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The Hardy-Weinberg Model Advanced
Douglas Wilkin, Ph.D.
Barbara Akre
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Printed: December 21, 2015
AUTHORS
Douglas Wilkin, Ph.D.
Barbara Akre
www.ck12.org
C HAPTER
Chapter 1. The Hardy-Weinberg Model - Advanced
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The Hardy-Weinberg Model
- Advanced
• List the five conditions for genetic equilibrium according to the Hardy-Weinberg model.
• State and explain the generalized equation for Hardy-Weinberg equilibrium.
• Explain how to use the Hardy-Weinberg equation to solve for allele or genotype frequencies.
Why is balance important?
To these individuals, the importance of maintaining balance is obvious. If balance, or equilibrium, is maintained
within a population’s genes, can evolution occur? No. But maintaining this type of balance today is difficult.
Populations at Equilibrium: The Hardy-Weinberg Model
Like Darwin and Wallace, who independently developed similar ideas about evolution and natural selection, mathematician Godfrey Hardy and physician Wilhelm Weinberg independently developed a model of populations at
equilibrium. The Hardy-Weinberg model (sometimes called a law) states that a population will remain at genetic
equilibrium - with constant allele and genotype frequencies and no evolution - as long as five conditions are met:
1.
2.
3.
4.
5.
No mutation.
No migration.
Very large population size.
Random mating.
No natural selection.
We will consider the above in more detail in later sections of the concept because deviations from these conditions
are the causes of evolutionary change. For now, let’s look more closely at the Hardy-Weinberg equilibrium model.
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We’ll use another hypothetical rabbit population to make the model concrete: 9 albino rabbits and 91 brown rabbits
(49 homozygous and 42 heterozygous). The gene pool contains 140 B alleles and 60 b alleles –gene frequencies of
0.7 and 0.3, respectively. The Figure 1.1 summarizes the data for this parent population.
FIGURE 1.1
The coat color gene pool for a hypothetical rabbit population includes two alleles.
Genotype and allele frequencies are calculated given the number of individuals
having each of the three possible genotypes.
If we assume that alleles sort independently and segregate randomly as sperm and eggs form, and we also assume
that mating and fertilization are random, then the probability that an offspring will receive a particular allele from
the gene pool is identical to the frequency of that allele in the population. A Punnett Square ( Figure 1.2) using these
frequencies predicts the probability of each genotype (and phenotype) in the next generation:
FIGURE 1.2
A Punnett Square predicts the probability
of each genotype and phenotype for the
offspring of the population described in
the Figure .
Row A summarizes the
frequencies of each genotype among offspring, and row B calculates allele frequencies for the next generation. Comparing these to the parent generation
shows that the gene pool remains constant. The population is stable; it is at a
Hardy-Weinberg equilibrium.
If we calculate the frequency of genotypes among the offspring predicted by the Punnett square, we see that they are
identical to the genotype frequencies of the parents. Allele frequency remains constant as well. The population is
stable –at a Hardy-Weinberg genetic equilibrium.
A useful equation generalizes the calculations we’ve just completed. The variables are as follows: p = the frequency
of one allele (we’ll use allele B here) and q = the frequency of the second allele ( b, in this example). We will use
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Chapter 1. The Hardy-Weinberg Model - Advanced
only two alleles, but similar, valid equations can be written for more alleles.
Recall that the allele frequency equals the probability of any particular gamete receiving that allele. Therefore,
when egg and sperm combine, the probability of any genotype (as in the Punnett square above) is the product of the
probabilities of the alleles in that genotype:
• Probability of genotype BB = p X p = p2 .
• Probability of genotype Bb= ( p X q) + ( q X p) = 2 pq.
• Probability of genotype bb = q X q = q2 .
We have included all possible genotypes, so the probabilities must add to 1.0. Our equation becomes this:
TABLE 1.1:
p2
frequency of
genotype BB
+
2 pq
frequency of
genotype Bb
+
q2
frequency of
genotype bb
=
1
This is the Hardy-Weinberg equation, which describes the relationship between allele frequencies and genotype
frequencies for a population at equilibrium.
