Genotypic frequencies General formula:

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Genotypic frequencies -- General formula f(AA)
NAA/N -- gt 50/100 0.5 f(Aa) NAa/N -- gt
30/100 0.3 f(aa) Naa/N -- gt 20/100 0.2
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Allele Frequencies AA 50, Aa 30, aa
20 Note, every individual carries two copies of
the gene thus, the total number of alleles is
2N. p frequency of A and q frequency of
a. The frequency of A is p (50 50
30)/200 0.65
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Frequency of a is q (20 20 30)/200
0.35 Note p q 1 therefore, an equivalent
formula is p f(AA) 0.5f(Aa) and q
0.5f(Aa) f(aa)
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• Hardy-Weinberg Equilibrium under certain
• conditions, allele and genotypic frequencies will
• remain constant in a population from one
• generation to the next.
• Assumptions of Hardy-Weinberg Equilibrium
• Organism in question is diploid
• Reproduction is sexual
• Generations are non-overlapping
• Panmixia
• Population size is infinitely large, or at least
• large enough to avoid stochastic errors

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• Migration (immigration/emigration) is negligible
• No mutation
• Natural selection does NOT affect the gene
• under consideration
• Hardy-Weinberg equilibrium is simple but provides
• the basis for detecting deviations from random
• mating, testing for selection, modeling the
  effects
• of inbreeding and selection, and estimating
• allele frequencies.

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Single autosomal locus in a diploid organism
with discrete generations. Initially consider a
locus with only two alleles A and a with
initial frequencies p and q. Designate
frequencies of genotypes AA, Aa, and aa as P, H,
and Q, respectively. Random Union of Gametes
Many marine invertebrates release their gametes
into the sea and the gametes find one another and
combine at random.
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Testing for deviations from H.W.E H.W.E serves
as a null hypothesis and tells us what to expect
if nothing interesting is happening. If we
sample a population and find that the predictions
of H.W.E are not met, then we can conclude that
one or more of the assumptions is violated.
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Chi-square test of Goodness of Fit ?2
?(observed - expected)2/expected Example You
are studying a population of African elephants
and assay the entire population (N 260) for the
ADH locus and find that the population contains
only two alleles (F and f) with the
following genotypic counts FF 65, Ff 125,
ff 70
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Step 1 Determine allele frequencies p F
(65 65 125)/520 0.4904 q f 1 - p 1 -
0.4904 0.5096 Step 2 Calculate Expected
genotypic freq. P p2 (0.4904)2
0.2405 H 2pq 2(0.4904)(0.5096) 0.4998 Q
q2 (0.5096)2 0.2597
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Step 3 Calculate chi-square statistic
O E (O-E)2/E P 65 0.2405 X 260
62.53 0.098 H 125 0.4998 X 260
129.95 0.189 Q 70 0.2597 X 260
67.52 0.091 ?2 0.378 Step 4 Compare
calculated ?2 with tabled ?2 Degrees of
freedom 3( of genotypes) - 1(constant) - 1(
parameters) 1
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• Look up critical values for ?2 statistic
• Level of Significance
• D.f. 0.05 0.01 0.001
• 1 3.84 6.64 10.83
• 2 5.99 9.21 13.82
• 3 7.82 11.34 16.27
• Calculated ?2 (0.378) is less than tabled value
• therefore we fail to reject the null hypothesis.

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Cautionary notes about testing for
deviations from H.W.E Caution 1 If we find a
population does not deviate from Hardy-Weinberg
Equilibrium, we cannot conclude that no
evolutionary forces are operating. Caution 2
The ability of the chi-square test to detect
significant deviations from Hardy-Weinberg equilib
riums is very weak.
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Caution 3 Deviations from Hardy-Weinberg expecta
tions gives us not information about the kinds
or directions of the evolutionary
forces operating.
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Deviations from H.W.E There are two types of
non-random mating, those Where mate choice is
based on ancestry (inbreeding and crossbreeding)
and those whose Choice is based upon genotypes at
a particular Locus (assortative and
disassortative mating).
