Evolutionary operator: p`= p 2 +pq; q` = q 2 +pq. For lethal alleles: p`= p 2 +pq; q` = q 2 +pq....
32
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 2 2 2 2 2 2 2 . 0 2 89 . 0 2 . 0 ' ; 2 . 0 2 89 . 0 89 . 0 ' q pq p pq q q q pq p pq p p For Thalassemia evolutionary operator 2 2 11 22 2 2 2 2 11 22 11 22 ' ; ' 2 2 Wp pq Wq pq p q Wp pq Wq Wp pq Wq or pq p pq q pq p pq p p 2 ' ; 2 ' 2 2 2
-
Upload
gary-sparks -
Category
Documents
-
view
223 -
download
3
Transcript of Evolutionary operator: p`= p 2 +pq; q` = q 2 +pq. For lethal alleles: p`= p 2 +pq; q` = q 2 +pq....
Evolutionary operator: p`= p2 +pq; q` = q2 +pq.
For lethal alleles: p`= p2 +pq; q` = q2 +pq.
Allele frequenses A-p, a - q
22
2
22
2
2.0289.0
2.0';
2.0289.0
89.0'
qpqp
pqqq
qpqp
pqpp
For Thalassemia evolutionary operator
2 211 22
2 2 2 211 22 11 22
' ; '2 2
W p pq W q pqp q
W p pq W q W p pq W q
or
pqp
pqq
pqp
pqpp
2';
2'
22
2
Evolutionary operator with selection
2 2AA Aa aa AaW p W pq W q W pq
p ' ; q ' ;W W
2 2AA aa AaW W p W q 2W pq - mean fitness
Selection in case Thalassemia: WAA=0.89; WAa=1; Waa=0.2
Selection recessive lethal gene: WAA=1; WAa=1; Waa=0
2 2AA aa AaW W p W q 2W pq - mean fitness
WAA, WAa, Waa –individual fitnesses
Why mean?
2 2 2p q 2pq (p q) 1.
2 2AA Aa aa AaW p W pq W q W pq
p ' ; q ' ;W W
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.
2 2AA Aa aa AaW p W pq W q W pq
p ; q ;W W
2 2AA aa AaW W p W q 2W pq
AA Aa aa Aa
AA Aa aa Aa
AA Aa aa Aa
pW p(W p W q); qW q(W q W p);
if p,q 0
W W p W q; W W q W p;
W p W q W q W p;
q 1 p
2 2AA aa AaW W p W q 2W pq
aa Aa AA Aa
AA aa Aa AA aa Aa
W W W Wp ; q ;
W W 2W W W 2W
2 2AA Aa aa AaW p W pq W q W pq
p ; q ;W W
Equation for equilibria points
aa Aa AA aa Aa
aa Aa AA aa Aa
AA Aa AA aa Aa
AA Aa AA aa Aa
p 0
q
if
W W 0and W W 2W 0
or W W 0and W W 2W 0
if
W W 0and W W 2W 0
or W W 0and W W 0
0
2W
0 , 1p q
1p q aa Aa AA Aa
AA aa Aa AA aa Aa
W W W Wp ; q
W W 2W W W 2W
Condition of the polymorphic state
aa Aa
AA Aa
aa Aa
AA Aa
AA aa Aa
W W 0
W W 0
W W
W W
max(W , W ) W
Superdominance, when a heterozygote is fitter than both homozygotes
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
aa Aa AA Aa
AA aa Aa AA aa Aa
W W W Wp ; q ;
W W 2W W W 2W
Heterozygote equilibrium states: p>0, q>0
AA aa Aamin(W , W ) W
AA aa Aamax(W , W ) W
Lethal allele
AA Aa aa Aa
AA aa Aa AA aa Aa
W W W Wp ; q ;
W W 2W W W 2W
Let Waa=0
AA aa Aamax(W , W ) W
AA aa Aamin(W , W ) W
If WAa > max(WAA,Waa) =WAA
Equilibrium point is polymorphic
Previous conditions: WAA=WAa=1, Waa=0
Condition not so realistic
Selection against a recessive allele.
1, 1 .AA Aa aaW W W s
AA Aa aa Aa
AA aa Aa AA aa Aa
W W W Wp ; q ;
W W 2W W W 2W
AA aa Aamax(W , W ) W
AA aa Aamin(W , W ) W
Selection in case Thalassemia
AA Aa aa Aa
AA aa Aa AA aa Aa
W W W Wp ; q ;
W W 2W W W 2W
AA aa Aamax(W , W ) W
AA aa Aamin(W , W ) W
WAA=0.89, WAa=1, Waa=0.2
If WAa > max(WAA,Waa) =WAA
Equilibrium point is polymorphic
Good condition. Note, that can be WAA<1
Dominant selection(also selection against a recessive allele)
aa Aa AA Aa
AA aa Aa AA aa Aa
W W W Wp ; q ;
W W 2W W W 2W
AA aa Aamax(W , W ) W
AA aa Aamin(W , W ) W
Two different phenotypes
{AA, Aa}, {aa} WAA=WAa=1, Waa =1-s
WAA=WAa=1-s, Waa =1
No polymorphic equilibria point
Haploid selection
; ;AA A A Aa A a aa a aW W W
2 2AA aa A amax( , )
aa Aa AA Aa
AA aa Aa AA aa Aa
W W W Wp ; q ;
W W 2W W W 2W
AA aa Aamax(W , W ) W
AA aa Aamin(W , W ) W
2 2AA aa A amin( , )
Equilibria points and trajectories
Selection against a recessive allele.Trajectories.
