adrian gonz´ alez casanova & maite wilke...

The seed-bank coalescent

an Gonz

alez Casanova & Maite Wilke Berenguer

joint work (in progress) with

Jochen Blath and Noemi Kurt

(all TU Berlin)

Mini-workshopDuality of Markov processes and applications to spatial population models

November 6th-7th, 2014

AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 1 / 35

1 Dynamics

2 The Frequency Process

3 Duality

4 The Seed-bank Coalescent

5 ...and its properties

Dynamics

The Wright-Fisher model with geometric seed-bankConsider a haploid population reproducing in discrete generationsi = 0, 1, 2...

and type-space {a,A}

N Mc=2

N plants produce N-c plants by multinomial samplingc seeds are selected uniformly to germinate:they produce exactly one plant eachN plants produce c seeds by multinomial samplingN-c seeds stay in the seed-bankOffspring inherit the type of their parents

Dynamics

N Mc=2

Dynamics

N Mc=2

Dynamics

N Mc=2

Dynamics

N Mc=2

Dynamics

N Mc=2

Dynamics

N Mc=2

Dynamics

N Mc=2

Dynamics

N Mc=2

Dynamics

N Mc=2

Dynamics

N Mc=2

Dynamics

N Mc=2

Dynamics

N Mc=2

Dynamics

N Mc=2

Dynamics

N Mc=2

Dynamics

N Mc=2

Dynamics

N Mc=2

Dynamics

The Wright-Fisher model with geometric seed-bankConsider a haploid population reproducing in discrete generationsi = 0, 1, 2... and type-space {a,A}

N Mc=2

Dynamics

N Mc=2

a AA A A Aa a aa

Dynamics

N Mc=2

a AA A A Aa a aa

A Aa a a aA A A

Dynamics

The Wright-Fisher model with geometric seed-bank

N Mc=2

Dynamics

N Mc=2

Dynamics

N Mc=2

Dynamics

N Mc=2

Dynamics

N Mc=2

Dynamics

N Mc=2

a A A A A Aa a a

Dynamics

N Mc=2

a A A A A Aa a a

a a a a

a a a aa

a a a a a

a aaaa a

aa a aaaa

A AAAA

A A A A

Dynamics

Formal Definition

Definition 1 (Wright-Fisher model with geometric seed-bank)

Fix population-size N 2 N, seed-bank size M = M(N), seed-bankintensity c 2 N, c N,M genetic type space E . Given initial typeconfigurations ⇠0 2 EN and ⌘0 2 EM , let

⇠i :=�

⇠i(j)�

j2[N], and ⌘i :=

⌘k (j)�

j2[M], i 2 N,

be the random genetic type configuration in EN of the plants ingeneration k , resp. that of the seeds in EM (obtained from thedynamics above).We call the discrete-time Markov-chain (⇠k , ⌘k )k2N0 with values inEN ⇥ EM the type configuration process of the Wright-Fisher modelwith geometric seed-bank component .

The Frequency Process

The frequency process

Consider the bi-allelic case E = {a,A}.

Definition 2 (The frequency process)Let

X Nk :=

1{⇠k (i)=a} and Y Mk :=

1{⌘k (j)=a}, k 2 N0.

be the fraction of type a plants, resp. seeds.

Both are (discrete-time) Markov chains with values in

2N, . . . , 1

resp. IM =n

, . . . , 1o

X Nk :=

1{⇠k (i)=a} and Y Mk :=

1{⌘k (j)=a}, k 2 N0.

N Mc=2

a AA A A Aa a aa

A Aa a a aA A A

U: # of “a” plants whose parents are plants ) U ⇠ Bin(N � c, x)Z: # of “a” plants whose parents are seeds ) Z ⇠ Hyp(M, c, yM)

V: # of “a” seeds whose parents are plants ) V ⇠ Bin(c, x)yM-Z: # of “a” seeds whose parents were seeds

X Nk :=

1{⇠k (i)=a} and Y Mk :=

1{⌘k (j)=a}, k 2 N0.

N Mc=2

a AA A A Aa a aa

A Aa a a aA A A

X Nk :=

1{⇠k (i)=a} and Y Mk :=

1{⌘k (j)=a}, k 2 N0.

N Mc=2

a AA A A Aa a aa

A Aa a a aA A A

X Nk :=

1{⇠k (i)=a} and Y Mk :=

1{⌘k (j)=a}, k 2 N0.

N Mc=2

a AA A A Aa a aa

A Aa a a aA A A

X Nk :=

1{⇠k (i)=a} and Y Mk :=

1{⌘k (j)=a}, k 2 N0.

N Mc=2

a AA A A Aa a aa

A Aa a a aA A A

X Nk :=

1{⇠k (i)=a} and Y Mk :=

1{⌘k (j)=a}, k 2 N0.