Cystic Fibrosis
The equation can be used to determine genotype frequencies if allele frequencies are known or allele frequencies if
genotype frequencies are known. Let’s use a common human genetic disease as an example. Cystic fibrosis (CF) is
caused by a recessive allele ( f ) which makes a non-functional chloride ion channel, leading to excessive mucus in
the lungs, inadequate enzyme secretion by the pancreas, and early death.
Knowing that 1 in every 3,900 children in the United States is born with CF, we can use the Hardy-Weinberg equation
to ask what proportion of the population unknowingly carries the allele for cystic fibrosis. An individual who has
CF must have the genotype ff , because the allele is recessive. Using the value for the frequency of the homozygous
recessive genotype, we can calculate q, the frequency of the recessive allele, f:
1/3900 = 0.0002564 = frequency of ff genotypes = q 2
√
To find q, the frequency of allele f , we must take the square root of the frequency of ff genotypes: q = 0.0002564
= 0.016 = frequency of the f allele. Because p + q = 1 (the sum of the frequency of allele f and the frequency of
allele F must equal 1.0), p = 1 – q = 1.0 –0.0160 = 0.9840 = frequency of the F allele.
According to the Hardy-Weinberg equation, for a population at equilibrium the frequency of carrier genotypes, Ff ,
is 2pq = 2 X 0.016 X 0.984 = 0.0315.
In other words, if the population is indeed at equilibrium for this gene, just over 3% of the population carries the
recessive allele for cystic fibrosis. Of course, this calculation holds true only if the US population meets the five
conditions we listed at the beginning of this section. In nature, populations seldom satisfy all five criteria. Let’s
consider how well each condition describes the US population for the cystic fibrosis gene:
Very Large Population Size
Although the equation ideally describes an infinitely large population –never found in nature, of course –the US
population is probably large enough that this condition alone does not significantly disrupt equilibrium.
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FIGURE 1.3
Cystic fibrosis is an inherited disease of
the lungs and pancreas.
A recessive
allele of a gene located on chromosome
7 (the gene locus is colored red, right),
which codes for a chloride ion channel,
causes the disease. Treatment includes
using a ventilator (left) and antibiotic therapy.
No Mutation
Mutations happen constantly, if at a low rate, so “no mutation” is a second unrealistic condition. However, mutations
affecting any one particular gene are rare, so their effect on an otherwise large, stable population is small.
No Migration
This condition assumes no net additions or losses of either allele to the gene pool through immigration or emigration.
For the US population, immigration is probably more significant than emigration. Gene flow, in essence is the flow
of alleles into or out of a population, and may be the most significant problem for this particular gene, because
the frequency of the allele for cystic fibrosis varies greatly according to ancestry. Although 1 in 25 Europeans
carry the f allele, the frequency is just 1 in 46 among Hispanics, 1 in 60 among Africans, and 1 in 90 among
Asians. Therefore, disproportionate immigration by certain groups changes allele frequencies, destabilizing the
Hardy-Weinberg equilibrium.
Random Mating
Here is another assumption which is probably not realistic. Marriage between individuals of similar ancestry and
culture is still more common than intermarriage, although both occur. If marriages are not random, then the HardyWeinberg equilibrium does not apply.
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Chapter 1. The Hardy-Weinberg Model - Advanced
No Natural Selection
The final condition is that all genotypes must have an equal chance to survive and reproduce. Victims of cystic
fibrosis (genotype ff ) have shorter lifespans, which inevitably reduce reproduction compared to individuals without
the disease (genotypes FF and Ff ). Although medical care is improving, differential survival and reproduction
among genotypes means the gene pool is not at equilibrium.
With respect to the cystic fibrosis gene, the U.S. population fails to meet at least three of the criteria for equilibrium.
Therefore, the actual frequencies of alleles and genotypes probably deviate somewhat from those we calculated.
In nature, very few populations meet the Hardy-Weinberg requirements for equilibrium. If we look at this fact
from a different angle, we see that any of these conditions can destabilize equilibrium, causing a change in allele
frequencies. In other words, these five conditions are five major causes of microevolution. The remaining sections
of this lesson will explore each cause in detail.
