Extensions to Basic Coalescent Chapter 4, Part 2

Extension 2

• Relax another assumption – > Coalescent with population structure 3/3/2009 COMP 790-Extensions to Basic Coalescent 2

Extension 2

• • Finite island model Assume population is divided into d islands (demes) of equal size 2N, with total population of 2 N d genes.

• Each island contributes with a fraction m to a pool of migrants from which the island in return receives an equal proportion of migrants.

• Thus all demes can be treated equally 3/3/2009 COMP 790-Extensions to Basic Coalescent 3

Extension 2

• • Coalescent process with time scaled in units of 2 N d generations. Time until first coalescent event is exponentially distributed with parameter 3/3/2009 COMP 790-Extensions to Basic Coalescent 4

• Migration rate

Extension 2

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Extension 2

• In total this leads to a rate of I coal + I migr This event is coalescent with probability until the first event. • And migration event with probability 3/3/2009 COMP 790-Extensions to Basic Coalescent 6

Extension 2

• If coalescent event occurs, deme j is chosen with probability 3/3/2009 COMP 790-Extensions to Basic Coalescent 7

Extension 2

• Sample genealogy 3/3/2009 COMP 790-Extensions to Basic Coalescent 8

Extension 2

• • • • Coalescent tree in finite island model T(2,0) is the time for two genes in two different demes to find a MRCA T(0,1) is the time for two genes in the same demes to find MRCA Two genes cannot find a MRCA until they are in the same deme and they must do so before one of them migrates 3/3/2009 COMP 790-Extensions to Basic Coalescent 9

• Thus

Extension 2

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Extension 2

• When these equations are solved we have • Variances 3/3/2009 COMP 790-Extensions to Basic Coalescent 11

Extension 2

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Extension 2

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Extension 2

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Extension 2

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Basic coalescent (multimodal)

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Extension 2

• • • General models of subdivision 1. Stepping stone models Island model assume that a gene from one island or deme is equally likely to migrate to any other island • In stepping stone models Equally sized demes are repplaces by demes of arbitrary sizes and rate of migration from deme I to deme j is given 3/3/2009 COMP 790-Extensions to Basic Coalescent 16

Extension 2

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Extension 2

• Splitting of populations 3/3/2009 COMP 790-Extensions to Basic Coalescent 19

Extension 2

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Thank you

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