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.
Autosomal locus
m
AA
BB
AB
AA
AA
f
BB
AB
AB
½AA+½AB
AB
½AA+½AB
BB
½BB+½AB
½BB+½AB
¼AA+¼BB+½AB
Let state of population is xAA,xBB,xAB
(AA,AA) - xAAxAA ; (AA,BB) – xAAxBB; (AA,AB) - xAA xAB; (BB,BB) - xBB xBB; (BB,AB) - xBB xAB; (AB,AB) - xAB xAB
(xAA ) ´ = (xAA)2 + xAAxAB + ¼(xAB ) 2
(xBB ) ´ = (xBB)2 + xBBxAB + ¼(xBB ) 2
(xAB ) ´ = 2xAAxBB + xAAxAB + xBBxAB + ½(xAB ) 2
(xAA )´ = (xAA)2 + xAAxAB + ¼(xAB ) 2 = (xAA+ ½xAB )2
(xBB )´ = (xBB)2 + xBBxAB + ¼(xBB ) 2 = (xBB+ ½xAB )2
(xAB )´ = 2xAAxBB + xAAxAB + xBBxAB + ½(xAB ) 2 = 2(xAA+ ½xAB )(xBB+ ½xAB )
p = (xAA+ ½xAB ); q = (xBB+ ½xAB );
p+q=1
p and q is the frequencies of alleles A and B in the population.
p`=(xAA )´ + 1/2(xAB )´ = p2 +pq=p(p+q)=p
q`=(xBB )´ + 1/2(xAB )´ = q2 +pq=q(p+q)=q
p`= p2 +pq;
(xAA )´ = p2; (xBB ) ´ = q2 ;
(xAB )´ = 2pq;
q` = q2 +pq.
II. X-linkage
Let distributions genotypes A1A1, A2A2, A1A2 in female part of
current generation are (x11,x22,x12) accordingly, and distributions
genotypes A1, A2 in male part of current generation are (y1,y2). As
usual x and y nonnegative and x11+x22+x12=1; y1+y2=1.
Evolutionary equations of male part of population
y1’=x11y1+x11y2+ ½x12y1+ ½x12y2
y2’=x22y1+x22y2+ ½x12y1+ ½x12y2
II. X-linkage
Evolutionary equations of female part of population
x11= x11y1+ ½x12y1
x22= x22y2+ ½x12y2
x12=x11y2+x22y1+ ½ x12y1+ ½ x12y2
II. X-linkage
Evolutionary operator of the population
y1’=x11y1+x11y2+ ½x12y1+ ½x12y2;
x11’= x11y1+ ½x12y1;
y2’=x22y1+x22y2+ ½x12y1+ ½x12y2
x22’= x22y2+ ½x12y2
x12’=x11y2+x22y1+ ½ x12y1+ ½ x12y2
Let pf= x11+ ½x12; qf= x22+ ½x12; pm=y1; qm=y2
pf+qf=x11+x22+x12=1; pm+qm=y1+y2=1
pf, qf -frequencies A1 and A2 in female part of population; pm, qm -frequencies A1 and A2 in male part of
population
Then
y1’=pf, y2’=qf
x11’=pfpm, x22’=qfqm, x12’=pfqm+pmqf
genotype-gene
connection
pf ’= x11’+ ½x12’ = pfpm+ ½ (pfqm+pmqf)=
½ pf (pm+qm)+½ pm (pf+qf)=½ (pf + pm);
pm’ = y1’= pf.
II. X-linkage
Evolutionary operator of the population (on gene level)
pf ’= ½ (pf + pm); qf ’= ½ (qf + qm);
pm’ = pf; qm’ = qf
Two level populations
Generation N
Diploid organizms
Genotypes
AA, aa, Aa
Haplod hamete
Alleles
A, a
Generation N+1
Diploid organizms
Genotypes
AA, aa, Aa
One-locus multiallele systems
Let A1, A2,…, As - set of alleles
Let p1, p2,…, ps –frequency alleles
in current generation (of the gametes)
(p1+ p2 + p3+… + ps=1)
p1p2
p1p1
(p1, p2,…, ps)
p2p1 p5p2
p7p11
?
Set of possible
zygotes {pipj}
One-locus multiallele systems
Zygote pipj produce gamete pi or pj
Evolutionary equation
pi’ = pi p1+ pi p2 + pi p3+… + pi ps
pi’ = pi (p1+ p2 + p3+… + ps)
pi’ = pi
Zygote frequences:
xij=pipj
Multiallele X-linkage system
Let A1, A2,…, As - set of alleles by X-linkage loci
pf1, pf2,…, pfs –frequency gametes of the female origins
pm1, pm2,…, pms –frequency gametes of the male origins
pf1, pf2,…, pfs
pfi pmj
p’f1, p’f2,…, p’fs
p’fi=
½(pfi+pmi)
Gametes level
pm1, pm2,…, pms (pY)
Zygotes level
pfi
p’m1, p’m2,…, p’ms
Gametes level
p’mi= pfi
Equilibira conditions pfi=pmi (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
A
O
B dominance
AA, AO, = A
BB, BO, = B
AB =
AB
OO =
O
allel
A
B
O
enzyme
A
B
-
The ABO Blood Group
A, B, O –alleles A and B dominant to O
Blood type Genotypes Frequency
A
AA,AO
RA (= pA2 +2pApO)
B
BB,BO
RB (= pB2 +2pBpO )
AB
AB
RAB (= 2pApB )
O
OO
R0
(= pO2)
HW
If RA, RB, RAB,and RO – are the observed frequencies
of the blood type, we have pO= (RO) ½, (pA+ pO)2 =
RA+ RO; …
Selection
Example. Selection against recessive lethal gene
p`= p2 +pq;
p`= p2 +pq;
q` = q2 +pq.
q` = q2 +pq.
p 2  pq
pq
p'  2
; q'  2
p  2 pq
p  2 pq
p 2  pq
1 p
p  p' 
 p  2
q

