the coalescent model - hits ggmbhrescaling time - exponential growth 2500 5000 7500 10000 12500...

The Coalescent Model

Florian Weber

23. 7. 2016

coalescent = „zusammenwachsend“

Outline

I Population Genetics and the Wright-Fisher-modelI The CoalescentI Non-constant population-sizesI Further extensionsI Summary

Population Genetics

(Shamelessly stealing Alexis’ slides)I Study of polymorphisms in a population

I What are the processes that introduce polymorphisms in thepopulation?

I If a polymorphism exists in a population, will it be there forever?

I Is there some process that removes polymorphisms from thepopulation?

I Do the polymorphisms exhibit patterns?I . . .

Motivation

I The coalescent is basically the Wright-Fisher-model with a lotof analysis.

I It can easily do calculations about the past

I It is very fast to compute

I Is can easily be extended to represent a more complex reality

Motivation

Hardy-Weinberg

I Assuming an infinite population size, random mating, diploidpopulation, no selection. . .the allele-frequencies are constant

I Infinity is weird. . . 0.3×∞ =∞I . . . and unrealistic

Hardy-Weinberg

I Assuming an infinite population size, random mating, diploidpopulation, no selection. . .the allele-frequencies are constant

I Infinity is weird. . . 0.3×∞ =∞I . . . and unrealistic

Wright-Fisher

I Assuming a finite but constant population size, randommating, non-overlapping generations, no selection. . .all alleles except for one will disappear over time.

I The likelihood for an allele to prevail is equal to it’s initialfrequency

Wright-Fisher

I Assuming a finite but constant population size, randommating, non-overlapping generations, no selection. . .all alleles except for one will disappear over time.

I The likelihood for an allele to prevail is equal to it’s initialfrequency

Wright-Fisher

Figure 1: A simulation of three alleles under the model

Wright-Fisher (Individuals)past

present

Figure 2: An evolutionary history in the model

present

Figure 3: Extinct alleles removed

present

Figure 4: Surviving Tree

Wright-Fisher (MRCA)

present

Figure 5: Most Recent Common Ancestor marked

Wright-Fisher (Coalescence-Events)

Coalescence-Events

present

Figure 6: Coalescence-Events of the green individuals

P(B)P(A)

Figure 7: Two individuals and their parents

I Likelihood for two nodes to coalesce in the previous generation:p(P(A) = P(B)) = 1

I In the previous two generations:1− (N−1

N · N−1N ) = 1−

(N−1

I In the previous three generations:1−

((N−1

)2· N−1

)= 1−

(N−1

I In the previous t generations1−

(N−1

N · N−1N ) = 1−

(N−1

((N−1

)2· N−1

)= 1−

(N−1

N · N−1N ) = 1−

(N−1

((N−1

)2· N−1

)= 1−

(N−1

N · N−1N ) = 1−

(N−1

((N−1

)2· N−1

)= 1−

(N−1

* without loss of generality

(N-1) / N

Figure 8: Likelihood of coalescence

I Likelihood of coalescence in the previous t generations:

1−(N − 1

I Likelihood for lineages to remain distinct for t generations:(N − 1N

I Expected time for coalescence: E (t) = N

I Rescale: τ = tN : (N − 1

)dNτe−−−−→N→∞

e−τ

1−(N − 1

)dNτe−−−−→N→∞

e−τ

1−(N − 1

)dNτe−−−−→N→∞

e−τ

1−(N − 1

)dNτe−−−−→N→∞

e−τ

⇒ The likelihood for two lineages to stay distinct over timeis exponentially small!

0.50.25

1 2 time t

lineage 1

lineage 2

Moar Lineages!!

lineages

Figure 9: http://what-if.xkcd.com/13/

More Lineages

I Likelihood of no coalescence in one generation and threelineages:

N − 1N × N − 2

I One generation, k lineages:

N − 1N × N − 2

N × · · · × N − k + 1N =

k−1∏i=1

N − iN

More Lineages

I Likelihood of no coalescence in one generation and threelineages:

N − 1N × N − 2

I One generation, k lineages:

N − 1N × N − 2

N × · · · × N − k + 1N =

k−1∏i=1

N − iN

More Lineages

I For some reason this is equal to:

k−1∏i=1

N − iN = 1−

N + O( 1N2

)≈ 1−

2)is the binomial coefficient and equates to k·(k−1)

I There are(k

2)ways to pick two lineages from a set of k lineages.

