Two-locus systems
description
Transcript of Two-locus systems
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Two-locus systems
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Scheme of genotypes
genotype genotype
Two-locus
genotypes
Multilocus genotypesgenotype
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Two-locus two allele population
Gamete
p1 p2 p3 p4
Independent combination of randomly chosen parental gametes
Next generation on zygote level
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Table gametes from genotypes I
(1-r) –no cross-over (r) – cross-over
Zygote
gamete
0.5(1-r)
Type zygote- one locus is homozygotes
0.5(1-r) 0.5(r) 0.5(r)
Zygote (AB,Ab) have gamete (AB) with frequency
0.5(1-r)+0.5r=0.5
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Table gametes from genotypes II
(1-r) –no cross-over (r) – cross-over
0.5(1-r) 0.5(1-r) 0.5(r) 0.5(r)
Zygote
gamete
Type zygote- both loci is heterozygotes
Zygote (AB,ab) have gamete (AB) with frequency
0.5(1-r)
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gamete
).,(
);,();,(
);,(;),();,(
;),();,();,();,(
abab
abaBaBaB
abAbaBAbAbAb
abABaBABAbABABAB
zygote
Position effect
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Table zygote productions
AB: p1’=p1
2+p1p2+p1p3+(1-r)p1p4+rp2p3
Evolutionary equation for genotype AB
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p1’=p1
2+p1p2+p1p3+(1-r)p1p4+rp2p3
p2’=p2
2+p1p2+p2p4+rp1p4+(1-r)p2p3
p3’=p3
2+p3p4+p1p3+rp1p4+(1-r)p2p3
p4’=p4
2+p3p4+p2p4+(1-r)p1p4+rp2p3
r is probabilities of cross-over (coefficient of recombination).
Usually 0 r 0.5. If r=0.5 then loci are called unlinked (or independent). If r=0 then population transform to one loci population with four alleles.
AB Ab aB ab
p1 p2 p3 p4
1
1
1
' 21 1 2 1 3 1 4 2 3
' 21 1 2 1 3 1 4 1
1 4
4 2 3
' 21 1 2 1 3 2 31 4
1 4 2 3
p =p +p p +p p +(1-r)p p +rp p
p =p +p p +p p +p p -rp p +rp p
p = p p -p pp +p p +p p +p p -r( )
p p -p pLet D
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Measure of disequilibriaD= p1p4-p2p3
1
1
1
' 21 1 2 1 3 1 4
'1 1 2 3 4
'1
.
p =p +p p +p p +p p -rD
p =p (p +p +p +p )-rD
p =p -rD
Then
2
2
2
2
' 22 1 2 2 3 2 4 1 4
' 22 1 2 2 3 2 4 1 4 2 3
'2 2 1 3 4 1 4 2 3
'2
p =p +p p +(1-r)p p +p p +rp p
p =p +p p +p p +p p +rp p -rp p
p =p (p +p +p +p )+r(p p -p p )
p =p +rD
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p1’=p1- rD ; p2
’=p2 +rD;
p3’=p3+ rD; p4
’=p4 - rD.
Gene Conservation Low
p1’+ p2
’ = p1+ p2=p(A); p1’+ p3
’ = p1+ p3=p(B)
AB Ab aB ab
p1 p2 p3 p4
p1+p2=p(A)
p1+p3=p(B)
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Two-locus two allele population. Equilibria.
p1=p1- rD ; p2=p2 +rD;
p3=p3+ rD; p4=p4 - rD.
Measure of disequilibriaD= p1p4-p2p3
D=0; p1p4 = p2p3
21 1 1 2 3 4 1 1 2 1 3 1 4
21 1 1 2 2 33 4 1 1 2 1 3
1 1 1 2 3 1 2 3
1
2 1 1
p =p (p + p +p + p )= p +p p +p p +p p
p =p (p + p +p + p )= p +p p +p +p pp
p =p (p +p )+p (p +p ) (p +p )
p =p(A
(p +p )
)p(B)
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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
'1 1 1 2 1 3
21 1 1 2 1 3 2 3
21 1 1 1 4 1 42 1 3 2 3
1
'1
1
( ) ( ) ( )( )
( )
(
( )
)
(1 ) .
.( ) (1 )
p p p p
p p A p B p rD p p p p
p rD p p p p p p p
p rD p p p p p p p
p rD p D r D
p p A p B r D
Measure of disequilibriaD= p1p4-p2p3
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Convergence to equilibrium
D’=p1’p4
’- p2’p3
’;p1
’=p1- rD ; p2’=p2 +rD;
p3’=p3+ rD; p4
’=p4 - rD.
D’=(p1- rD )(p4 - rD)-(p2 +rD)(p3+ rD)
D’= p1 p4- p2p3 -rD(p1+p2+p3+p4) +(rD)2-(rD)2
D’=D-rD=(1-r)D;
D(n)=(0.5)nD(0);
Maximal speed convergence to equilibrium for r=0.5
D(n)=(1-r)nD(0);
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p1= p(A) p(B); p2= p(A) p(b); p3= p(a) p(B); p4= p(a) p(b).
