intensive course on population genetics · intensive course on population genetics ville mustonen...

Intensive Course on Population Genetics

Ville Mustonen

Thursday, 1 September 2011

• Population genetics: ”The study of the distribution of inherited variation among a group of organisms of the same species” [Oxford Dictionary of Biology]

http://www.exploratorium.edu/exhibits/mutant_flies/normal.gif http://www.emacswiki.org/emacs/EmacsIcons


http://www.exploratorium.edu/exhibits/mutant_flies/normal.gif

http://www.exploratorium.edu/exhibits/mutant_flies/normal.gif

http://www.emacswiki.org/emacs/EmacsIcons

http://www.emacswiki.org/emacs/EmacsIcons

AACTGTCCACGTTCTTCCGATGCAGCCTGAAACTGTCCACGTTCTTCCGATGCAGCCTGAAACTGTCCACGTTCTTCCGATGCAGCCTGAAACTGTCCACGTTCTTCCGATGCAGCCTGAAACTGTCCACGTTCTTCCGATGCAGCCTGAAACTGTCCAGGTTCTTCCGATGCAGCCTGAAACTGTCCAGGTTCTTCCGATGCAGCCTGAAACTGTCCAGGTTCTTCCGATGCAGCCTGA

AACTGTCCAGGTTCTTCCGACGCAGCCTGA

Cross- and intra-species comparisons at the molecular level

m in

grou

p sa

mpl

es

outgroup

polymorphic locus substitution event

frequency counts 000000000500000000008000000000


• Let’s study fundamental evolutionary forces:

1. genetic drift (noise in reproduction)

2. selection (differential reproductive success of individuals)

3. mutation (process providing variation)

4. ...


Genetic drift: noise in reproduction time in generations

indiv

idual

s


5

10

15

20

10

20

30

400.0

0.1

0.2

0.3

0.4


Wright - Fisher process

pb = Nb/N = x

pa = Na/N = 1 ! x

P (m, N, t + 1) =

!

N

m

"

xmt (1 ! xt)

N!m

!m" = Nxt

!m2" # !m"2 = Nxt(1 # xt)

!xt+1" = xt

!x2t+1" # !xt+1"

2 =xt(1 # xt)

N


Wright - Fisher process

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0 N=100

xt

t

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0N=10

xt

t

!d ! N


Selection: differential reproductive success of individuals

2 4 6 8 10

2

3

4

5

6

7

t

Na = FaNa

Nb = FbNb

x =d

dt

Nb(t)

Na(t) + Nb(t)=

Nb

Na + Nb

!

Nb

Na + Nb

Nb + Na

Na + Nb

= (Fb ! Fa)x(1 ! x)


Selection

x = Fbax(1 ! x)

x(t) =x0 exp(Fbat)

1 + x0(exp(Fbat) ! 1)

0 2 4 6 8 10 12 140.0

0.2

0.4

0.6

0.8

1.0

t

xt!s ! 1/Fba


Mutation: source of variationtime in generations

indiv

idual

s


Mutation: source of variation

µ

µ

Na = µNb ! µNa

Nb = µNa ! µNb

x =d

dt

Nb(t)

Na(t) + Nb(t)=

Nb

Na + Nb

!

Nb

Na + Nb

Nb + Na

Na + Nb

= µ(1 ! 2x)


Mutation

x = µ(1 ! 2x)

x(t) =1

2(1 + exp(!2µt)(2x0 ! 1))

0 2 4 6 8 10 12 140.0

0.2

0.4

0.6

0.8

1.0

t

xt

!m ! 1/µ


Drift, Selection and Mutation

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

N=10

xt

t

!d ! N

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

N=10

xt

t

!d ! N



0 2 4 6 8 10 12 140.0

0.2

0.4

0.6

0.8

1.0

t

xt

!s ! 1/Fba

0 2 4 6 8 10 12 140.0

0.2

0.4

0.6

0.8

1.0

t

xt

!s ! 1/Fba



t

xt

!m ! 1/µ

0 2 4 6 8 10 12 140.0

0.2

0.4

0.6

0.8

1.0

t

xt

!m ! 1/µ

0 2 4 6 8 10 12 140.0

0.2

0.4

0.6

0.8

1.0



x(t) = Fba x(t)[1 ! x(t)] + µ[1 ! 2x(t)] + !x(t)

!!x(t)!x(t!)" = (x(1 # x)/N) "(t # t!)

