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 2/26/2009COMP 790-Extensions to Basic Coalescent2

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 2/26/2009COMP 790-Extensions to Basic Coalescent3

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 2/26/2009COMP 790-Extensions to Basic Coalescent4

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 2/26/2009COMP 790-Extensions to Basic Coalescent5

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 2/26/2009COMP 790-Extensions to Basic Coalescent6

Intuition 2/26/2009COMP 790-Extensions to Basic Coalescent7 Time in basic coalescent Time in extended model Each of the intervals between dashed lines represents 2N generations

Formulation 2/26/2009COMP 790-Extensions to Basic Coalescent8 Accumulated coalescent rate over time measured relative to the rate at time t=0 where

Formulation Let T 2,… T n be the waiting times while there are 2,…,n ancestors of the sample and let V k = T n + … +T k be the accumulated waiting times from there are n genes until there are k-1 ancestors The distribution of T k conditiona on V k+1 is 2/26/2009COMP 790-Extensions to Basic Coalescent9

Formulation T k * : Waiting times in basic coalescent T k : Waiting times in extended model Algorithm 1.Simulate T 2 *, … T n * according to the basiccoalescent, where T k * is exponentially distributed with parameter C(k,2). Denote the simulated values by t k * 2.Solve 3. The values t k = v k- v k+1 are an outcome of the process, T 2, …,T n 2/26/2009 COMP 790-Extensions to Basic Coalescent10

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? 2/26/2009COMP 790-Extensions to Basic Coalescent11

Exponential Growth For this specific population change function we can derive the following: Using the algorithm: 2/26/2009COMP 790-Extensions to Basic Coalescent12

Characterizations of Exponential Growth Now lets have a look at various characterizations of this population growth Characterization 1 – Waiting times, T 2, …,T n 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 2/26/2009COMP 790-Extensions to Basic Coalescent13

Characterizations of Exponential Growth 2/26/2009COMP 790-Extensions to Basic Coalescent14

Characterizations of Exponential Growth Characterization 2: Genealogy 2/26/2009COMP 790-Extensions to Basic Coalescent15 Characterization 2: Genealogy Basic coalescentExponential growthBasic coalescentExponential growth

Characterizations of Exponential Growth With high levels of exponential growth, tree becomes almost star shaped. 2/26/2009COMP 790-Extensions to Basic Coalescent16

Characterizations of Exponential Growth 2/26/2009COMP 790-Extensions to Basic Coalescent17

Characterizations of Exponential Growth Pairwise distances between all pairs of sequences. 2/26/2009COMP 790-Extensions to Basic Coalescent18 Basic coalescent (multimodal) Exponential growth (unimodal)

Characterizations of Exponential Growth Frequency spectrum of mutants 2/26/2009COMP 790-Extensions to Basic Coalescent19

Characterizations of Exponential Growth Percentage of contribution of kth waiting time to the mean and variance of total waiting time 2/26/2009COMP 790-Extensions to Basic Coalescent20

Population Bottlenecks Now we move on to the next type of population size change function: bottlenecks A way to model ice age 2/26/2009COMP 790-Extensions to Basic Coalescent21

Population Bottlenecks 2/26/2009COMP 790-Extensions to Basic Coalescent22 4 parameters. Strength of the bottleneck is determined by its length (tb) and severity(f)

Effective Population Size 2/26/2009COMP 790-Extensions to Basic Coalescent23 We defined effective population size in very first lectures as:

Effective Population Size 2/26/2009COMP 790-Extensions to Basic Coalescent24

Next Time Relax another assumption – > Coalescent with population structure 2/26/2009COMP 790-Extensions to Basic Coalescent25