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A population is in gametic phase equilibrium if and only if the gametic frequencies are the product of their gene frequencies

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Title: A population is in gametic phase equilibrium if and only if the gametic frequencies are the product of their gene frequencies


1
A population is in gametic phase equilibrium if
and only if the gametic frequencies are the
product of their gene frequencies
If D 0 then
  • In one locus genetics, one has to consider only
    zygotic phase equilibrium
  • Multiple loci genetics must consider both zygotic
    and gametic phase equilibrium

2
Approach to gametic equilibrium
  • Under the usual simplifying assumptions, each
    locus is at equilibrium with respect to its own
    alleles immediately upon random mating.
  • The gametic combinations of two loci may not be
    at equilibrium initially, but under random mating
    will approach equilibrium rapidly if they are
    independently assorting.

No MMSD, equal gene frequencies between the
sexes, no sex-linkage and random mating
3
Example
Starting condition AABB 50, aaBB 20, AAbb
10, aabb 20
To find the genotypic frequencies and the gametic
frequencies
The zygote matrix
P(AABB) 0.5 P(aaBB) 0.2 P(AAbb) 0.1 P(aabb)
0.2
The gametic matrix
4
The genotypic frequencies in the progeny
The expected genotypic frequencies in the progeny
AA Aa aa
BB Bb bb
5
The allelic frequencies are now found as usual
AA Aa aa
BB Bb bb
P(AB) 0.25 ½(0.2 0.1) ¼(0.24)
0.46 P(Ab) 0.04 ½(0.2 0.08) ¼(0.24)
0.24 P(aB) 0.01 ½(0.04 0.1) ¼(0.24)
0.14 P(ab) 0.04 ½(0.04 0.08) ¼(0.24)
0.16
A a
B b
p(A) 0.46 0.24 0.7 p(B) 0.14 0.46
0.6
6
Last Finding Gametic disequilibrium
Parental gametic disequilibrium
Progeny gametic disequilibrium
7
Zygotic and gametic equilibrium
  • Under the usual simplifying assumptions, each
    locus is at equilibrium with respect to its own
    alleles immediately upon random mating Zygotic
    equilibrium
  • The gametic combinations of two loci may not be
    at equilibrium initially, but under random mating
    will approach equilibrium rapidly if they are
    independently assorting
  • Gametic equilibrium When the square of the
    gametic output produces expected zygotic values
    that are in conformity with the observed zygotic
    values for that generation, the gametic set is at
    equilibrium

8
Random association of alleles as defined in Hartl
and Clark using the Spiess notation
When the alleles of genes are in random
association, the frequency of a gamete carrying
any particular combination of alleles equals the
product of the frequency of those alleles.
Genes in random association are in linkage
equilibrium. Genes in not random association are
in linkage disequilibrium.
Physical proximity is ONE cause of linkage
disequilibrium.
9
Given the starting configuration of AB and ab and
the recombination fraction r
Nonrecombinant gametes
The frequency of AB (1-r)/2 The frequency of
ab (1-r)/2
Recombinant gametes
The frequency of Ab r/2 The frequency of aB
r/2
What is recombinant is defined by the starting
configuration.
10
The recombination fraction r
  • In book problems we can calculate r directly from
    the information given
  • If the genotype AB/ab produces these gametes in
    these frequencies
  • then the recombination fraction r 0.24.
  • In most situations in real life, the
    recombination fraction must be inferred because
    the gametic frequencies are not known.

AB 0.38 Ab 0.12 aB 0.12 Ab 0.38
11
The Detection of Linkage
For any two mendelian traits p and t
H
no genetic linkage between the p and t loci
o
H
the p and t loci are statistically independent
o
These two hypotheses are not equivalent.
12
Goodness-of-Fit Test For Independence
P
p
T
t
p
p
t
t
n 467
x
t
t
T
t
116.75
116.75
P
p
191
37
116.75
116.75
p
p
36
203
Philp (1934)
13
P
p
T
t
p
p
t
t
p
p
t
t
P
p
T
t
2
2
2
2
(191-
116.75
)
(37 -
116.75
)
(36 -
116.75
)
(203 -
116.75
)




116.75
116.75
116.75
116.75
221.266
df 3
14
The meaning of X2 values in tests of independence
between the segregation ratios of two Mendelian
genes
  • Big values of chi-squared may arise from
    deviation in any cell
  • Certain patterns of deviation are consistent with
    genetic linkage
  • Other patterns of deviation can have systematic
    biological causes yet NOT be due to genetic
    linkage

15
t
t
T
t
P
p
191
37
p
p
36
203
  • If cell 1 cell 4 cell 3 cell 2, then the
    genes are statistically independent.
  • If cell 1 cell 4 and cell 3 cell 2 and (cell
    1 cell 4) gt (cell 3 cell 2), then the genes
    may be linked.
  • If cell 3 cell 2 0 and cell 1 cell 4, then
    linkage is complete.

