Section 3
Characterizing Genetic Diversity:
Single Loci
Gene with 2 alleles designated “A” and “a”.
Three genotypes: AA, Aa, aa
Population of 100 individuals with the following
Genotypes: AA = 50, Aa = 30, aa = 20
Genotypic frequencies -- General formula:
f(AA) = NAA/N -- > 50/100 = 0.5
f(Aa) = NAa/N -- > 30/100 = 0.3
f(aa) = Naa/N -- > 20/100 = 0.2
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
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)
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:
1. Organism in question is diploid
2. Reproduction is sexual
3. Generations are non-overlapping
4. Panmixia
5. Population size is infinitely large, or at least
large enough to avoid stochastic errors
6. Migration (immigration/emigration) is negligible
7. No mutation
8. 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.
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.
Sperm
Allele
Frequency
A p
E
G
G
a q
A
p
a
q
AA
p2
Aa
pq
Aa
pq
aa
q2
Note: p2 + 2pq + q2 = (p + q)2 = 1
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.
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
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
H = 2pq = 2(0.4904)(0.5096)
Q = q2 = (0.5096)2
=
=
=
0.2405
0.4998
0.2597
Step 3: Calculate chi-square statistic:
P
H
Q
O
E
65 0.2405 X 260 = 62.53
125 0.4998 X 260 = 129.95
70 0.2597 X 260 = 67.52
2 =
(O-E)2/E
0.098
0.189
0.091
0.378
Step 4: Compare calculated 2 with tabled 2:
Degrees of freedom
3(# of genotypes) - 1(constant) - 1(# parameters)
=1
Look up critical values for 2 statistic:
D.f.
1
2
3
Level of Significance
0.05
0.01
0.001
3.84
6.64
10.83
5.99
9.21
13.82
7.82
11.34
16.27
Calculated 2 (0.378) is less than tabled value
therefore we fail to reject the null hypothesis.
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
equilibriums is very weak.
Caution 3: Deviations from Hardy-Weinberg
expectations gives us not information about the
kinds or directions of the evolutionary forces
operating.
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).
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.
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.
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
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.
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.
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.
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.
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.
Consider an example where two different
monomorphic 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.
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.
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.
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
Actual freq.
Equil. freq.
Disequilibrium:
r
s
t
pAqA pAqB qApB
D = ru - st
u
qAqB
1.0
1.0
Numerical Example: pA = 0.70, qA = 0.30, pB = 0.70, qB = 0.30
Actual freq.
Equil. freq.
0.70
0.00
0.00
0.30
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
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.
Many different measures of disequilibrium.
Lewontin (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.
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.
A1B1
0.5
0.4
0.25
0.1
0.0
Gamete Freq.
A1B2
A2B1
0.0
0.0
0.1
0.1
0.25
0.25
0.4
0.4
0.5
0.5
A2B2
0.5
0.4
0.25
0.1
0.0
D
0.25
0.15
0.0
-0.15
-0.25
D’
1.0
0.6*
0.0
-0.6
-1.0
A1=B1=0.9
0.9
0.85
0.81
0.0
0.05
0.09
0.0
0.05
0.09
0.1
0.05
0.01
0.09
0.04
0.0
1.0
0.44
0.0
A1=B2=0.9
0.0
0.05
0.09
0.9
0.85
0.81
0.1
0.05
0.01
0.0
0.05
0.09
-0.09
-0.04
0.0
-1.0
-0.44*
0.0
0.1
0.05
0.0
0.05
0.4
0.45
0.5
0.45
0.05
0.0
1.0
0.0
Allele Freq.
A1=B1=0.5
A1=0.1,B1=0.5
Example 1:A1=B1=0.5
Actual Gametic Freq:
Equilib. Gametic Freq:
A1B1 A1B2 A2B1 A2B2
0.4 0.1 0.1 0.4
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
Example 2:A1=B2=0.9
Actual Gametic Freq:
Equilib. Gametic Freq:
A1B1 A1B2 A2B1 A2B2
0.05 0.85 0.05 0.05
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
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.
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
genetic drift
mutation
gene flow
inbreeding
Expected heterozygosity (He) = Gene diversity:
For a single locus with two alleles, He = 2pq
When more than two alleles, it is simpler to
Calculate He as:
k
He = 1 - pi2
i=1
Where k = number of alleles
If sample sizes are smaller than 50 individuals:
k
He = 2N(1 - pi2)/(2N - 1)
i=1
Where N is the number of individuals sampled.
Gene diversity (He) is usually reported in
Preference to observed heterozygosity as it is
Less affected by sampling.
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.
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.