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BI 3010 H08
Population genetics
Halliburton Kap 1-3
• Population genetics concerns
the study of gene frequencies
and their variation in time and
space. In other words; the study
of evolution.
• Population genetics includes a
qualitative branch (treating
traits determined by genes at
single loci) and a quantitative
branch (treating traits
influenced by genes at many
loci (breeding genetics).
•
The most sentral theoreme is
Hardy-Weinberg’s law, which in a sense
is the "population version" of the
Mendelian laws of inheritance.
•
The materials studied are genetic
polymorfisms, and the methods leans
heavily towards mathematical models
and statistics tools.
•
Our starting question is: How are the
gene (allele) frequencies in populations
influenced by the 4 evolutionary forces
[mutations, genetic drift, gene flow
(immigration), and selection].
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Population genetics
Halliburton Kap 1-3
Genetic nomenclature (jargon):
Synonyms: Gene frequency = allele frequency = allelic proportion
The frequency of an allele (e.g. A,B,C) is often abbreviated qA, qB, qC, etc,
or alternatively, pA, qB, rC etc.
In general, the phenotypes (e.g. active isozymes, proteins) are written in normal
font (e.g. AA phenotype) while genotype is written in Italic (AA genotype). However,
there is no general consistency between textbooks in how this is handled.
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Population genetics is in itself a theoretical science based on mathematical models and
statistic analyses. However, it has a wide range of practical applications, such as:
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Plant- and animal breeding
DNA fingerprinting in forensic science
Habitat fragmentation and conservation of endangered species
Understanding the mechanisms of genetic differentiation and speciation
HUGO (the Human Genome Project)
Identification of genes related to disease in humans
The ancient history and evolution in man
Estimating effects of releasing GMO in nature
Identification and delimitation of biological units in resource management
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Population genetics
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The theoretical foundation was laid down by a.a.:
• Charles Darwin (natural selection theory)
• Gregor Mendel (pea crossings and segregation ratios)
• Hardy and Weinberg (HW-principle) [no photo]
• Ronald A. Fisher (Anova; Analysis of Variance)
• Sewall Wright (shifting balance, adaptive landscapes, FST, inbreeding coeff.)
• Theodosius Dobzhansky (Drosophila, speciation, hybrides/backcrossing)
• Richard Lewontin (elektrophoresis as a tool)
• Masatoshi Nei (genetic distance, (GST (multilocus FST))
• Motoo Kimura (neutral theory of genetic differentiation and evolution)[no photo]
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Some important themes:
1.
Types of genetic
polymorphisms
5. How are new traits formed?
2.
Variation within and between
populations
6. Adaptive landscapes
3.
Variation in qualitative and
quantitative traits
7. Interaction between the four
evolutionary forces
4.
Variation due to sexual
reproduction
8. Speciation; it's genetic basis
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Population genetics
Halliburton Kap 1-3
Populations are the real evolutionary units.
The raw material of evolution are mutations, which can accumulate with time
in populations and species, and result in multiple alleles at many loci.
Evolution can thus be defined as "any change in allele frequencies".
The frequencies of different alleles at a locus can be changed by the
4 evolutionary forces, which are:
1.
2.
3.
4.
Mutations
Random genetic drift
Gene flow (immigration)
Selection
If these forces are nullified, we have what is called a "Hardy-Weinberg population"; a panmictic (random mating),
statistically ideal population where the allele frequencies, and thereby the genotype frequencies, are constant and
do not change over generations (a socalled H-W equilibrium). The population genetic approach is to assume a
H-W equilibrium situation, and then study how the 4 evolutionary forces each, and in combinations, influence the
allele- and genotypic frequencies within and between
populations.
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Population genetics
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Underlying assumptions for the H-W law about the relation between
allele frequencies and genotypic proportions:
1.
2.
3.
4.
5.