16
Any estimation of genetic linkage is based on the
supposition
that the lack of independence between loci is
DUE TO LINKAGE ALONE
17
Is Deviation From Expectation Is Due To Linkage ?
Partition X2 into single degree of freedom
orthogonal contrasts
2
(191 37 - 36 - 203
)
P
A test for the segregation of

000.259
467
2
(191 - 37 36 - 203
)
T
A test for the segregation of

000.362
467
2
(191 - 37- 36 203
)
T
P
A test for the linkage of

220.645
and
467
221.266
2
C
221.266
3df
18
What is the most likely value for recombination
given the observed phenotypes?
19
The Estimation of Linkage
MAXIMUM LIKELIHOOD
provides an estimate
of r, (the recombination fraction), which has
smallest variance
the data will allow
the
20
THE MAXIMUM LIKELIHOOD CALCULATION
p
p
t
t
T
t
P
p
P
p
T
t
t
t
p
p
m
m
m
m
2
1
4
3
a
a
a
a
2
1
4
3
Let
m
...m
represent the EXPECTED proportion

1
i
of individuals in segregate classes.
Let
a
...a
represent the OBSERVED numbers.

1
i
21
This maximum likelihood expression and its
logarithm will both be maxima at the same value of
r
.
Set the first derivative to zero and solve.
22
CALCULATING THE RECOMBINATION FRACTION
Express the expected fraction in terms of
r

n
n
n
n
m
m
m
m
(
1-
r
)
(
1-
r
)
r
r




1
4
2
3
2
2
2
2
Find the logarithm of the likelihood expression L
log (
r
)

log (
r
)


L C
log (1/2
(
1-
r
))
/2
log (1/2
(
1-
r
))
/2
23
p
p
t
t
T
t
P
p
P
p
T
t
t
t
p
p
203
37
36
191
d
L
191
203
36
37


0



r
d
r
1-
r
1-
r
r
73
r
0.1563


467
24
has a variance.
The recombination fraction
p
This variance depends on the population size and
structure.
For a backcross, the variance of r is
The value of n is critical to map creation.
25
(No Transcript)
26
The linear order of map is inferred from
overlapping sets of two and/or three point
linkages.
27
(No Transcript)
28
Given the starting configuration of AB and ab and
the recombination fraction r
Nonrecombinant gametes
The frequency of AB (1-r)/2 The frequency of
ab (1-r)/2
Recombinant gametes
The frequency of Ab r/2 The frequency of aB
r/2
What is recombinant is defined by the starting
configuration.
29
Calculation of of gamete frequencies when r is
given
The MN and Ss blood groups are genetically
linked. The recombination fraction r 0.01.
MS/Ns genotype produces these gametes in these
frequencies
MS (1-0.01)/2 0.495 Ns (1-0.01)/2
0.495 Ms 0.01/2 0.005 NS 0.01/2 0.005
The recombination fraction r is also called the
crossover rate.
30
The effect of linkage on the approach to
equilibrium
  • The rate of change in the gametic set will be
    slower than with independently assorting loci.
  • The tighter the linkage, the slower the rate of
    change.
  • Linkage does not affect the gametic output of
    homozygotes (AB/AB, ab/ab) or single
    heterozygotes (AB/aB, AB/Ab)
  • -recombination occurs, but the arrangement of
    alleles cannot change
  • Linkage changes the gametic output of the double
    heterozygotes (AB/ab, Ab/aB)
  • -In independent assortment, the double
    heterozygote produces all four gametes in EQUAL
    proportions
  • -In linkage, the proportion of nonrecombinant
    gametes is higher until gametic equilibrium is
    reached many generations later

31
The frequency of gametes produced by generation
0, when the recombination fraction r 0.1
The frequency of AB (1-r)/2 0.45 The
frequency of ab (1-r)/2 0.45
The frequency of Ab r/2 0.05 The frequency
of aB r/2 0.05
32
  • The inbred parents produced only coupling
    gametes (1/2 AB1/2ab)
  • The recombination fraction between the A and B
    loci 0.1
  • After random mating, the F1 progeny will produce
    F2 gametes like this

¼(ab)
½(0.45AB0.05Ab0.05aB0.45ab)
¼(AB)
0.25(AB) 0.225(AB)
0.25(ab) 0.225(ab)
0.025(Ab)
0.025(aB)
0.475(AB)
0.475(ab)
Recombinant gametes
33
Approach to equilibrium under different degrees
of linkage
34
Recombination and linkage disequilibrium
  • The recombination fraction is estimated using the
    progeny of a controlled cross in which the gene
    frequencies in the parents are known in advance.
  • Linkage disequilibrium is estimated by comparing
    the observed frequency of the gametes in the
    population with the expected frequency
  • Genetic linkage CANNOT be detected by an analysis
    of gametic frequencies in a random sample of a
    population

A population is in gametic phase equilibrium if
and only if the gametic frequencies are the
product of their gene frequencies
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