1 / 25

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

logan-hansen
Download Presentation

Extensions to Basic Coalescent Chapter 4, Part 1

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related