I Therefore a coalescence-event is(k

2)-times as likely with k

lineages than with 2

I The number of coalescence-events grows quadraticallywith the number of lineages!

More Lineages

k−1∏i=1

N − iN = 1−

N + O( 1N2

)≈ 1−

I There are(k

More Lineages

k−1∏i=1

N − iN = 1−

N + O( 1N2

)≈ 1−

I There are(k

More Lineages

k−1∏i=1

N − iN = 1−

N + O( 1N2

)≈ 1−

I There are(k

More Lineages

coalescence-rateEvents gettingexponentially rare

Figure 10: More lineages = faster coalscence

Properties

I Few deep furcations

I Likelihood: Everything is possible but maybe unlikely

I Calculation is backward in times (Wright-Fisher: forward)

I Efficient: no calculation per individual or for extinct lineages

Properties

Non-constant population-sizesWorldpopulation,billions

10,000 BC 8000 6000 4000 2000 AD 1 1000 2000

Figure 11: Wordpopulation - not very constant [Wikimedia]

Non-constant population-sizes

I Non-constant, but known population-sizeI Coalescence is more likely in small populationsI ⇒ Coalescence-rate changes over time

I Simply rescale time.

Non-constant population-sizes

I Non-constant, but known population-sizeI Coalescence is more likely in small populationsI ⇒ Coalescence-rate changes over time

I Simply rescale time.

Rescaling Time

I Before: t Generations corresponded to t/N units ofcoalescence-time

I Now: t Generations correspond to

t∑i=1

units of coalescence-timeI Note: for a constant population both formulas are equal

Rescaling Time - Example

I 5 Generations, with on average 5 individuals:

I For constant 5 individuals: τ = tN = 5

5 = 1 unit of coalescencetime

I For non-constant {4, 4, 5, 6, 6} individuals:

τ =t∑

= 14 + 1

4 + 15 + 1

6 + 16 = 31

note the lesser influence of the larger generations

I A generation with twice the size, will get halve thecoalescence-time

τ =t∑

= 14 + 1

4 + 15 + 1

6 + 16 = 31

τ =t∑

= 14 + 1

4 + 15 + 1

6 + 16 = 31

τ =t∑

= 14 + 1

4 + 15 + 1

6 + 16 = 31

Rescaling Time - Exponential Growth

population-size

Generation

10 516

Figure 12: Exponentially growing population versus coalescence-time

Rescaling Time - Exponential Growth

2500 5000 7500 10000 12500 15000

constant size

exponential growth

2500 5000 7500 10000 12500 15000

real time (generations)

Figure 13: Exponentially growing and constant opulations. Note thereverse time-scale! [Nordborg]

Rescaling Time - Applicability

I Approximation converges against theory for growing NI Close enough for most purposes

Further Extensions

I Separated PopulationsI Diploid PopulationsI Males and FemalesI SelectionI Multiple SpeciesI . . .

Wright-Fisher:Assuming a finite but constant population size, randommating, non-overlapping generations, no selection. . .

Coalescent:Assuming non-overlapping generations. . .

Further Extensions

An actual example

Figure 14: Coalescent vs. Anthropological Estimates [Atkinson et al.]

Software

Software that uses the coalescent model1:BEAST, COAL, CoaSim, DIYABC, DendroPy, GeneRecon, genetree,GENOME, IBDSim, IMa, Lamarc, Migraine, Migrate, MaCS, ms &msHOT, msms, Recodon and NetRecodon, SARG, simcoal2,TreesimJ

1Source: https://en.wikipedia.org/wiki/Coalescent_theory

Summary

I The coalescent is the Wright-Fisher-model plus mathI Coalescent-events are, with exponential likelihood, relatively

recentI The more lineages there are, the more coalescence-events occurI Non-Constant populations can be simulated by rescaling timeI The simulated time for a generation is anti-proportional to it’s

References

ContentI Magnus Nordborg, “Coalescent Theory”, March 2000

Software-listI en.wikipedia.org/wiki/Coalescent_theory

ImagesI Fig. 09: Randal Munroe: what-if.xkcd.com/13/I Fig. 11: El T:

commons.wikimedia.org/wiki/File:Population_curve.svgI Fig. 13: Magnus Nordborg: “Coalescent Theory”, 2000I Fig. 14: Atkinson et al.: “mtDNA variation predicts population

size in humans and reveals a major Southern Asian chapter inhuman prehistory”, 2008

the coalescent model - hits ggmbhrescaling time - exponential growth 2500 5000 7500 10000 12500...

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