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Coalescent Module- Faro July 26th-28th 04 www.coalescentwww.coalescent.dk Monday H: The Basic Coalescent W: Forest Fire W: The Coalescent + History, Geography & Selection H: The Coalescent with Recombination Tuesday H: Recombination cont. W: The Coalescent & Combinatorics HW: Computer Session H: The Coalescent & Human Evolution Wednesday H: The Coalescent & Statistics HW: Linkage Disequilibrium Mapping

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globin Exon 2 Exon 1 Exon 3 5’ flanking 3’ flanking (chromosome 11) Zooming in! (from Harding + Sanger) *5.000 *20 6*10 4 bp 3*10 9 bp *10 3 3*10 3 bp ATTGCCATGTCGATAATTGGACTATTTTTTTTTT30 bp

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From Cavalli-Sforza,2001 Human Migrations

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Data: -globin from sampled humans. From Griffiths, 2001 Assume: 1. At most 1 substitution per position. 2.No recombination Reducing nucleotide columns to bi- partitions gives a bijection between data & unrooted gene trees. C G

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Africa Non-Africa 0.2 Mutation rate: 2.5 Rate of common ancestry: 1 Past Present Simplified model of human sequence evolution. Wait to common ancestry: 2N e

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From Griffiths, 2001

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Models and their benefits. Models + Data 1. probability of data (statistics...) 2. probability of individual histories 3. hypothesis testing 4. parameter estimation

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Fixed Parameters: Population Structure, Mutation, Selection, Recombination,... Reproductive Structure Genealogies of non-sequenced data Genealogies of sequenced data Parameter Estimation Model Testing Coalescent Theory in Biology www. coalescent.dk TGTTGT CATAGT CGTTAT

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Haploid Model Diploid Model Wright-Fisher Model of Population Reproduction 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 Individuals are made by sampling a chromosome from the female and one from the male previous generation with replacement Assumptions 1.Constant population size 2.No geography 3.No Selection 4.No recombination

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10 Alleles’ Ancestry for 15 generations

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Mean, E(X 2 ) = 2N. Ex.: 2N = 20.000, Generation time 30 years, E(X 2 ) = 600000 years. Waiting for most recent common ancestor - MRCA P(X 2 = j) = (1-(1/2N)) j-1 (1/2N) Distribution until 2 alleles had a common ancestor, X 2 ?: P(X 2 > j) = (1-(1/2N)) j P(X 2 > 1) = (2N-1)/2N = 1-(1/2N) 1 2N 1 1 1 2 j 1 1 2 j

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P(k):=P{k alleles had k distinct parents} 1 2N 1 2N *(2N-1) *..* (2N-(k-1)) =: (2N) [k] (2N) k k -> any k -> k k -> k-1 Ancestor choices: k -> j For k << 2N: S k,j - the number of ways to group k labelled objects into j groups.(Stirling Numbers of second kind.

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Geometric/Exponential Distributions The Geometric Distribution: {1,..} Geo(p): P{Z=j)=p j (1-p) P{Z>j)=p j E(Z)=1/p. The Exponential Distribution: R+ Exp (a) Density: f(t) = ae -at, P(X>t)= e -at Properties: X ~ Exp(a) Y ~ Exp(b) independent i. P(X>t 2 |X>t 1 ) = P(X>t 2 -t 1 ) (t 2 > t 1 ) ii. E(X) = 1/a. iii. P(Z>t)=(≈)P(X>t) small a (p=e -a ). iv. P(X < Y) = a/(a + b). v. min(X,Y) ~ Exp (a + b).

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2 563 0.0 1.0 1.0 corresponds to 2N generations 1 4 0 2N 0 6 6/2N e t c :=t d /2N e Discrete Continuous Time

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Probability for two genes being identical: P(Coalescence < Mutation) = 1/(1+ ). m mutation pr. nucleotide pr.generation. L: seq. length µ = m*L Mutation pr. allele pr.generation. 2N e - allele number. := 4N*µ -- Mutation intensity in scaled process. Adding Mutations sequence time Discrete time Discrete sequence Continuous time Continuous sequence 1/L 1/(2N e ) time sequence /2 mutation coalescence Note: Mutation rate and population size usually appear together as a product, making separate estimation difficult. 1

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The Standard Coalescent Two independent Processes Continuous: Exponential Waiting Times Discrete: Choosing Pairs to Coalesce. 12345 WaitingCoalescing 4--5 3--(4,5) (1,2)--(3,(4,5)) 1--2 {1}{2}{3}{4}{5} {1,2}{3,4,5} {1,2,3,4,5} {1}{2}{3}{4,5} {1}{2}{3,4,5}

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Expected Height and Total Branch Length Expected Total height of tree: H k = 2(1-1/k) i.Infinitely many alleles finds 1 allele in finite time. ii. In takes less than twice as long for k alleles to find 1 ancestors as it does for 2 alleles. Expected Total branch length in tree, L k : 2*(1 + 1/2 + 1/3 +..+ 1/(k-1)) ca= 2*ln(k-1) 1 2 3 k 1/3 1 2 1 2/(k-1) Time Epoch Branch Lengths

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B. The Paint Box & exchangable distributions on Partitions. C. All coalescents are restrictions of “The Coalescent” – a process with entrance boundary infinity. D. Robustness of “The Coalescent”: If offspring distribution is exchangeable and Var( 1 ) --> 2 & E( 1 m ) < M m for all m, then genealogies follows ”The Coalescent” in distribution. E. A series of combinatorial results. Kingman (Stoch.Proc. & Appl. 13.235-248 + 2 other articles,1982) A. Stochastic Processes on Equivalence Relations. ={(i,i);i= 1,..n} ={(i,j);i,j=1,..n} 1 if s < t q s,t = 0 otherwise This defines a process, R t, going from to through equivalence relations on {1,..,n}.

