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

Recovering Migration Rates Using a Deterministic Approach. Applied to Human Data. Yosef E. Maruvka 1* , Nadav M. Shnerb 1 Yaneer Bar-Yam 2 , Jonh Wakeley 3 1 Department of Physics Bar-Ilan University. * http://yosi.maruvka.googlepages.com/ 2. New England Complex System institution Boston

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

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  1. 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

  2. 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.

  3. Master Equation The probability to have n lineages at time t in the past The time dependence of is given by the equation:

  4. Replacing Stochastic Description With Deterministic Description Deterministic ODE This replacement is valid when

  5. 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:

  6. 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 population’s size: In the common notation:

  7. N1 n1 N2 n2 m2 m1 Migration-Coalescence Forward in time process:

  8. Mean Field Approximationfor Two Demes Backward in time process Number of lineages as function of time:

  9. Uniqueness – External Branch Length Probability to not coalesce: Uniqueness, probability to coalesce at time t: U=2 U=1 U=4 Erik M. Rauch and Yaneer Bar-Yam. Nature431, 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

  10. 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.

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

  12. 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):

  13. Recovering Growth Rate of a Population Recovering growth rate. Growth estimation 0.0054+-0.0003 Real growth rate 0.0056

  14. Migration Rate Between Two Growing Populations Number of lineages as a function of time: The probability to not coalesce, as function of time:

  15. 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 -> China 0.01 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.

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

  17. 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

  18. Thank you for listening

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