Panmixi (random mating)
No mutations
No random genetic drift (i.e. infinitely large population)
No gene flow between populations (i.e. no immigration)
No selection (same fitness for all genotypes)
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Population genetics
Halliburton Kap 1-3
The Hardy-Weinberg law:
"In a panmictic, statistically ideal population, the genotypic
proportions is determined by the allele frequencies (p and q in
formula I below) at the locus according to the binomial
formula". [multinomial if more than two alleles; e.g. 3 in formula II)]
I (p+q)2 = p2 + 2pq + q2 (if only two alleles)
II (p+q+r)2 = p2 + 2pq + 2pr + q2 + 2qr + r2 (if 3 alleles)
[ The number of possible genotypes is n(n+1)/2 where n=number of alleles ]
The allele frequencies will be constant over generations and restore the same
genotypic proportions in each new generation. Allele- and genotype frequencies
can thus serve as population characteristics.
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Population genetics
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Methods for the study of frequencies of single genes
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Population genetics
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separation page
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Chi-squared ( 2 ) test for goodness-of-fit to H-W proportions
Assume that we have studied a locus with two alleles by electrophoresis. We have named the alleles
F and S according to their electrophoretic migration distance (Fast and Slow). There are three possible
genotypes (FF, SF og SS). In a sample of N=50 individuals from a natural population we observed 10 with
genotype FF, 28 with genotype SF, and 12 with genotype SS. We want to test if this genotype distribution
is reasonably close to the H-W expected values calculated from the observed allele frequencies in the
sample. For this we use a so-called chi-squared Goodness-of-fit test (table below). We use a table of
critical values of the chi-square distribution in some text book to check the significance level P of the
chi-square value calculated. The degrees of freedom (df) is caclulated as the number of genotypes minus
the number of alleles (i.e. df = 3-2 =1).
Observed
Expected
FF
10
(11.5)
FS
28
(25.0)
SS
12
(13.5)
N
50
(50.0)
Freq. F
0.48
Freq. S
0.52
Goodness-of-fit to Hardy-Weinberg expected distribution: chi-square = 0.742, df=1, P = 0.389,
(or P>0.10 according to the table on the next slide) i.e. not significant deviation between observed and
expected genotype distribution.
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Critical values of the
2 distribution
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Population genetics
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Effects of deviations from the assumptions of the H-W law
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1. Effect from the assumption of panmixia (random mating)
Deviations from random mating accur in natural populations. The most common effect of nonrandom mating is an increase in homozygosity ( = reduced heterozygosity) in the population
compared to the expected value under H-W equilobrium. Per definition this is inbreeding.
The probability for homozygosity by inheritance is called the Inbreeding coefficient.
(jfr path-analysis in Fig. 24-6 and Box 24-5). In principle two types of situations:
a) – related individuals prefer each other or avoid each other. If two parents are more related
than the average in the population, inbreeding is taking place. If less related than the average,
we have outbreeding.
b) – Phenotypically similar individuals for a specific trait (which have a genetic component) prefer
each other at mating. Examples of this in humans can be skin colour or body length (too many
homozygotes is formed each generation compared to H-W expectations. This is called positive
assortative mating, and will increase the inbreeding for the trait. The opposite tendency, negative
assortative mating, will increase outbreeding (reduce inbreeding) for the trait in the population.
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Populasjonsgenetikk
Halliburton Kap 1-3
2. Effect of deviation from the assumption of no mutation
Point mutations (basepair changes which cause amino acid substitutions) are the real source of genetic
variation, and the raw material of evolution. They occur randomly but with an average frequency which
appears to vary between loci. However, it is difficult to evaluate how large (or small) the difference
betwen loci is. For example, mutations in particularly important proteins (or parts of proteins) will less
likely ”survive” until they can be detected in a functional phenotype. This would lead to an underestimation of the true mutation rate at such loci. However, mutation rates are for the most so low that
they rarely interfere with the practical application of the Hardy-Weinberg principle on natural
populations, e.g. the use of observed allele frequency to characterize populations.
