Introduction to Coalescent Models. Biostatistics 666
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1 Introducton to Coalescent Models Bostatstcs 666
2 Prevously Allele frequences Hardy Wenberg Equlbrum Lnkage Equlbrum Expected state for dstant markers Lnkage Dsequlbrum Assocaton between neghborng alleles Expected to decrease wth dstance Measures of lnkage dsequlbrum D, D and ² or r 2
3 Makng predctons What allele frequences do we expect? How much varaton n a gene? How are neghborng varants related? Are these predctons unversal? Do they depend on natural selecton or the hstory of a populaton? How can we use genetc varaton to buld models of the past?
4 1000 Genomes Data: Varants per Genome Type Varant stes / genome SNPs ~3,800,000 Indels ~570,000 Moble Element Insertons ~1000 Large Deletons ~1000 CNVs ~150 Inversons ~11
5 1000 Genomes Data: Demographc Models
6 Smple Approach: Smulaton 1. N startng sequences 2. Sample N offsprng sequences Apply mutatons accordng to µ 3. Increment tme 4. If enough tme has passed Generate fnal sample Stop. 5. Otherwse, return to step 1.
7 Smulatng a Populaton Sequences Tme
8 Today Introduce coalescent approach Framework for studyng genetc varaton Provdes ntuton on patterns of varaton Provdes analytcal solutons
9 Am Gene genealoges: Descrptons of relatedness between sequences Analogous to phylogenetc trees for speces The shape of the genealogy depends on populaton hstory, selecton, etc. Together wth mutaton rate, genealogy predcts DNA varaton
10 Genealogy Hstory of a partcular set of sequences Descrbes ther relatedness Specfes dvergence tmes Includes only a subset of the populaton Most Recent Common Ancestor (MRCA)
11 Coalescent approach Generate genealogy for a sample of sequences. Introduces computatonal and analytcal convenence. Instead of proceedng forward through tme, go backwards!
12 Hstory of the Populaton
13 Genealogy of Fnal Populaton
14 Levels of Complexty Hstory of the populaton Includes sequences that are extnct Hstory of all modern sequences Includes sequences that we haven t sampled Hstory of a subset of modern sequences Mnmalst approach!
15 Examples of Typcal Coalescent Trees
16 Parameters we wll focus on Mutaton rate (µ) Populaton Sze Haplod populaton (N chromosomes) Dplod populaton (2N chromosomes) Tme (t) Sample sze (n) Recombnaton rate (r)
17 Other Parameters Selecton For gene of nterest For neghborng gene Demographc parameters Mgraton Populaton Structure Populaton Growth
18 Mutaton Model The mutaton process s complex Rate depends on surroundng sequence Reverse mutatons are possble Two smple models are popular Infnte alleles Every mutaton generates a dfferent allele Infnte stes Every mutaton occurs at a dfferent ste
19 Mutaton Model Focus on nfnte stes model Mutaton rate n genomc DNA s ~10-8 / bp Recurrent mutatons should be very rare Scaled mutaton rate parameter, e.g.: 1000 bp sequence 10-8 mutatons per base par per generaton μ 10-5 per sequence per generaton
20 Neutral Varants Varants that do not affect ftness Accumulate nexorably through tme Lost through genetc drft Do not affect genealogy
21 Example: Modelng Accumulaton of Mutatons Populaton of dentcal sequences Sample one descendant after t generatons How many mutatons have accumulated? Hnt: depends on mutaton rate μ and tme t Tougher questons How many mutatons have been fxed? How much varaton n the total populaton?
22 So far Dvergence of a sngle sequence Accumulaton of mutatons Depends on tme t Depends on mutaton rate μ Does not depend on populaton sze N Does not depend on populaton growth Next: A par of sequences!
