Week 9: Coalescents, part 2 Genome 562 March, 2015 Week 9: Coalescents, part 2 – p.1/71
Fixation probabilities with multiplicative fitnesses 1.0 100 0.9 0.8 10 Fixation probability 0.7 1 0.6 0.1 0.5 −0.1 −1 0.4 0.3 −10 0.2 0.1 −100 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Initial gene frequency U ( p ) = 1 − e − 4 Nsp 1 − e − 4 Ns Week 9: Coalescents, part 2 – p.2/71
Gene copies in a population of 10 individuals A random−mating population Time Week 9: Coalescents, part 2 – p.3/71
Going back one generation A random−mating population Time Week 9: Coalescents, part 2 – p.4/71
... and one more A random−mating population Time Week 9: Coalescents, part 2 – p.5/71
... and one more A random−mating population Time Week 9: Coalescents, part 2 – p.6/71
... and one more A random−mating population Time Week 9: Coalescents, part 2 – p.7/71
... and one more A random−mating population Time Week 9: Coalescents, part 2 – p.8/71
... and one more A random−mating population Time Week 9: Coalescents, part 2 – p.9/71
... and one more A random−mating population Time Week 9: Coalescents, part 2 – p.10/71
... and one more A random−mating population Time Week 9: Coalescents, part 2 – p.11/71
... and one more A random−mating population Time Week 9: Coalescents, part 2 – p.12/71
... and one more A random−mating population Time Week 9: Coalescents, part 2 – p.13/71
... and one more A random−mating population Time Week 9: Coalescents, part 2 – p.14/71
The genealogy of gene copies is a tree Genealogy of gene copies, after reordering the copies Time Week 9: Coalescents, part 2 – p.15/71
Ancestry of a sample of 3 copies Genealogy of a small sample of genes from the population Time Week 9: Coalescents, part 2 – p.16/71
Here is that tree of 3 copies in the pedigree Time Week 9: Coalescents, part 2 – p.17/71
Kingman’s coalescent Random collision of lineages as go back in time (sans recombination) Collision is faster the smaller the effective population size u9 u8 In a diploid population of u7 Average time for u6 effective population size N, u5 k copies to coalesce to 4N u4 k−1 = Average time for n k(k−1) u3 copies to coalesce = 4N ( 1 − 1 ( generations n Average time for u2 two copies to coalesce = 2N generations What’s misleading about this diagram: the lineages that coalesce are random pairs, not necessarily ones that are next to each other in a linear order. Week 9: Coalescents, part 2 – p.18/71
The Wright -Fisher model This is the canonical model of genetic drift in populations. It was invented in 1930 and 1932 by Sewall Wright and R. A. Fisher. In this model the next generation is produced by doing this: Choose two individuals with replacement (including the possibility that they are the same individual) to be parents, Each produces one gamete, these become a diploid individual, Repeat these steps until N diploid individuals have been produced. The effect of this is to have each locus in an individual in the next generation consist of two genes sampled from the parents’ generation at random, with replacement. Week 9: Coalescents, part 2 – p.19/71
Sir John Kingman J. F . C. Kingman in about 1983 Currently Emeritus Professor of Mathematics at Cambridge University, U.K., and former head of the Isaac Newton Institute of Mathematical Sciences. Week 9: Coalescents, part 2 – p.20/71
The coalescent – a derivation The probability that k lineages becomes k − 1 one generation earlier turns out to be (as each lineage “chooses” its ancestor independently): k ( k − 1 ) / 2 × Prob (First two have same parent , rest are different) � k � (since there are = k ( k − 1 ) / 2 different pairs of copies) 2 We add up terms, all the same, for the k ( k − 1 ) / 2 pairs that could coalesce; the sum is: � � 1 1 k ( k − 1 ) / 2 × 1 × 2N × 1 − 2N � � � � 2 1 − k − 2 × 1 − × · · · × 2N 2N so that the total probability that a pair coalesces is = k ( k − 1 ) / 4N + O ( 1 / N 2 ) Week 9: Coalescents, part 2 – p.21/71
Can probabilities of two or more lineages coalescing Note that the total probability that some combination of lineages coalesces is 1 − Prob (Probability all genes have separate ancestors) � � � � � � �� 1 − 1 1 − 2 1 − k − 1 = 1 − 1 × . . . 2N 2N 2N � � 1 − 1 + 2 + 3 + · · · + ( k − 1 ) + O ( 1 / N 2 ) = 1 − 2N and since 1 + 2 + 3 + . . . + ( n − 1 ) = n ( n − 1 ) / 2 the quantity � � 1 − k ( k − 1 ) / 4N + O ( 1 / N 2 ) ≃ k ( k − 1 ) / 4N + O ( 1 / N 2 ) = 1 − Week 9: Coalescents, part 2 – p.22/71
