Introduction On the ancestral process of long-range seed bank models Application to biology

The Seed Bank Model

Joint work with J. Blath, N. Kurt and D. Spano

RTG1845Berlin Mathematical School

Technische Universitat BerlinUNAM

05-04-2013

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 1/25

Introduction On the ancestral process of long-range seed bank models Application to biology

Content

1 Introduction

2 On the ancestral process of long-range seed bank models

3 Application to Biology

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 2/25

Introduction On the ancestral process of long-range seed bank models Application to biology

Classical Models

The Wright-Fisher Model

Description

I A generation consists of N individuals. Each individual in generation iselects a parent uniformly in the generation i − 1.

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 3/25

Introduction On the ancestral process of long-range seed bank models Application to biology

Classical Models

The Wright-Fisher Model

Description

I A generation consists of N individuals. Each individual in generation iselects a parent uniformly in the generation i − 1.

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 3/25

Introduction On the ancestral process of long-range seed bank models Application to biology

Classical Models

The Wright-Fisher Model

Description

I A generation consists of N individuals. Each individual in generation iselects a parent uniformly in the generation i − 1.

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 3/25

Introduction On the ancestral process of long-range seed bank models Application to biology

Classical Models

The Wright-Fisher Model

Description

I A generation consists of N individuals. Each individual in generation iselects a parent uniformly in the generation i − 1.

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 3/25

Introduction On the ancestral process of long-range seed bank models Application to biology

Classical Models

The Wright-Fisher Model

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 4/25

Introduction On the ancestral process of long-range seed bank models Application to biology

Classical Models

The Wright-Fisher Model

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 4/25

Introduction On the ancestral process of long-range seed bank models Application to biology

Classical Models

The Wright-Fisher Model

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 4/25

Introduction On the ancestral process of long-range seed bank models Application to biology

Classical Models

The Wright-Fisher Model

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 4/25

Introduction On the ancestral process of long-range seed bank models Application to biology

Classical Models

The Wright-Fisher Model

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 4/25

Introduction On the ancestral process of long-range seed bank models Application to biology

Classical Models

The Wright-Fisher Model

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 4/25

Introduction On the ancestral process of long-range seed bank models Application to biology

Classical Models

The Wright-Fisher Model

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 4/25

Introduction On the ancestral process of long-range seed bank models Application to biology

Classical Models

The Wright-Fisher Model

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 4/25

Introduction On the ancestral process of long-range seed bank models Application to biology

Classical Models

The Wright-Fisher Model

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 4/25

Introduction On the ancestral process of long-range seed bank models Application to biology

Classical Models

The Wright-Fisher Model

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 4/25

Introduction On the ancestral process of long-range seed bank models Application to biology

Classical Models

Scaling

Fix a sample of size n. Let the number of individuals go to infinity.Measure the time in terms of the number of individuals per generation.

BN([Nt])⇒ K (t)

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 5/25

Introduction On the ancestral process of long-range seed bank models Application to biology

Classical Models

(The block counting process of) The Kingman coalescent

Description

Each pair of blocks coalesce at rate 1, independently of the others.

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 6/25

Introduction On the ancestral process of long-range seed bank models Application to biology

Classical Models

(The block counting process of) The Kingman coalescent

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 7/25

Introduction On the ancestral process of long-range seed bank models Application to biology

Classical Models

(The block counting process of) The Kingman coalescent

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 7/25

Introduction On the ancestral process of long-range seed bank models Application to biology

Classical Models

Time to the most recent common ancestor

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 8/25

Introduction On the ancestral process of long-range seed bank models Application to biology

Classical Models

Time to the most recent common ancestor

E [TMRCA] =n−1∑i=1

E [Ti ] =n−1∑i=1

1(n+1−i2

) =n∑

u=2

2

u(u − 1)= 2(1− 1

n

)

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 9/25

Introduction On the ancestral process of long-range seed bank models Application to biology

The Seed Bank Model

The work of Kaj, Krone and Lascoux (2001)

I The seed bank model introduced by Kaj, Krone and Lascoux is ageneralization of the Wright-Fisher model. Its biological motivationare species that reproduce using seeds. (Like a cactus.)

I Dynamics: Let µ be a bounded measure on N. Each individual selectsits parent independently by the following 2 steps:

1 Select the generation of the parent by performing a µ distributed jump.2 Select a parent uniformly among the members of the selected

generation.

I Problem: We lose the Markov property.

