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Noble names, branching processes, and fixation probabilities

Noble Names, Branching Processes, and Fixation Probabilities

Joachim Hermisson

Mathematics & MFPL, University of Vienna

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Noble names, branching processes, and fixation probabilities

Sir Francis Galton (1822-1911)

The fate of aristocratic family names

Henry William Watson (1827 - 1903)

A problem of inheritance inspires new mathematics

“The decay of the families of men who occupied conspicuous positions in the past times has been a subject of frequent remark and has given rise to various conjectures …“ [Galton and Watson 1874]

Conjecture:

• Aristocrats (or “other men of genius“) have reduced fertility → trade-off ?

• Population only maintained by proletarians

Galton:

• It may also be just chance: Need a model ! Degradation risk !

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Noble names, branching processes, and fixation probabilities

Sir Francis Galton (1822-1911)

Henry William Watson (1827 - 1903)

Z0 → Z1

k2 = ?

Galton‘s branching model

k1 = ?

k3 = ?

• Each founder j can have kj = 0,1,2,3, … sons

→ independently and with identical probability pk

Z0 founders of noble families in generation n = 0

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Noble names, branching processes, and fixation probabilities

Sir Francis Galton (1822-1911)

Henry William Watson (1827 - 1903)

• Each founder j can have kj = 0,1,2,3, … sons

→ independently and with identical probability pk

Galton‘s branching model

Z0 founders of noble families in generation n = 0

• Iterate with offspring generation

Z0 → Z1 → Z2 → … Zn

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Noble names, branching processes, and fixation probabilities

in and transition probabilities

Sir Francis Galton (1822-1911)

Henry William Watson (1827 - 1903)

is a Markov chain with values

Galton‘s branching model

 

Zn n0

A Galton-Watson process with offspring distribution

 

pk k0

Z

i

k Z

i

mp

km

P

1

 |  

where is the m-fold convolution of

(i.e., the distribution of the sum of m i.i.d. random variables, each with distribution )

 

pk

 pk

 

pkm

Due to independence, we can use Z0 = 1 as default initial state (“fate of one family“)

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Noble names, branching processes, and fixation probabilities

Watson‘s insights

0

) (

k

k kt p

t

use generating function of offspring distribution

Henry William Watson (1827 - 1903)

Sir Francis Galton (1822-1911)

probability for extinction by generation n

n :

Recursion:

n1

 

n

n (monotonic and bounded)

Thus:

 

fixed point of

 (t )

( continuous)

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Noble names, branching processes, and fixation probabilities

Fixed points of

1 t 1

0

) (

k

k k

t p

t

1 )

1 ( 

'(1) 

k kpk

) 0

0

(  p

( ) 0

0 )

(

 

  t t

Assume:

p0 > 0

p0 + p1 < 1

)

 (t

 1

: Case

average offspring number)

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Noble names, branching processes, and fixation probabilities

Fixed points of

1 t 1

0

) (

k

k k

t p

t

1 )

1 ( 

'(1) 

k kpk

0 )

(

0 )

(

 

  t t

Assume:

p0 > 0

p0 + p1 < 1

 1

: Case

)

 (t

average offspring number)

(9)

Noble names, branching processes, and fixation probabilities

Fixed points of

1 t

0

) (

k

k k

t p

t

1 )

1 ( 

'(1) 

k kpk

0 )

(

0 )

(

 

  t t

Assume:

p0 > 0

p0 + p1 < 1

 1

: Case

)

 (t

0 0 p

1

average offspring number)

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Noble names, branching processes, and fixation probabilities

Extinction probability

Thus:

 

smallest fixed point of

 (t )

For

k

k p

k average offspring number:

1 .

1   1 .

2   1 .

3  

subcritical critical supercritical

1

1

• Galton and Watson overlooked the smaller fixed point and concluded that all family names must die out because of chance alone

• Lotka (1931): for US white males (1920 data) 0.82

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Noble names, branching processes, and fixation probabilities

Fixation probability

 1  p

fix

Ronald A. Fisher J. B. S. Haldane

The spread of a rare beneficial mutant through a population can be described

as a supercritical branching process

[Fisher 1922, Haldane 1927]

The fate of a beneficial mutant is decided while it is rare

When frequent: loss very unlikely → eventual fixation (frequency 1)

While rare: independent reproduction!

Mutant population can be described by a branching process

Fixation probability follow as:

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Noble names, branching processes, and fixation probabilities

Fixation probability

 

(1)

2 ) 1 1 ( )

1 ( 1

1

  

2

  pfix   pfix   pfix   pfix 

Average offspring number Wildtype:

Mutant:

1

wt

m 1 s

(constant population size)

(typical s : 10-4 – 10-2 → “slightly supercitical“)

m

(1) 1 ; (1) 

 

2

2 ( 1)

) 1 (

) 1 (

k

m m

m

pk

k

k

  

Taylor expansion of the fixed point equation:

where:

( variance of the offspring distribution)m2 :

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Noble names, branching processes, and fixation probabilities

(all mutants in heterozygotes)

Fixation probability p

fix

Solve for :

Typical

s

: 10-4 – 10-2

In particular, Wright-Fisher model (~ Poisson offspring distribution):

m

hs

m2

   1 

 

 

2

 

2

2

2 1

1

2 s s

p

m m

m m

m

fix  

 

pfix  2hs

almost all beneficial mutations in a population are lost because of random fluctuations (genetic drift)

(Haldane 1927)

Referenzen

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