Separation of timescales for the seed bank diffusion and its jump-diffusion limit [original]

Journal of Mathematical Biology (2021) 82:53

https://doi.org/10.1007/s00285-021-01596-0

Mathematical Biology

Separation of timescales for the seed bank diffusion and its

jump-diffusion limit

Jochen Blath1·Eugenio Buzzoni1·Adrián González Casanova2·

Maite Wilke Berenguer3

Received: 6 July 2018 / Revised: 1 October 2020 / Accepted: 27 October 2020 / Published online: 28 April 2021

Abstract

We investigate scaling limits of the seed bank model when migration (to and from

the seed bank) is ‘slow’ compared to reproduction. This is motivated by models for

bacterial dormancy, where periods of dormancy can be orders of magnitude larger

than reproductive times. Speeding up time, we encounter a separation of timescales

phenomenon which leads to mathematically interesting observations, in particular

providing a prototypical example where the scaling limit of a continuous diffusion

will be a jump diffusion. For this situation, standard convergence results typically fail.

While such a situation could in principle be attacked by the sophisticated analytical

scheme of Kurtz (J Funct Anal 12:55–67, 1973), this will require significant technical

efforts. Instead, in our situation, we are able to identify and explicitly characterise a

well-definedlimitviadualityinasurprisinglynon-technicalway. Indeed, we show that

momentdualityisinasuitablesensestableunderpassagetothelimitandallowsadirect

and intuitive identification of the limiting semi-group while at the same time providing

a probabilistic interpretation of the model. We also obtain a general convergence

strategy for continuous-time Markov chains in a separation of timescales regime,

which is of independent interest.

Keywords Strong seed bank ·Two-island model ·Separation of timescales ·

Diffusion limits ·Jump-diffusion ·Duality

Mathematics Subject Classification Primary 60K35; Secondary 92D10

BMaite Wilke Berenguer

maite.wilkeberenguer@ruhr-uni-bochum.de

1Institut für Mathematik, Technische Universität Berlin, Berlin, Germany

2Instituto de Matemáticas, Universidad Nacional Autónoma de México, Mexico City, Mexico

3Fakultät für Mathematik, Ruhr-Universität Bochum, Bochum, Germany

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53 Page 2 of 34 J. Blath et al.

1 Motivation and main results

In this extended introductory section, we first provide some background on the bio-

logical concept of dormancy and its relevance in particular in microbial communities.

This is followed by a short review of modelling approaches for dormancy in population

genetics, where we think that dormancy might be seen as an additional evolutionary

force, interacting with other forces such as genetic drift in complex ways. Since dor-

mancy periods vary over several orders of magnitude (depending on the underlying

species and environmental conditions), we aim for a systematic classification of rel-

evant timescales, leading to the distinction of three separate scaling regimes. While

the first two regimes have been modelled and analysed in population genetics before,

the last one, leading to a separation of timescales between genetic drift and dormancy

periods, is new, and completes the picture (at least on the level of ‘toy models’) of

modelling scenarios. Our results for this regime will be presented in this introduction

both for the forward-in time population model as well as for the dual genealogical

processes, leading to novel scaling limits, which are interesting also from a purely

mathematical perspective.

The proofs of these results can be found in Sects. 2and 3for the results going

backwards and forwards in time, respectively. We believe that our rather direct method

of proof to obtain and characterise these limits, making extensive use of duality for

Markov processes, can be applied in a variety of situations, so that in each section, we

first present the corresponding methodology in a general set-up and then discuss its

application to our concrete motivation.

Background on dormancy Dormancy is a complex trait that has developed indepen-

dently in many species across the tree of life and comes in many different guises.

Originally, theory for dormancy and the resulting seed banks has be developed in the

context of bet-hedging strategies for plants Cohen (1966). However, dormancy is also

a highly common trait in microbial communities, with important consequences for

their evolutionary, ecological and pathogenic properties.

Here, we define dormancy as the ability of (micro-) organisms to enter and leave

a state of vanishing metabolic activity. It has been observed for many habitats that at

any given time a large fraction of micro-organisms can be in such a dormant state. For

example, more than 80% of bacteria in soil are reported to be metabolically inactive,

forming large ‘seed banks’ comprised of dormant individuals, see Lennon and Jone

(2011). While dormancy seems to be an efficient and wide-spread strategy, e.g. to

withstand unfavourable environmental conditions, competitive pressure, or antibiotic

treatment, it is at the same time a costly trait whose maintenance involves energy and

a sophisticated ‘switching machinery’.