Can calculate how many coalescences are of pairs This shows, since the terms of order 1 / N are the same, that the events involving 3 or more lineages simultaneously coalescing are in the terms of order 1 / N 2 and thus become unimportant if N is large. Here are the probabilities of 0, 1, or more coalescences with 10 lineages in populations of different sizes: 0 1 > 1 N 100 0.79560747 0.18744678 0.01694575 1000 0.97771632 0.02209806 0.00018562 10000 0.99775217 0.00224595 0.00000187 Note that increasing the population size by a factor of 10 reduces the coalescent rate for pairs by about 10 -fold, but reduces the rate for triples (or more) by about 100-fold. Week 9: Coalescents, part 2 – p.23/71
The coalescent To simulate a random genealogy, do the following: 1. Start with k lineages 2. Draw an exponential time interval with mean 4N / ( k ( k − 1 )) generations. 3. Combine two randomly chosen lineages. 4. Decrease k by 1. 5. If k = 1 , then stop 6. Otherwise go back to step 2. Week 9: Coalescents, part 2 – p.24/71
An accurate analogy: Bugs In A Box There is a box ... Week 9: Coalescents, part 2 – p.25/71
An accurate analogy: Bugs In A Box with bugs that are ... Week 9: Coalescents, part 2 – p.26/71
An accurate analogy: Bugs In A Box hyperactive, ... Week 9: Coalescents, part 2 – p.27/71
An accurate analogy: Bugs In A Box indiscriminate, ... Week 9: Coalescents, part 2 – p.28/71
An accurate analogy: Bugs In A Box voracious ... Week 9: Coalescents, part 2 – p.29/71
An accurate analogy: Bugs In A Box Gulp! (eats other bug) ... Week 9: Coalescents, part 2 – p.30/71
An accurate analogy: Bugs In A Box and insatiable. Week 9: Coalescents, part 2 – p.31/71
Random coalescent trees with 16 lineages M O C S M L P K E J I T R H Q F B N D G A M J B F G C E R A S Q K N L H T P D O B G T M L Q D O F K P E A J S C H R N F R N L D H B T C Q S O G P I A K J E I I O M G M M Q C A J L S G P F O D H B M E T R K N R C L D K H Q F B S I T P A J E N N P R H L E S O F B G J D C I T K Q A N H C R P G L T E D S O I K J Q F A B I Week 9: Coalescents, part 2 – p.32/71
Coalescence is faster in small populations Change of population size and coalescents N e time the changes in population size will produce waves of coalescence the tree time Coalescence events time The parameters of the growth curve for Ne can be inferred by likelihood methods as they affect the prior probabilities of those trees that fit the data. Week 9: Coalescents, part 2 – p.33/71
Migration can be taken into account Time population #2 population #1 Week 9: Coalescents, part 2 – p.34/71
Recombination creates loops Recomb. Different markers have slightly different coalescent trees Week 9: Coalescents, part 2 – p.35/71
Cann, Stoneking, and Wilson Becky Cann Mark Stoneking the late Allan Wilson Cann, R. L., M. Stoneking, and A. C. Wilson. 1987. Mitochondrial DNA and human evolution. Nature 325:a 31-36. Week 9: Coalescents, part 2 – p.36/71
Mitochondrial Eve Week 9: Coalescents, part 2 – p.37/71
We want to be able to analyze human evolution "Out of Africa" hypothesis Europe Asia (vertical scale is not time or evolutionary change) Africa Week 9: Coalescents, part 2 – p.38/71
coalescent and “gene trees” versus species trees Consistency of gene tree with species tree Week 9: Coalescents, part 2 – p.39/71
coalescent and “gene trees” versus species trees Consistency of gene tree with species tree Week 9: Coalescents, part 2 – p.40/71
coalescent and “gene trees” versus species trees Consistency of gene tree with species tree Week 9: Coalescents, part 2 – p.41/71
coalescent and “gene trees” versus species trees Consistency of gene tree with species tree Week 9: Coalescents, part 2 – p.42/71
coalescent and “gene trees” versus species trees Consistency of gene tree with species tree Week 9: Coalescents, part 2 – p.43/71
coalescent and “gene trees” versus species trees Consistency of gene tree with species tree coalescence time Week 9: Coalescents, part 2 – p.44/71
If the branch is more than N e generations long ... Gene tree and Species tree N2 N1 t1 N3 N4 t2 N5 Week 9: Coalescents, part 2 – p.45/71
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