2
2
2
2
2
2
22( 2 )
( 2 )
) ( )(
aa aa
aa
aa
aa Aa
aa Aa
Aa AA Aa
Aa AA Aa
Aa AA Aa
W q W pqq q q
W
W q W pq Wqq q
W
W pq W p W pq qq q
W
W pq W p q W pqq q
W
W p W p W qq q
W
W q W q
W q p
pq W q
2 2AA Aa aa AaW p W pq W q W pq
p ' ; q ' ;W W
2 2AA aa AaW W p W q 2W pq
2
1, 1 .
) (1 ) 1 ; ( )( 1; ;
AA Aa aa
Aa A Aa A aa
W W W s
W p s qW q p sq W p W qpq s
q qW
2 2
2
(1 )
1
pq s q q sq q q
W sq
Example. Selection against recessive lethal gene
Fisher’s Fundamental Theorem of Natural Selection
Mean fitness increase along the trajectory
Lethal allele2
2
2
( );
;
2 ( 2 ) (1 ).
(1 )(1 ) .
1; .
1 1
(1 )
p pq p p q
W W
W p pq p p q p q
q q
qp q
q q
pp
Wpq
qW
W q
2 2
2 2 2 2
(1 ) (1 ),
, , .
1, 1 0, 1 1 ,
1 1.
If W q W q
then q q q q q q
qq q q
q
2(1 )W q
0
0.2
0.4
0.6
0.8
1
1.20
0.07
0.14
0.21
0.28
0.35
0.42
0.49
0.56
0.63 0.7
0.77
0.84
0.91
0.98
q
Wmax
Selection against a recessive allele.2 2
AA Aa aa AaW p W pq W q W pqp ' ; q ' ;
W W
2 2
AA aa AaW W p W q 2W pq
1, 1 .AA Aa aaW W W s 2 2 2
2 2 2
2 2
2
2 2
(1 ); ;
(1 ) (1 ) ( ) 1.
( ) 1 ( ) 1.
.
1
1 .
1 ( ) 1 .
p pq s q pq sq qp q
W W W
W q s q s q
W W s q s q
q q
Then
sq qq sq W
W
sq s qq q q
2( ) 1W s q
0
0.2
0.4
0.6
0.8
1
1.2
0
0.0
6
0.1
2
0.1
8
0.2
4
0.3
0.3
6
0.4
2
0.4
8
0.5
4
0.6
0.6
6
0.7
2
0.7
8
0.8
4
0.9
0.9
6
s=0.5
s=0.7
s=0.9
s=0.1
Mean fitness in case Thalassemia
WAA=0.89, WAa=1, Waa=0.2
2 2AA aa Aa
2 2AA aa Aa
2AA aa Aa Aa AA AA
W W p W q 2W pq
W W (1 q) W q 2W (1 q)q
W q (W W 2W ) q(2W 2W ) W
2W 0.91q 0.22q 0.89
00.10.20.30.40.50.60.70.80.9
1
0
0.07
0.14
0.21
0.28
0.35
0.42
0.49
0.56
0.63 0.7
0.77
0.84
0.91
0.98
q
W
WAA=0.50, WAa=1, Waa=0.2
2W 1.3q q 0.50
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.80
0.07
0.14
0.21
0.28
0.35
0.42
0.49
0.56
0.63 0.7
0.77
0.84
0.91
0.98
q
W
Mean fitness calculation and dynamics
Convergence to equilibria2 2
AA aa AaW W p W q 2W pq
In intermediate cases:
Waa WAa WAA (or WAA WAa Waa)
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
AA aa Aamin(W , W ) W
AA aa Aamax(W , W ) W
2 2AA aa AaW W p W q 2W pq
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
1 1 1 1 2 1 3
2 2 1 2 2 2 3
3 3
1 2 3
1 3 2 3 3
; ;0
p p p p p p p
p p p
A p B
p p p p
p p p p
p
p p p
p
A O B dominance A AAA, AO, = A B BBB, BO, = B O -AB = ABOO = O
12
12
1 1 1 1 2 1 3
2 2 2
1 2 3
11 11
1 2 2 2 3
3 3 1 311
2 2
33
2
2 2 3 32
; ;0
p p p p p p p
p p p p p p p
p p
A p B p
W W
W
W W
p p W p p
p
pW
W
W
12
12
1 1 1 1 2 1 3
2 2 2
1 2 3
11 11
1 2 2 2 3
3 3 1 311
2 2
33
2
2 2 3 32
; ;0
p p p p p p p
p p p p p p p
p p
A p B p
W W
W
W W
p p W p p
p
pW
W
W
1 1 1 1 2 1 3
2 2 1 2 2 2 3
3 3 1 3 2 3 3
1 1 1 2
12
12
12
22 22
22
22 21 3 2 2 2 3 3 3
11 11
11
11 11 2
33
33
( ) /
( ) /
( ) /
2 2 2 .
p p p p p p p W
p p p p pW p p W
p p p p p p p
W W
W W
W p p p p p p p p p
W
W
W
W
W
p p pW W WW
W
W
Evolutionary operator
Simulation
One-locus multiallele autosomal systems
Fishers Fundamental Theorem of Natural Selection
2 211 1 12 1 2 nn nW W P W P P ... W P
Mean fitness
increase along the trajectory
![bw)># >] 8 PS ])...WfG 2Wf WfcWf q) %)Pq J 3fWvcP Wv P EW9P cf9 q) %)Pq J 3fWvc c fqP)f9P3 Wf v 9P3 P ) 9jq9P3 vj9P)jjU #v f Pq))% %) )Pq c 7) G 2Wf Wv vqP WvYf) WfG9P3 2Wf cf9 q)](https://static.fdocuments.in/doc/165x107/61345670dfd10f4dd73baaa2/bw-8-ps-wfg-2wf-wfcwf-q-pq-j-3fwvcp-wv-p-ew9p-cf9-q-pq.jpg)