Dormancy also plays a role in various (human) diseases. So-called persister cells,

that may evade antibiotic treatment by remaining in a state of low activity, play a

major role in chronic infections, cf. Fisher et al. (2017), and individual cell dormancy

is linked to relapses in cancer, cf. Marx (2018), Endo and Inoue (2019).

In this paper, we will focus on microbial seed banks. Lennon and Jone (2011)

and Shoemaker and Lennon (2018) provide a broad overview of this rich and fas-

cinating field and serve as a motivation in the present paper. Given the relevance of

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Separation of timescales for the seed bank diffusion… Page 3 of 34 53

biological systems exhibiting dormancy, investigating the mathematical implications

of dormancy in large populations seems to be a timely and interesting task.

Classification of the duration of dormancy: Known models and motivation for this

paper As indicated above, dormancy comes in many different forms, specific to the

involved species and environments. One variation lies in the duration of dormancy

periods: While in some microbial species dormancy periods last at most a few days,

others stay dormant for prolonged periods of time, and some, e.g. bacterial endospores,

have been reported to successfully resuscitate from dormancy after millions of years

(Shoemaker and Lennon 2018; Cano and Borucki 1995; Johnson et al. 2007; Morono

et al. 2020). The theoretical derivation and analysis of mathematical models may

help to identify, understand and classify the different effects of dormancy, on suitable

timescales, on the population dynamics and genealogical processes of the underlying

populations.

Hence, in this paper, we consider the consequences of dormancy and seed banks

in the framework of population genetics. More precisely, we are interested in the

interplay of dormancy and the classical evolutionary force of random genetic drift,in

particular with respect to its sensitivity to the duration of dormancy periods.

In a bi-allelic, haploid population that reproduces according to the Wright-Fisher

model, the frequency of a given allele converges to the Wright-Fisher diffusion,given

as the solution to

dZ(t)=Z(t)(1−Z(t))dB(t),

where (B(t))t≥0is a standard Brownian motion, if one measures time in the coalescent

timescale (alsoknown as the evolutionary timescale),i.e. on the order of the population

size as this tends to infinity. This diffusion is dual to the block-counting process of the

Kingman coalescent which in turn describes the genealogy of the population. These

objects serve as a reference for populations without dormancy and are widely studied

and applied in biology and mathematics alike. See e.g. Wakeley (2009) or Etheridge

(2011) for an overview. We will consider suitable extensions incorporating dormancy.

We propose to distinguish three regimes comparing the duration of dormancy peri-

ods to the coalescent timescale, i.e. the scale at which the random genetic drift acts.

1. Dormancy periods are small compared to the coalescent timescale.

In 2001, Kaj et al. (2001) introduced a model for dormancy in the following fashion:

insteadof alwayschoosing theancestorinthe preceding generations likein theWright-

Fisher model, individuals are allowed to choose an ancestor several generations in the

past. Their lineages thus ‘jump’ this number of generations and can be interpreted as

dormant during that time. If we denote by B≥1 the expected size of the ‘jump’, the

genealogy of the model converges on the coalescent timescale to a delayed Kingman

coalescent, depicted in Fig. 1b, where coalescences occur at rate β2, where β:= 1/B,

instead of at rate 1, cf. Kaj et al. (2001), Blath et al. (2013). This in turn is dual to the

delayed Wright-Fisher diffusion

d˜

Z(t)=β2˜

Z(t)(1−˜

Z(t))dB(t), (1)

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53 Page 4 of 34 J. Blath et al.

(a) (b) (c)

Fig. 1 Typical realisations of athe Kingman coalescent, where lineages merge at rate 1 per pair, ba delayed

Kingman coalescent, where lineages merge at rate β2<1 per pair, and cthe seed bank coalescent, see Def.

