Seed bank Cannings graphs How dormancy smoothes random genetic drift Adrián González Casanova1 Lizbeth Peñaloza2 and Arno Siri-Jégousse3

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Seed bank Cannings graphs: How dormancy smoothes random

genetic drift

Adrián González Casanova1, Lizbeth Peñaloza2, and Arno Siri-Jégousse3

1Unidad Cuernavaca del Instituto de Matemáticas de la Universidad Nacional Autónoma de México, Av. Universidad s/n

Periferica, 62210 Cuernavaca, Morelos, México. Email: adrian.gonzalez@im.unam.mx

2Instituto de Investigación de Matemáticas y Actuaría, Universidad del Mar, campus Huatulco. Carretera Federal No 200,

Km 250, Santa María Huatulco 70989, Oaxaca, México. Email: lizbeth@huatulco.umar.mx

3Instituto de Investigaciones en Matemáticas Aplicadas y Sistemas, Universidad Nacional Autónoma de México, Circuito

Escolar 3000, C.U., 04510 Coyoacán, CDMX, México. Email: arno@sigma.iimas.unam.mx

May 23, 2023

Abstract

In this article, we introduce a random (directed) graph model for the simultaneous forwards and

backwards description of a rather broad class of Cannings models with a seed bank mechanism. This

provides a simple tool to establish a sampling duality in the ﬁnite population size, and obtain a path-

wise embedding of the forward frequency process and the backward ancestral process. Further, it

allows the derivation of limit theorems that generalize celebrated results by Möhle to models with seed

banks, and where it can be seen how the eﬀect of seed banks aﬀects the genealogies. The explicit

graphical construction is a new tool to understand the subtle interplay of seed banks, reproduction

and genetic drift in population genetics.

Keywords: Seed bank, Moment duality, Weak convergence, Mixing time.

2020 Mathematics Subject Classiﬁcation: 92D10, 60F05, 60G10, 60J05, 60J90, 92D25.

1 Introduction

Cannings models and their modiﬁcations, along with their multiple merger genealogies are a major topic in math-

ematical population genetics [29, 28, 31, 1, 14, 16, 32]. Also, in the last decade, the study of dormancy (also called

seed bank eﬀect) received signiﬁcant attention [22, 4, 5, 6]. One of the unifying themes in both modeling areas is

that they arise from extensions of the Wright-Fisher model, and that classical evolutionary forces such as genetic

drift and selection are aﬀected in important ways. While the theory of Cannings models is now robust, the study

of models with dormancy is still work in progress. The main goal of this paper is to stabilize a framework in which

seed banks can be combined with Cannings models, and to generalize the known limiting results for models without

dormancy to Cannings models with seed bank.

An important tool in population genetics is the moment duality for Markov processes. This technique establishes

a mathematical relation between forward and backward in time processes. The celebrated duality between the

Wright-Fisher diﬀusion and the Kingman coalescent was gradually generalized to a wide class of neutral population

genetics models, including some ﬁnite size discrete populations such as Cannings-type models [17]. In this situation,

the duality leads to asymptotic results for both forward frequency and genealogical processes. In the context of

dormancy, duality was established for the seed bank diﬀusion, which arises as a limit in models with geometric

seed bank [5]. However, for discrete seed bank models, the duality relation is the ﬁrst open gap that this article

aims to close.

There are two main models for dormancy phenomena.

arXiv:2210.05819v2 [math.PR] 19 May 2023

•Kaj et al. The model deﬁned in [22] is based on the Wright-Fisher model with additional multi-generational

jumps of (bounded) size, the system has been extended to geometric jump sizes of bounded expected range

in [25] (which also provide some insight into the forward in time frequency diﬀusion), to the general ﬁnite

expectation case in [4], and even to unbounded (heavy-tailed) jump sizes in [3].

