flies in 1000 that would exhibit the recessive phenotype.
e. The number of flies, in 1000, that would be homozygous dominate.
f. The number of flies, in 1000, that would carry the recessive allele.
Equilibrium and Speciation
Questions:
1. What is the Hardy-Weinberg equilibrium?
2. What is the relationship between population, genes and gene pools?
3. What are the five assumptions of the Hardy-Weinberg equilibrium?
4. If all of these assumptions are met, a population is in equilibrium. What does this mean in
terms of evolution?
5. Give the equations that Hardy and Weinberg generated and describe what each of the terms
denotes and what each equation function is.
6. If the frequency of allele A = .4 and the frequency of the allele a = .6 in the population where
A yields dark hair and a yields light hair, what will the genotypic frequencies of homozygous
recessives, heterzygotes, homozygous recessives be in the next generation? What will the
frequency of dominant phenotypes be? Recessive?
7. Define genetic drift.
8. Define the phrase variation in traits of a population.
9. Why is variation so important to the success of any population?
10. What is the bottleneck effect? Give an example. Explain why the bottleneck effect is an
example of genetic drift and how it causes evolution to occur.
11. Explain how a population bottleneck decreases variation within a population.
12. What is the founder effect? Give an example. Explain why the founder effect is an example
of genetic drift and how it can cause evolution to occur.
13. Explain how the founder effect decreases variation within a population.
14. Define gene flow. Explain how gene flow can cause evolution to occur.
15. Describe what a mutation is. Where in an animal must a mutation occur in order for it to be
passed on to offspring? When does a mutation cause evolution to occur?
16. Why is mutation in and of itself not evolution? What must happen for evolution to occur?
17. Is a mutation a response to environmental pressure? Explain
18. Why is mating never completely random? What is assortative mating?
19. What are sexual dimorphism and sexual selection? Using peacocks as an example,
explain how sexual dimorphism evolved in many species of animals and has directed the
evolution of that species. Do humans display sexual dimorphisms? Why? Do seastars?
Why?
20. Define natural selection and explain how it causes evolution to occur.
21. What are the three types of natural selection? Define them and give an example. How do
they differ from one another? Which one do you think would lead to two new species being
created? Explain.
22. What happens to gene frequencies in a population when the environment is changing?
When it is stable?
23. What does it mean to be selected for or selected against in an evolutionary sense?
24. What is a species?
25. The process of speciation is an example of divergent evolution. Using the steps described
in your text start with on ancestral species and describe how it may become separated into
two or more new species. (Be sure to include both geographic and reproductive isolation in
your description of the process).
26. What two things must happen for a new species to form?
27. How do species become isolated?
28. How does genetic divergence take place?
29. Compare and contrast micro and macro-evolution.
30. Your book says speciation is at the boundary between microevolution and macroevolution.
Explain.
Questions on Populations
1. Define population, community, ecosystem, and biosphere, indicating how each is related to the
others.
2. Define population density. Give two methods biologists use to estimate population densities
and distinguish between uniform, clumped, and random distributions, and indicate the conditions
under which one is the most common.
3. Draw an exponential growth curve(J-shaped curve). Write the equation for the intrinsic rate of
increase, defining all the terms used in the equation.
4. Draw a logistic growth curve(S-shaped curve), and label the carrying capacity, the inflection
point, the portion of the curve showing an accelerating rate of population growth, and the portion
showing a decelerating rate. Compare this curve with the exponential curve. Then explain what is
meant by zero population growth, and describe how this condition is reached.
5. Explain how density-dependent and density-independent factors operate in limiting population
growth.
6. On a single graph draw type I, type II, and type III survivorship curves. Explain each curve
briefly at the bottom of the graph.
7. Construct a table showing the differences between r-selected species and K-selected species
with respect to body size, life-span, number of offspring, relative time of reproduction (earlier or
later in life), type of survivorship curve, type of growth curve (S-shaped or boom-and-bust).
8. Contrast r and K species, give examples of each to illustrate your points.
9. Using examples, discuss the ways in which parasitism, predation, intraspecific competition,
emigration, mutualism, and physiological and behavioral mechanisms can act as densitydependent limitations on population growth. Explain, using an example, how destroying the
balance between predator and prey in a community can upset the ecology of an area.