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 10/25

Introduction On the ancestral process of long-range seed bank models Application to biology

The Seed Bank Model

The work of Kaj, Krone and Lascoux (2001)

I The seed bank model introduced by Kaj, Krone and Lascoux is ageneralization of the Wright-Fisher model. Its biological motivationare species that reproduce using seeds. (Like a cactus.)

I Dynamics: Let µ be a bounded measure on N. Each individual selectsits parent independently by the following 2 steps:

1 Select the generation of the parent by performing a µ distributed jump.2 Select a parent uniformly among the members of the selected

generation.

I Problem: We lose the Markov property.

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 10/25

Introduction On the ancestral process of long-range seed bank models Application to biology

The Seed Bank Model

m=4

N=10

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 11/25

Introduction On the ancestral process of long-range seed bank models Application to biology

The Seed Bank Model

1/10

1/10

1/10

1/10

1/10

1/10

1/10

1/10

1/10

1/10

m=4

N=10

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 11/25

Introduction On the ancestral process of long-range seed bank models Application to biology

The Seed Bank Model

1/10

1/10

1/10

1/10

1/10

1/10

1/10

1/10

1/10

1/10

m=4

N=10

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 11/25

Introduction On the ancestral process of long-range seed bank models Application to biology

The Seed Bank Model

1/10

1/10

1/10

1/10

1/10

1/10

1/10

1/10

1/10

1/10

m=4

N=10

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 11/25

Introduction On the ancestral process of long-range seed bank models Application to biology

The Seed Bank Model

1/10

1/10

1/10

1/10

1/10

1/10

1/10

1/10

1/10

1/10

m=4

N=10

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 11/25

Introduction On the ancestral process of long-range seed bank models Application to biology

The Seed Bank Model

The work of Kaj, Krone and Lascoux (2001)

I The ancestral process can be described in terms of a finite stateMarkov Chain.

I Main result: The scaling limit is the Kingman coalescent, under aconstant time change.

I Limitation: µ must be bounded.

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 12/25

Introduction On the ancestral process of long-range seed bank models Application to biology

The model

The seed bank model with long-range dependence

I What happens if we remove the boundedness condition of the jumpmeasure µ in the seed bank model?

I Motivation.

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 13/25

Introduction On the ancestral process of long-range seed bank models Application to biology

The model

The seed bank model with long-range dependence

I Answer: It depends on µ.

I We say that µ ∈ Γα, if

µ({n, ...}) = n−αL(n), α > 0,

where L(n) is a slowly varying function.

I The qualitative behaviour of the model changes drastically dependingon α.

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 14/25

Introduction On the ancestral process of long-range seed bank models Application to biology

Results

Construction of the Renewal Process

The ancestral line A(v) of an individual v is given by a renewal processwith interarrival law µ and an additional uniform choice of an individual.The renewal times correspond to the generation of an ancestor. Ancestrallines of a sample of individuals are coupled renewal processes.

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 15/25

Introduction On the ancestral process of long-range seed bank models Application to biology

Results

J. Blath, AGC, N. Kurt, D. Spano (2011)

Let µ,N be fixed and let v ,w denote two individuals living at time 0.

(a) If α > 1, then E[TMRCA] <∞(b) If α ∈ (1/2, 1), then P(A(v) ∩ A(w) 6= ∅) = 1 and E[T ] =∞(c) If α ∈ (0, 1/2), then P(A(v) ∩ A(w) 6= ∅) < 1

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 16/25

Introduction On the ancestral process of long-range seed bank models Application to biology

Results

J. Blath, AGC, N. Kurt, D. Spano (2011)

If Eµ[ν] <∞ the block counting process induced by our model convergesweakly to the Kingman coalescent (constantly time changed),i.e. BN(Nt)⇒ K (γ(1)2t)

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 17/25

Introduction On the ancestral process of long-range seed bank models Application to biology

Results

Ingredient 1: a Markov process

n=10

m=4

X10

(0)=(10, 0, 0, 0)

(X1(0), X2(0), X3(0), X4(0))

The configuration process in level 10

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 18/25

Introduction On the ancestral process of long-range seed bank models Application to biology

Results

Ingredient 1: a Markov process

n=10

m=4

X10

(0)=(10, 0, 0, 0)

X10

(1)=(3, 3, 2, 2)

(X1(1), X2(1), X3(1), X4(1))

The configuration process in level 10

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 18/25

Introduction On the ancestral process of long-range seed bank models Application to biology

Results

Ingredient 1: a Markov process

n=10

m=4

X10

(0)=(10, 0, 0, 0)

The configuration process in level 10

X10

(1)=(3, 3, 2, 2)

X10

(2)=(4, 3, 2, 1)

Finite States Irreducible Markov

Chain

(X1(2), X2(2), X3(2), X4(2))

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 18/25

Introduction On the ancestral process of long-range seed bank models Application to biology

Results

Ingredient 2: stationary distribution

Proof

Let X n be the configuration process in level n.