A video explanation of the Hardy-Weinberg model can be viewed at http://www.youtube.com/watch?v=4Kbruik_L
Oo (14:57).
MEDIA
Click image to the left or use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/175
Vocabulary
• gene flow: The net movement of genes into or out of a population through immigration or emigration.
• genetic equilibrium: State of a population in which allele and genotype frequencies remain constant from
one generation to the next –a non-evolving population.
• Hardy-Weinberg model: Describes a population at genetic equilibrium, meeting five conditions: no mutation, no migration, very large population size, random mating, and no natural selection.
Summary
• Non-evolving populations have constant frequencies of alleles and genotypes –genetic equilibrium.
• In nature very few populations meet the Hardy-Weinberg requirements for equilibrium, due to constant mutation, gene flow, small populations, nonrandom mating, and environmental change.
• The Hardy-Weinberg model holds for equilibrium populations under 5 conditions:
1.
2.
3.
4.
5.
No mutation.
No migration.
Very large population size.
Random mating.
No natural selection.
• A generalized form of the Hardy-Weinberg equation for a gene pool of two alleles at equilibrium is:
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TABLE 1.2:
p2
frequency of
genotype BB
+
2 pq
frequency of
genotype Bb
+
q2
frequency of
genotype bb
=
1
where p is the frequency of one allele and q the frequency of the other allele.
Practice
Use this resource to answer the questions that follow.
• Hardy-Weinberg practice questions at http://www.k-state.edu/parasitology/biology198/hardwein.html
1. You have sampled a population in which you know that the percentage of the homozygous recessive genotype
(aa) is 36%. Using that 36%, calculate the frequency of the "A" allele.
2. There are 100 students in a class. Ninety-six did well in the course whereas four blew it totally and received
F’s. In the highly unlikely event that these traits are genetic rather than environmental, assume that these traits
involve dominant and recessive alleles. If the four (4%) represent the frequency of the homozygous recessive
condition, calculate the frequency of heterozygous individuals.
3. Within a population of butterflies, the color brown (B) is dominant over the color white (b). 9% of all
butterflies are white. Given this information, calculate the percentage of butterflies in the population that
are heterozygous.
4. A very large population of randomly-mating laboratory mice contains 36% white mice. White coloring is
caused by the homozygous recessive genotype, "aa". Calculate allelic and genotypic frequencies for this
population.
5. Cystic fibrosis is a recessive condition that affects about 1 in 2,500 babies in the Caucasian population of the
United States. Calculate the frequency of the dominant allele in the population.
6. In a given population, only the "A" and "B" alleles are present in the ABO system; there are no individuals
with type "O" blood or with O alleles in this particular population. If 200 people have type A blood, 75 have
type AB blood, and 25 have type B blood, what are the alleleic frequencies of this population (i.e., what are p
and q)?
Review
1. Describe genetic equilibrium, including the Hardy-Weinberg conditions.
2. In the United States, about 1 in every 12 black people of African descent carry one copy of the allele for sicklecell anemia. Assuming the five conditions of Hardy-Weinberg equilibrium hold, calculate the probability that
a member of the next generation will suffer from sickle-cell anemia (have the homozygous Hemoglobin S
genotype).
3. Discuss the reasons why the 5 conditions for Hardy-Weinberg equilibrium are probably not met by the black
population for the Hemoglobin alleles discussed in the question above.
References
1. Brown rabbit: Dean Sebourn; White rabbit: Benny Mazur. http://www.flickr.com/photos/dsebourn/591099
2556; http://www.flickr.com/photos/benimoto/603735193 . CC BY 2.0
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Chapter 1. The Hardy-Weinberg Model - Advanced
2. Brown rabbit: Dean Sebourn; White rabbit: Benny Mazur. http://www.flickr.com/photos/dsebourn/591099
2556; http://www.flickr.com/photos/benimoto/603735193 . CC BY 2.0
3. Army Medicine; Ensembl. http://www.flickr.com/photos/armymedicine/5126027490; http://commons.wik
imedia.org/wiki/File:Mucoviscidose.PNG . CC BY 2.0; The creator of this work allows anyone to use it
for any purpose including unrestricted redistribution, commercial use, and modification.
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