Upload
others
View
10
Download
0
Embed Size (px)
Citation preview
The seed-bank coalescent
Adri
´
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
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 2 / 35
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
0
1
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
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 3 / 35
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
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 3 / 35
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
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 3 / 35
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
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 3 / 35
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
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 3 / 35
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
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 3 / 35
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
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 3 / 35
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
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 3 / 35
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
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 3 / 35
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
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 3 / 35
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
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 3 / 35
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
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 3 / 35
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
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 3 / 35
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
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 3 / 35
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
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 3 / 35
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
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 3 / 35
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
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 3 / 35
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
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 3 / 35
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
a AA A A Aa a aa
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
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 3 / 35
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
a AA A A Aa a aa
A Aa a a aA A A
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
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 3 / 35
Dynamics
The Wright-Fisher model with geometric seed-bank
N Mc=2
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 4 / 35
Dynamics
The Wright-Fisher model with geometric seed-bank
N Mc=2
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 4 / 35
Dynamics
The Wright-Fisher model with geometric seed-bank
N Mc=2
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 4 / 35
Dynamics
The Wright-Fisher model with geometric seed-bank
N Mc=2
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 4 / 35
Dynamics
The Wright-Fisher model with geometric seed-bank
N Mc=2
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 4 / 35
Dynamics
The Wright-Fisher model with geometric seed-bank
N Mc=2
a A A A A Aa a a
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 4 / 35
Dynamics
The Wright-Fisher model with geometric seed-bank
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
A A A A
A A A
A A
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 4 / 35
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 .
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 5 / 35
The Frequency Process
The frequency process
Consider the bi-allelic case E = {a,A}.
Definition 2 (The frequency process)Let
X Nk :=
1N
X
i2[N]
1{⇠k (i)=a} and Y Mk :=
1M
X
j2[M]
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
IN =n
0,1N,
2N, . . . , 1
o
resp. IM =n
0,1M
,2M
, . . . , 1o
.
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 6 / 35
The Frequency Process
The frequency process
X Nk :=
1N
X
i2[N]
1{⇠k (i)=a} and Y Mk :=
1M
X
j2[M]
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 y
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
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 7 / 35
The Frequency Process
The frequency process
X Nk :=
1N
X
i2[N]
1{⇠k (i)=a} and Y Mk :=
1M
X
j2[M]
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 y
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
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 7 / 35
The Frequency Process
The frequency process
X Nk :=
1N
X
i2[N]
1{⇠k (i)=a} and Y Mk :=
1M
X
j2[M]
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 y
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
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 7 / 35
The Frequency Process
The frequency process
X Nk :=
1N
X
i2[N]
1{⇠k (i)=a} and Y Mk :=
1M
X
j2[M]
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 y
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
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 7 / 35
The Frequency Process
The frequency process
X Nk :=
1N
X
i2[N]
1{⇠k (i)=a} and Y Mk :=
1M
X
j2[M]
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 y
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
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 7 / 35
The Frequency Process
The frequency process
X Nk :=
1N
X
i2[N]
1{⇠k (i)=a} and Y Mk :=
1M
X
j2[M]
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 y
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
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 7 / 35
The Frequency Process
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.
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 8 / 35
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!
8
>
<
>
:
(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 .
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 9 / 35
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 ].
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 10 / 35
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)
dHdy
(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) +
✓
n2
◆
(xn�1ym � xnym)
+ cKm(xn+1ym�1 � xnym)
= AH((x , y), ·)(n,m)
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 11 / 35
The Seed-bank Coalescent
The genealogy of a sample
N Mc=2
1 2 3
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 12 / 35
The Seed-bank Coalescent
The genealogy of a sample
N Mc=2
1 2 3
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 12 / 35
The Seed-bank Coalescent
The genealogy of a sample
N Mc=2
1 2 3
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 12 / 35
The Seed-bank Coalescent
The genealogy of a sample
N Mc=2
1 2 3
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 12 / 35
The Seed-bank Coalescent
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 .
