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Population Genetic
Calculations
Mr. Nichols
PHHS
Fitness
• A fundamental concept in evolutionary theory is “fitness”,
which can defined as the ability to survive and
reproduce. Reproduction is key: to be evolutionarily fit,
an organism must pass its genes on to future
generations.
• Basic idea behind evolution by natural selection: the
more fit individuals contribute more to future generations
than less fit individuals. Thus, the genes found in more
fit individuals ultimately take over the population.
• Natural selection requires 3 basic conditions:
– 1. there must be inherited traits.
– 2. there must be variation in these traits among members of the
species.
– 3. some inherited traits must affect fitness
Genetics of Populations
• A “population” is a group of organisms of the same
species that reproduce with each other. There is only
one human population: we all interbreed.
• The “gene pool” is the collection of all the alleles
present within a population.
• We are mostly going to look at frequencies of a single
gene, but population geneticists generally examine
many different genes simultaneously.
Allele and Genotype Frequencies
• Each diploid individual in the population has 2 copies of each
gene. The allele frequency is the proportion of all the genes in
the population that are a particular allele.
• The genotype frequency of the proportion of a population that is a
particular genotype.
• For example: consider the MN blood group. In a certain
population there are 60 MM individuals, 120 MN individuals, and
20 NN individuals, a total of 200 people.
• The genotype frequency of MM is 60/200 = 0.3.
• The genotype frequency of MN is 120/200 = 0.6
• The genotype frequency of NN is 20/200 = 0.1
• The allele frequencies can be determined by adding the
frequency of the homozygote to 1/2 the frequency of the
heterozygote.
• The allele frequency of M is 0.3 (freq of MM) + 1/2 * 0.6 (freq of
MN) = 0.6
• The allele frequency of N is 0.1 + 1/2 * 0.6 = 0.4
Heterozygosity and Polymorphism
• A gene is called “polymorphic” if there is more than 1
allele present in at least 1% of the population. Genes
with only 1 allele in the population are called
“monomorphic”. Some genes have 2 alleles: they are
“dimorphic”.
• In a study of white people from New England, 122
human genes that produced enzymes were examined.
Of these, 51 were monomorphic and 71 where
polymorphic. On the DNA level, a higher percentage of
genes are polymorphic.
• Heterozygosity is the percentage of heterozygotes in a
population. Averaged over the 71 polymorphic genes
mentioned above, the heterozygosity of this population
of humans was 0.067.
Hardy-Weinberg Equilibrium
• Early in the 20th century G.H. Hardy and Wilhelm Weinberg
independently pointed out that under ideal conditions you could
easily predict genotype frequencies from allele frequencies, at
least for a diploid sexually reproducing species such as humans.
• For a dimorphic gene (two alleles, which we will call A and a), the
Hardy-Weinberg equation is based on the binomial distribution:
p2 + 2pq + q2 = 1
where p = frequency of A and q = frequency of a, with p + q = 1.
• p2 is the frequency of AA homozygotes
• 2pq is the frequency of Aa heterozygotes
• q2 is the frequency of aa homozygotes
Hardy-Weinberg Example
• Taking our previous example population, where
the frequency of M was 0.6 and the frequency of
N was 0.4.
• p2 = freq of MM = (0.6)2 = 0.36
• 2pq = freq of MN - 2 * 0.6 * 0.4 = 0.48
• q2 = freq of NN = (0.4)2 = 0.16
• These H-W expected frequencies don’t match
the observed frequencies. We will examine the
reasons for this soon.
Necessary Conditions for HardyWeinberg Equilibrium
• The relationship between allele frequencies and genotype
frequencies expressed by the H-W equation only holds if these
5 conditions are met. None of them is completely realistic, but
all are met approximately in many populations.
• If a population is not in equilibrium, it takes only 1 generation
of meeting these conditions to bring it into equilibrium. Once
in equilibrium, a population will stay there as long as these
conditions continue to be met.
–
–
–
–
–
1. no new mutations
2. no migration in or out of the population
3. no selection (all genotypes have equal fitness)
4. random mating
5. very large population
Testing for H-W Equilibrium
• If we have a population where we can distinguish all
three genotypes, we can use the chi-square test once
again to see if the population is in H-W equilibrium. The
basic steps:
– 1. Count the numbers of each genotype to get the observed
genotype numbers, then calculate the observed genotype
frequencies.
– 2. Calculate the allele frequencies from the observed genotype
frequencies.
– 3. Calculate the expected genotype frequencies based on the HW equation, then multiply by the total number of offspring to get
expected genotype numbers.
– 4. Calculate the chi-square value using the observed and
expected genotype numbers.
– 5. Use 1 degree of freedom (because there are only 2 alleles).
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Example
• Wild oats is a common plant in California, the cause of the goldenbrown hillsides all summer out there.