With respect to loss of genetic variability due to random genetic drift in isolated populations, mutations
can reduce or prevent such loss. As a rule of thumb, the mutation rate ( µ ) and the population size
N must satisfy the following equiation in order for level of genetic variation to be maintained:
Nµ≥1
If the mutation rate is 0.000001, the effective population size must thus be at least one million to prevent
that the loss by genetic drift (measured as lowered heterozygosity) is not larger than the gain by new
mutations.
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3. Effect of deviation from no random genetic drift (i.e. infinitely large population size)
It can be useful to look at sexual reproduction as a process where eggs and sperm from all potential
parents are coloured marbles in a bucket. They are then drawn two by two to determine each offspring's
genotype at a specific locus.
If all matings give the same number of offspring the frequency of each allele in the offspring (F1) generation
will tend to be the same as in the parental generation (P0). However, the individuals which become parents
are usually a subsample of the whole gene pool, which introduces an inaccuracy on their representability
for the population. Repeated subsampling may give different outcomes although their mean in a large number
of outcomes is expected to be equal to the parent generation's allele frequencies.
Mathematically this process is called binomial sampling (when two alleles), and the spread around the mean
frequency is called the binomial variance [(p(1-p)/2N in the formula below]. The Standard Error (SE) for an allele
frequency p in the F1 generation is the square root of the binomial variance and hence calculated as:
which shows that both the allele frequency and the mating population size have effect on the stability (or the
variability) of allele frequencies over generations. The formula shows that allele frequencies (particularly those
around 0.5) in small populations will tend to fluctuate considerably between generations.
This phenomenon is called random genetic drift.
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Evolutionary forces: Examples
Drosophila sp.: Genetic drift.
Ne = 16 (8 males & 8 females)
Random drift of allele frequencies during 19 generations
in a large set of small populations (effective population
size (Ne) is 16 individuals) of the fruit fly Drosophila sp.
The locus under study had two alleles at intermediate
allele frequencies in all populations at the start of the
experiment. The portion of populations which become
fixed for one or the other of the two alleles increased
steadily with time.
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4. Effect of deviation from the assumption of no gene flow (no immigration)
Immigrants from populations with different allele frequencies will (when genetically effective) cause disturbance to the stability of the local H-W
frequencies. The frequencies in the local population (the recipient) will change towards that of the immigrant (the donor), to a degree that is
proportional to the relative number of immigrants. The effect will accumulate over generations. Let the local allele frequency be p and that of the
immigrant q. If the immigrant constitute a proportion m in the local reproduction, the change in local allele frequency is described by:
p = m(q-p)
which shows that the amount of change depends both on the relative number of immigrants and on the difference in allele frequency between
donor and recipient. This change will, according to H-W, result in a change in local genotypic proportions as well.
Sometimes a sample may contain a physical mixture of individuals from two or more populations with different allelic and genotypic frequencies.
When counting up genotypes in such a mixture there will be too few heterozygotes compared to what is expected from the allele frequencies and
the H-W law. This phenomenon (i.e. the deficiency) is called the "Wahlund effekt", and is actually a tool for detecting such physical mixtures.
The magnitude of the deficiency is positively correlated with the difference in allele frequencies and the relative size of the involved groups.
With respect to genetic differentiation between populations, immigration (gene flow) will counteract the effect of genetic drift in local populations.
As a rule-of-thumb, the magnitude of immigration sufficient to counteract substantial local genetic differentiation must in each generation be:
m
≥
1/N
which, re-arranged as mN ≥ 1, shows that it is in fact the absolute number of immigrants (Nm) which is important (because it is the differentiating
effect of genetic drift which is to be counteracted, and that effect is largest when N is small and vice versa). However, even though an
immigration of one individual per generation is sufficient to counteract fixations of different alleles in the populations, it will not prevent that
substantial differences in allele frequencies can arise and sustain between populations.
