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Constructing coalescent processes from branching processes Dragana Radojiˇci´c

February 24, 2017

Dragana Radojiˇci´c (BMS)

BMS student conference 2017

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Outline

1

Introduction

2

Coalescent processes obtained from supercritical Galton-Watson processes (by Jason Schweinsberg)

3

An individual-based model for the Lenski experiment, and the deceleration of the relative fitness (by Adrián González Casanova, Noemi Kurt, Anton Wakolbinger and Linglong Yuan)

4

Conclusion

Dragana Radojiˇci´c (BMS)

BMS student conference 2017

2 / 27

Outline

1

Introduction

2

Coalescent processes obtained from supercritical Galton-Watson processes (by Jason Schweinsberg)

3

An individual-based model for the Lenski experiment, and the deceleration of the relative fitness (by Adrián González Casanova, Noemi Kurt, Anton Wakolbinger and Linglong Yuan)

4

Conclusion

Dragana Radojiˇci´c (BMS)

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Introduction

Two different mathematical approaches in the population genetics forward simulation (Galton-Watson processes, Wright-Fisher model, etc.) backward simulation (coalescent processes)

Dragana Radojiˇci´c (BMS)

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Galton-Watson process Introduced in 1889 by English scientist Francis Galton (1822-1911) Xn = the number of individuals in n-th generation X0 = 1 (l)

(ξn )n∈N , l ∈ N i.i.d with offspring distribution (l) pk = P(ξn = k ) Xn+1 =

(l) l=1 ξn

PXn (1)

µ = E[ξo ] = E[X1 ] Dragana Radojiˇci´c (BMS)

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Important notation for the coalescent processes Pn the set of partitions of {1, ..., n} P∞ the set of partitions of N ξ, η ∈ Pn ξ ≺ η if from η we can get block ξ by merging two blocks of η.

Example Let η and ξ be two partitions from P7 : η = {{1, 3}, {2}, {4, 5, 6}, {7}} and ξ = {{1, 3, 7}, {2}, {4, 5, 6}} then we can write ξ ≺ η, i.e. {{1, 3, 7}, {2}, {4, 5, 6}} ≺ {{1, 3}, {2}, {4, 5, 6}, {7}} Dragana Radojiˇci´c (BMS)

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Coalescent processes coalesce = to come together to form one group or mass i ∼r j, if i and j have common ancestor in r generations ago

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Ancestral processes

Definition (Ancestral process) Sample n individuals at generation 0 and denote them by l1 , l2 , ..., ln . Let Pn be set of partitions of {1, 2, ..., n} and Ψ(N,n) = (Ψn,N g )g∈N0 be the process taking values in Pn such that any j, k being in the same (N,n) block in Ψg if and only if there is a common ancestor at generation −g for individuals lj , lk . Then Ψ(N,n) is the ancestral process of the chosen sample.

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Kingman’s n-coalescent

the simplest version of coalescent processes (Πn (t) : t ≥ 0) is a continuous time Markov process taking values in the space Pn Πn (0) = {{1}, {2}, ..., {n}} each pair of blocks merges at rate 1, independently of their size the transition rates q(ξ, η) are as follows: ( 1 lim t −1 P(Πn (s + t) = ξ|Πn (s) = η) = t↓0 0

Dragana Radojiˇci´c (BMS)

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if ξ ≺ η otherwise

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Λ-coalescents

coalescents with multiple collisions introduced by Pitman(1999) λb,k be the transition rate of any k blocks (out of b blocks) merge in one block it is not possible that two such mergers happens at once Pitman showed that: Z 1 λb,k = x k −2 (1 − x)b−k λ(dx), (2 ≤ k ≤ b) 0

Dragana Radojiˇci´c (BMS)

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Outline

1

Introduction

2

Coalescent processes obtained from supercritical Galton-Watson processes (by Jason Schweinsberg)

3

An individual-based model for the Lenski experiment, and the deceleration of the relative fitness (by Adrián González Casanova, Noemi Kurt, Anton Wakolbinger and Linglong Yuan)

4

Conclusion

Dragana Radojiˇci´c (BMS)

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Model presented in the paper N individuals in every generation probability that each individual has k or more children ∼ Ck −a each individual has random number of offspring with mean >1

Cannings’s model (1974,1975) fixed, discrete and non-overlapping generations infinity many generations backwards in past r = 0 is the present generation, and r = −1 is the previous generation, and so on... (m)

νi,N be the number of offspring in −m-th generation from i-th individual from −(m + 1)-th generation (m)

(m)

(m)

ν = (ν1,N , ν2,N , · · · , νN,N ) describe the distribution of the offspring number for generation m PN (m) i=1 νi,N = N for each m ∈ Z Dragana Radojiˇci´c (BMS)

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Notation

Notation: (x)k = x(x − 1)(x − 2) · · · (x − k + 1). The coalescent probability: cN =

Dragana Radojiˇci´c (BMS)

E[(ν1,N )2 ] N −1

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Motivation for this paper

Proposition Suppose E[(ν1,N )3 ] = 0. N→∞ N 2 cN lim

(1)

Then, as N → ∞, the processes (Ψn,N (bt/cN c), t ≥ 0) converge to the n-coalescent. -> establish the convergence argument for n-coalescent, which is result of Section 4 in the paper "Total variation distances and rates of convergence for ancestral coalescent processes in exchangeable population models" by Martin Möhle

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Proposition Suppose lim cN = 0.

