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

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Recovering Migration RatesUsing a Deterministic
Approach.Applied to Human Data.

• Yosef E. Maruvka1, Nadav M. Shnerb1
• Yaneer Bar-Yam2, Jonh Wakeley3
• 1 Department of Physics Bar-Ilan University.
• http//yosi.maruvka.googlepages.com/
• 2. New England Complex System institution Boston
• 3. Department of Organismic Evolutionary
  Biology Harvard University

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Coalescence 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.

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Master Equation
The probability to have n lineages at time t in
the past
The time dependence of is given by the
equation
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Replacing Stochastic Description With
Deterministic Description
Deterministic ODE
This replacement is valid when
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Time Dependence of Number of Lineages
Number of lineages as function of time
Time to most recent common ancestor
Average n(t) for 50 realizations
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Population Size Estimation
Fitting the simulation to the formula gives a
very good estimation of the population size.
For 40 realizations we get this estimation of
the populations size In the common notation
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Migration-Coalescence
Forward in time process
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Mean Field Approximationfor Two Demes
Backward in time process
Number of lineages as function of time
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Uniqueness External Branch Length
Probability to not coalesce
Uniqueness, probability to coalesce at time t
U2
U1
U4
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
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Relative Uniqueness
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 Uniqueness
The probability that an individual sampled from
deme i coalesced at time t in the past with any
of the n individuals sampled at deme j.
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Recovering Demographic Parameters
Using the previous equation for the relative
uniqueness, we recovered the parameters of the
simulation using the best fit.
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Recovering Growth Rate of a Population
The time dependence of the number of lineages as
a function of time

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

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Recovering Growth Rate of a Population

• Recovering growth
• rate.

Growth estimation 0.0054-0.0003 Real growth rate
0.0056
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Migration Rate Between Two Growing Populations
Number of lineages as a function of time
The probability to not coalesce, as function of
time
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Migration between India China
The migration from India to China was much
stronger than the migration from China to India
during the years -50ky -2ky Estimated
migration rates India -gt China 0.01 i/g China
-gt India 0.001 i/g
Data obtained from www.hvrbase.org. Nucleic Acids
Res.1998 Jan 126(1)126-9. Distance calculated
using DNAdist, Kimura 2 model.
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Advantages

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

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Summary

• Moved from a stochastic process to a
  deterministic process
• Demonstrated this method for one deme
• Used the Relative Uniqueness to estimate
  migration rates for fixed populations
• Expanded to include growing populations
• Applied this method to estimate migration rates
  between China and India

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Thank you for listening