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Population genetics
Halliburton Kap 1-3
5. Effect of deviation from the assumption of no selection
Selection is capable of causing large changes in allele frequencies between generations,
and give rise to genotypic proportions that deviate substantially from H-W expectations.
In order to treat selection, we shall first define and show how to estimate two different population
parametres, the fitness-coefficient (w) and the selection coefficient (s).
For this purpose we assume a very simple scenario with three genotypes at one locus, and fitness
calculated from relative survival only (there are of course many other possible components in
individual fitness).
On this locus, we count the number of the different genotype before and after selection
(e.g. by birth and just before reproducion). Cf table next page.
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Box 1
Population genetics
Halliburton Kap 1-3
Selection
Calculation of fitness- and selection coefficients for survival (i.e. partial coeffs.)
Before selection
After selection
Survival
Relative survival
Fitness coefficient (w)
Selection coefficient (s=1-w)
FF
250
180
180/250=0.72
0.72/0.80=0.9
0.9
1.0 - 0.9 = 0.1
SF
500
400
400/500=0.80
0.80/0.80=1.0
1
1.0 - 1.0 = 0.0
SS
250
120
120/250=0.48
0.48/0.80=0.6
0.6
1.0 - 0.6 = 0.4
N
1000
700
The table shows an example of selection of the type called "Balanced selection" or "Balanced polymorphism",
or "Overdominance", in which the heterozygote has the highest fitness coefficient (survives best).
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Population genetics
Halliburton Kap 1-3
Selection con't.
The efficiency of selection, measured as the change in allele frequency per generation, depends not only
on the size of the selection coefficients, but also on the allele frequency itself (the change per generation is largest
for allele frequencies around 0.5). This can be seen from the formula for the average fitness for the population
for a single locus trait, where the allele frequencies are incorporated as follows:
Wmean = p2WFF + 2pqWFS + q2WSS
From this formula we can derive the "mean fitness" for each allele as:
WF-mean = pWFF + qWFS, and WS-mean = pWFS + qWSS
After some algebra the above formulas can be combined to give p; the change in allele frequency per generation
due to selection:
p = pq[WF-mean – WS-mean] / Wmean
which states that the speed of change in allele frequency per generation is proportional to the frequency (pq)
of heterozygotes, which in turn is largest at allele frequencies around 0.5. At extreme allele frequencies
(approaching 0 or 1), the change per generation will be small.
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The three main type of selection:
Selection is often divided in three types based on the effect is has on the genotypic and
allelic frequencies at a locus (or on the mean value and variance for quantitative traits).
1. directional selection
2. stabilizing selection
3. disruptive selection
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Relevant computer simulations of the effect of selection on population allele frequencies
can be performed e.g. with the PopG.exe Windows programme, or alternatively with the
P14.exe DOS programme available at It's learning. The following slides are made with the
latter.
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Halliburton Kap 1-3
Type 1: Directional selection (single locus):
("Screen-dump" from the computer programme P14.exe (J. Mork, TBS). X-axix: number of generations, Y-axis: frequency
of the F allele). The line graph describes the change in frequency of the favoured allele (qF) during a directional selection
under the assumptions for e.g. relative fitness coefficients given in the panel above the graph.
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Population genetics
Halliburton Kap 1-3
Type 2: Stabilizing selection (balanced polymorphism, overdominance)
("Screen-dump" from the computer programme P14.exe (J. Mork, TBS). X-axis: number of generations, Y-axis: frequency of the
favoured allele F). The line graph shows the development of qF with overdominance (i.e. the heterozygote SF has the highest
fitness). The selection coefficients s2 and s1 (i.e. 1 minus the fitness coefficients) for the homozygotes FF and SS are 0.1 og 0.4.
respectively- The allele frequency of F (which is 1 minus qS) stabilizes at qF = 0.8 as expected by the formula for its equilibrium
value qF = s1/(s1 + s2) = 0.4/(0.4 + 0.1) = 0.8. NB! Random genetic drift is omitted in the simulation, hence the smooth curve.
