Two-locus systems

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Two-locus systems
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Scheme of genotypes
Two-locus genotypes
genotype
genotype
genotype
Multilocus genotypes
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Two-locus two allele population
Gamete
p1 p2 p3 p4
Next generation on zygote level
Independent combination of randomly chosen
parental gametes
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Table gametes from genotypes I
(1-r) no cross-over
(r) cross-over
Type zygote- one locus is homozygotes
Zygote
Zygote (AB,Ab) have gamete (AB) with frequency
0.5(1-r)0.5r0.5
gamete
0.5(1-r)
0.5(1-r)
0.5(r)
0.5(r)
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Table gametes from genotypes II
(r) cross-over
(1-r) no cross-over
Type zygote- both loci is heterozygotes
Zygote
Zygote (AB,ab) have gamete (AB) with frequency
0.5(1-r)
gamete
0.5(1-r)
0.5(1-r)
0.5(r)
0.5(r)
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gamete
Position effect
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Table zygote productions
AB p1p12p1p2p1p3(1-r)p1p4rp2p3
Evolutionary equation for genotype AB
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p1p12p1p2p1p3(1-r)p1p4rp2p3 p2p22p1p2p2p
4rp1p4(1-r)p2p3 p3p32p3p4p1p3rp1p4(1-r)p2p
3 p4p42p3p4p2p4(1-r)p1p4rp2p3
r is probabilities of cross-over (coefficient of
recombination). Usually 0? r ? 0.5. If r0.5 then
loci are called unlinked (or independent). If r0
then population transform to one loci population
with four alleles.
AB Ab aB ab p1 p2 p3 p4
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Measure of disequilibria D p1p4-p2p3
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p1p1- rD p2p2 rD p3p3 rD
p4p4 - rD.
p1p2p(A) p1p3p(B)
AB Ab aB ab p1 p2 p3 p4
Gene Conservation Low p1 p2 p1 p2p(A)
p1 p3 p1 p3p(B)
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Two-locus two allele population. Equilibria.
p1p1- rD p2p2 rD p3p3 rD p4p4 - rD.
Measure of disequilibria D p1p4-p2p3
D0 p1p4 p2p3
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p1 p(A) p(B) p2 p(A) p(b) p3 p(a) p(B) p4
p(a) p(b).
In equilibria point the genes are statistically
independence.
But the genes are dependent physically, because
are in pairs on chromosome
Measure of disequilibria D p1p4-p2p3
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Convergence to equilibrium
p1p1- rD p2p2 rD p3p3 rD p4p4 - rD.
Dp1p4- p2p3
D(p1- rD )(p4 - rD)-(p2 rD)(p3 rD)
p1 p4- p2p3
-rD(p1p2p3p4)
(rD)2-(rD)2
D
D(n)(1-r)nD(0)
DD-rD(1-r)D
Maximal speed convergence to equilibrium for r0.5
D(n)(0.5)nD(0)
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Gene Conservation Low p1 p2 p1 p2p(A)
p1 p3 p1 p3p(B)
p1 p(A) p(B) p2 p(A) p(b) p3 p(a) p(B) p4
p(a) p(b).
Infinite set of equilibrium points
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p1p12p1p2p1p3(1-r)p1p4rp2p3 p2p22p1p2p2p
4rp1p4(1-r)p2p3 p3p32p3p4p1p3rp1p4(1-r)p2p
3 p4p42p3p4p2p4(1-r)p1p4rp2p3
r0
p1p12p1p2p1p3p1p4 p1 p2p22p1p2p2p4p2p3
p2 p3p32p3p4p1p3p2p3 p3 p4p42p3p4p2p
4p1p4 p4
p1p1- rD p2p2 rD p3p3 rD p4p4 - rD.
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p1p12p1p2p1p3(1-r)p1p4rp2p3 p2p22p1p2p2p
4rp1p4(1-r)p2p3 p3p32p3p4p1p3rp1p4(1-r)p2p
3 p4p42p3p4p2p4(1-r)p1p4rp2p3
r1
p1p12p1p2p1p3p2p3 (p1p2)(p1p3)
p(A)p(B) p2p22p1p2p2p4p1p4 (p1p2)(p2p4)
p(A)p(b) p3p32p3p4p1p3p1p4
(p3p4)(p1p3) p(a)p(B) p4p42p3p4p2p4p2p3
(p3p4)(p2p4) p(a)p(b)
p1p1- rD p2p2 rD p3p3 rD p4p4 - rD.
D(n)(1-r)nD(0)
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simulation
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Multilocus multiallele population
Three loci
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Equilibrium point
Equilibrium pointlimiting point of trajectories
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General case
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M loci and L alleles in each locus
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Problem definition of the linkage
distribution. Nonrandom crossovers.
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definition of the linkage distribution.
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Equilibrium point for multilocus population
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Polyploids systems
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4-ploids
2-ploids (diploids)
Chromatid dabbling
Four gamete produced
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Problem definition of the coefficients.
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Polyploids systems
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