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Two-locus systems
Scheme of genotypes
Two-locus
genotypes
genotype
genotype
Multilocus genotypes
genotype
Two-locus two allele population
Gamete
p1
p2
p3
p4
Next generation on zygote level
Independent combination of randomly chosen parental gametes
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)+0.5r=0.5
gamete
0.5(1-r)
0.5(1-r)
0.5(r)
0.5(r)
Table gametes from genotypes II
(1-r) –no cross-over
(r) – 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)
gamete
zygote
( AB, AB); ( AB, Ab); ( AB, aB); ( AB, ab);
( Ab, Ab); ( Ab, aB); ( Ab, ab);
(aB, aB); (aB, ab);
(ab, ab).
Position effect
Table zygote productions
AB: p1’=p12+p1p2+p1p3+(1-r)p1p4+rp2p3
Evolutionary equation for genotype AB
p1’=p12+p1p2+p1p3+(1-r)p1p4+rp2p3
r is probabilities of cross-over
(coefficient of recombination).
p2’=p22+p1p2+p2p4+rp1p4+(1-r)p2p3
p3’=p32+p3p4+p1p3+rp1p4+(1-r)p2p3
p4’=p42+p3p4+p2p4+(1-r)p1p4+rp2p3
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
p'1 =p12 +p1p 2 +p1p3 +(1-r)p1p 4 +rp 2 p3
p'1 =p12 +p1p 2 +p1p3 +p1p 4 -rp1p 4 +rp 2 p3
p'1 =p12 +p1p 2 +p1p3 +p1p 4 -r(p1p 4 -p 2 p3 )
Let
D  p1p 4 -p 2 p3
. Then
'
2
1
p 1 =p +p1p 2 +p1p3 +p1p 4 -rD
'
p 1 =p1 (p1 +p 2 +p3 +p 4 )-rD
'
p 1 =p1 -rD
Measure of disequilibria
D= p1p4-p2p3
p'2 =p 22 +p1p 2 +(1-r)p 2 p3 +p 2 p 4 +rp1p 4
'
2
2
p 2 =p +p1p 2 +p 2 p3 +p 2 p 4 +rp1p 4 -rp 2 p3
p'2 =p 2 (p 2 +p1 +p3 +p 4 )+r(p1p 4 -p 2 p3 )
p'2 =p 2 +rD
p1’=p1- rD ;
p2’=p2 +rD;
p3’=p3+ rD; p4’=p4 - rD.
AB Ab aB ab
p1+p2=p(A)
p1 p2 p3 p4
p1+p3=p(B)
Gene Conservation Low
p1’+ p2’ = p1+ p2=p(A);
p1’+ p3’ = p1+ p3=p(B)
Two-locus two allele population. Equilibria.
p1=p1- rD ; p2=p2 +rD;
p3=p3+ rD; p4=p4 - rD.
D=0;
Measure of disequilibria
D= p1p4-p2p3
p1p4 = p2p3
p1 =p1 (p1 + p 2 +p3 + p 4 )= p12 +p1p 2 +p1p3 +p1p 4
p1 =p1 (p1 + p 2 +p3 + p 4 )= p12 +p1p 2 +p1p3 +p 2 p3
p1 =p1 (p1 +p 2 )+p3 (p1 +p 2 )  (p1 +p 2 )(p1 +p3 )
p1 =p(A)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
p1'  p( A) p( B)  p1  rD  ( p1  p2 )( p1  p3 ) 
 p1  rD  ( p  p1 p2  p1 p3  p2 p3 ) 
2
1
p1  rD  ( p12  p1 p2  p1 p3  p1 p4  p1 p4  p2 p3 )
 p1  rD  p1  D  (1  r ) D.
p1'  p( A) p( B)  (1  r ) D.
Measure of disequilibria
D= 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)=(1-r)nD(0);
Maximal speed convergence to equilibrium for r=0.5
D(n)=(0.5)nD(0);
Gene Conservation Low
p1’+ p2’ = p1+ p2=p(A);
p1’+ p3’ = p1+ p3=p(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
p1’=p12+p1p2+p1p3+(1-r)p1p4+rp2p3
p2’=p22+p1p2+p2p4+rp1p4+(1-r)p2p3
p3’=p32+p3p4+p1p3+rp1p4+(1-r)p2p3
p4’=p42+p3p4+p2p4+(1-r)p1p4+rp2p3
r=0
p1’=p12+p1p2+p1p3+p1p4 = p1
p2’=p22+p1p2+p2p4+p2p3 = p2
p3’=p32+p3p4+p1p3+p2p3 = p3
p4’=p42+p3p4+p2p4+p1p4 = p4
p1’=p1- rD ; p2’=p2 +rD;
p3’=p3+ rD; p4’=p4 - rD.
