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Preview What does Recombination do to Sequence Histories. Probabilities of such histories. Quantities of interest. Detecting & Reconstructing Recombinations.

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Haploid Reproduction Model (i.e. no recombination) 122N3 12 3 i. Individuals are made by sampling with replacement in the previous generation. ii. The probability that 2 alleles have same ancestor in previous generation is 1/2N. iii. The probability that k alleles have less than k-1 ancestors in previous generation is vanishing.

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0 recombinations implies traditional phylogeny 4 32 1

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Diploid Model with Recombination An individual is made by: 1.The paternal chromosome is taken by picking random father. 2.Making that father’s chromosomes recombine to create the individuals paternal chromosome. Similarly for maternal chromosome.

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A recombinant sequence will have have two different ancestor sequences in the grandparent. The Diploid Model Back in Time.

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The ancestral recombination graph N 1 Time

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1- recombination histories I: Branch length change 4 3 12 4 3 1 2 4 3 1 2

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1- recombination histories II: Topology change 4 3 12 4 3 1 2 4 3 1 2

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1- recombination histories III: Same tree 4 3 12 4 3 1 2 4 3 1 2

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1- recombination histories IV: Coalescent time must be further back in time than recombination time. 3 412 c r

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Recombination Histories V: Multiple Ancestries.

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Recombination Histories VI: Non-ancestral bridges

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Summarising new phenomena in recombination-genealogies Consequence of 1 recombination Branch length change Topology change No change Time ranking of internal nodes Multiple Ancestries Non-ancestral bridges Recombination genealogies are called ”ancestral recombination graphs - ARGs” What is the probability of different histories?

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r recombination pr. Nucleotide pair pr.generation. L: seq. length R = r*(L-1) Recombination pr. allele pr.generation. 2N e - allele number := 4N*R -- Recombination intensity in scaled process. Adding Recombination sequence time Discrete time Discrete sequence Continuous time Continuous sequence 1/(L-1) 1/(2N e ) time sequence Recombination Event: /2 Waiting time exp( /2) Position Uniform Recombination versus Mutation: As events, they are identically position and time wise. Mutations creates a difference in the sequence Recombination can create a shift in genealogy locally

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Recombination-Coalescence Illustration Copied from Hudson 1991 Intensities Coales. Recomb. 1 2 3 2 6 2 3 (2+b) 1 (1+b) 0 b

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Age to oldest most recent common ancestor From Wiuf and Hein, 1999 Genetics Scaled recombination rate - Age to oldest most recent common ancestor 0 kb 250 kb

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Properties of Neighboring Trees. (partially from Hudson & Kaplan 1985) Leaves Topo-Diff Tree-Diff 2 0.0.666 3 0.0.694 4 0.073.714 5 0.134.728 6 0.183.740 10 0.300.769 15 0.374.790 500 0.670 1 23 4 1 23 4

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Grand Most Recent Common Ancestor: GMRCA (griffiths & marjoram, 96) i. Track all sequences including those that has lost all ancestral material. ii. The G-ARG contains the ARG. The graph is too large, but the process is simpler. Sequence number - k. Birth rate: *k/2 Death rate: 123k E(events until {1}) = (asymp.) exp( ) + log(k)

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Old +Alternative Coalescent Algorithm Adding alleles one-by-one to a growing genealogy 1 12 3 12 3 12 Old

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Spatial Coalescent-Recombination Algorithm (Wiuf & Hein 1999 TPB) 1. Make coalescent for position 0.0. 2. Wait Exp(Total Branch length) until recombination point, p. 3. Pick recombination point (*) uniformly on tree branches. 4. Let new sequence coalesce into genealogical structure. Continue 1-4 until p > L.

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Properties of the spatial process i. The process is non-Markovian ii. The trees cannot be reduced to Topologies * = *

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Compatibility 1 2 3 4 5 6 7 1 A T G T G T C 2 A T G T G A T 3 C T T C G A C 4 A T T C G T A i i i i. 3 & 4 can be placed on same tree without extra cost. ii. 3 & 6 cannot. 1 4 3 2 Definition: Two columns are incompatible, if they are more expensive jointly, than separately on the cheapest tree. Compatibility can be determined without reference to a specific tree!!

