Two-locus systems

Scheme of genotypes

genotype genotype

Two-locus

genotypes

Multilocus genotypesgenotype

Two-locus two allele population

Gamete

p1 p2 p3 p4

Independent combination of randomly chosen parental gametes

Next generation on zygote level

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

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)

gamete

).,(

);,();,(

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

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

abab

abaBaBaB

abAbaBAbAbAb

abABaBABAbABABAB

zygote

Position effect

Table zygote productions

AB: p1’=p1

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

Evolutionary equation for genotype AB

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

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

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)

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)

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

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);

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

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.

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

simulation

Multilocus multiallele population

genotypespossibleall

aBcAbCrr

abCABcrr

AbcaBCrr

abcABCrr

gametesyprobabilit

1

_____________________

,)1(

,)1(

,)1)(1(

21

21

21

21

Three loci

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

)();(

)();(

)();(

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

...

...

...)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

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

ondistributiLinkage

iesprobabilitofsetsij }{ ,

Problem: definition of the linkage distribution.

Nonrandom crossovers.

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

definition of the linkage distribution.

partitionthisyprobabilitvup )|(

Equilibrium point for multilocus population

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

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

point equilibria toeconvergenc theof Speed

vuwherevu

Polyploids systems

4-ploids 2-ploids (diploids)

Chromatid dabbling

Four gamete produced

ncombinatiopossibleall

pppppppp

ppppppppp

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

411,14311,13211,12111,111

Problem: definition of the coefficients.

Polyploids systems