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Effective Populations Size, N e. In an idealised Wright-Fisher model: i. loss of variation per generation is 1-1/(2N). ii. Waiting time for random alleles to find a common ancestor is 2N. Factors that influences N e : i. Variance in offspring. WF: 1. If variance is higher, then effective population size is smaller. ii. Population size variation - example k cycle: N 1, N 2,..,N k. k/N e = 1/N 1 +..+ 1/N k. N 1 = 10 N 2 = 1000 => N e = 50.5 iii. Two sexes N e = 4N f N m /(N f +N m )I.e. N f - 10 N m -1000 N e - 40

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6 Realisations with 25 leaves Observations: Variation great close to root. Trees are unbalanced.

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Sampling more sequences The probability that the ancestor of the sample of size n is in a sub-sample of size k is Letting n go to infinity gives (k-1)/(k+1), i.e. even for quite small samples it is quite large.

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Three Models of Alleles and Mutations. Infinite Allele Infinite Site Finite Site acgtgctt acgtgcgt acctgcat tcctgcat acgtgctt acgtgcgt acctgcat tcctggct tcctgcat i. Only identity, non-identity is determinable ii. A mutation creates a new type. i. Allele is represented by a line. ii. A mutation always hits a new position. i. Allele is represented by a sequence. ii. A mutation changes nucleotide at chosen position.

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1 2 3 45 Infinite Allele Model

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Final Aligned Data Set: Infinite Site Model

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0 1 2 3 4 5 6 7 8 1 1 4 2 5 3 1 5 5

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0 1 2 3 4 5 6 7 8 1 1 4 2 5 3 1 5 5 Number of paths: 2 2 2 3 4 4 6 2 7 7 14 8 2 28 22 10 50 32 82 2

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1 3 4 5 2 1 3 4 5 2 {},, Ignoring mutation position Ignoring sequence label Ignoring mutation position Ignoring sequence label Labelling and unlabelling:positions and sequences 9 coalescence events incompatible with data 4 classes of mutation events incompatible with data The forward-backward argument

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Infinite Site Model: An example Theta=2.12 2 3 2 3 5 5 4 9 10 5 19 14 33

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Impossible Ancestral States

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Final Aligned Data Set: acgtgctt acgtgcgt acctgcat tcctgcat s s s Finite Site Model

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1) Only substitutions. s1 TCGGTA s1 TCGGA s2 TGGT-T s2 TGGTT 2) Processes in different positions of the molecule are independent. 3) A nucleotide follows a continuous time Markov Chain. 4) Time reversibility: I.e. π i P i,j (t) = π j P j,i (t), where π i is the stationary distribution of i. This implies that Simplifying assumptions 5) The rate matrix, Q, for the continuous time Markov Chain is the same at all times. = a N1N1 N2N2 l 2 +l 1 l1l1 l2l2 N2N2 N1N1

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Evolutionary Substitution Process t1t1 t2t2 C C A P i,j (t) = probability of going from i to j in time t.

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Jukes-Cantor 69: Total Symmetry. -3* -3* -3* -3* TO A C G T FROM A.Stationary Distribution: (.25,.25,.25,.25) B. Expected number of substitutions: 3 t ACGTACGT 0 t Higher Cells ChimpMouse Fish E.coli ATTGTGTATATAT….CAG ATTGCGTATCTAT….CCG

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History of Coalescent Approach to Data Analysis 1930-40s: Genealogical arguments well known to Wright & Fisher. 1964: Crow & Kimura: Infinite Allele Model 1968: Motoo Kimura proposes neutral explanation of molecular evolution & population variation. So does King & Jukes 1971: Kimura & Otha proposes infinite sites model. 1972: Ewens’ Formula: Probability of data under infinite allele model. 1975: Watterson makes explicit use of “The Coalescent” 1982: Kingman introduces “The Coalescent”. 1983: Hudson introduces “The Coalescent with Recombination” 1983: Kreitman publishes first major population sequences.

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History of Coalescent Approach to Data Analysis 1987-95: Griffiths, Ethier & Tavare calculates site data probability under infinite site model. 1994-: Griffiths-Tavaré + Kuhner-Yamoto-Felsenstein introduces highly computer intensitive simulation techniquees to estimate parameters in population models. 1996- Krone-Neuhauser introduces selection in Coalescent 1998- Donnelly, Stephens, Fearnhead et al.: Major accelerations in coalescent based data analysis. 2000-: Several groups combines Coalescent Theory & Gene Mapping. 2002: HapMap project is started.

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Basic Coalescent Summary i. Genealogical approach to population genetics. ii. ”The Coalescent” - generic probability distribution on allele trees. iii. Combining ”The Coalescent” with Allele/Mutation Models allows the calculation the probability of data.

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