1.2. In the seed bank coalescent there are two kinds of lines: blue lines are active lineages, while purple

lines are dormant lineages. The differences can be seen in the (asymptotic) expected time to the most recent

ancestor when started with a sample of n(active and mdormant) individuals given on the time-axis (colour

figure online)

that again describes the frequency of a given allele in the population, cf. Fig. 2a. Note

that βdoes not depend on the population size, whence its qualitatively weak impact

on the coalescent timescale.

2. Dormancy periods on the order of the coalescent timescale

For microbial species, however, dormancy times can be much longer than just a

few ‘generations’, In this set-up, Lennon and Jone (2011) proposed a model based on

two reservoirs, the ‘active’ and the ‘dormant’ population, between which individuals

‘migrate/switch’ via initiation of and resuscitation from dormancy, at fixed rates. A

mathematical model for ‘spontaneous/stochastic’ switching (observed in nature under

stable environmental conditions, cf. Epstein 2009; Shoemaker and Lennon 2018), was

introduced and studied in Blath et al. (2016). This is reminiscent of the ‘two-island

model’ (Wright 1931; Moran 1959) with the notable difference of the absence of

reproduction on the second island.

If the size of the active and dormant population are proportional with the ratio

given by some K>0, the frequencies X(t)and Y(t)of a given allele in the active and

dormant population, respectively, when time is measured on the coalescent timescale,

are described by the seed bank diffusion, cf. Fig. 2b. This diffusion was first introduced

in Corollary 2.5 in Blath et al. (2016). The existence of a unique strong solution that

is Feller follows from Theorem 3.2 and Remark 3.2 in Shiga and Shimizu (1980), see

also Greven et al. (2020) for a more general seed bank diffusion.

Definition 1.1 (Seed bank diffusion)Let(B(t))t≥0be a standard Brownian motion and

c,Kfinite positive constants. The [0,1]2-valued continuous strong Markov process

(X(t), Y(t))t≥0given as the unique strong solution of the initial value problem

dX(t)=c(Y(t)−X(t))dt+√X(t)(1−X(t))dB(t),

dY(t)=Kc(X(t)−Y(t))dt,

(2)

with (X(0), Y(0)) =(x,y)∈[0,1]2, is called seed bank diffusion with parameters

c,K, starting at (x,y)∈[0,1]2.

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Separation of timescales for the seed bank diffusion… Page 5 of 34 53

(a)

(b)

(c)

Fig. 2 Typical realisations of the trajectory of aa time-changed Wright-Fisher diffusion, where the time-

change is an effect of a weak seed bank, bthe seed bank diffusion, with the frequency of a given allele in

the active population displayed in blue and in the dormant population, in purple, cthe frequency process

(˜

X(t), ˜

Y(t)), using the same colour code (colour figure online)

The genealogy of such a population is given by the seed bank coalescent, introduced

in Definition 3.2 in Blath et al. (2016). Here, lineages can switch between an active and

a dormant state independently (hence ‘spontaneous’ switching) at a given rate c>0.

While the active lineages behave like the Kingman coalescent, dormant lineages are

prohibited from coalescing, as depicted in Fig. 1c.

That dormancy appears in such a prominent form in the coalescent and in the

diffusion and therefore is visible on the coalescent timescale is due to the underlying

scaling assumptions of the model. These imply that dormancy times are of the order of

the population size and therefore on the coalescent timescale. Here, many population

genetic quantities and statistics are affected in non-trivial ways, see Blath et al. (2015),

Blath et al. (2016) and Blath et al. (2020b) for a discussion of the scaling assumptions

and further extensions of the model. Since the seed bank here has a major qualitative

effect on both the diffusion and the coalescent, this is sometimes referred to as the

strong seed bank model.

As in the previous models, an important mathematical tool in our analysis will

be the formal duality relation between the seed bank diffusion (X(t), Y(t))t≥0and

the block-counting process of the seed bank coalescent (N(t), M(t))t≥0. Note that

the notion of a ‘block’ comes from the mathematical definition of a coalescent as a

partition-valued process. In thebiologicalcontext,the process could aswellbe denoted

the line-counting process, keeping track of the number of ancestral lines presents at

each time in the past.

Definition 1.2 (Block-counting process of the seed bank coalescent)LetE:= N0×

N0.Letc,K>0. We define (N(t), M(t))t≥0to be the continuous-time Markov chain

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