•Blath et al. A second modeling frame is given by an external seed bank in terms of a “second island” (in

the spirit of Wright’s island model), eﬀectively leading to geometric jump sizes on the evolutionary scale.

Here, forward and backward limits have been constructed, giving rise to the seed bank diﬀusion and the

seed bank coalescent [5] (see more analysis and generalization in [15, 6] and an interesting connection with

metapopulations in [26]).

Both modeling frames (generational jumps and second island) have their advantages and disadvantages. For

the Wright-Fisher model with multi-generational jumps, one typically loses the Markov property. For the island

version, one retains the Markov property but then needs to investigate two-dimensional frequency processes, which

in the limit are harder to analyze than one-dimensional diﬀusions, since e.g. the Feller theory is missing (this can

in part be replaced by recent theory for polynomial diﬀusions [2]). Interestingly, it turns out that for the limiting

frequency processes, both approaches can be two sides of the same medal.

In none of the above approaches, more general reproductive mechanisms, such as based on Cannings models,

have been analyzed. This paper’s second aim is to close this gap. We present an extended framework for the

simultaneous construction of seed bank models with general multi-generational jump distributions and Cannings-

type reproductive laws satisfying a paintbox construction. We are also able to obtain forward and backward

convergence results (extending [22], [25] and [4]) and to provide an explicit sampling duality, which is valid already

in the ﬁnite individual models.

More precisely, we show that if a sequence of Cannings models (with no seed bank eﬀect) is in the universality

class of the Kingman coalescent, meaning that its ancestral process converges in the evolutionary scale to the

Kingman coalescent, then the ancestry of the same sequence with a seed bank eﬀect will converge to the Kingman

coalescent delayed by a constant β2, where β < ∞is the expected number of generations that separates an

individual from its ancestor. This extends the results of [22] and [4]. Convergence of the frequency process to the

solution of the Wight-Fisher diﬀusion with the same delay is also proved. We go further and study how sequences

of seed bank models with divergent expectations can make sequences of Cannings models that originally were

not in the Kingman class, converge to the Kingman coalescent. This is achieved using the mixing time of some

auxiliary Markov chains introduced in [22]. If instead of considering Cannings processes in the Kingman class we

consider that their genealogy converges to a Ξ-coalescent, we show that their seed bank modiﬁcation converges to

aΞβ-coalescent. Heuristically, the transformation Ξ→Ξβconsists in dividing by βall the non-dust boxes in a Ξ

paintbox event to obtain a Ξβpaintbox event. Similar asymptotics are shown for the forward process. All those

results are extended for models in the presence of mutations.

Note that the interplay of general reproduction and seed banks with other evolutionary forces can be subtle,

and we provide a framework for its analysis (also regarding the real-time embedding of coalescent-based estimates,

see e.g. [6]).

The paper is organized as follows. In section 2 we construct a random graph that allows us to embed the

ancestry and the frequency processes of both Cannings and dormancy models simultaneously and study the duality

relation of the processes forward and backward in time. Furthermore, we analyze the scaling limits of the ancestral

process in presence of skewed reproduction mechanisms and dormancy. We give conditions for convergence to

the Kingman coalescent and study scenarios beyond this universality class, where we can describe how seed bank

phenomena reduce the typical size coalescence events when combining seed banks with Cannings models that would,

in absence of the seed bank component, converge to a Λ- or a Ξ- coalescent. Section 3 uses the moment duality

to formally prove convergence of the frequency process to a Wright-Fisher diﬀusion. This intuitively clear result

was missing in the literature, probably since the lack of Markov property for the frequency process makes usual

techniques fail. In section 4 we study a variant of the seed bank random graph where mutations are added and we

extend the results obtained in sections 2 and 3.