p  2 pq
1
p

 2 p  p 2  1  0 
 (1  p ) 2
2 p
Trajectory calculator
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q
generation
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p 2  pq
pq
p'  2
; q'  2
p  2 pq
p  2 pq
TRAJECTORY CALCULATION AND VIZUALIZATION
Letal1.exe
p 2  pq
pq
p'  2
; q'  2
p  2 pq
p  2 pq
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`= p2 +pq;
q` = q2 +pq
p`= 0.89p2 +pq;
q` = 0.2q2 +pq
0.89 p 2  pq
0.2q 2  pq
p' 
; q' 
2
2
0.89 p  2 pq  0.2q
0.89 p 2  2 pq  0.2q 2
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?
W11 p 2  pq
W22 q 2  pq
p' 
; q' 
2
2
W11 p  2 pq  W22 q
W11 p 2  2 pq  W22 q 2
if p, q  0 then in equilibria point
W  W11 p  q; W  W22 q  p;
W11 p  q  W22 q  p;
W11 p  (1  p )  W22 (1  p )  p
p (W11  2  W22 )  W22  1
W22  1
p
W11  2  W22
0.2  1
0.8
p

 0.879; q  0.121
0.89  2  0.2 0.91
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; ( W11  1?
•1/100 in areas where malaria is endemic.
)
W22  1
W11  1
p
;q 
;
W11  2  W22
W11  2  W22
W11  1
q  1/10 
;
W11  2  W22
W22  9W11  8
Evolutionary operator with selection
WAA p2  WAa pq
Waa q 2  WAa pq
p' 
; q' 
;
W
W
W  WAA p 2  Waa q 2  2WAa pq
- mean fitness
Selection recessive lethal gene: WAA=1; WAa=1; Waa=0
Selection in case Thalassemia: WAA=0.89; WAa=1; Waa=0.2
WAA p2  WAa pq
Waa q 2  WAa pq
p' 
; q' 
;
W
W
WAA, WAa, Waa –individual fitnesses
W  WAA p 2  Waa q 2  2WAa pq
- mean fitness
Equilibria points
WAA p2  WAa pq
Waa q 2  WAa pq
p
; q
;
W
W
W  WAA p 2  Waa q 2  2WAa pq
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 AA.
Homozygote equilibria states
Heterozygote equilibrium states: p>0, q>0
Waa  WAa
WAA  WAa
p
; q
;
WAA  Waa  2WAa
WAA  Waa  2WAa
max(WAA , Waa )  WAa
Superdominance, when a heterozygote is fitter than both homozygotes
min(WAA , Waa )  WAa
Superrecessivity , when a heterozygote is les fit than either homozygotes
In intermediate cases:
WAA Waa  WAa (if WAA < WAa) or
WAa Waa  WAA (if WAa < WAA)
The population has no polymorphic equilibria
Lethal allele
Waa  WAa
WAA  WAa
p
; q
;
WAA  Waa  2WAa
WAA  Waa  2WAa
max(WAA , Waa )  WAa
min(WAA , Waa )  WAa
Let WAA=0
If WAa > max(WAA,Waa) =Waa
Equilibrium point
is polymorphic
Dominant selection
p
Waa  WAa
WAA  WAa
; q
;
WAA  Waa  2WAa
WAA  Waa  2WAa
max(WAA , Waa )  WAa
min(WAA , Waa )  WAa
Two different phenotypes
{AA, Aa}, {aa}
WAA=WAa=1, Waa =1-s
No polymorphic equilibria point
Selection against a recessive allele
.
WAA=WAa=1, Waa =1-s;
qWaa+pWAa=q(1-s)+p=1-sq;
qWAa+pWAA=q+p=1;
W  WAA p 2  Waa q 2  2WAa pq = p2+ q2+ 2pq- sq2 =1- sq2
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
W  WAA p 2  Waa q 2  2WAa pq
In intermediate cases:
Waa WAa  WAA (or WAA WAa  Waa)
The population has no polymorphic equilibria
Convergence to equilibria
2
2
W  WAA p  Waa q  2WAa pq
max(WAA , Waa )  WAa
Superdominance (overdominance), when a heterozygote is fitter than
both homozygotes
min(WAA , Waa )  WAa
Superrecessivity (underdominance), when a heterozygote is les fit than
either homozygotes
One-locus multiallele autosomal systems
Fishers Fundamental
Theorem of Natural Selection
Mean fitness
W  W P  W12 P1 P2  ...  W P
2
11 1
increase along the trajectory
2
nn n