I There exists a stationary distribution for X n if and only if Eµ[X ] <∞.

The stationary distribution is

γ = mult( 1

Eµ[X ],µ(i > 1)

Eµ[X ],µ(i > 2)

Eµ[X ], ...)

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 19/25

Introduction On the ancestral process of long-range seed bank models Application to biology

Results

Ingredient 2: stationary distribution

{ball 1 visits urn k} = {X 1k = (1, 0, 0, ...)}

Pγ({ball 1 visits urn k}) = 1Eµ[X ]

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 20/25

Introduction On the ancestral process of long-range seed bank models Application to biology

Results

Ingredient 2: stationary distribution

{ball 1 visits urn k} = {X 1k = (1, 0, 0, ...)}

Pγ({ball 1 visits urn k}) = 1Eµ[X ]

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 20/25

Introduction On the ancestral process of long-range seed bank models Application to biology

Results

Ingredient 2: stationary distribution

{ball 1 visits urn k} = {X 1k = (1, 0, 0, ...)}

Pγ({ball 1 visits urn k}) = 1Eµ[X ]

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 20/25

Introduction On the ancestral process of long-range seed bank models Application to biology

Results

Ingredient 2: stationary distribution

{ball 1 visits urn k} = {X 1k = (1, 0, 0, ...)}

Pγ({ball 1 visits urn k}) = 1Eµ[X ]

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 20/25

Introduction On the ancestral process of long-range seed bank models Application to biology

Results

Ingredient 2: stationary distribution

{ball 1 visits urn k} = {X 1k = (1, 0, 0, ...)}

Pγ({ball 1 visits urn k}) = 1Eµ[X ]

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 20/25

Introduction On the ancestral process of long-range seed bank models Application to biology

Results

Ingredient 2: stationary distribution

{ball 1 visits urn k} = {X 1k = (1, 0, 0, ...)}

Pγ({ball 1 visits urn k}) = 1Eµ[X ]

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 20/25

Introduction On the ancestral process of long-range seed bank models Application to biology

Results

Ingredient 2: stationary distribution

{ball 1 visits urn k} = {X 1k = (1, 0, 0, ...)}

Pγ({ball 1 visits urn k}) = 1Eµ[X ]

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 20/25

Introduction On the ancestral process of long-range seed bank models Application to biology

Results

Ingredient 2: stationary distribution

{ball 1 visits urn k} = {X 1k = (1, 0, 0, ...)}

Pγ({ball 1 visits urn k}) = 1Eµ[X ]

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 20/25

Introduction On the ancestral process of long-range seed bank models Application to biology

Results

Ingredient 2: stationary distribution

{ball 1 visits urn k} = {X 1k = (1, 0, 0, ...)}

Pγ({ball 1 visits urn k}) = 1Eµ[X ]

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 20/25

Introduction On the ancestral process of long-range seed bank models Application to biology

Results

Ingredient 2: stationary distribution

{ball 1 visits urn k} = {X 1k = (1, 0, 0, ...)}

Pγ({ball 1 visits urn k}) = 1Eµ[X ]

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 20/25

Introduction On the ancestral process of long-range seed bank models Application to biology

Results

Ingredient 2: stationary distribution

{ball 1 visits urn k} = {X 1k = (1, 0, 0, ...)}

Pγ({ball 1 visits urn k}) = 1Eµ[X ]

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 20/25

Introduction On the ancestral process of long-range seed bank models Application to biology

Results

Ingredient 2: stationary distribution

{ball 1 visits urn k} = {X 1k = (1, 0, 0, ...)}

Pγ({ball 1 visits urn k}) = 1Eµ[X ]

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 20/25

Introduction On the ancestral process of long-range seed bank models Application to biology

Results

Ingredient 2: stationary distribution

{ball 1 visits urn k} = {X 1k = (1, 0, 0, ...)}

Pγ({ball 1 visits urn k}) = 1Eµ[X ]

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 20/25

Introduction On the ancestral process of long-range seed bank models Application to biology

Results

Ingredient 2: stationary distribution

{ball 1 visits urn k} = {X 1k = (1, 0, 0, ...)}

Pγ({ball 1 visits urn k}) = 1Eµ[X ]