2 A random graph version of the model of Kaj, Krone and Lascoux

Consider a discrete-time haploid population of constant size N≥1at each generation. The vertex set VN=Z×[N]

represents the whole population. For each individual v∈VN, denote by g(v)its generation and by `(v)its label

so that v= (g(v), `(v)). We denote the g-th generation of the population by VN

g:= {v∈VN:g(v) = g}. Set

a probability measure WNon the exchangeable probability measures on [N]. Let {¯

g}g∈Zbe a sequence of

independent WN-distributed random variables with ¯

g={WN

v}v∈VN

g. Each variable WN

vgives the reproductive

weight of the individual vin the population graph. This multinomial setting can be extended to some more general

Cannings models (as in [28]) or non-exchangeable reproductive success (as in [32]). Also, consider a sequence

{mN}N≥1of integers and set a probability measure µNon [mN]. Let {JN

v}v∈VNbe a collection of independent

µN-distributed random variables. The variable JN

vsays how many generations ago an individual v’s mother is

living. Finally, set a collection of random variables in [N],{UN

v}v∈VNsuch that UN

vis the label of the mother of

v. Its conditional distribution is

P(UN

v=k|JN

v=j, {¯

g}g∈Z) = WN

(g(v)−j,k).

Deﬁnition 1. (The seed bank random di-graph) Consider the random set of directed edges

EN={(v, (g(v)−JN

v, UN

v)),for all v∈VN}.

The seed bank random di-graph with parameters N,WNand µNis given by GN:= (VN, EN).

Two classical examples are

•the Kaj, Krone and Lascoux (KKL) seed bank graph [22], in this case µNhas ﬁnite support [m], i.e. mN=m,

and WN=δ(1/N,...,1/N).

•the Cannings model with parameter WN[8, 9, 28], in this case µN=δ1.

For every u, v ∈VNwe denote by δ(u, v)the distance of uand vin the graph GN, i.e. the number of vertices

in a path from uto vor from vto u. Now let us deﬁne the ancestral process associated with this graph.

Deﬁnition 2 (The ancestral process).Fix a generation g0and Sg0consisting in a sample of individuals living

between generation g0and g0−mN+ 1, i.e. Sg0⊂ ∪mN

i=1 VN

g0+1−i. For every g≥0, let AN

gbe the set composed by

the most recent ancestors of the individuals of Sg0that live at a generation g0−g0for some g0≥g, that is

g={v∈ ∪∞

g0=gVN

g0−g0:∃u∈Sg0such that δ(u, v)≤δ(u, v0)for all v0∈ ∪∞

g0=gVN

g0−g0}.

Deﬁne, for all i∈[mN],

AN,i

g=|AN

g∩VN

g0−g+1−i|

and ¯

g= (AN,1

g, . . . , AN,mN

g). We call {¯

g}g≥0the ancestral process. In the sequel, we consider the initial

conﬁguration Sg0(¯n), for ¯n= (n1, . . . , nmN), such that ni≥0individuals are uniformly sampled (with repetition)

from generation g0+ 1 −i. We denote the law of the ancestral process of this sample by P¯n. See Figure 1 for an

illustration.

For simplicity, we suppose that sup{i≥1 : ni>0}does not depend on N. This model was introduced, for

reproductions as in the Wright-Fisher model, by Kaj et al. [22] directly, in the sense that they construct a random

graph only implicitly. Our construction permits to provide a transparent relation between the ancestral process

and the forward frequency process deﬁned in section 3. Observe that {¯

g}g≥0is a Markov chain. We start our

results by formalizing the remark on p. 290 in [22]. This illustrative result is established when the Cannings model

is in the domain of attraction of the Kingman coalescent, although it can be easily generalized to any type of

reproduction law. Here we use the classical notations cN(resp. dN) that denote the probability that two (resp.

three) given individuals choose the same parent in a Cannings model. Those notations will be helpful all along the

paper. Recall, e.g. from [28], that the genealogies of a Cannings model fall into the domain of attraction of the

Kingman coalescent when cN→0and dN=o(cN)while multiple merger coalescents arise when dNand cNare of

the same order.

