Evolutionary operator of the population. Selection

Two level populations From the point of view of genetics population has two levels of organization, namely zygote and gamete levels, tied together by the processes of meiosis and fertilization. Usually we consider a population as consisting of zygotes or, more precisely, of organisms which are genetically identified with zygotes. In parallel with the population of zygotes there exists and evolves the population of gametes, or the gamete pool of a given population.

f BBAAAB m AA AB BB ½AA+½AB ½BB+½AB ¼AA+¼BB+½AB AA BB AB½AA+½AB Let state of population is x AA,x BB,x AB (AA,AA) - x AA x AA ; (AA,BB) – x AA x BB; (AA,AB) - x AA x AB; (BB,BB) - x BB x BB; (BB,AB) - x BB x AB; (AB,AB) - x AB x AB (x AA ) ´ = (x AA ) 2 + x AA x AB + ¼(x AB ) 2 (x BB ) ´ = (x BB ) 2 + x BB x AB + ¼(x BB ) 2 (x AB ) ´ = 2x AA x BB + x AA x AB + x BB x AB + ½(x AB ) 2 Autosomal locus

(x AA )´ = (x AA ) 2 + x AA x AB + ¼(x AB ) 2 = (x AA + ½x AB ) 2 (x BB )´ = (x BB ) 2 + x BB x AB + ¼(x BB ) 2 = (x BB + ½x AB ) 2 (x AB )´ = 2x AA x BB + x AA x AB + x BB x AB + ½(x AB ) 2 = 2(x AA + ½x AB )(x BB + ½x AB ) p = (x AA + ½x AB ); q = (x BB + ½x AB ); p+q=1 p and q is the frequencies of alleles A and B in the population. (x AA )´ = p 2 ; (x BB ) ´ = q 2 ; (x AB )´ = 2pq; p`=(x AA )´ + 1/2 (x AB )´ = p 2 + pq=p(p+q)=p q`=(x BB )´ + 1/2 (x AB )´ = q 2 + pq=q(p+q)=q p`= p2 +pq; q` = q2 +pq.

II. X-linkage Let distributions genotypes A 1 A 1, A 2 A 2, A 1 A 2 in female part of current generation are (x 11,x 22,x 12 ) accordingly, and distributions genotypes A 1, A 2 in male part of current generation are (y 1,y 2 ). As usual x and y nonnegative and x 11 +x 22 +x 12 =1; y 1 +y 2 =1. Evolutionary equations of male part of population y 1 ’ =x 11 y 1 +x 11 y 2 + ½x 12 y 1 + ½x 12 y 2 y 2 ’ =x 22 y 1 +x 22 y 2 + ½x 12 y 1 + ½x 12 y 2

II. X-linkage Evolutionary equations of female part of population x 11 = x 11 y 1 + ½x 12 y 1 x 22 = x 22 y 2 + ½x 12 y 2 x 12 =x 11 y 2 +x 22 y 1 + ½ x 12 y 1 + ½ x 12 y 2

II. X-linkage Evolutionary operator of the population y 1 ’ =x 11 y 1 +x 11 y 2 + ½x 12 y 1 + ½x 12 y 2 ;y 2 ’ =x 22 y 1 +x 22 y 2 + ½x 12 y 1 + ½x 12 y 2 x 11 ’ = x 11 y 1 + ½x 12 y 1 ;x 22 ’ = x 22 y 2 + ½x 12 y 2 x 12 ’ =x 11 y 2 +x 22 y 1 + ½ x 12 y 1 + ½ x 12 y 2 Let p f = x 11 + ½x 12 ; q f = x 22 + ½x 12 ; p m =y 1 ; q m =y 2 Then y 1 ’ =p f, y 2 ’ =q f genotype-gene x 11 ’ =p f p m, x 22 ’ =q f q m, x 12 ’ =p f q m +p m q f connection p f ’ = x 11 ’ + ½x 12 ’ = p f p m + ½ (p f q m +p m q f )= ½ p f (p m +q m )+½ p m (p f +q f )=½ (p f + p m ); p f +q f =x 11 +x 22 +x 12 =1; p m +q m =y 1 +y 2 =1 p m ’ = y 1 ’ = p f. p f, q f -frequencies A 1 and A 2 in female part of population; p m, q m -frequencies A 1 and A 2 in male part of population

II. X-linkage Evolutionary operator of the population (on gene level) p f ’ = ½ (p f + p m ); q f ’ = ½ (q f + q m ); p m ’ = p f ; q m ’ = q f

Two level populations Diploid organizms Genotypes AA, aa, Aa Haplod hamete Alleles A, a Diploid organizms Genotypes AA, aa, Aa Generation N Generation N+1

