Two-locus systems. Scheme of genotypes genotype Two-locus genotypes Multilocus genotypes genotype.

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Two-locus systems

Transcript of Two-locus systems. Scheme of genotypes genotype Two-locus genotypes Multilocus genotypes genotype.

Page 1: Two-locus systems. Scheme of genotypes genotype Two-locus genotypes Multilocus genotypes genotype.

Two-locus systems

Page 2: Two-locus systems. Scheme of genotypes genotype Two-locus genotypes Multilocus genotypes genotype.

Scheme of genotypes

genotype genotype

Two-locus

genotypes

Multilocus genotypesgenotype

Page 3: Two-locus systems. Scheme of genotypes genotype Two-locus genotypes Multilocus genotypes genotype.

Two-locus two allele population

Gamete

p1 p2 p3 p4

Independent combination of randomly chosen parental gametes

Next generation on zygote level

Page 4: Two-locus systems. Scheme of genotypes genotype Two-locus genotypes Multilocus genotypes genotype.

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

Page 5: Two-locus systems. Scheme of genotypes genotype Two-locus genotypes Multilocus genotypes genotype.

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)

Page 6: Two-locus systems. Scheme of genotypes genotype Two-locus genotypes Multilocus genotypes genotype.

gamete

).,(

);,();,(

);,(;),();,(

;),();,();,();,(

abab

abaBaBaB

abAbaBAbAbAb

abABaBABAbABABAB

zygote

Position effect

Page 7: Two-locus systems. Scheme of genotypes genotype Two-locus genotypes Multilocus genotypes genotype.

Table zygote productions

AB: p1’=p1

2+p1p2+p1p3+(1-r)p1p4+rp2p3

Evolutionary equation for genotype AB

Page 8: Two-locus systems. Scheme of genotypes genotype Two-locus genotypes Multilocus genotypes genotype.

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

Page 9: Two-locus systems. Scheme of genotypes genotype Two-locus genotypes Multilocus genotypes genotype.

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

Page 10: Two-locus systems. Scheme of genotypes genotype Two-locus genotypes Multilocus genotypes genotype.

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)

Page 11: Two-locus systems. Scheme of genotypes genotype Two-locus genotypes Multilocus genotypes genotype.

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)

Page 12: Two-locus systems. Scheme of genotypes genotype Two-locus genotypes Multilocus genotypes genotype.

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

Page 13: Two-locus systems. Scheme of genotypes genotype Two-locus genotypes Multilocus genotypes genotype.

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);

Page 14: Two-locus systems. Scheme of genotypes genotype Two-locus genotypes Multilocus genotypes genotype.

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

Page 15: Two-locus systems. Scheme of genotypes genotype Two-locus genotypes Multilocus genotypes genotype.

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.

Page 16: Two-locus systems. Scheme of genotypes genotype Two-locus genotypes Multilocus genotypes genotype.

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

Page 17: Two-locus systems. Scheme of genotypes genotype Two-locus genotypes Multilocus genotypes genotype.

simulation

Page 18: Two-locus systems. Scheme of genotypes genotype Two-locus genotypes Multilocus genotypes genotype.

Multilocus multiallele population

genotypespossibleall

aBcAbCrr

abCABcrr

AbcaBCrr

abcABCrr

gametesyprobabilit

1

_____________________

,)1(

,)1(

,)1)(1(

21

21

21

21

Three loci

Page 19: Two-locus systems. Scheme of genotypes genotype Two-locus genotypes Multilocus genotypes genotype.

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

Page 20: Two-locus systems. Scheme of genotypes genotype Two-locus genotypes Multilocus genotypes genotype.

)();(

)();(

)();(

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

Page 21: Two-locus systems. Scheme of genotypes genotype Two-locus genotypes Multilocus genotypes genotype.

...

...

...)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

Page 22: Two-locus systems. Scheme of genotypes genotype Two-locus genotypes Multilocus genotypes genotype.

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

Page 23: Two-locus systems. Scheme of genotypes genotype Two-locus genotypes Multilocus genotypes genotype.

ondistributiLinkage

iesprobabilitofsetsij }{ ,

Problem: definition of the linkage distribution.

Nonrandom crossovers.

Page 24: Two-locus systems. Scheme of genotypes genotype Two-locus genotypes Multilocus genotypes genotype.

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

Page 25: Two-locus systems. Scheme of genotypes genotype Two-locus genotypes Multilocus genotypes genotype.

definition of the linkage distribution.

partitionthisyprobabilitvup )|(

Page 26: Two-locus systems. Scheme of genotypes genotype Two-locus genotypes Multilocus genotypes genotype.

Equilibrium point for multilocus population

)()...()()()...( 321321 mm apapapapaaaap

0,)),|(-max(1 is

point equilibria toeconvergenc theof Speed

vuwherevu

Page 27: Two-locus systems. Scheme of genotypes genotype Two-locus genotypes Multilocus genotypes genotype.

Polyploids systems

Page 28: Two-locus systems. Scheme of genotypes genotype Two-locus genotypes Multilocus genotypes genotype.

4-ploids 2-ploids (diploids)

Chromatid dabbling

Four gamete produced

Page 29: Two-locus systems. Scheme of genotypes genotype Two-locus genotypes Multilocus genotypes genotype.

ncombinatiopossibleall

pppppppp

ppppppppp

...221,22811,18711,17611,16

411,14311,13211,12111,111

Problem: definition of the coefficients.

Page 30: Two-locus systems. Scheme of genotypes genotype Two-locus genotypes Multilocus genotypes genotype.

Polyploids systems

Page 31: Two-locus systems. Scheme of genotypes genotype Two-locus genotypes Multilocus genotypes genotype.