N Mc=2

a AA A A Aa a aa

A Aa a a aA A A

The scaling-limit of the allele frequency process

Assume: N 2 N, M=M(N)=KN (for K 2 N), c 2 N fix,

Theorem 3Under these conditions,

(X NbNtc,Y

NbNtc)t�0 ) (Xt ,Yt)t�0

on D[0,1)([0, 1]2), where (Xt ,Yt)t�0 is a 2-dimensional diffusion solving(

dXt = c(Yt � Xt)dt +p

Xt(1 � Xt)dBt ,

dYt = cK (Xt � Yt)dt .

We call (Xt ,Yt)t�0 the seed-bank diffusion.

Duality

The dual of the seed-bank diffusion

Definition 4 (The block-counting process of the seed-bank coalescent)Let (Nt ,Mt)t�0 be the continuous time Markov chain taking values inN2

0 with transitions

(n,m) 7!

(n � 1,m + 1) at rate cn(n + 1,m � 1) at rate cKm(n � 1,m) at rate

�n2�

We call this the block counting process of the seed-bank coalescent .

Duality

Moment Duality

The block-counting process of the seedbank-coalescent is the momentdual of the seed-bank diffusion!

Theorem 5For every x , y 2 [0, 1]2,every n,m 2 N2

0 and every t � 0 we have

Ex ,y [X nt Y m

t ] = En,m[xNt yMt ].

Duality

Proof of moment duality

Let H : [0, 1]2 ⇥ N20 ! R be given by H((x , y), (n,m)) := xnym and

denote by A and A the generators of (Xt ,Yt)t�0 and (Nt ,Mt)t�0.

AH(·, (n,m))(x , y)

= c(y � x)dHdx

(x , y) +12

x(1 � x)d2Hdx2 (x , y) + cK (x � y)

(x , y)

= c(y � x)nxn�1ym +12

x(1 � x)n(n � 1)xn�2ym + cK (x � y)xnmym�1

= cn(xn�1ym+1 � xnym) +

(xn�1ym � xnym)

+ cKm(xn+1ym�1 � xnym)

= AH((x , y), ·)(n,m)

The Seed-bank Coalescent

The genealogy of a sample

N Mc=2

Partitions with flags

Let Pk be the set of partitions of [k ] := {1, . . . , k}.For ⇣ 2 Pk , let |⇣| be the number of clocks of ⇣.Space of marked partitions:

P{p,s}k := {(⇣,~u) | ⇣ 2 Pk ,~u 2 {s, p}|⇣|}.

Example: k=5

{{1, 3}p, {2}s, {4, 5}p}.

Can trace whether an ancestral line is currently a seed or a plant .

The seed-bank coalescentDefinition 6

Let k 2 N, c,K 2 (0,1). The seed-bank k-coalescent⇣

⇧(k)t

t�0with

seed-bank intensity c and relative seed-bank size K is the(continuous-time) Markov chain with values in P{s,p}

k and transitions

⇡ 7! ⇡ at rate8

1 if exactly 2 p-blocks coalesce,c if blocks stay the same, but one p is replaced by an s,cK if blocks stay the same, but one s is replaced by a p.

Definition 7

We define the seed-bank coalescent (⇧t)t�0 (with values in P{p,s}1 ) as

the projective Kolmogoroff limit as k ! 1 of the laws of the seed-bankk -coalescent.

The seed-bank coalescentDefinition 6

Let k 2 N, c,K 2 (0,1). The seed-bank k-coalescent⇣

⇧(k)t

t�0with

seed-bank intensity c and relative seed-bank size K is the(continuous-time) Markov chain with values in P{s,p}

k and transitions

⇡ 7! ⇡ at rate8

1 if exactly 2 p-blocks coalesce,c if blocks stay the same, but one p is replaced by an s,cK if blocks stay the same, but one s is replaced by a p.

Definition 7

We define the seed-bank coalescent (⇧t)t�0 (with values in P{p,s}1 ) as

the projective Kolmogoroff limit as k ! 1 of the laws of the seed-bankk -coalescent.

A possible realisation of theseed-bank 8-coalescent :

At the red line:{{1}p, {2, 3, 4, 5}s, {6, 7, 8}p}

1 2 3 4 5 6 7 8

A possible realisation of theseed-bank 8-coalescent :

At the red line:{{1}p, {2, 3, 4, 5}s, {6, 7, 8}p}

1 2 3 4 5 6 7 8

The seed-bank coalescent as scaling limit

Theorem 8Again, assuming M = M(N) = KM (for some K 2 N), c fix, and k 2 N,⇣

⇧(N,k)Nt

t�0converges weakly as N ! 1 to the seed-bank

k-coalescent⇣

⇧(k)t

t�0.

...and its properties

Not coming down from infinity

Theorem 9The Seed-Bank coalescent does not come down from infinity.