One-locus multiallele systems Let A 1, A 2,…, A s - set of alleles Let p 1, p 2,…, p s –frequency alleles in current generation (of the gametes) (p 1 + p 2 + p 3 +… + p s =1) ? (p 1, p 2,…, p s ) Set of possible zygotes {p i p j } p 1 p 2 p 1 p 1 p 2 p 1 p 5 p 2 p 7 p 11

One-locus multiallele systems Evolutionary equation p i ’ = p i p 1 + p i p 2 + p i p 3 +… + p i p s p i ’ = p i (p 1 + p 2 + p 3 +… + p s ) p i ’ = p i Zygote p i p j produce gamete p i or p j Zygote frequences: x ij =p i p j

Multiallele X-linkage system p f1, p f2,…, p fs p m1, p m2,…, p ms Let A 1, A 2,…, A s - set of alleles by X-linkage loci p f1, p f2,…, p fs –frequency gametes of the female origins p m1, p m2,…, p ms –frequency gametes of the male origins (p Y ) Gametes level p fi p mj p fi Zygotes level p ’ m1, p ’ m2,…, p ’ ms Gametes levelp ’ f1, p ’ f2,…, p ’ fs p ’ fi = ½(p fi +p mi ) p ’ mi = p fi Equilibira conditions p fi =p mi (i=1,2,…,s)

An ideal population 1. Discrete non-overlapping generations 2. Allele frequencies are identical in males and females 3. Panmictic population: Mating of individuals is made at random 4. Population size is very large (infinite) 5. There is no migration (closed population) 6. Mutations can be ignored 7. Selection does not affect allele frequencies (neutral alleles) Properties of an ideal diploid population studied at a single autosomal locus with Mendelian inheritance

Predictions from Hardy-Weinberg : IF… – No selection – No mutation – No migration – Random mating THEN… – Allele frequencies remain constant – Genotype frequencies predictable

HW for locus with dominant alleles

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

The ABO Blood Group A, B, O –alleles A and B dominant to O Blood type Genotypes Frequency A AA,AO R A (= p A 2 +2p A p O ) B BB,BO R B (= p B 2 +2p B p O ) AB AB R AB (= 2p A p B ) O OO R 0 ( = p O 2 ) If R A, R B, R AB,and R O – are the observed frequencies of the blood type, we have p O = (R O ) ½, ( p A + p O ) 2 = R A + R O ; … HW

Selection

Example. Selection against recessive lethal gene

p`= p 2 +pq; q` = q 2 +pq.

Current state (point)Next state (point)

TRAJECTORY CALCULATION AND VIZUALIZATION Letal1.exe

Dominant lethal allele

Thalassemia A very large number of different mutations induce either a-thalassemia (a reduction in the synthesis rate of Hb a-chains) or b-thalassemia (a reduction in the synthesis rate of Hb b-chains). Both of these classes of mutations can induce malarial resistance in heterozygotes, but once again at the expense of anemia (which depending upon the exact nature of the thalassemia, can vary from virtually none to lethal) in the homozygotes. Thalassemia is found in high frequency in many historically malarial regions of the world in Africa, the Mediterranean, and Asia.

Selection in case Thalassemia p`= p 2 +pq; q` = q 2 +pq p`= 0.89p 2 +pq; q` = 0.2q 2 +pq

TRAJECTORY CALCULATION AND VIZUALIZATION Equilibrium point Talas1.exe

Sravnit s nature population. Vibor coefficientov, chtobi poluchit nature chastotu. Odnoznachno li eto mozno cdelat- ved dva coefficient?

1 in 25 heterozygote alpha-thalassemia SE Asians,Chinese 1 in 30 heterozygote beta-thalassemia Greeks, Italians B-thalassemia 1/20,000 in general population; ( ) 1/100 in areas where malaria is endemic.

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

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. Homozygote equilibria states

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

Dominant selection Two different phenotypes {AA, Aa}, {aa} W AA =W Aa =1, W aa =1-s No polymorphic equilibria point

Selection against a recessive allele. W AA =W Aa =1, W aa =1-s; qW aa +pW Aa =q(1-s)+p=1-sq;qW Aa +pW AA =q+p=1; = p 2 + q 2 + 2pq- sq 2 =1- sq 2 When q is very small, a homozygotes are very rare. When q is small, q2 is very small. So the recessive allele is hardly ever expressed. The same logic applies to the case where the recessive allele is favored (1+s). The disfavored dominant can be eliminated easily even when scarce, but when the recessive is rare, even though it is favored it is very hard for selection to "see" it and build it up.

Example. Selection against recessive lethal gene

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

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

One-locus multiallele autosomal systems

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