Embed Size (px)
Citation preview
Extensions to Basic CoalescentChapter 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
Extension 2
• Migration rate
3/3/2009 COMP 790-Extensions to Basic Coalescent 5
Extension 2
• In total this leads to a rate of Icoal + I migr until the first event. This event is coalescent with probability
• 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
Extension 2
• Thus
3/3/2009 COMP 790-Extensions to Basic Coalescent 10
Extension 2
• When these equations are solved we have
• Variances
3/3/2009 COMP 790-Extensions to Basic Coalescent 11
Extension 2
3/3/2009 COMP 790-Extensions to Basic Coalescent 12
Extension 2
3/3/2009 COMP 790-Extensions to Basic Coalescent 13
Extension 2
3/3/2009 COMP 790-Extensions to Basic Coalescent 14
Extension 2
3/3/2009 COMP 790-Extensions to Basic Coalescent 15
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
3/3/2009 COMP 790-Extensions to Basic Coalescent 16
Extension 2
3/3/2009 COMP 790-Extensions to Basic Coalescent 18
Extension 2
• Splitting of populations
3/3/2009 COMP 790-Extensions to Basic Coalescent 19
Extension 2
3/3/2009 COMP 790-Extensions to Basic Coalescent 20
Thank you
3/3/2009 COMP 790-Extensions to Basic Coalescent 21