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Inbreeding Is of major importance in
conservation genetics as it leads to reduced
reproductive fitness. When related individuals
mate at a rate greater then expected by random
mating, the frequency of heterozygotes is reduced
relative to H.W.E. Avoidance of inbreeding and
cross-breeding can lead to higher than expected
heterozygosities.
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Assortative and Disassortative Mating
the preferential mating of like-with-like
genotype is called assortative mating. The
mating of unlike genotypes is referred to
as disassortative mating. In general,
assortative mating leads to increased homozygosity
, while disassortative mating increases
heterozygosity, relative to H.W. expectations.
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Fragmented populations Allele
frequencies diverge in isolated populations due
to chance and selection. This results in an
overall deficiency of heterozygotes, even when
individual populations are themselves in H.W.E
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Linkage Disequilibrium In large, randomly
mating populations at equilibrium, alleles at
different loci are expected to be randomly
associated. Consider loci A and B with alleles
A1, A2, and B1, B2, and frequencies pA, qA, pB,
qB, respectively. These loci and alleles form
gametes A1B1, A1B2, A2B1, and A2B2. Under random
mating and independent assortment, These gametes
will have frequencies that are the Product of
their allele frequencies, A1B2 pAqB.
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Random association of alleles at different
loci is referred to as Linkage
Equilibrium. Non-random association of alleles
among loci is referred to as Linkage
Disequilibrium. Chance events in small
populations, population bottlenecks, recent
mixing of different populations, and selection
all may cause non-random associations among loci.
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Loci that show deviations from linkage
equilibrium in large randomly mating populations
are often subject to strong forces of natural
selection. In small populations, neutral alleles
that have no selective differences between
genotypes may behave as if they are under
selection due to non-random association with
alleles at nearby loci that are being strongly
selected.
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Linkage disequilibrium is of importance in
populations of conservation concern as Linkage
disequilibrium will be common in
threatened species as their population sizes are
small. Population bottlenecks frequently cause
linkage disequilibrium. Evolutionary processes
are altered when there is linkage disequilibrium.
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Functionally important gene clusters
exhibiting linkage disequilibrium (such as MHC)
are of major importance to the persistence of
threatened species. Linkage disequilibrium is
one of the signals that can be used to detect
admixture of differentiated populations. Linkage
disequilibrium can be used to estimate genetically
effective population sizes.
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Consider an example where two different monomorphi
c populations with genotypes A1A1B1B1 and
A2A2B2B2 are combined and allowed to mate at
random. Each autosomal locus is expected to
attain individual H.W.E. in one
generation. However, alleles at different loci
do not attain linkage equilibrium frequencies in
one generation, they only approach is
asymptotically at a rate dependent on the
recombination frequency between the two loci.
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In this example of the pooled population,
assume 70 of pooled population isA1A1B1B1 30
of pooled population is A2A2B2B2 equal number of
females males of both genotypes. Only two
gametic types are produced A1B1, A2B2 Next
generation A1A1B1B1, A1A2B1B2, A2A2B2B2 These
loci are clearly in linkage disequilibrium.
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In subsequent generations, two other possible
gametic types A1B2 and A2B1 are generated
by recombination in the multiply heterozygous
genotype. For example, A1B1//A2B2 heterozygotes
produce recombinant gametes A1B2 and A2B1 at
frequencies of 1/2c, where c is the rate of
recombination and non-recombinant A1B1,
A2B2 gametes in frequencies 0.5(1-c). Eventually,
all 9 possible genotypes will be formed and
attained at equilibrium frequencies.
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Until equilibrium is reached, genotypes will
deviate from their expected frequencies. Linkage
disequilibrium is the deviation of
gametic frequencies from their equilibrium
frequencies. The measure of linkage
disequilibrium D is the difference between the
product of the frequencies of the A1B1 and A2B2
gametes (referred to as r and u) and the product
of the frequencies of the A1B2 and A2B1 gametes
(s and t) D ru - st
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Actual freq. r s t u 1.0 Equil.
freq. pAqA pAqB qApB qAqB 1.0 Disequilibrium D
ru - st Numerical Example pA 0.70, qA
0.30, pB 0.70, qB 0.30 Actual freq. 0.70
0.00 0.00 0.30 Equil.
freq. 0.7X0.7 0.7X0.3 0.3X0.7
0.3X0.3 0.49 0.21 0.21
0.09 Disequilibrium D (0.7 X 0.3) - (0.0 X
0.0) 0.21
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Dmax 0.25 and occurs when r 0.5, s 0.0, t
0.0, u 0.5 Dmin -0.25 and occurs when r
0.0, s 0.5, t 0.5, u 0.0 Under
equilibrium, ru st and D 0.
