Two-locus systems

1. Two-locus systems

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

3. Two-locus two allele population Gamete p1 p2 p3 p4 Next generation on zygote level Independent combination of randomly chosen parental gametes

4. 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)

5. Table gametes from genotypes II (r) – cross-over (1-r) –no 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)

6. gamete Position effect

7. Table zygote productions AB: p1’=p12+p1p2+p1p3+(1-r)p1p4+rp2p3 Evolutionary equation for genotype AB

8. 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 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

9. Measure of disequilibria D= p1p4-p2p3

10. p1’=p1- rD;p2’=p2 +rD; p3’=p3+ rD; p4’=p4 - rD. p1+p2=p(A) p1+p3=p(B) AB Ab aB ab p1 p2 p3 p4 Gene Conservation Low p1’+ p2’ = p1+ p2=p(A); p1’+ p3’ = p1+ p3=p(B)

11. Two-locus two allele population. Equilibria. p1=p1- rD;p2=p2 +rD; p3=p3+ rD; p4=p4 - rD. Measure of disequilibria D= p1p4-p2p3 D=0; p1p4 = p2p3

12. 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 Measure of disequilibria D= p1p4-p2p3

13. Convergence to equilibrium p1’=p1- rD;p2’=p2 +rD; p3’=p3+ rD; p4’=p4 - rD. D’=p1’p4’- p2’p3’; D’=(p1- rD)(p4 - rD)-(p2 +rD)(p3+ rD) p1 p4-p2p3 -rD(p1+p2+p3+p4) +(rD)2-(rD)2 D’= D(n)=(1-r)nD(0); D’=D-rD=(1-r)D; Maximal speed convergence to equilibrium for r=0.5 D(n)=(0.5)nD(0);

14. 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

15. 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.

16. 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; p3’=p3+ rD; p4’=p4 - rD. D(n)=(1-r)nD(0);

17. simulation

18. Multilocus multiallele population Three loci

20. Equilibrium point Equilibrium point=limiting point of trajectories

21. General case

22. M loci and L alleles in each locus

23. Problem: definition of the linkage distribution. Nonrandom crossovers.

25. definition of the linkage distribution.

26. Equilibrium point for multilocus population

27. Polyploids systems

28. 4-ploids 2-ploids (diploids) Chromatid dabbling Four gamete produced

29. Problem: definition of the coefficients.

30. Polyploids systems