Fixed Parameters: Population Structure, Mutation, Selection, Recombination,... Reproductive Structure Genealogies of non- sequenced data Genealogies of sequenced data Parameter Estimation Model Testing Coalescent Theory in Biology www. coalescent.dk TGTTGT CATAGT CGTTAT
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Fixed Parameters: Population Structure, Mutation, Selection, Recombination,... Reproductive Structure Genealogies of non-sequenced data Genealogies of.
P(k):=P{k alleles had k distinct parents} 1 2N 1 2N *(2N-1) *..* (2N-(k-1)) =: (2N) [k] (2N) k k -> any k -> k k -> k-1 Ancestor choices: k -> j For k
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Fixed Parameters: Population Structure, Mutation, Selection, Recombination,...
Reproductive Structure
Genealogies of non-sequenced data
Genealogies of sequenced data
Parameter Estimation
Model Testing
Coalescent Theory in Biologywww. coalescent.dk
TGTTGT CATAGTCGTTAT
Haploid Model
Diploid Model
Wright-Fisher Model of Population Reproduction
i. Individuals are made by sampling with replacement in the previous generation.
ii. The probability that 2 alleles have same ancestor in previous generation is 1/2N
Individuals are made by sampling a chromosome from the female and one from the male previous generation with replacement
Assumptions
1. Constant population size
2. No geography
3. No Selection
4. No recombination
P(k):=P{k alleles had k distinct parents}
1 2N
1
2N *(2N-1) *..* (2N-(k-1)) =: (2N)[k]
(2N)k
k -> any k -> k k -> k-1
Ancestor choices:
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P(k) =2N[k ]
(2N)k ≈ (k 2 < 2N) 1−k2 ⎛ ⎝ ⎜
⎞ ⎠ ⎟/2N ≈ e
−k2 ⎛ ⎝ ⎜
⎞ ⎠ ⎟/ 2N
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k2 ⎛ ⎝ ⎜
⎞ ⎠ ⎟(2N)[k−1]
k -> j
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Sk, j (2N)[ j ]
For k << 2N:
Sk,j - the number of ways to group k labelled objects into j groups.(Stirling Numbers of second kind.
Mean, E(X2) = 2N.
Ex.: 2N = 20.000, Generation time 30 years, E(X2) = 600000 years.
Waiting for most recent common ancestor - MRCA
P(X2 = j) = (1-(1/2N))j-1 (1/2N)
Distribution until 2 alleles had a common ancestor, X2?:
1. Simultaneous Events 2. Multifurcations.3. Underestimation of Coalescent Rates
Multiple and Simultaneous Coalescents
2 56 3 0.0
1.0
1.0 corresponds to 2N generations
1 40
2N
0
6 6/2Ne
tc:=td/2Ne
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Xk is exp[k2 ⎛ ⎝ ⎜
⎞ ⎠ ⎟] distributed. E(Xk ) =1/
k2 ⎛ ⎝ ⎜
⎞ ⎠ ⎟
Discrete Continuous Time
The Standard CoalescentTwo independent Processes
Continuous: Exponential Waiting Times
Discrete: Choosing Pairs to Coalesce.
1 2 3 4 5
Waiting Coalescing
4--5
3--(4,5)
(1,2)--(3,(4,5))
1--2
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Exp52
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
€
Exp42
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
€
Exp22
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
€
Exp32
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
{1}{2}{3}{4}{5}
{1,2}{3,4,5}
{1,2,3,4,5}
{1}{2}{3}{4,5}
{1}{2}{3,4,5}
)1(2
2/1
−=⎟⎟⎠
⎞⎜⎜⎝⎛
kkk
Expected Height and Total Branch Length
Expected Total height of tree: Hk= 2(1-1/k)
i.Infinitely many alleles finds 1 allele in finite time. ii. In takes less than twice as long for k alleles to find 1 ancestors as it does for 2 alleles.
Expected Total branch length in tree, Lk:
2*(1 + 1/2 + 1/3 +..+ 1/(k-1)) ca= 2*ln(k-1)
1
2
3
k
1/3
1 2
1
2/(k-1)
Time Epoch Branch Lengths
Effective Populations Size, Ne.
In an idealised Wright-Fisher model:
i. loss of variation per generation is 1-1/(2N).
ii. Waiting time for random alleles to find a common ancestor is 2N.
Factors that influences Ne:
i. Variance in offspring. WF: 1. If variance is higher, then effective population size is smaller.