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 13 / 35
The Seed-bank 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.
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 14 / 35
The Seed-bank 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.
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 14 / 35
The Seed-bank Coalescent
The seed-bank 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
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 15 / 35
The Seed-bank Coalescent
The seed-bank 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
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 15 / 35
The Seed-bank Coalescent
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.
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 16 / 35
...and its properties
Not coming down from infinity
Theorem 9The Seed-Bank coalescent does not come down from infinity.
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 17 / 35
...and its properties
Non-monotonicity of Nt
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 18 / 35
...and its properties
Non-monotonicity of Nt
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 18 / 35
...and its properties
Non-monotonicity of Nt
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 18 / 35
...and its properties
Non-monotonicity of Nt
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 18 / 35
...and its properties
Non-monotonicity of Nt
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 18 / 35
...and its properties
Non-monotonicity of Nt
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 18 / 35
...and its properties
Non-monotonicity of Nt
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 18 / 35
...and its properties
Non-monotonicity of Nt
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 18 / 35
...and its properties
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✓
n2
◆
.
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 19 / 35
...and its properties
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
E⇥
⌧11⇤
= E⇥
1X
i=1
⌧1i � ⌧1i�1⇤
=1X
i=1
1� j
2�
+ cj< 2
In particular ⌧11 < 1 a.s. We say that the kingman phase is over at therandom time ⌧11
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 20 / 35
...and its properties
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
E⇥
⌧11⇤
= E⇥
1X
i=1
⌧1i � ⌧1i�1⇤
=1X
i=1
1� j
2�
+ cj< 2
In particular ⌧11 < 1 a.s. We say that the kingman phase is over at therandom time ⌧11
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 20 / 35
...and its properties
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
E⇥
⌧11⇤
= E⇥
1X
i=1
⌧1i � ⌧1i�1⇤
=1X
i=1
1� j
2�
+ cj< 2
In particular ⌧11 < 1 a.s. We say that the kingman phase is over at therandom time ⌧11
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 20 / 35
...and its properties
Infinitely many plants become seeds (escape from theKingman Phase)Note that
P�
deactivation at ⌧nj�1
�
=cj
� j2�
+ cj=
2cj + 2c � 1
,
independently of the number of inactive blocks.Then
E⇥
nX
j=1
1deactivation at ⌧nj�1
⇤
=nX
j=1
2cj + 2c � 1
⇠ 2c log(n)
and
E⇥
1X
j=1
1deactivation at ⌧1j�1
⇤
⇠1X
j=1
2cj + 2c � 1
= 1
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 21 / 35
...and its properties
Infinitely many plants become seeds (escape from theKingman Phase)Note that
P�
deactivation at ⌧nj�1
�
=cj
� j2�
+ cj=
2cj + 2c � 1
,
independently of the number of inactive blocks.Then
E⇥
nX
j=1
1deactivation at ⌧nj�1
⇤
=nX
j=1
2cj + 2c � 1
⇠ 2c log(n)
and
E⇥
1X
j=1
1deactivation at ⌧1j�1
⇤
⇠1X
j=1
2cj + 2c � 1
= 1
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 21 / 35
...and its properties
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.
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 22 / 35
...and its properties
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.
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 22 / 35
...and its properties
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
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 23 / 35
...and its properties
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}
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 24 / 35
...and its properties
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}
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 24 / 35
...and its properties
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.
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 25 / 35
...and its properties
A coloured coalescent
1 2 3 4 5 6 7 8
Seedbank-coalescent:“colourblind”
Artificial process:sees only white
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 26 / 35
...and its properties
A coloured coalescent
1 2 3 4 5 6 7 8
Seedbank-coalescent:“colourblind”
Artificial process:sees only white
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 26 / 35
...and its properties
A coloured coalescent
1 2 3 4 5 6 7 8
Seedbank-coalescent:“colourblind”
Artificial process:sees only white
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 26 / 35
...and its properties
A coloured coalescent
1 2 3 4 5 6 7 8
Seedbank-coalescent:“colourblind”
Artificial process:sees only white
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 26 / 35
...and its properties
A coloured coalescent
1 2 3 4 5 6 7 8
Seedbank-coalescent:“colourblind”
Artificial process:sees only white
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 26 / 35
...and its properties
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
> 0,
andlim sup
n!1
E[TMRCA[n]]log log n
< 1.
