Extensions to Basic CoalescentChapter 4, Part 1

Extension 1

• One of the assumptions of basic coalescent (Wright-Fisher) model:

Population size is constant

• We will relax this assumption

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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

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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

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Intuition

• Let p=probability by which two genes find a common ancestor

• Wright Fisher modelp = 1/2N

• Extended modelp(t) = 1/2N(t)

• E.g. when N(t) < N (declining population size)Probability of a coalescence event increases and a

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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

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Intuition

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Time in basic coalescent

Time in extended model

Each of the intervals between dashed lines represents 2N generations

Formulation

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Accumulated coalescent rate over time measured relative to the rate at time t=0

where

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

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Formulation

• Tk* : Waiting times in basic coalescent

• Tk : Waiting times in extended modelAlgorithm1. Simulate T2

*, … Tn* according to the basiccoalescent,

where Tk* is exponentially distributed with parameter

C(k,2). Denote the simulated values by tk*

2. Solve3. The values tk = vk- vk+1 are an outcome of the process,

T2, … ,Tn

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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?

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Exponential Growth

• For this specific population change function we can derive the following:

• Using the algorithm:

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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

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Characterizations of Exponential Growth

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Characterizations of Exponential Growth

• Characterization 2: Genealogy

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• Characterization 2: Genealogy

Basic coalescent Exponential growthBasic coalescent Exponential growth

Characterizations of Exponential Growth

• With high levels of exponential growth, tree becomes almost star shaped.

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Characterizations of Exponential Growth

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Characterizations of Exponential Growth

• Pairwise distances between all pairs of sequences.

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Basic coalescent (multimodal)

Exponential growth (unimodal)

Characterizations of Exponential Growth

• Frequency spectrum of mutants

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Characterizations of Exponential Growth

• Percentage of contribution of kth waiting time to the mean and variance of total waiting time

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Population Bottlenecks

• Now we move on to the next type of population size change function: bottlenecks

• A way to model ice age

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Population Bottlenecks

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4 parameters. Strength of the bottleneck is determined by its length (tb) and severity(f)

Effective Population Size

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• We defined effective population size in very first lectures as:

Effective Population Size

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Next Time

• Relax another assumption– > Coalescent with population structure

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