Gene Conservation Low
p1’+ p2
’ = p1+ p2=p(A); p1’+ p3
’ = p1+ p3=p(B)
Infinite set of equilibrium points
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p1’=p1
2+p1p2+p1p3+(1-r)p1p4+rp2p3
p2’=p2
2+p1p2+p2p4+rp1p4+(1-r)p2p3
p3’=p3
2+p3p4+p1p3+rp1p4+(1-r)p2p3
p4’=p4
2+p3p4+p2p4+(1-r)p1p4+rp2p3
r=0
p1’=p1
2+p1p2+p1p3+p1p4 = p1
p2’=p2
2+p1p2+p2p4+p2p3 = p2
p3’=p3
2+p3p4+p1p3+p2p3 = p3
p4’=p4
2+p3p4+p2p4+p1p4 = p4
p1’=p1- rD ; p2
’=p2 +rD;
p3’=p3+ rD; p4
’=p4 - rD.
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p1’=p1
2+p1p2+p1p3+(1-r)p1p4+rp2p3
p2’=p2
2+p1p2+p2p4+rp1p4+(1-r)p2p3
p3’=p3
2+p3p4+p1p3+rp1p4+(1-r)p2p3
p4’=p4
2+p3p4+p2p4+(1-r)p1p4+rp2p3
r=1
p1’=p1
2+p1p2+p1p3+p2p3 = (p1+p2)(p1+p3) = p(A)p(B)
p2’=p2
2+p1p2+p2p4+p1p4 = (p1+p2)(p2+p4) = p(A)p(b)
p3’=p3
2+p3p4+p1p3+p1p4 = (p3+p4)(p1+p3) = p(a)p(B)
p4’=p4
2+p3p4+p2p4+p2p3 = (p3+p4)(p2+p4) = p(a)p(b)
p1’=p1- rD ; p2
’=p2 +rD;
p3’=p3+ rD; p4
’=p4 - rD.
D(n)=(1-r)nD(0);
0.0 10 DD
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simulation
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Multilocus multiallele population
genotypespossibleall
aBcAbCrr
abCABcrr
AbcaBCrr
abcABCrr
gametesyprobabilit
1
_____________________
,)1(
,)1(
,)1)(1(
21
21
21
21
Three loci
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aBcAbCrr
abCABcrr
AbcaBCrr
abcABCrr
zygoteforgametesyprobabilit
abcabCaBcaBC
AbcAbCABcABC
21
21
21
21
,)1(
,)1(
,)1)(1(
)8,1(
8,7,6,5
,4,3,2,1
...
...
...)1(
...)1)(1(
81213
81212
81211
pprrp
pprrp
pprrp
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)();(
)();(
)();(
8,7,6,5
,4,3,2,1
86427531
87436521
87654321
cpppppBppppp
bpppppBppppp
apppppAppppp
abcabCaBcaBC
AbcAbCABcABC
Equilibrium point
...
)()()(
)()()(
)()()(
3
2
1
CpbpAPp
cpBpAPp
CpBpAPp
Equilibrium point=limiting point of trajectories
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...
...
...)1(
...)1)(1(
81213
81212
81211
pprrp
pprrp
pprrp
ncombinatiopossibleall
pppppppp
ppppppppp
...221,22811,18711,17611,16
411,14311,13211,12111,111
1... 8,2,1,,, ijijijsjisij
ondistributiLinkage
iesprobabilitofsetsij }{ ,
General case
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ncombinatiopossibleall
pppppp
ppppppppp
...811,18711,17611,16
411,14311,13211,12111,111
1... ,2,1,,, Mijijijsjisij
ondistributiLinkage
iesprobabilitofsetsij }{ ,
M loci and L alleles in each locus
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ondistributiLinkage
iesprobabilitofsetsij }{ ,
Problem: definition of the linkage distribution.
Nonrandom crossovers.
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31233
32222
21211
2
2
2
pppp
pppp
pppp
)1(
)1(
)1(
2133
1322
3211
pppp
pppp
pppp
)2()1(
10
113211
213
pppppp
ppp
1321 ppp
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definition of the linkage distribution.
partitionthisyprobabilitvup )|(
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Equilibrium point for multilocus population
)()...()()()...( 321321 mm apapapapaaaap
0,)),|(-max(1 is
point equilibria toeconvergenc theof Speed
vuwherevu
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Polyploids systems
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4-ploids 2-ploids (diploids)
Chromatid dabbling
Four gamete produced
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ncombinatiopossibleall
pppppppp
ppppppppp
...221,22811,18711,17611,16
411,14311,13211,12111,111
Problem: definition of the coefficients.
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Polyploids systems
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