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Population genetics
Halliburton Kap 1-3
Type 3: Disruptive selection (underdominance)
(("Screen-dump" from the computer programme P14.exe (J. Mork, TBS). X-axis: number of generations, Y-axis: frequency of the
favoured allele F). The graph shows outcomes from 10 independent simulations of allelic loss due to disruptive selection (i.e.
when heterozygote fitness is lower than for both the homozygotes). Here the simulations incorporated effects of random genetic
drift (hence the ragged curve).
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A well-known example of a balanced polymorphism:
The maps show the similarity of the geographical
distribution of the human haemoglobin allele causing
sickle-cell anemia, and the distribution of the malaria
blood parasite.
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Population genetics Halliburton Kap 1-3
Box 2
Chi-squared test of genetic differences between samples
To test for differences in allelic or genotypic proportions between samples we use an RxC (rows by columns) chi-square contingency test. We
usually test for both genotypic and allelic heterogeneity between samples. The latter is statistically the most powerfulI one. Consider two samples
of N=100 each, and one polymorphic locus with two alleles (A og B):
Genotypes_______
Samples
AA
AB
BB
N
------------------------------------------------------------------------------------------------Sample 1
36 (26)
48 (48)
16 (26)
100
Sample 2
16 (26)
48 (48)
36 (26)
100
------------------------------------------------------------------------------------------------Total
52
96
52
200
=======================================================
Our "Null hypothesis" for test is that the samples are drawn from one and the same population. If so, our best estimate of the true genotype and
allele distribution in the materials is found in the "Total". We therefore use the distribution in the "Total" to estimate what "should have been" in the
two single samples (we "forget" about the H-W distributions in this test).The expected number of AA genotypes in Sample 1 is then, e.g.,
((52/200)*100)=26. As in all chi-square tests we find the test observator in this way: Take the square of the difference between observed and
expected value, and divide it by the expected value. Do this for all genotypes in both samples, and sum the results. The number of degrees of
fredom in a contingency table (RxC) is calculated differently from the "H-W Goodness-of-fit". Degrees of fredom is here calculated as (R-1)(C-1).
For the genotypic values in the table above, we find a chi-square value of 15.38, and 2 degrees of freedom. In a textbook table of the chi-square
distribution we find that this corresponds to a significance level of P < 0.001. (actually the exact P-value is 0.00046). For testing for homogeneity
of allelic proportions in the same two samples, the RxC table will look like this:
Allele
_
Sample
A
B
N
------------------------------------------------------------------------------------------------Sample 1
120 (100)
80 (100)
200
Sample 2
80 (100)
120 (100)
200
------------------------------------------------------------------------------------------------Total
200
200
400
=======================================================
The chi-square for this table is 16.00. However, here we have only one degree of freedom. Therefore the P-value for this outcome (P=0.00006) is
lower than in the genotype test. In both tests, the null hypothesis can be safely rejected. Our samples are not drawn from the same population.
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Halliburton Kap 1-3
Genetic differentiation and genetic structure
Sewall Wright’s Fst (a relative measure of differentiation):
Evolution can be defined as any change in population allele frequencies. The evolutionary forces that can bring about allele frequency changes
(mutations, random genetic drift, gene flow, and selection) can, given time, lead to different allele frequencies in the populations. Several models
have been suggested to describe this process. One of the best known and most frequently employed is Sewall Wright's "Mainland-Island" model.
It assumes a start ("Mainland") population which is split in a large number of sub-populations ("Island populations"), and describes the genetic
differentiation which takes between these in formulas which considers e.g. sub-population sizes, migration rates, and number of generations.
Wright uses a specific measure – FST – for the degree of differentiation. The calculation of FST is based on heterozygosities (H), which can be
calculated for groups at different hierarchical levels (sub- and total population).