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 27 / 35
...and its properties
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
> 0,
andlim sup
n!1
E[TMRCA[n]]log log n
< 1.
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 27 / 35
...and its properties
Idea
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 28 / 35
...and its properties
Idea
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 28 / 35
2
n plants
~ log n
?
...and its properties
Idea
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 28 / 35
2
n plants
~ log n
log log n~
...and its properties
Relation to other models
KKL Seed Bank ModelBolthausen-Sznitman CoalescentStructured Coalescent (Two Island Model)
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 29 / 35
...and its properties
Relation to other models
KKL Seed Bank ModelBolthausen-Sznitman CoalescentStructured Coalescent (Two Island Model)
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 29 / 35
...and its properties
Relation to other models
KKL Seed Bank ModelBolthausen-Sznitman CoalescentStructured Coalescent (Two Island Model)
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 29 / 35
...and its properties
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)
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 30 / 35
...and its properties
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)
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 30 / 35
...and its properties
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)
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 30 / 35
...and its properties
Ignore the problem
Recall,
dXt = c(Yt � Xt)dt +p
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 ,
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 31 / 35
...and its properties
Ignore the problem
Recall,
dXt = c(Yt � Xt)dt +p
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 ,
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 31 / 35
...and its properties
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
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 32 / 35
...and its properties
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
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 32 / 35
...and its properties
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
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 32 / 35
...and its properties
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).
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 33 / 35
...and its properties
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
.
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 34 / 35
...and its properties
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
.
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 34 / 35
...and its properties
Duality saves the day
limt!1
Ex ,y⇥
X nt Y m
t⇤
= limt!1
En,mh
xNt yMti
= limt!1
En,mh
xNt yMt | T ti
Pn,m (T t)
+ limt!1
En,mh
xNt yMt
�
�
�
T > ti
Pn,m�T > t�
= limt!1
⇣
xPn,m�Nt = 1�
+ yPn,m�Mt = 1�
⌘
=xK
1 + K+
y1 + K
,
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 35 / 35
...and its properties
Duality saves the day
limt!1
Ex ,y⇥
X nt Y m
t⇤
= limt!1
En,mh
xNt yMti
= limt!1
En,mh
xNt yMt | T ti
Pn,m (T t)
+ limt!1
En,mh
xNt yMt
�
�
�
T > ti
Pn,m�T > t�
= limt!1
⇣
xPn,m�Nt = 1�
+ yPn,m�Mt = 1�
⌘
=xK
1 + K+
y1 + K
,
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 35 / 35
...and its properties
Duality saves the day
limt!1
Ex ,y⇥
X nt Y m
t⇤
= limt!1
En,mh
xNt yMti
= limt!1
En,mh
xNt yMt | T ti
Pn,m (T t)
+ limt!1
En,mh
xNt yMt
�
�
�
T > ti
Pn,m�T > t�
= limt!1
⇣
xPn,m�Nt = 1�
+ yPn,m�Mt = 1�
⌘
=xK
1 + K+
y1 + K
,
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 35 / 35
...and its properties
Duality saves the day
limt!1
Ex ,y⇥
X nt Y m
t⇤
= limt!1
En,mh
xNt yMti
= limt!1
En,mh
xNt yMt | T ti
Pn,m (T t)
+ limt!1
En,mh
xNt yMt
�
�
�
T > ti
Pn,m�T > t�
= limt!1
⇣
xPn,m�Nt = 1�
+ yPn,m�Mt = 1�
⌘
=xK
1 + K+
y1 + K
,
AGC & MWB (TU Berlin) The seed-bank coalescent November 2014 35 / 35