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 20/25

Introduction On the ancestral process of long-range seed bank models Application to biology

Results

Ingredient 2: stationary distribution

{ball 1 visits urn k} = {X 1k = (1, 0, 0, ...)}

Pγ({ball 1 visits urn k}) = 1Eµ[X ]

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 20/25

Introduction On the ancestral process of long-range seed bank models Application to biology

Results

Ingredient 2: stationary distribution

{ball 1 visits urn k} = {X 1k = (1, 0, 0, ...)}

Pγ({ball 1 visits urn k}) = 1Eµ[X ]

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 20/25

Introduction On the ancestral process of long-range seed bank models Application to biology

Results

Ingredient 3: coupling argument

Idea

If particles are always in the stationary distribution

P(coalesce in generation k) =1

N(Eµ[X ])2=

1

Nγ(1)2

Then consider an artificial system where particles are always in thestationary distribution and couple it with the ancestral process of the seedbank model.

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 21/25

Introduction On the ancestral process of long-range seed bank models Application to biology

Results

Ingredient 3: Coupling argument

The coupling is fast

Let τ be the first time particles labeled 1 in each system are in the samegeneration.

E [τ ] <∞

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 22/25

Introduction On the ancestral process of long-range seed bank models Application to biology

Current work

J. Blath, AGC, N. Kurt, D. Spano (2011)

Let µ,N be fixed and let v ,w denote two individuals living at time 0.

(a) If α > 1, then E[TMRCA] <∞(b) If α ∈ (1/2, 1), then P(A(v) ∩ A(w) 6= ∅) = 1 and E[T ] =∞(c) If α ∈ (0, 1/2), then P(A(v) ∩ A(w) 6= ∅) < 1

Common ancestor will be close to the seed-mutation.Genetic drift will not cause fixation nor extinction.

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 23/25

Introduction On the ancestral process of long-range seed bank models Application to biology

Current work

Azotobacter vinelandii

I The seed effect is very important in evolution of bacteria.

I Azotobacter vinelandii makes very big jumps. (endospores)

I Azotobacter vinelandii has too many genes of Pseudomonas not to bea Pseudomonas and too less to be one.

I Our model partially explains this phenomena.

I Challenges.

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 24/25

Introduction On the ancestral process of long-range seed bank models Application to biology

Current work

Azotobacter vinelandii

I The seed effect is very important in evolution of bacteria.

I Azotobacter vinelandii makes very big jumps. (endospores)

I Azotobacter vinelandii has too many genes of Pseudomonas not to bea Pseudomonas and too less to be one.

I Our model partially explains this phenomena.

I Challenges.

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 24/25

Introduction On the ancestral process of long-range seed bank models Application to biology

Current work

Azotobacter vinelandii

I The seed effect is very important in evolution of bacteria.

I Azotobacter vinelandii makes very big jumps. (endospores)

I Azotobacter vinelandii has too many genes of Pseudomonas not to bea Pseudomonas and too less to be one.

I Our model partially explains this phenomena.

I Challenges.

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 24/25

Introduction On the ancestral process of long-range seed bank models Application to biology

Current work

Azotobacter vinelandii

I The seed effect is very important in evolution of bacteria.

I Azotobacter vinelandii makes very big jumps. (endospores)

I Azotobacter vinelandii has too many genes of Pseudomonas not to bea Pseudomonas and too less to be one.

I Our model partially explains this phenomena.

I Challenges.

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 24/25

Introduction On the ancestral process of long-range seed bank models Application to biology

Current work

Azotobacter vinelandii

I The seed effect is very important in evolution of bacteria.

I Azotobacter vinelandii makes very big jumps. (endospores)

I Azotobacter vinelandii has too many genes of Pseudomonas not to bea Pseudomonas and too less to be one.

I Our model partially explains this phenomena.

I Challenges.

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 24/25

Introduction On the ancestral process of long-range seed bank models Application to biology

References

References

I Kaj, I., Krone, S., Lascoux, M. 2001. Coalescent theory for seed bankmodels. J. Appl. Prob. 38:285-300

I J. Blath, A. Gonzalez Casanova, N. Kurt and D. Spano. On theancestral process of long-range seed bank models. To appear in J. ofAppl. Prob.

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 25/25

Introduction On the ancestral process of long-range seed bank models Application to biology

References

Adrian Gonzalez Casanova On the Ancestral process of the Seed Bank Model 26/25