Non-monotonicity of Nt

The Artificial Process

Definition 5.1

We define the artificial process (Nt ,Mt)t�0 to be the continuous timeMarkov chain started in (N0,M0) 2 N0 ⇥ N0 with transitions

(n,m) 7! (n � 1,m + 1) at rate cn,(n,m) 7! (n,m � 1) at rate cKm,

(n,m) 7! (n � 1,m) at rate✓

The Kingman phase

For n 2 N [ {1} let

⌧nj := inf{t � 0 : N(n,0)

t = j}, 1 j n � 1, j < 1

be the first time that the number of active blocks of an n-samplereaches j .Then

⌧11⇤

= E⇥

⌧1i � ⌧1i�1⇤

1� j

+ cj< 2

In particular ⌧11 < 1 a.s. We say that the kingman phase is over at therandom time ⌧11

The Kingman phase

For n 2 N [ {1} let

⌧nj := inf{t � 0 : N(n,0)

t = j}, 1 j n � 1, j < 1

⌧11⇤

= E⇥

⌧1i � ⌧1i�1⇤

1� j

+ cj< 2

The Kingman phase

For n 2 N [ {1} let

⌧nj := inf{t � 0 : N(n,0)

t = j}, 1 j n � 1, j < 1

⌧11⇤

= E⇥

⌧1i � ⌧1i�1⇤

1� j

+ cj< 2

Infinitely many plants become seeds (escape from theKingman Phase)Note that

deactivation at ⌧nj�1

� j2�

2cj + 2c � 1

independently of the number of inactive blocks.Then

1deactivation at ⌧nj�1

2cj + 2c � 1

⇠ 2c log(n)

1deactivation at ⌧1j�1

2cj + 2c � 1

Infinitely many plants become seeds (escape from theKingman Phase)Note that

deactivation at ⌧nj�1

� j2�

2cj + 2c � 1

independently of the number of inactive blocks.Then

1deactivation at ⌧nj�1

2cj + 2c � 1

⇠ 2c log(n)

1deactivation at ⌧1j�1

2cj + 2c � 1

The seed phase

Each ancestral line that jumped to the seed bank, stay in the seedbank until time t with probability at least e�cKt . So if infinitely manyancestral lines become seeds, infinitely many stay seeds.

We conclude:The artificial process does not come down from infinity.

The seed phase

Each ancestral line that jumped to the seed bank, stay in the seedbank until time t with probability at least e�cKt . So if infinitely manyancestral lines become seeds, infinitely many stay seeds.

We conclude:The artificial process does not come down from infinity.

Back to the Seed Bank Coalescent

To conclude that the Seed Bank Coalescent does not come down frominfinity, we use that almost surely

Nt + Mt � Nt + Mt

The coupling: a coloured coalescent

In addtion to the flags indicating state of a lineage, we add colours(blue and white) to the particles:The coloured partitions with marks:

P{p,s},{w ,b}k := {(⇣,~u,~v) | ⇣ 2 Pk ,~u 2 {s, p}|⇣|,~v 2 {w , b}k}.

Example: k=8

{{1w , 2b}p, {3b, 4b, 5b}s, {6b, 7b, 8b}p}

The coupling: a coloured coalescent

In addtion to the flags indicating state of a lineage, we add colours(blue and white) to the particles:The coloured partitions with marks:

P{p,s},{w ,b}k := {(⇣,~u,~v) | ⇣ 2 Pk ,~u 2 {s, p}|⇣|,~v 2 {w , b}k}.

Example: k=8

{{1w , 2b}p, {3b, 4b, 5b}s, {6b, 7b, 8b}p}

A coloured coalescent

The coloured seed-bank k -coalescent⇣

⇧(k)t

t�0is the

(continuous-time) Markov chain with values in P{s,p},{w ,b}k and

transitions

⇡ 7! ⇡ at rate8

1 if exactly 2 p-blocks coalesce, colours stay the samec if blocks stay the same, but one p is replaced by an s,

colours stay the samecK if blocks stay the same, but one s is replaced by a p

and all particles in that block are coloured blue.

1 2 3 4 5 6 7 8

Seedbank-coalescent:“colourblind”

Artificial process:sees only white

1 2 3 4 5 6 7 8

Time to the most recent common ancestor

Theorem 5.2

For all c,K 2 (0,1), the seed bank coalescent satisfies

E[TMRCA[n]] ⇣ log log n.

Here, the symbol ⇣ denotes weak asymptotic equivalence ofsequences meaning that we have

lim infn!1

E[TMRCA[n]]log log n

andlim sup

Time to the most recent common ancestor

Theorem 5.2

For all c,K 2 (0,1), the seed bank coalescent satisfies

E[TMRCA[n]] ⇣ log log n.