10. What are the five properties of communities?
11. Carefully define the concept of ecological niche, and explain its significance with respect to
the competitive exclusion principle. Specify the three possible results of intense interspecific
competition.
12. Discuss, using an example, the relationship between species diversity and complexity and
community stability. Describe the effect of human intervention in biological communities.
13. Describe the process of ecological succession, indicating why the species in a given area
change over time. Distinguish between primary and secondary successions, and give an example
of each. Also, summarize the trends seen in many successions, and explain what is meant by a
climax community.
14. Explain the types of interspecies relationships and tell how each member of the pair is
affected by the interaction (include predation, parasitism, commensalism and mutualism).
Hardy-Weinberg Problem Set 2
1. List conditions for determining the stability of a gene pool:
2 Define
a) Allelic Frequency:
b) Individual Frequency:
3. People with type dominant trait p is 84% while 16% of the people have the recessive q. Find
the allelic frequency and the frequency of individuals that are homozygous dominant and
heterozygotes (or carriers).
4. In the plant Phlox the alcohol dehydrogenase (ADH) locus shows two alleles a and b. The
following genotypic frequencies were found in a population; aa = 0.05, ab = 0.35, bb = 0.60.
Calculate the allele frequencies (assume a is dominant.)
p=
q=
5. a)What is the frequency of heterozygotes (Aa) in a population if the frequency of recessive
phenotypes (aa) is 0.8 and the population is in H-W equilibrium? Assume equilibrium, so
expected heterozygote frequency = observed. First, find q (freq of a) by noting that under H-W
conditions, observed frequency of aa, 0.8, is equal to q^2.
Therefore, q = _____ and p = _____. And the Heterozygous or 2pq = _____.
b) Assuming the population is in H-W equilibrium, explain how two evolutionary mechanisms
could be operating to counter one another to keep the population in equilibrium.
6. What is the frequency of heterozygotes (Aa) in a population in which the frequency of all
dominant phenotypes is 0.25 and the population is in H-W equilibrium?
7. Seventy percent of a population have the dominant trait while 30% are recessive. What
percentage of the population are carriers?
8. For foxes there exists a single gene that controls coat thickness. Allele C confers a thick coat
while allele c a thin coat. In a certain population of 540 foxes, 49 have a thin coat. What are the
allelic and individual frequencies?
Challenge Questions For Fun
In a sample of minnows from a local stream, three genotypes controlled by 2 alleles (a1 & a2) at
one esterase locus showed the following numbers in the population; a1a1 = 10, a1a2 = 75, and
a2a2 = 15. Are these numbers what you would expect if this population were in Hardy-Weinberg
equilibrium? If not, is one microevolutionary mechanism more likely to be acting than others,
based on the data?
Hardy-Weinberg Problem Set 3
1. Consider a locus A with two alleles, A and a. If the frequency of AA = 0.36, what are the
expected frequencies of the three genotypes under Hardy-Weinberg assumptions?
What are the expected gene frequencies?
2. Suppose 25 out of 750 students are redheads. What is the frequency of redheads?
If a random student is chosen, what is the probability that he/she is a redhead?
3. If the genotypes AA, Aa, and aa have frequencies of 0.5, 0.25, and 0.25 (respectively), what is
p = f(A) =? What is q = f(a) = ? After a single generation of random mating, what is the expected
frequency of AA, AA, and aa?
4. In a human population, 16% of the people tested were found to be Rh-/Rh-, a recessive
condition. What fraction of the Rh+/ population are homozygous?
What fraction of children from a large group of families where both parents are Rh+ would be
expected to be Rh-?
5. What is the frequency of the recessive gene in a Hardy-Weinberg population in which there are
10 times as many heterozygotes as homozygous recessives?
6. Calculate the gene frequency of the recessive gene a (q) at which crosses between two
heterozygotes would produce the same proportion of recessives as crosses between two
recessives (assume Hardy-Weinberg conditions).
7. If 160 PTC tasters (tasting is dominant; non-tasting is recessive) are discovered in a population
of 200 individuals, is this population in Hardy-Weinberg equilibrium?