Two-locus systems

Scheme of genotypes genotype Two-locus genotypes Multilocus genotypes genotype

Two-locus two allele population Gamete p 1 p 2 p 3 p 4 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) 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(r) Zygote gamete Type zygote- both loci is heterozygotes Zygote (AB,ab) have gamete (AB) with frequency 0.5(1-r)

gamete Position effect

Table zygote productions AB: p 1 ’ =p 1 2 +p 1 p 2 +p 1 p 3 +(1-r)p 1 p 4 +rp 2 p 3 Evolutionary equation for genotype AB

p 1 ’ =p 1 2 +p 1 p 2 +p 1 p 3 +(1-r)p 1 p 4 +rp 2 p 3 p 2 ’ =p 2 2 +p 1 p 2 +p 2 p 4 +rp 1 p 4 +(1-r)p 2 p 3 p 3 ’ =p 3 2 +p 3 p 4 +p 1 p 3 +rp 1 p 4 +(1-r)p 2 p 3 p 4 ’ =p 4 2 +p 3 p 4 +p 2 p 4 +(1-r)p 1 p 4 +rp 2 p 3 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 p 1 p 2 p 3 p 4

Measure of disequilibria D= p 1 p 4 -p 2 p 3

p 1 ’ =p 1 - rD ; p 2 ’ =p 2 +rD; p 3 ’ =p 3 + rD; p 4 ’ =p 4 - rD. Gene Conservation Low p 1 ’ + p 2 ’ = p 1 + p 2 =p(A); p 1 ’ + p 3 ’ = p 1 + p 3 =p(B) AB Ab aB ab p 1 p 2 p 3 p 4 p 1 +p 2 =p(A) p 1 +p 3 =p(B)

Two-locus two allele population. Equilibria. p 1 =p 1 - rD ; p 2 =p 2 +rD; p 3 =p 3 + rD; p 4 =p 4 - rD. Measure of disequilibria D= p 1 p 4 -p 2 p 3 D=0; p 1 p 4 = p 2 p 3

p 1 = p(A) p(B); p 2 = p(A) p(b); p 3 = p(a) p(B); p 4 = p(a) p(b). In equilibria point the genes are statistically independence. But the genes are dependent physically, because are in pairs on chromosome Measure of disequilibria D= p 1 p 4 -p 2 p 3

Convergence to equilibrium D ’ =p 1 ’ p 4 ’ - p 2 ’ p 3 ’ ; p 1 ’ =p 1 - rD ; p 2 ’ =p 2 +rD; p 3 ’ =p 3 + rD; p 4 ’ =p 4 - rD. D ’ =(p 1 - rD )(p 4 - rD)-(p 2 +rD)(p 3 + rD) D’=D’= p 1 p 4 - p 2 p 3 -rD(p 1 +p 2 +p 3 +p 4 )+(rD) 2 -(rD) 2 D ’ =D-rD=(1-r)D; D (n) =(0.5) n D (0) ; Maximal speed convergence to equilibrium for r=0.5 D (n) =(1-r) n D (0) ;

p 1 = p(A) p(B); p 2 = p(A) p(b); p 3 = p(a) p(B); p 4 = p(a) p(b). Gene Conservation Low p 1 ’ + p 2 ’ = p 1 + p 2 =p(A); p 1 ’ + p 3 ’ = p 1 + p 3 =p(B) Infinite set of equilibrium points

p 1 ’ =p 1 2 +p 1 p 2 +p 1 p 3 +(1-r)p 1 p 4 +rp 2 p 3 p 2 ’ =p 2 2 +p 1 p 2 +p 2 p 4 +rp 1 p 4 +(1-r)p 2 p 3 p 3 ’ =p 3 2 +p 3 p 4 +p 1 p 3 +rp 1 p 4 +(1-r)p 2 p 3 p 4 ’ =p 4 2 +p 3 p 4 +p 2 p 4 +(1-r)p 1 p 4 +rp 2 p 3 r=0 p 1 ’ =p 1 2 +p 1 p 2 +p 1 p 3 +p 1 p 4 = p 1 p 2 ’ =p 2 2 +p 1 p 2 +p 2 p 4 +p 2 p 3 = p 2 p 3 ’ =p 3 2 +p 3 p 4 +p 1 p 3 +p 2 p 3 = p 3 p 4 ’ =p 4 2 +p 3 p 4 +p 2 p 4 +p 1 p 4 = p 4 p 1 ’ =p 1 - rD ; p 2 ’ =p 2 +rD; p 3 ’ =p 3 + rD; p 4 ’ =p 4 - rD.

p 1 ’ =p 1 2 +p 1 p 2 +p 1 p 3 +(1-r)p 1 p 4 +rp 2 p 3 p 2 ’ =p 2 2 +p 1 p 2 +p 2 p 4 +rp 1 p 4 +(1-r)p 2 p 3 p 3 ’ =p 3 2 +p 3 p 4 +p 1 p 3 +rp 1 p 4 +(1-r)p 2 p 3 p 4 ’ =p 4 2 +p 3 p 4 +p 2 p 4 +(1-r)p 1 p 4 +rp 2 p 3 r=1 p 1 ’ =p 1 2 +p 1 p 2 +p 1 p 3 +p 2 p 3 = (p 1 +p 2 )(p 1 +p 3 ) = p(A)p(B) p 2 ’ =p 2 2 +p 1 p 2 +p 2 p 4 +p 1 p 4 = (p 1 +p 2 )(p 2 +p 4 ) = p(A)p(b) p 3 ’ =p 3 2 +p 3 p 4 +p 1 p 3 +p 1 p 4 = (p 3 +p 4 )(p 1 +p 3 ) = p(a)p(B) p 4 ’ =p 4 2 +p 3 p 4 +p 2 p 4 +p 2 p 3 = (p 3 +p 4 )(p 2 +p 4 ) = p(a)p(b) p 1 ’ =p 1 - rD ; p 2 ’ =p 2 +rD; p 3 ’ =p 3 + rD; p 4 ’ =p 4 - rD. D (n) =(1-r) n D (0) ;

simulation

Multilocus multiallele population Three loci

Equilibrium point Equilibrium point=limiting point of trajectories

General case

M loci and L alleles in each locus

Problem: definition of the linkage distribution. Nonrandom crossovers.

definition of the linkage distribution.

Equilibrium point for multilocus population

Polyploids systems

4-ploids2-ploids (diploids) Chromatid dabbling Four gamete produced

Problem: definition of the coefficients.

Polyploids systems