23 A tougher example Sample of two sequences 100 bp each How many dfferences are expected? Populaton of sze, N 1000 Mutaton rate µ 10-8 / bp / generaton µ 10-6 / 100 bp / generaton
24 Genealogy of two sequences MRCA Tme T(2) Sequence 1 Sequence 2 Mutatons between MRCA and Sequence 1?
25 Genealogy of two sequences MRCA Tme T(2) Sequence 1 Sequence 2 Total mutatons n genealogy?
26 Number of mutatons S Dstrbuted as Posson, condtonal on total tree length E(S) µe(t tot ) Var(S) µe(t tot ) + µ²var(t tot ) T tot s the total length of all branches
27 Estmatng Coalescence Tme Probablty that two sequences have dstnct ancestors n prevous generaton PP 2 NN 1 NN 1 1 NN Probablty of dstnct ancestors for t generatons s P(2) t
28 Probablty of MRCA at tme t+1 P(2) t (1 P(2)) 1 N N 1 N t 1 N 1 1 N t 1 N e 1 t N
29 For n > 2 Coalescence when two sequences have common ancestor For smplcty, consder the possblty of multple smultaneous coalescent events to be neglgble Requrements for no coalescence: Pck one ancestor for sequence 1 Pck dstnct ancestor for sequence 2 Pck yet another ancestor for sequence 3
30 Estmatng P(n) Probablty that n sequences have n dstnct ancestors n prevous generaton P( n) n N N n 2 N Assume: N s large n s small Terms of order N -2 can be gnored
31 Probablty of Coalescence at Tme t+1 t N n t t e N n N n N n n P n P )) ( (1 ) (
32 Tme to next coalescent event Use an exponental dstrbuton to approxmate tme to next coalescent event Decay Rate Mean λ 1 λ n 2 N N n 2
33 T(j) For convenence, measure tme to next coalescent event n unts: N generatons for haplods 2N generatons for dplods E( T j ) 1/ j 2 How would you calculate tme to MRCA of n sequences?
34 Total Tme n Tree Sum of all the branch lengths Total evolutonary tme avalable e.g. for mutatons to occur ) ( 2 ) ( ) ( n n n n tot T T E
35 T MRCA vs. T TOT Relatve Tme to MRCA T MRCA Relatve Sum of Branch Lengt T TOT Number of Sequences Number of Sequences
36 Number of Segregatng Stes Commonly named S Total number of mutatons n genealogy Assumng no recurrent mutaton A functon of the total length of the genealogy T tot
37 Expected number of mutatons Factor N for haplods, 2N for dplods Populaton genetcsts defne θ4nµ (for dplods) For gene mappers, θ s usually the recombnaton rate For populaton genetcsts, r s the recombnaton rate ( ) / 1/ 4 ) ( 2 ) ( n n n N T E N S E θ µ µ
38 Expected number of mutatons Factor N for haplods, 2N for dplods Populaton genetcsts defne θ4nµ (for dplods) For gene mappers, θ s usually the recombnaton rate For populaton genetcsts, r s the recombnaton rate ( ) / 1/ 4 ) ( 2 ) ( n n n N T E N S E θ µ µ
39 E(S) as a functon of n Expected Number of Segregatng Stes Parameters N 10,000 ndvduals μ 10-4 θ Sample Sze
40 More about S Very large varance Var( S) θ n 1 1 1/ + θ Most of the varance contrbuted by early coalescent events (.e. wth small n) 2 n 1 1 1/ 2
41 Var(S) as a functon of n Varance n Number of Segregatng Stes Parameters N 10,000 ndvduals μ 10-4 θ Sample Sze
42 Inferences about θ Could be estmated from S Dvde by expected length of genealogy ˆ θ n 1 1 S 1/ Could then be used to: Estmate N, f mutaton rate µ s known Estmate µ, f populaton sze N s known
43 ^ Var(θ) as a functon of n Varance n Estmate of Theta Parameters N 10,000 ndvduals μ 10-4 θ Sample Sze
44 Alternatve Estmator for θ Count parwse dfferences between sequences Compute average number of dfferences ~ θ n 2 1 n n S j 1 j + 1
45 Today Probablty of coalescence events Length of genealogy and ts branches Expected number of mutatons Smple estmates of θ
46 Recommended Readng Rchard R. Hudson (1990) Gene genealoges and the coalescent process Oxford Surveys n Evolutonary Bology, Vol. 7. D. Futuyma and J. Antonovcs (Eds). Oxford Unversty Press, New York.
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