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Many different measures of disequilibrium. Lewont
in (1964) suggested D, which is D D /
Dmax Where, Dmax is the maximum D possible for a
given set of allele frequencies at the two loci.
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Dmax is equal either to the lesser of A1B2 (s)
or A2B1 (t) if D is positive or to the lesser of
A1B1 (r) or A2B2 (u) if D is negative. The
advantage of this measure is that it ranges from
-1.0 to 1.0, regardless of the allele frequencies
at the two loci.
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Gamete Freq. Allele Freq. A1B1 A1B2 A2B1 A2B2
D D 0.5 0.0 0.0 0.5 0.25 1.0 0.4 0.1 0.1 0.4
0.15 0.6 A1B10.5 0.25 0.25 0.25 0.25 0.0 0.0
0.1 0.4 0.4 0.1 -0.15 -0.6 0.0 0.5 0.5 0.0
-0.25 -1.0 0.9 0.0 0.0 0.1 0.09 1.0 A1B10.9
0.85 0.05 0.05 0.05 0.04 0.44 0.81 0.09 0.09 0.
01 0.0 0.0 0.0 0.9 0.1 0.0 -0.09 -1.0 A1B20
.9 0.05 0.85 0.05 0.05 -0.04 -0.44 0.09 0.81 0
.01 0.09 0.0 0.0 A10.1,B10.5 0.1 0.0 0.4 0.5
0.05 1.0 0.05 0.05 0.45 0.45 0.0 0.0
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Example 1A1B10.5 A1B1 A1B2 A2B1 A2B2 Actua
l Gametic Freq 0.4 0.1 0.1 0.4 Equilib. Gametic
Freq 0.25 0.25 0.25 0.25 D (A1B1 X A2B2) -
(A1B2 X A2B1) D (0.4 X 0.4) - (0.1 X 0.1)
0.16 - 0.01 0.15 D D/Dmax 0.15/0.25
0.6
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Example 2A1B20.9 A1B1 A1B2 A2B1 A2B2 Actua
l Gametic Freq 0.05 0.85 0.05 0.05 Equilib.
Gametic Freq 0.09 0.81 0.01 0.09 D (A1B1 X
A2B2) - (A1B2 X A2B1) D (0.05 X 0.05) - (0.85
X 0.05) 0.0025 - 0.0425 -0.04 D
D/Dmax -0.04/0.09 -0.44
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Linkage disequilibrium decays as
recombination produces underrepresented
gametes. The rate of decay depends upon
recombination frequency as follows Dt D0(1
- c)t Linkage disequilibrium declines rapidly
for unlinked loci, with approximate linkage
equilibrium reached in five generations.
Conversely, decay of disequilibrium is slow for
closely linked loci.
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When linkage disequilibrium has been observed in
a population, it has often been attributed to
some type of multilocus selection. This
assumption may not be valid because a number of
other factors can affect linkage
disequilibrium including recombination gene
tic drift mutation gene flow inbreeding
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Conservation biologists are often concerned
with changes in levels of genetic diversity over
time, as loss of genetic diversity is one
indication that the population is undergoing
inbreeding and losing its evolutionary
potential. Heterozygosity is often expresses as
the proportion of heterozygosity retained over
time. Ht/H0 where Ht is level of heterozygosity
at generation t and H0 is the level at some
time earlier, referred to as time 0.
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For example, H0 may be the heterozygosity before
a population crash and Ht after the crash. Then
1 - (Ht/H0) reflects the proportion
of heterozygosity lost as a result of the crash.