Here, the symbol ⇣ denotes weak asymptotic equivalence ofsequences meaning that we have

lim infn!1

andlim sup

n plants

~ log n

n plants

~ log n

log log n~

Relation to other models

KKL Seed Bank ModelBolthausen-Sznitman CoalescentStructured Coalescent (Two Island Model)

Long term behavior

A = {(0, 0), (1, 1)} is the absorbing set of (Xt ,Yt).Let

⌧ = inf{t > 0 : (Xt ,Yt) 2 A}

What is the distribution of (X⌧ ,Y⌧ )?Problem: There is no reason to believe that ⌧ < 1 a.s. ((X⌧ ,Y⌧ ) is noteven well defined)

Long term behavior

⌧ = inf{t > 0 : (Xt ,Yt) 2 A}

Long term behavior

⌧ = inf{t > 0 : (Xt ,Yt) 2 A}

Ignore the problem

Recall,

Xt(1 � Xt)dBt ,

dYt = cK (Xt � Yt)dt ,

Let Wt = K Xt + Yt . Then Wt is a Mattingale

dWt = Kp

Xt(1 � Xt)dBt ,

Ignore the problem

Recall,

Xt(1 � Xt)dBt ,

dYt = cK (Xt � Yt)dt ,

Let Wt = K Xt + Yt . Then Wt is a Mattingale

dWt = Kp

Xt(1 � Xt)dBt ,

Keep ignoring the problem

Do not forget Wt = KXt + Yt .Note that by Doob´ s optional stopping theorem (That we can NOTapply)

Ex ,y [W0] = Ex ,y [W⌧ ]

ThenKx + y = (K + 1)Px ,y

(X⌧ ,Y⌧ ) = (1, 1)�

So we conclude that

Px ,y�

(X⌧ ,Y⌧ ) = (1, 1)�

=Kx + yK + 1

Ex ,y [W0] = Ex ,y [W⌧ ]

(X⌧ ,Y⌧ ) = (1, 1)�

So we conclude that

Px ,y�

(X⌧ ,Y⌧ ) = (1, 1)�

=Kx + yK + 1

Ex ,y [W0] = Ex ,y [W⌧ ]

(X⌧ ,Y⌧ ) = (1, 1)�

So we conclude that

Px ,y�

(X⌧ ,Y⌧ ) = (1, 1)�

=Kx + yK + 1

The result is (almost) true

Corollary 10 (Fixation in law)

Given c,K and (X0,Y0) = (x , y) 2 [0, 1]2, a.s., we have that

limt!1

L(x ,y)�

Xt ,Yt�

=y + xK1 + K

�(1,1) +1 + (1 � x)K � y

1 + K�(0,0).

Duality saves the day

Proposition 5.3

All mixed moments of (Xt ,Yt)t�0 converge to the same finite limitdepending only on x , y ,K .More precisely, for each fixed n,m 2 N [ {0}, n + m � 1 we have

limt!1

Ex ,y [X nt Y m

t ] =y + xK1 + K

T := TMRCA := inf�

t > 0 : Nt + Mt = 1

Proposition 5.3

All mixed moments of (Xt ,Yt)t�0 converge to the same finite limitdepending only on x , y ,K .More precisely, for each fixed n,m 2 N [ {0}, n + m � 1 we have

limt!1

Ex ,y [X nt Y m

t ] =y + xK1 + K

T := TMRCA := inf�

t > 0 : Nt + Mt = 1

limt!1

Ex ,y⇥

X nt Y m

= limt!1

xNt yMti

= limt!1

xNt yMt | T ti

Pn,m (T t)

+ limt!1

xNt yMt

T > ti

Pn,m�T > t�

= limt!1

xPn,m�Nt = 1�

+ yPn,m�Mt = 1�

1 + K+

y1 + K

limt!1

Ex ,y⇥

X nt Y m

= limt!1

xNt yMti

= limt!1

xNt yMt | T ti

Pn,m (T t)

+ limt!1

xNt yMt

T > ti

Pn,m�T > t�

= limt!1

xPn,m�Nt = 1�

+ yPn,m�Mt = 1�

1 + K+

y1 + K

limt!1

Ex ,y⇥

X nt Y m

= limt!1

xNt yMti

= limt!1

xNt yMt | T ti

Pn,m (T t)

+ limt!1

xNt yMt

T > ti

Pn,m�T > t�

= limt!1

xPn,m�Nt = 1�

+ yPn,m�Mt = 1�

1 + K+

y1 + K

limt!1

Ex ,y⇥

X nt Y m

= limt!1

xNt yMti

= limt!1

xNt yMt | T ti

Pn,m (T t)

+ limt!1

xNt yMt

T > ti

Pn,m�T > t�

= limt!1

xPn,m�Nt = 1�

+ yPn,m�Mt = 1�

1 + K+

y1 + K

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