• Wild oats can pollinate itself, but the pollen also blows in the wind
so it can cross fertilize. The task is to estimate the relative
proportions of these two types of mating.
• Data for the phosphoglucomutase (Pgm) gene:
– 104 AA, 9 AB, 42 BB = 155 total individuals
• H-W calculations:
– freq of A = 104 + 1/2 * 9 = 108.5 / 155 = 0.7
– freq of B = 1 - freq(A) = 0.3
–
–
–
–
exp heterozygotes = 2pq = 2 * 0.7 * 0.3 = 0.42 (freq) * 155 = 65.1
F = 1 -(obs hets) / (exp hets) = 1 - 9 / 65.1 = 1 - 0.14
F = 0.84
This is a very inbred population: most matings are self-pollination.
Inbreeding Depression and Genetic
Load
•
•
•
For most species, including humans,
too much inbreeding leads to weak
and sickly individuals, as seen in this
example of mice inbred by brothersister matings.
Inbreeding depression is caused by
homozygosity of genes that have slight
deleterious effects. It has been
estimated that on the average, each
human carries 3 recessive lethal
alleles. These are not expressed
because they are covered up by
dominant wild type alleles. This
concept is called the “genetic load”.
However, it has been argued that
some amount of inbreeding is good,
because it allows the expression of
recessive genes with positive effects.
The level of inbreeding in the US has
been estimated (from Roman Catholic
parish records) at about F = 0.0001,
which is approximately equivalent to
each person mating with a fifth cousin.
gen litter
size
0
7.50
% dead
by 4
weeks
3.9
6
7.14
4.4
12
7.71
5.0
18
6.58
8.7
24
4.58
36.4
30
3.20
45.5
Mutation
• Mutation is unavoidable. It happens as a result of
radiation in the environment: cosmic rays, radioactive
elements in rocks and soil, etc., as well as mutagenic
chemical compounds, both natural and artificially made,
and just as a chance event inherent in the process of
DNA replication.
• However, the rate of mutation is quite low: for any given
gene, about 1 copy in 104 - 106 is a new mutation.
• Mutations provide the necessary raw material for
evolutionary change, but by themselves new mutations
do not have a measurable effect on allele or genotype
frequencies.
Migration
• Migration is the movement of individuals in or
out of a population. Migration is necessary to
keep a species from fragmenting into several
different species. Even as low a level as one
individual per generation moving between
populations is enough to keep a species unified.
• Migration can be thought of as combining two
populations with different allele frequencies and
different numbers together into a single
population. After one generation of random
mating, the combined population will once again
be in H-W equilibrium.
Migration Examples
•
•
Population X has 20 individuals with frequency of the A allele = 0.8.
Population Y has 10 individuals with frequency of the A allele = 0.2. The
two populations mix. What is the frequency of A in the final population?
There are 20 + 10 = 30 individuals in the final population, for a total of 60
copies of the gene.
– For population X, 40 * 0.8 = 32 copies are A, and 8 are a.
– For population Y, 20 * 0.2 = 4 copies are A, and 16 are a.
– Adding these together, the final population has 32 + 4 = 36 A alleles and 8 + 16 =
24 a alleles. Out of 60 alleles, the frequency of A is 36/60 = 0.6
•
A real example: African Americans have a large proportion of African
ancestry, but also some European ancestry. The Duffy blood group has an
allele with a frequency of 0 among West African populations, and an
average frequency of 0.43 among European populations. Other blood
groups can also be used in this technique: very little assortative mating
occurs on the basis of blood group.
– In Oakland CA, African-Americans are reported to have about 22% European
ancestry
– In Charleston South Carolina, the proportion is about 3.7%
Selection
• Selection is the primary factor driving evolution. Genes that
confer increased fitness tend to take over a population. Note
that random events also play a big factor: sometimes a “good”
gene is lost due to chance events. Also, a gene that confers
increased fitness in one environment may confer decreased
fitness in another environment.
• Selection can occur at many places in the life cycle: the embryo
might be defective, the fetus might not survive to birth, the
immature offspring might be killed, the individual might not be
able to find a mate or might be sterile.
• We will simplify all of this by assuming that the gametes are
produced at random and combine at random, to produce a
population of zygotes in H-W equilibrium. Then, we will apply
selection to the zygotes, killing off different proportions of the
different genotypes.
• Fitness is a function of the genotype. We will define the “relative
fitness” of the best genotype as equal to 1.0, and the fitnesses of
the two other genotypes as equal to or less than 1.
Selection Against Recessive
Homozygote
•
•
•
•
•
•
•
This situation is what happens with a recessive genetic disease.
Heterozygotes and dominant homozygotes are indistinguishable and have
the same relative fitness: 1.0. The recessive homozygote has the genetic
disease and a fitness less than 1. The exact fitness depends on the nature
of the disease.
