Patterns of Evolution

(1)

Patterns of Evolution

Summary statistics based on segregating sites

Site Frequency Spectrum

4 3 2 1 1 1

0 1 2 3

1 2 3 4 5

(2)

Patterns of Evolution

Summary statistics based on segregating sites

Site Frequency Spectrum

total number of segregating sites in an sample of size n

number of mutants that appear in i copies

in the sample

:

1



^



ⁿ

i

S

i

S

0 1 2 3

1 2 3 4 5

i

:

S S

i

: )

1

¹

(

2 1



^

 



 









ⁿ

i

n

i n i S

i



average number of pairwise differences

i

(3)

: )

1

¹

(

2 1



^

 



 









ⁿ

i

n

i n i S

i



Patterns of Evolution

Summary statistics based on segregating sites

Site Frequency Spectrum

total number of segregating sites in an sample of size n

number of mutants that appear in i copies

in the sample

:

1



^



ⁿ

i

S

i

S

i

: S

Each mutation of size

i

contributes to divergence in

i (n – i)

sequence pairs

0 1 2 3

1 2 3 4 5

S

i

(4)

Coalescent Theory

Estimators

Unbiased estimators of the mutation parameter q = 4Nu:



^







¹

1 1

1

ˆ

ⁿ

1

i n

i

i n

W

S i

a q S

singleton estimator:

^ˆ

_s

^ ^ ¹  ^S

₁

^ ^S

_n_₁



n q n

Watterson‘s estimator:

 -based estimator:

i n

i

n

i ( n i ) S

ˆ

¹

1 1

2



^





















  q

_

singletons of the folded spectrum

i n

H



^

i S

















 ¹ 1 1 2 2

q ˆ

Fay and Wu‘s estimator:

(equal weights)

(intermediate frequencies)

(high frequencies)

(extreme frequencies)

(5)

Coalescent Theory

Test statistics

Test statistics for the deviation from neutrality :



W



W

D

T

q q



ˆ Var ˆ

ˆ ˆ



 

Tajima‘s D:

Fay and Wu‘s H:

Fu and Li‘s D:



W S



S W

D

FL

q q

ˆ Var ˆ

ˆ ˆ



 



H



H

FW

q q



ˆ Var ˆ

ˆ ˆ



 

⁰

0,2 0,4 0,6 0,8 1

1 2 3 4 5 6

 0

D

T

(6)

Coalescent Theory

Test statistics

Test statistics for the deviation from neutrality :



W



W

D

T

q q



ˆ Var ˆ

ˆ ˆ



 

Tajima‘s D:

Fay and Wu‘s H:

Fu and Li‘s D:



W S



S W

D

FL

q q

ˆ Var ˆ

ˆ ˆ



 



H



H

FW

q q



ˆ Var ˆ

ˆ ˆ



 

⁰

0,2 0,4 0,6 0,8 1

1 2 3 4 5 6

 0

D

T

(7)

Coalescent Theory

Test statistics

Test statistics for the deviation from neutrality :



W



W

D

T

q q



ˆ Var ˆ

ˆ ˆ



 

Tajima‘s D:

Fay and Wu‘s H:

Fu and Li‘s D:



W S



S W

D

FL

q q

ˆ Var ˆ

ˆ ˆ



 



H



H

FW

q q



ˆ Var ˆ

ˆ ˆ



 

⁰

0,2 0,4 0,6 0,8 1

1 2 3 4 5 6

 0

D

FW

(8)

Coalescent Theory

Test statistics

Test statistics for the deviation from neutrality :



W



W

D

T

q q



ˆ Var ˆ

ˆ ˆ



 

Tajima‘s D:

Fay and Wu‘s H:

Fu and Li‘s D:



W S



S W

D

FL

q q

ˆ Var ˆ

ˆ ˆ



 



H



H

FW

q q



ˆ Var ˆ

ˆ ˆ



 

⁰

0,2 0,4 0,6 0,8 1

1 2 3 4 5 6

 0

H

FW

Patterns of Evolution