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Extensions to Basic Coalescent Chapter 4, Part 1

Extensions to Basic Coalescent Chapter 4, Part 1. Extension 1. One of the assumptions of basic coalescent (Wright-Fisher) model: Population size is constant We will relax this assumption. Outline. Intuition behind extension Formal definition of the extended model

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Extensions to Basic Coalescent Chapter 4, Part 1

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  1. Extensions to Basic CoalescentChapter 4, Part 1

  2. Extension 1 • One of the assumptions of basic coalescent (Wright-Fisher) model: Population size is constant • We will relax this assumption COMP 790-Extensions to Basic Coalescent

  3. Outline • Intuition behind extension • Formal definition of the extended model • Compare extended model to basic model for 2 different population change functions • Exponential growth (more emphasis on this) • Population bottlenecks • Effective population size 2/26/2009 COMP 790-Extensions to Basic Coalescent 3

  4. Intuition • We will only consider deterministic population changes • Population size at time t is given by N(t), a function of t only • N(0) = N • We assume N(t) is given in terms of continuous time (in units of 2N generations) and N(t) need not to be an integer 2/26/2009 COMP 790-Extensions to Basic Coalescent 4

  5. Intuition • Let p=probability by which two genes find a common ancestor • Wright Fisher model p = 1/2N • Extended model p(t) = 1/2N(t) • E.g. when N(t) < N (declining population size) Probability of a coalescence event increases and a MRCA is found more rapidly than if N(t) is constant 2/26/2009 COMP 790-Extensions to Basic Coalescent 5

  6. Intuition • If p(t) is smaller than p(0) by factor if two (for example) then time should be stretched locally by a factor if two to accommodate this 2/26/2009 COMP 790-Extensions to Basic Coalescent 6

  7. Intuition Time in basic coalescent Time in extended model Each of the intervals between dashed lines represents 2N generations 2/26/2009 COMP 790-Extensions to Basic Coalescent 7

  8. Formulation Accumulated coalescent rate over time measured relative to the rate at time t=0 where 2/26/2009 COMP 790-Extensions to Basic Coalescent 8

  9. Formulation • Let T2,… Tn be the waiting times while there are 2,…,n ancestors of the sample • and let Vk = Tn + … +Tk be the accumulated waiting times from there are n genes until there are k-1 ancestors • The distribution of Tk conditiona on Vk+1 is 2/26/2009 COMP 790-Extensions to Basic Coalescent 9

  10. Formulation • Tk* : Waiting times in basic coalescent • Tk : Waiting times in extended model Algorithm • Simulate T2*, … Tn* according to the basiccoalescent, where Tk* is exponentially distributed with parameter C(k,2). Denote the simulated values by tk* • Solve 3. The values tk = vk- vk+1 are an outcome of the process, T2, … ,Tn 2/26/2009 COMP 790-Extensions to Basic Coalescent 10

  11. Exponential growth • Now lets have a look at specific population size change function: exponential growth • Question: This is a declining function. How come this can be a growth? 2/26/2009 COMP 790-Extensions to Basic Coalescent 11

  12. Exponential Growth • For this specific population change function we can derive the following: • Using the algorithm: 2/26/2009 COMP 790-Extensions to Basic Coalescent 12

  13. Characterizations of Exponential Growth • Now lets have a look at various characterizations of this population growth • Characterization 1 • Waiting times, T2, … ,Tn are no longer independent of each other as in basic coalescent but negatively correlated • If one of them is large the others are more likely to be small 2/26/2009 COMP 790-Extensions to Basic Coalescent 13

  14. Characterizations of Exponential Growth 2/26/2009 COMP 790-Extensions to Basic Coalescent 14

  15. Characterizations of Exponential Growth • Characterization 2: Genealogy Basic coalescent Basic coalescent Exponential growth Exponential growth • Characterization 2: Genealogy 2/26/2009 COMP 790-Extensions to Basic Coalescent 15

  16. Characterizations of Exponential Growth • With high levels of exponential growth, tree becomes almost star shaped. 2/26/2009 COMP 790-Extensions to Basic Coalescent 16

  17. Characterizations of Exponential Growth 2/26/2009 COMP 790-Extensions to Basic Coalescent 17

  18. Characterizations of Exponential Growth Basic coalescent (multimodal) Exponential growth (unimodal) • Pairwise distances between all pairs of sequences. 2/26/2009 COMP 790-Extensions to Basic Coalescent 18

  19. Characterizations of Exponential Growth • Frequency spectrum of mutants 2/26/2009 COMP 790-Extensions to Basic Coalescent 19

  20. Characterizations of Exponential Growth • Percentage of contribution of kth waiting time to the mean and variance of total waiting time 2/26/2009 COMP 790-Extensions to Basic Coalescent 20

  21. Population Bottlenecks • Now we move on to the next type of population size change function: bottlenecks • A way to model ice age 2/26/2009 COMP 790-Extensions to Basic Coalescent 21

  22. Population Bottlenecks 4 parameters. Strength of the bottleneck is determined by its length (tb) and severity(f) 2/26/2009 COMP 790-Extensions to Basic Coalescent 22

  23. Effective Population Size • We defined effective population size in very first lectures as: 2/26/2009 COMP 790-Extensions to Basic Coalescent 23

  24. Effective Population Size 2/26/2009 COMP 790-Extensions to Basic Coalescent 24

  25. Next Time • Relax another assumption • > Coalescent with population structure COMP 790-Extensions to Basic Coalescent

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