(2)

E[(ν1,N )2 (ν2,N )2 ] = 0. N→∞ N 2 cN

(3)

N→∞

and

lim

Also, assume that for some probability measure Λ on [0, 1], we have N P(ν1,N > Nx) = N→∞ cN

Z

1

lim

y −2 Λ(dy )

x

for all x ∈ (0, 1) at which the limit function is continuous. Then, as N → ∞, the processes (Ψn,N (bt/cN c), t ≥ 0) converge to a process (Ψn,∞ (t), t ≥ 0) that has the same law as the restriction to {1, ..., n} of the Λ-coalescent. -> from "A Classification of Coalescent Processes for Haploid Exchangeable Population Models", by Martin Möhle and Serik Sagitov. Dragana Radojiˇci´c (BMS)

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A model involving supercritical Galton-Watson processes and the main result GOAL: (m)

(m)

to obtain the family sizes vectors (ν1,N , · · · , νN,N ) from the supercritical Galton-Watson process Xi be the number of offspring of the i-th individal in generation −(m + 1), such that E[X1 ] > 1

(4)

and P(X1 ≥ k ) ∼ Ck −a

(5)

SN = X1 + X2 + · · · + XN

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Theorem Assume that E[X1 ] > 1 is satisfied. (a) If E[X12 ] < ∞ (in particular, if P(X1 ≥ k ) ∼ Ck −a holds and a > 2), then the processes (Ψn,N (bt/cN c), t ≥ 0) converge as N → ∞ to the Kingman n-coalescent. (b) If P(X1 ≥ k ) ∼ Ck −a holds and a = 2, then the processes (Ψn,N (bt/cN c), t ≥ 0) converge as N → ∞ to the Kingman n-coalescent. (c) When P(X1 ≥ k ) ∼ Ck −a holds with 1 ≤ a < 2, the processes (Ψn,N (bt/cN c), t ≥ 0) converge as N → ∞ to a continuous-time process (Ψn,∞ (t), t ≥ 0) that has the same law as the restriction to {1, ..., n} of the Λ-coalescent, where Λ is the probability measure associated with the Beta(2 − a, a) distribution. The transition rates are given by B(k − a, b − k + a) λb,k = (6) B(2 − a, a) where B(α, β) = Dragana Radojiˇci´c (BMS)

Γ(α)Γ(β) Γ(α+β)

is the beta function.

BMS student conference 2017

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Outline

1

Introduction

2

Coalescent processes obtained from supercritical Galton-Watson processes (by Jason Schweinsberg)

3

An individual-based model for the Lenski experiment, and the deceleration of the relative fitness (by Adrián González Casanova, Noemi Kurt, Anton Wakolbinger and Linglong Yuan)

4

Conclusion

Dragana Radojiˇci´c (BMS)

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Relevant information on the experiment

cycles of one day each Beginning of the day: N individuals Reproduction stops after the glucose is consumed – at that time there are γN individuals, for some γ > 1 Out of the γN individuals, N are sampled (uniformly and without replacement) to form the population at the begging of the next day

−→ continuous part: population growth during a day discrete part: sampling between the days

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Additional theory (1)

Definition (Yule process) A Yule process with rate r is a continuous-time Markov process taking values in N such that the transition rates are given by: ( n → n + 1 at rate rn (7) n → others at rate 0.

Dragana Radojiˇci´c (BMS)

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Additional theory (2)

Lemma Let Z r be a Yule process with rate r . a) If Z r (0) = 1, for t > 0, Z r (t) follows a geometric distribution with parameter e−rt . b) If Z r (0) = n0 ∈ N, then Z r (t) follows a negative binomial distribution with parameters n0 and e−rt . In particular, E[Z r (t)] = n0 ert and var (Z r (t)) = ert (ert − 1)n0 .

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Mathematical model for the continuous part (N)

σ (N) = inf{t > 0 : E[Zt

] ≥ γN}

Yti = the size of the family of individual i at time t. (N)

(Zt )t≥0 is a Yule process with parameter r > 0, and that the process (N) starts with N individuals, i.e. Z0 = N, and ZtN = Yt1 + Yt2 + · · · + YtN , with (Yti )1≤i≤N i.i.d. as a geometric distribution with parameter e−rt (N)

The stopping time σ (N) is deterministic since E[Zt from Nert = γN we get: log γ . σ (N) = r Dragana Radojiˇci´c (BMS)

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] = Nert , and thus

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Theorem (Convergence to Kingman’s coalescent) For all n ∈ N, the sequence of ancestral processes (N,n) ) bNt/2(1− γ1 )c t≥0

(B

converges weakly on the space of càdlàg paths as N → ∞ to Kingman’s n-coalescent.

Dragana Radojiˇci´c (BMS)

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Outline

1

Introduction

2

Coalescent processes obtained from supercritical Galton-Watson processes (by Jason Schweinsberg)

3

An individual-based model for the Lenski experiment, and the deceleration of the relative fitness (by Adrián González Casanova, Noemi Kurt, Anton Wakolbinger and Linglong Yuan)

4

Conclusion

Dragana Radojiˇci´c (BMS)

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A brief conclusion

Relatively new area...

Important fields of applications...

Dragana Radojiˇci´c (BMS)

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My prospective work

Natural to try to relate these convergence results from obtained papers. The question to be asked is which conditions we need to reach convergence to Λ- coalescent in the settings of Lenski experiment.

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Thank You!

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