Extensions to Basic Coalescent Chapter 4, Part 2.

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Presentation transcript:

Extensions to Basic Coalescent Chapter 4, Part 2

Extension 2 Relax another assumption – > Coalescent with population structure 3/3/2009COMP 790-Extensions to Basic Coalescent2

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 3/3/2009COMP 790-Extensions to Basic Coalescent3

Extension 2 Coalescent process with time scaled in units of 2 N d generations. Time until first coalescent event is exponentially distributed with parameter 3/3/2009COMP 790-Extensions to Basic Coalescent4

Extension 2 Migration rate 3/3/2009COMP 790-Extensions to Basic Coalescent5

Extension 2 In total this leads to a rate of I coal + I migr until the first event. This event is coalescent with probability And migration event with probability 3/3/2009COMP 790-Extensions to Basic Coalescent6

Extension 2 If coalescent event occurs, deme j is chosen with probability 3/3/2009COMP 790-Extensions to Basic Coalescent7

Extension 2 Sample genealogy 3/3/2009COMP 790-Extensions to Basic Coalescent8

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 3/3/2009COMP 790-Extensions to Basic Coalescent9

Extension 2 Thus 3/3/2009COMP 790-Extensions to Basic Coalescent10

Extension 2 When these equations are solved we have Variances 3/3/2009COMP 790-Extensions to Basic Coalescent11

Extension 2 3/3/2009COMP 790-Extensions to Basic Coalescent12

Extension 2 3/3/2009COMP 790-Extensions to Basic Coalescent13

Extension 2 3/3/2009COMP 790-Extensions to Basic Coalescent14

Extension 2 3/3/2009COMP 790-Extensions to Basic Coalescent15 Basic coalescent (multimodal)

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 3/3/2009COMP 790-Extensions to Basic Coalescent16

Extension 2 3/3/2009COMP 790-Extensions to Basic Coalescent18

Extension 2 Splitting of populations 3/3/2009COMP 790-Extensions to Basic Coalescent19

Extension 2 3/3/2009COMP 790-Extensions to Basic Coalescent20

Thank you 3/3/2009COMP 790-Extensions to Basic Coalescent21