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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
MRCA is found more rapidly than if N(t) is constant2/26/2009 COMP 790-Extensions to Basic Coalescent 5
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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