Start with a population where p = 0.6 and q = 0.4, and assume that the aa
homozygote has a relative fitness of 0.1 (i.e. 90% of the aa offspring die
without reproducing).
The zygotes produces (in H-W equilibrium) are 0.36 AA, 0.48 Aa, and 0.16
aa.
Selection on the zygotes reduces the aa’s by 90%, to 0.016.
However, proportions must add to 1.0, so we divide each proportion by a
correction factor. The correction factor is the sum of the remaining
proportions: 0.36 + 0.48 + 0.016 = 0.856.
So, after selection, the frequency of AA is 0.36 / 0.856 = 0.42. The
frequency of Aa is 0.48 / 0.856 = 0.56. The frequency of aa is 0.016 / 0.856
= 0.019.
Final allele frequencies: A = 0.42 + 1/2 * 0.56 = 0.70. a = 1 - freq(A) = 0.3.
Selection Favoring the
Heterozygote
• Some genes maintain 2 alleles in the population by
having the heterozygote more fit than either
homozygote.
• An example is HbS, the sickle cell hemoglobin allele. In
rural West Africa, where malaria is endemic and medical
support is rudimentary, the relative fitness of the HbA
homozygote is estimated at 0.85, due to susceptibility to
malaria. The relative fitness of the HbS homozygote is
estimated at approximately 0, with almost none reaching
reproductive age due to sickle cell disease. The
heterozygote is the most fit, so it given a relative fitness
of 1.0. Under these conditions, it is possible to predict
an equilibrium frequency of the HbS allele of about 0.13.
This is approximately what is seen in various West
African countries.
Genetic Drift
• Genetic drift is the random changes in allele frequencies. Genetic
drift occurs in all populations, but it has a major effect on small
populations.
• For Darwin and the neo-Darwinians, selection was the only force
that had a significant effect on evolution. More recently it has been
recognized that random changes, genetic drift, can also significantly
influence evolutionary change. It is thought that most major events
occur in small isolated populations.
• Simple example: A population of 1 female and 2 males, where the
female chooses only 1 male to mate with. Assume that the female
has the Aa genotype, male #1 is AA, and male #2 is aa.
– initially the allele frequencies are 0.5 A and 0.5 a
– if male #1 gets to mate, the offspring will have a 0.75 A, 0.25 a
frequency
– if male #2 mates, the offspring will be 0.25 A and 0.75 a.
Fixation of Alleles
•
•
Genetic drift causes allele
frequencies to fluctuate randomly
each generation. However, if the
frequency of an allele ever
reaches zero, it is permanently
eliminated from the population.
The other allele, whose frequency
is now 1.0, is “fixed”, which means
that all individuals in the
population will be homozygous for
that allele. This continues for all
future generations (in the absence
of mutation).
The average rate at which alleles
become fixed is a function of the
population size. The larger the
population, the longer it takes for
fixation to occur.
Population Bottlenecks and
Founder Effect
• Bottlenecks and the founder effect are closely
related phenomena.
• Founder effect: If a small group of individuals
leaves a larger population and develops into a
separate, isolated population, the allele
frequencies in the new population are
determined by the allele frequencies in the
founders. Since these frequencies are probably
different from those found in the general
population, the new population will have a
different set of frequencies.
• This is especially true for rare alleles, which can
suddenly become prominent if one of the
founders has the rare allele.
Founder Effect Example
•
•
Founder effect example: the Amish are
a group descended from 30 Swiss
founders who renounced technological
progress. Most Amish mate within the
group. One of the founders had Ellisvan Crevald syndrome, which causes
short stature, extra fingers and toes,
and heart defects. Today about 1 in
200 Amish are homozygous for this
syndrome, which is very rare in the
larger US population.
Note the effect inbreeding has here:
the problem comes from this recessive
condition becoming homozygous due
to the mating of closely related people.
Bottlenecks
• A population bottleneck is essentially the same phenomenon as the
founder effect, except that in a bottleneck, the entire species is
wiped out except for a small group of survivors. The allele
frequencies in the survivors determines the allele frequencies in the
population after it grows large once again.
• Example: Pingalop atoll is an island in the South Pacific. A typhoon
in 1780 killed all but 30 people. One of survivors was a man who
was heterozygous for the recessive genetic disease achromatopsia.
This condition caused complete color blindness. Today the island
has about 2000 people on it, nearly all descended from these 30
survivors. About 10% of the population is homozygous for
achromatopsia This implies an allele frequency of about 0.26.
Human Bottleneck
• The human population is
thought to have gone through
a population bottleneck about
100,000 years ago. There is
more genetic variation among
chimpanzees living within 30
miles of each other in central
Africa than there is in the entire
human species.
• The tree represents mutational
differences in mitochondrial
DNA for various members of
the Great Apes (including
humans).