Coalescent Module- Faro July 26th-28th 04 www.coalescent .dk

1. Coalescent Module- Faro July 26th-28th 04 www.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

2. Zooming in!(from Harding + Sanger) 3*109 bp *5.000 b-globin (chromosome 11) 6*104 bp *20 Exon 3 Exon 1 Exon 2 3*103 bp 5’ flanking 3’ flanking *103 ATTGCCATGTCGATAATTGGACTATTTTTTTTTT 30 bp

3. Human Migrations From Cavalli-Sforza,2001

4. Data: b-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

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

6. From Griffiths, 2001

7. Models and their benefits. • Models+ Data • probability of data (statistics...) • probability of individual histories • hypothesis testing • parameter estimation

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

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

10. 10 Alleles’ Ancestry for 15 generations

11. Waiting for most recent common ancestor - MRCA Distribution until 2 alleles had a common ancestor, X2?: P(X2 > j) = (1-(1/2N))j P(X2 > 1) = (2N-1)/2N = 1-(1/2N) P(X2 = j) = (1-(1/2N))j-1 (1/2N) j j 2 2 1 1 1 1 1 1 2N 2N 2N Mean, E(X2) = 2N. Ex.: 2N = 20.000, Generation time 30 years, E(X2) = 600000 years.

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

13. Geometric/Exponential Distributions The Geometric Distribution: {1,..} Geo(p): P{Z=j)=pj(1-p) P{Z>j)=pj 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>t2|X>t1) = P(X>t2-t1) (t2 > t1) 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).

14. Discrete  Continuous Time tc:=td/2Ne 6 6/2Ne 0 2N 0 1 4 1.0 corresponds to 2N generations 1.0 0.0 2 6 5 3

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

16. 1 2 3 4 5 The Standard Coalescent Two independent Processes Continuous: Exponential Waiting Times Discrete: Choosing Pairs to Coalesce. Waiting Coalescing {1,2,3,4,5} (1,2)--(3,(4,5)) {1,2}{3,4,5} 1--2 {1}{2}{3,4,5} 3--(4,5) {1}{2}{3}{4,5} 4--5 {1}{2}{3}{4}{5}

17. Expected Height and Total Branch Length Branch Lengths Time Epoch 1 2 1 2 1 1/3 3 2/(k-1) k Expected Total height of tree: Hk= 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, Lk: 2*(1 + 1/2 + 1/3 +..+ 1/(k-1)) ca= 2*ln(k-1)

18. Kingman (Stoch.Proc. & Appl. 13.235-248 + 2 other articles,1982) A. Stochastic Processes on Equivalence Relations. D ={(i,i);i= 1,..n} Q ={(i,j);i,j=1,..n} 1 if s < t qs,t = 0otherwise This defines a process, Rt , going from to through equivalence relations on {1,..,n}. 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(n1) --> s2 & E(n1m) < Mm for all m, then genealogies follows ”The Coalescent” in distribution. E. A series of combinatorial results.

19. Effective Populations Size, Ne. 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 Ne: i.Variance in offspring. WF: 1. If variance is higher, then effective population size is smaller. ii.Population size variation - example k cycle: N1, N2,..,Nk. k/Ne= 1/N1+..+ 1/Nk. N1 = 10 N2= 1000 => Ne= 50.5 iii.Two sexesNe = 4NfNm/(Nf+Nm)I.e. Nf-10Nm -1000 Ne - 40

20. 6 Realisations with 25 leaves Observations: Variation great close to root. Trees are unbalanced.

21. 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.

22. Three Models of Alleles and Mutations. Finite Site Infinite Allele Infinite Site acgtgctt acgtgcgt acctgcat tcctgcat tcctgcat Q Q Q acgtgctt acgtgcgt acctgcat tcctggct tcctgcat i. Allele is represented by a sequence. ii. A mutation changes nucleotide at chosen position. 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.

23. Infinite Allele Model 4 5 1 2 3

24. Infinite Site Model Final Aligned Data Set:

25. 0 1 1 1 4 2 3 5 4 5 5 5 6 3 7 2 8 1

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

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

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

29. Impossible Ancestral States

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

31. Simplifying assumptions 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. πi Pi,j(t) = πj Pj,i(t), where πi is the stationary distribution of i. This implies that a l2+l1 = l1 l2 N1 N2 N2 N1 5) The rate matrix, Q, for the continuous time Markov Chain is the same at all times.

32. Evolutionary Substitution Process t1 e A t2 C C Pi,j(t) = probability of going from i to j in time t.

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

34. 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.

35. 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.

36. 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.