Evolutionary operator: p`= p 2 +pq; q` = q 2 +pq. For lethal alleles: p`= p 2 +pq; q` = q 2 +pq. Allele frequenses A-p, a - q For Thalassemia evolutionary operator or

Evolutionary operator with selection - mean fitness Selection in case Thalassemia: W AA =0.89; W Aa =1; W aa =0.2 Selection recessive lethal gene: W AA =1; W Aa =1; Waa=0

- mean fitness W AA, W Aa, W aa –individual fitnesses Why mean?

Equilibria points p=0, q=1 - population contains a allele only and on the zygote level the population consist of the homozygotes aa; p=1,q=0 - population contains A allele only and on the zygote level the population consist of the homozygotes A A.

Equation for equilibria points

Condition of the polymorphic state

Superdominance, when a heterozygote is fitter than both homozygotes Superrecessivity, when a heterozygote is les fit than either homozygotes In intermediate cases: W AA  W aa  W Aa (if W AA < W Aa ) or W Aa  W aa  W AA (if W Aa < W AA ) The population has no polymorphic equilibria Heterozygote equilibrium states: p>0, q>0

Lethal allele Let W aa =0 If W Aa > max(W AA,W aa ) =W AA Equilibrium point is polymorphic Previous conditions: W AA =W Aa =1, W aa =0 Condition not so realistic

Selection against a recessive allele.

Selection in case Thalassemia W AA =0.89, W Aa =1, W aa =0.2 If W Aa > max(W AA,W aa ) =W AA Equilibrium point is polymorphic Good condition. Note, that can be W AA <1

Dominant selection(also selection against a recessive allele) Two different phenotypes {AA, Aa}, {aa} W AA =W Aa =1, W aa =1-s W AA =W Aa =1-s, W aa =1 No polymorphic equilibria point

Haploid selection

Equilibria points and trajectories

Selection against a recessive allele. Trajectories.

Example. Selection against recessive lethal gene

Fisher’s Fundamental Theorem of Natural Selection Mean fitness increase along the trajectory

Lethal allele

max

Selection against a recessive allele.

Mean fitness in case Thalassemia W AA =0.89, W Aa =1, W aa =0.2

W AA =0.50, W Aa =1, W aa =0.2

Mean fitness calculation and dynamics

Convergence to equilibria In intermediate cases: W aa  W Aa  W AA (or W AA  W Aa  W aa ) The population has no polymorphic equilibria

Convergence to equilibria Superdominance (overdominance), when a heterozygote is fitter than both homozygotes Superrecessivity (underdominance), when a heterozygote is les fit than either homozygotes

Blood groups A,B,O –alleles allel enzyme A O B dominance A A AA, AO, = A B B BB, BO, = B O - AB = AB OO = O

A O B dominance A A AA, AO, = A B B BB, BO, = B O - AB = AB OO = O

Evolutionary operator

Simulation

One-locus multiallele autosomal systems

Fishers Fundamental Theorem of Natural Selection Mean fitness increase along the trajectory