The observed heterozygosity (HO) at a locus in a group is simply the proportion of heterozygote genotypes, while the expected heterozygosity is
calculated from the allele frequencies assuming H-W equilibrium (HE = 1- xi2). The calculated average H at a locus is usually denoted with a "bar"
above it, and is the arithmetic mean of H at all included loci (monomorphic as well as polymorphic):
Fst = 1 - [Hmean / Htotal ]
where Hmean is the arithmetic mean of the single locus heterozygosities in all subpopulations, while H total is the expected (H-W) heterozygosity
based on the joint allele frequency in the total material. It is apparent that F st is 0 when all the subpopulations are genetically identical, and 1 when
they are fixed for different alleles. Masatoshi Nei suggested later an analogous measure (G ST) which can be calculated fro sample allele frequencies
rather than genotype frequencies, assuming that all sub-populations are in H-W equilibrium. GST can be calculated as a mean over many loci.
Nei’s I and D (absolute measures of genetic similarity and differentation):
Masatoshi Nei also suggested another measure, D, (“Genetic Distance”), which gives an estimate of the absolute genetic differences between taxa
(" the mean number of amino acid substitutions per locus"). This measure utilises allele frequencies at many loci, and is calculated for each locus
via the quantity I ("Genetic Identity") according to this formula:
I = xiyi / SQRT[ (xi2)(yi2)], and so D = - ln(I)
where xi and yi are frequency of the i-th allele in population X and Y, respectively (SQRT means "square root"). The overall D is
calculated as the arithmetic mean over the actual number of loci genotyped.
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Practical calculation of FST
from genotype data
Population genetics
Halliburton Kap 1-3
Box 3
Consider a number of samples from different parts of the distribution area of a species. By electrophoretic analyses the individual
genotypes at a particular locus are determined. This locus has two alleles A and B| which segregates in three genotypes AA, AB, and
BB. After the analyses the follwing table can be constructed (only the F ST relevant figures are included).
Samples
AA
AB
BB
N
qA
qB
H
-----------------------------------------------------------------------------------------------------------------------------------------------------------------Sample 1
17
46
37
100
.46
Sample 2
89
62
9
160
.39
Sample3
48
104
48
200
.52
-----------------------------------------------------------------------------------------------------------------------------------------------------------------Mean Hobs
.46
-----------------------------------------------------------------------------------------------------------------------------------------------------------------Total
154
212
94
460
.57
.43
.49
============================================================================================
Mean observed heterozygosity (Hmean) for the three samples is calculated as [( 0.46+0.39+0.52 ) / 3)], giving 0.46.
Expected (Hardy-Weinberg) heterozygosity in the total material (Htotal) is calculated based on the allele frequencies, and is in this
case: (2*0.57*0.43 =) 0.49.
Fst = 1 - Hmean / Htotal
Inserting these values in the formula gives in this case an FST value of [ 1 – (.46/.49) ] = 0.06.
The measure, FST,comes from a decomposition of the total genetic heterogeneity into a "within" and a "between" component. In this
example, 6% (i.e. fraction 0.06) of the total genetic heterogeneity is due to differences between samples. The rest (94%) of the
heterogeneity is found within samples (i.e. between individuals). The calculations above follow Sewall Wright's original concept for
one locus. For multi-locus calculations (e.g. Nei's GST) one uses the expected (i.e. not observed) heterozygosity both for subsamples and the total. Masatoshi Nei's G-statistics (GST) is an extension of FST to many alleles and many loci. The main difference in
calculation is that it uses HW-expected heterozygosities in the subpopulations (here: samples). The overall GST is calculated as the
average over all loci.
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Useful software:
Hweq2.exe (Hardy-Weinberg chi-square goodness-of-fit test. 2 alleles, 3 genotypes)
ChiRxC.exe (chi-square contingency table test; Rows by Columns)
PopG.exe (interactive simulation of evolutionary forces, graphic display)
-------------------These programmes will be uploaded to It’s learning
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Last slide
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