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Hudson& Kaplan’s R M (k positions can at most have (k+1) types without recombination) ex. Data set: A underestimate for the number of recombination events: ------------------- --------------- ------- --------- ------- ----- If you equate R M with expected number of recombinations, this would be an analogue to Watterson’s estimators. Unfortunately, R M is a gross underestimate of the real number of recombinations.

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Myers-Griffiths’ R M Basic Idea: 1 S Define R: R j,k is optimal solution to restricted interval., then: B j,i R j,k R j,i j k i

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Recombination Parsimony 1 2 3 T i-1iL 21 Data Trees Recursion:W(T,i)= min T’ {W(T’,i-1) + subst(T,i) + d rec (T,T’)} Initialisation:W(T,1)= subst(T,1) W(T,i) - cost of history of first i columns if local tree at i is T subst(T,i) - substitution cost of column i using tree T. d rec (T,T’) - recombination distance between T & T’

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Metrics on Trees based on subtree transfers. Trees including branch lengths Unrooted tree topologies Rooted tree topologies Tree topologies with age ordered internal nodes Pretending the easy problem is the real problem, causes violation of the triangle inequality:

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Observe that the size of the unit-neighbourhood of a tree does not grow nearly as fast as the number of trees. Allen & Steel (2001) Song (2003) Explicit computation No known formula

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The 1983 Kreitman Data (M. Kreitman 1983 Nature from Hartl & Clark 1999)

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Methods # of rec events obtained Hudson & Kaplan (1985)5 Myers & Griffiths (2002)6 Song & Hein (2002). Set theory based approach.7 Song & Hein (2003). Current program using rooted trees.7 11 sequences of alcohol dehydrogenase gene in Drosophila melangaster. Can be reduced to 9 sequences (3 of 11 are identical). 3200 bp long, 43 segregating sites. We have checked that it is possible to construct an ancestral recombination graph using only 7 recombination events.

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Quality of the estimated local tree True ARG Reconstructed ARG 1 23 45 123 45 (1,2) - (3,4,5) (1,2,3) - (4,5) (1,3) - (2,4,5) (1,2,3) - (4,5) n=7 Rho=10 Theta=75 Due to Yun Song

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Actual, potentially detectable and detected recombinations 1 23 4 1 23 4 True ARG Minimal ARG 0 400 kb n=8 =15 =40 Leaves Topo-Diff Tree-Diff 2 0.0.666 3 0.0.694 4 0.073.714 5 0.134.728 6 0.183.740 10 0.300.769 15 0.374.790 500 0.670 Due to Yun Song

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s 1 = s 2 = s 3 = s 4 = s 5 = 0 0 0 0 1 1 0 1 1 1 0 0 1 1 1 2 3 0000 0011 0101 1100 1111 1 0000 00xx xx11 0101 1100 1111 2 0000 00xx 0101 1100 1111 3 Ancestral states Yun Song

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Ancestral configurations to 2 sequences with 2 segregating sites: k1k1 k2k2 (k 2 +1)*k 1 +1 possible ancestral columns.

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Asymptotic growth? Enumerating ancestral states in minimal histories? Branch and bound method for computing the likelihood? Enumeration of Ancestral States (via counting restricted non-negative integer matrices with given row and column sums) Due to Yun Song

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Ignoring recombination in phylogenetic analysis Mimics decelerations/accelerations of evolutionary rates. No & Infinite recombination implies molecular clock. General Practice in Analysis of Viral Evolution!!! RecombinationAssuming No Recombination 14321432

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Simulated Example

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Gene Conversion Recombination: Gene Conversion: Compatibilities among triples: + - -

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Gene Conversions & Treeness Pairwise Distances as sequences gets longer and longer Recombination Gene Conversion Coalescent: Star tree:

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Summary What does Recombination do to Sequence Histories. Probabilities of such histories. Quantities of interest. Detecting & Reconstructing Recombinations.

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