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