!!x(t)" = 0

• Langevin equation:


One locus two alleles model

• Wright-Fisher process with mutation and selection

• Ratio of time scales:

lighter shading indicates the fitter state

popu

latio

n fr

actio

n of

alle

le b

µt

N1/µ = µN ! 1


=

!p(x, t)g(x, !, dt) ! !

"

"xpg +

!2

2

"2

"x2pg + . . . d!

= p(x, t)

!g(x, !, dt)d! !

"

"xp

!!gd! +

1

2

"2

"x2p

!!2gd!

Diffusion equation

g(x, !, dt) x ! x + !, dt

p(x, t) probability density of populations

probability of change

p(x, t + dt) =

!p(x ! !, t)g(x ! !, !, dt)d!

[SH Rice, Evolutionary Theory (2004)]


= p(x, t)

!g(x, !, dt)d! !

"

"xp

!!gd! +

1

2

"2

"x2p

!!2gd!

= p(x, t) !!p(x, t)M(x)

!xdt +

1

2

!2p(x, t)V (x)

!x2dt

!!2g(x, !, dt)d! = !!"2 + var(!)

= V (x)dt

!!g(x, !, dt)d! = !!"

= M(x)dt

!g(x, !, dt)d! = 1

rate of directional change of allele frequency at point x

variance of allele frequency change due to nondirectional effects


p(x, t + dt) ! p(x, t)

dt= !

!p(x, t)M(x)

!x+

1

2

!2p(x, t)V (x)

!x2

!p(x, t)

!t= !

!p(x, t)M(x)

!x+

1

2

!2p(x, t)V (x)

!x2

Now we just need to plug in our earlier results to get Kimura’s diffusion equation for one

locus two alleles case:

!p(x, t)

!t=

1

2

!2

!x2

x(1 ! x)

Np(x, t) !

!

!x[Fbax(1 ! x) + µ(1 ! 2x)]p(x, t)


One locus two alleles model: looking at the averages

Equilibrium distributionsG

!t(x|x

0)

x

0.0 0.2 0.4 0.6 0.8 1.0

0.001

0.01

0.1

1G

!t(x|x

0)

x

0.0 0.2 0.4 0.6 0.8 1.0

0.001

0.01

0.1

1


One locus two alleles model: looking at the averages

Q(x

)

x

0.0 0.2 0.4 0.6 0.8 1.0

0.001

0.01

0.1

1Q

(x)

x

0.0 0.2 0.4 0.6 0.8 1.0

0.001

0.01

0.1

1

after infinite time


Substitution rate from Kimura’s diffusion equation

u/µ

!4 !2 0 2 4

1

2

3

4

2NFba

u(2NFba) = µ2NFba

1!exp(!2NFba)


Substitution dynamicsN

1/µ = µN ! 1

pa = ubapb ! uabpa

pb = uabpa ! ubapb


Substitution dynamics

!pa" =uba

uba + uab

!pb" =uab

uba + uab

!pa" = !pb" exp(#2NFba)

Ndeleterious = Nbeneficial exp(2NFba)


Conclusions

• Fundamental evolutionary forces:

1. genetic drift, diffusive force (!), effect strongest in small populations, changes allele frequencies at time-scale

2. selection, directed force, increases the fitter phenotypes, time-scale

3. mutation, directed force, “restarts” the process from monomorphic states, time-scale

1/F

1/µ

2N

• More complex scenarios, recombination, linkage, diploid genomes, demographics, frequency/time-dependent selection ...

• Further reading, Gillespie: Population Genetics, Hartl and Clark: Principles of Population Genetics


http://books.google.com/books?id=SB1vQgAACAAJ&dq=hartl+clark+population&hl=en&ei=pXNfTtXxDoOK4gTbjdFC&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCkQ6AEwAA




intensive course on population genetics · intensive course on population genetics ville mustonen...

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