Proposition 1 (Reformulation of Theorem 1 in [22]).Suppose that cN=NE[(WN

v)2]→0and dN=NE[(WN

v)3] =

o(cN). Let M(n)be a multinomial random variable with parameters nand {µN(i)}mN

i=1 . Also, for any ¯n=

(n1, . . . , nmN)∈[N]mN,let Z(¯n) = (n2, . . . , nmN,0) + M(n1). Then, the transitions of {¯

g}g≥0can be writ-

ten in terms of Mand Zas follows.

v5v1

0-1-2-3-4-5

Figure 1: In this case N= 8 and mN= 2. The gray circles represent the members of S0=

{v1, v2, v3, v4, v5}where, for example, v2= (0,4) and v5= (−1,3). The light gray circles represent

the ancestors of the sample. ¯

0= (4,1),¯

1= (2,2),¯

2= (3,0),¯

3= (1,1),¯

4= (1,0),¯

5= (1,0).

•P¯n(¯

1=Z(¯n)) = 1 −P∞

i=1 cN[n1

2µN(i)2+µN(i)n1ni+1] + o(cN)

•P¯n(¯

1=Z(¯n)−ei) = cN[n1

2µN(i)2+µN(i)n1ni+1] + o(cN)

where eiis the vector with the i-th coordinate equal to 1and the others equal to 0, for all i≥1.

Proof. We need to make two observations. First note that all the randomness in the transitions of the chain

{¯

g}g≥0lies in what happens to the ﬁrst coordinate. If for some g≥0,¯

g= (0, n2, . . . , nmN)it is easy to see

that ¯

g+1 = (n2, . . . , nmN,0) almost surely. On the other hand, if n1>0, the individuals that are in AN

g∩VN

g0−g

cannot belong to AN

g+1. Then, each of these individuals, if denoted by v, must be replaced by an individual which

lives JN

vgenerations in the past, that is

Pe1(¯

1=ei) = µN(i).

Further, if n1>1, one needs to ﬁnd n1new ancestors, but some of them could be the same due to some coalescence.

The complete picture is as follows. For i≥2and j, k ≥0, and by denoting e0for the null vector,

P2e1+ei(¯

1=ei−1+ej+ek) =











2µN(j)µN(k)if i−16=j6=k

(µN(j))2(1 −cN)if i−16=j, j =k

2µN(i−1)µN(k)(1 −cN)if i−1 = j, j 6=k

(2µN(i−1)µN(j)+(µN(j))2)cNif i−16=j, k = 0

(µN(i−1))2(cN−dN)if i−1 = j, k = 0

(µN(i−1))2dNif j=k= 0

.(2.1)

The proof follows easily after these observations.

We now construct a less natural backward process which will be very useful when establishing its moment

duality with the forward process in section 3. We start by deﬁning it in a graphical and intuitive way. More formal

deﬁnitions will follow all along the section.

Deﬁnition 3 (The window process).Fix a generation g0, and Sg0⊂ ∪mN

i=1 VN

g0+1−i. In the genealogical tree of

the sample Sg0, deﬁne the variable BN,1

gas the number of edges arriving to generation g0−g(plus the number

of individuals of Sg0living at this generation). For any i∈ {2, mN}, let BN,i

gbe the number of edges crossing

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SeedbankCanningsgraphs:HowdormancysmoothesrandomgeneticdriftAdriánGonzálezCasanova1,LizbethPeñaloza2,andArnoSiri-Jégousse31UnidadCuernavacadelInstitutodeMatemáticasdelaUniversidadNacionalAutónomadeMéxico,Av.Universidads/nPeriferica,62210Cuernavaca,Morelos,México.Email:adrian.gonzalez@im.unam.mx2Inst...

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Seed bank Cannings graphs How dormancy smoothes random genetic drift Adrián González Casanova1 Lizbeth Peñaloza2 and Arno Siri-Jégousse3

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