Recovering Migration Rates Using a Deterministic Approach. Applied to Human Data.

Recovering Migration RatesRecovering Migration RatesUsing a Deterministic Using a Deterministic

Approach.Approach.Applied to Human Data.Applied to Human Data.

Yosef E. MaruvkaYosef E. Maruvka1*1*, Nadav M. Shnerb, Nadav M. Shnerb11 Yaneer Bar-YamYaneer Bar-Yam22, Jonh Wakeley, Jonh Wakeley33 1 Department of Physics Bar-Ilan University.1 Department of Physics Bar-Ilan University.**http://yosi.maruvka.googlepages.com/http://yosi.maruvka.googlepages.com/

2. New England Complex System institution Boston2. New England Complex System institution Boston3. Department of Organismic & Evolutionary Biology Harvard 3. Department of Organismic & Evolutionary Biology Harvard

UniversityUniversity

Coalescence TheoryCoalescence Theory• One deme (well mixed) model.• Fixed mutation rate determines the time

scale• Genetics has no effect on fitness

(use uncoded DNA) • Population is well-mixed – no spatial

structure• The real history yields a phylogenetic tree• Dashed lines – lineages with no current

descendent • Full lines – lineages expressed in current

polymorphism data • “backward in time”: coalescence model

• Haploid mitochondrial DNA.• Wright Fisher Model.

Master EquationMaster Equation

00

1

2

1

2

1

N

tPnn

N

tnPntP nn

n

tPn

The probability to have n lineages at time t in the past

The time dependence of is given by the equation:

tPn

0Nn

Replacing Stochastic Replacing Stochastic Description With Deterministic Description With Deterministic

DescriptionDescription tnPtn n

0

2

2N

nn

dt

tndt

0

2

2N

nn

0

2

2N

n

0

2

2N

tntn

This replacement is valid when tntn

Deterministic ODE

Time Dependence of Number Time Dependence of Number of Lineagesof Lineages

0

0

0

22

nN

t

Ntn

Number of lineages as function of

time:

1 10 100 1000 10000

1

10

n(t)

t

00

112n

NTMRCATime to most recent common ancestor:

Average n(t) for 50 realizations:

Population Size EstimationPopulation Size Estimation

Fitting the simulation to the formula gives a very good estimation of the population size.

0

2

2

nNt

Ntn

For 40 realizations we get this estimation of the population’s size:

In the common notation: 0029.00201.02 0 N

1446100450 N

350100500 N

Migration-CoalescenceMigration-Coalescence

m1

m2

N2

n2

N1

n1

Forward in time process:

Mean Field ApproximationMean Field Approximationfor Two Demesfor Two Demes

Number of lineages as Number of lineages as function of time:function of time:

22112

22

2

22111

21

1

2

2

nmnmN

nn

nmnmN

nn

Backward in time process

Uniqueness – External Branch Uniqueness – External Branch LengthLength

0N

tntQtQ

3

00

02

2

80

Ntn

nN

dt

dQtU

Erik M. Rauch and Yaneer Bar-Yam. Nature 431, 449-452 (2004)Caliebe, A., et al On the Length Distribution of External Branches in Coalescence Trees:Genetic Diversity within Species. TPB, 72 (2007), 245-252

U=1 U=2U=4

Probability to not coalesce:

Uniqueness, probability to coalesce at time t:

2

222211

2

1

112211

1

N

tQtntQmtQm

dt

tdQ

N

tQtntQmtQm

dt

tdQ

The probability to not coalesce (survive) with any of the individuals from the other deme until time t:Rate equation for the probability not to coalesce until time t:

Relative UniquenessRelative Uniqueness

tQtQtQ 21

dt

tdQtPR

Relative UniquenessThe probability that an individual sampled from

deme i coalesced at time t in the past with any of the n n individuals sampled at deme j.

Recovering Demographic Recovering Demographic ParametersParameters

Using the previous equation for the relative uniqueness, we recovered the parameters of the simulation using the best fit.

.002 0.007=m0.005, 0.0038 =m 250, 2700=N 100,2400 =N1 :resultsfit Best

0.01=m 0.005,=m 2500, =N=N: parameters Real

212

21 21

Recovering Growth Rate of a Recovering Growth Rate of a PopulationPopulation

The time dependence of the The time dependence of the number of lineages as a number of lineages as a function of time:function of time:

teN

ntn

0

2

2

000

0

2

2

Nnen

Ntn

t

The number of lineages as a The number of lineages as a function of time (backward):function of time (backward):

Recovering Growth Rate of a Recovering Growth Rate of a PopulationPopulation

Recovering growthRecovering growth

rate.rate.

Growth estimation 0.0054+-0.0003Real growth rate 0.0056

Migration Rate Between Two Migration Rate Between Two Growing PopulationsGrowing Populations

22112

22

2

22111

21

1

2

1

2

2

nmnmeN

nn

nmnmeN

nn

t

t

t

t

eN

tQtntQmtQm

dt

tdQ

eN

tQtntQmtQm

dt

tdQ

2

1

2

222211

2

1

112211

1

Number of lineages as a function of time:

The probability to not coalesce, as function of time:

Migration between India & Migration between India & ChinaChina

The migration from India to China was The migration from India to China was muchmuch

stronger than the migration from China stronger than the migration from China to India during the years [-50ky -2ky ]to India during the years [-50ky -2ky ]

Estimated migration rates:Estimated migration rates: India -> China 0.01 i/g India -> China 0.01 i/g China -> India 0.001 i/g China -> India 0.001 i/g

Data obtained from www.hvrbase.org. Nucleic Acids Res.1998 Jan 1;26(1):126-9. Distance calculated using DNAdist, Kimura 2 model.

Advantages Advantages

Easy calculations.Easy calculations. Very fast estimations.Very fast estimations. Can handle large amount of data.Can handle large amount of data. No need for prior distribution.No need for prior distribution.

SummarySummary

Moved from a stochastic process to a Moved from a stochastic process to a deterministic processdeterministic process

Demonstrated this method for one demeDemonstrated this method for one deme Used the Relative Uniqueness to Used the Relative Uniqueness to

estimate migration rates for fixed estimate migration rates for fixed populationspopulations

Expanded to include growing populationsExpanded to include growing populations Applied this method to estimate Applied this method to estimate

migration rates between China and Indiamigration rates between China and India

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