p1’=p12+p1p2+p1p3+(1-r)p1p4+rp2p3
p2’=p22+p1p2+p2p4+rp1p4+(1-r)p2p3
p3’=p32+p3p4+p1p3+rp1p4+(1-r)p2p3
p4’=p42+p3p4+p2p4+(1-r)p1p4+rp2p3
r=1
p1’=p12+p1p2+p1p3+p2p3 = (p1+p2)(p1+p3) = p(A)p(B)
p2’=p22+p1p2+p2p4+p1p4 = (p1+p2)(p2+p4) = p(A)p(b)
p3’=p32+p3p4+p1p3+p1p4 = (p3+p4)(p1+p3) = p(a)p(B)
p4’=p42+p3p4+p2p4+p2p3 = (p3+p4)(p2+p4) = p(a)p(b)
p1’=p1- rD ; p2’=p2 +rD;
D(n)=(1-r)nD(0);
p3’=p3+ rD; p4’=p4 - rD.
D0  0. D1  0
simulation
Multilocus multiallele population
Three loci
probabilit y
gametes
(1  r1 )(1  r2 )
ABC , abc
r1 (1  r2 )
aBC ,
(1  r1 )r2
ABc , abC
r1r2
AbC aBc
Abc
_____________________
1
all possible genotypes
ABC  1, ABc  2, AbC  3, Abc  4,
aBC  5, aBc  6, abC  7, abc  8
probabilit y
gametes  for zygote (1,8)
(1  r1 )(1  r2 )
ABC , abc
r1 (1  r2 )
aBC ,
(1  r1 )r2
ABc , abC
r1r2
AbC aBc
Abc
p1  (1  r1 )(1  r2 ) p1 p8  ...
p2  (1  r1 )r2 p1 p8  ...
p3  r1r2 p1 p8  ...
...
ABC  1, ABc  2, AbC  3, Abc  4,
aBC  5, aBc  6, abC  7, abc  8
p1  p2  p3  p4  p( A); p5  p6  p7  p8  p(a)
p1  p2  p5  p6  p( B); p3  p4  p7  p8  p(b)
p1  p3  p5  p7  p( B); p2  p4  p6  p8  p(c)
Equilibrium point
p1  P ( A) p ( B ) p (C )
p2  P ( A) p ( B ) p (c)
p3  P ( A) p (b) p (C )
...
Equilibrium point=limiting point of trajectories
General case
p1  (1  r1 )(1  r2 ) p1 p8  ...
p2  (1  r1 )r2 p1 p8  ...
p3  r1r2 p1 p8  ...
...
p1  11,1 p1 p1  12,1 p1 p2  13,1 p1 p3  14,1 p1 p4 
 16,1 p1 p6  17,1 p1 p7  18,1 p1 p8   22,1 p2 p2  ... 
 all possible combination
{ij, s }  set of probabilit ies
Linkage distribution
ij, s   ji, s
ij,1  ij, 2  ...  ij,8  1
M loci and L alleles in each locus
p1  11,1 p1 p1  12,1 p1 p2  13,1 p1 p3  14,1 p1 p4 
 16,1 p1 p6  17,1 p1 p7  18,1 p1 p8  ... 
 all possible combination
{ij, s }  set of probabilit ies
Linkage distribution
ij, s   ji, s
ij,1  ij, 2  ...  ij, M  1
{ij, s }  set of probabilit ies
Linkage distribution
Problem: definition of the linkage distribution.
Nonrandom crossovers.
2

p1  p1  2 p1 p2
2

p2  p2  2 p2 p3
p1  p2  p3  1
p3  p32  2 p1 p3
p1  p1 (1  p2  p3 )
p2  p2 (1  p3  p1 )
p3  p3 (1  p1  p2 )
p3  0  p1  p2  1
p1  p1 (1  p2  p3 )  p1 (2  p1 )
definition of the linkage distribution.
p(u | v)  probabilit y this partition
Equilibrium point for multilocus population
p(a1a2 a3 ...am )  p(a1 ) p(a2 ) p(a3 )... p(am )
Speed of the convergenc e to equilibria point
is max(1 -  (u | v)), where u, v  0
Polyploids systems
4-ploids
Chromatid dabbling
Four gamete produced
2-ploids (diploids)
p1  11,1 p1 p1  12,1 p1 p2  13,1 p1 p3  14,1 p1 p4 
 16,1 p1 p6  17,1 p1 p7  18,1 p1 p8   22,1 p2 p2  ... 
 all possible combination
Problem: definition of the coefficients.
Polyploids systems