Pulled pushed or failed the demographic impact of a gene drive can change the nature of its spatial spread Léna Kläy1 Léo Girardin2 Vincent Calvez2 and Florence Débarre1

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Pulled, pushed or failed: the demographic impact of a gene

drive can change the nature of its spatial spread

Léna Kläy∗,1, Léo Girardin2, Vincent Calvez2, and Florence Débarre1

1Institute of Ecology and Environmental Sciences Paris (IEES Paris), Sorbonne Université, CNRS, IRD,

INRAE, Université Paris Est Creteil, Université de Paris, Paris Cedex 5, France.

2Institut Camille Jordan, UMR 5208 CNRS and Universite Claude Bernard Lyon 1, France

November 1, 2023

Abstract

Understanding the temporal spread of gene drive alleles – alleles that bias their own transmission –

through modelling is essential before any ﬁeld experiments. In this paper, we present a deterministic

reaction-diﬀusion model describing the interplay between demographic and allelic dynamics, in a one-

dimensional spatial context. We focused on the traveling wave solutions, and more speciﬁcally, on the

speed of gene drive invasion (if successful). We considered various timings of gene conversion (in the zygote

or in the germline) and diﬀerent probabilities of gene conversion (instead of assuming 100%conversion

as done in a previous work). We compared the types of propagation when the intrinsic growth rate of

the population takes extreme values, either very large or very low. When it is inﬁnitely large, the wave

can be either successful or not, and, if successful, it can be either pulled or pushed, in agreement with

previous studies (extended here to the case of partial conversion). In contrast, it cannot be pushed when

the intrinsic growth rate is vanishing. In this case, analytical results are obtained through an insightful

connection with an epidemiological SI model. We conducted extensive numerical simulations to bridge

the gap between the two regimes of large and low growth rate. We conjecture that, if it is pulled in the

two extreme regimes, then the wave is always pulled, and the wave speed is independent of the growth

rate. This occurs for instance when the ﬁtness cost is small enough, or when there is stable coexistence of

the drive and the wild-type in the population after successful drive invasion. Our model helps delineate

the conditions under which demographic dynamics can aﬀect the spread of a gene drive.

∗Corresponding author: lena.klay@sorbonne-universite.fr

arXiv:2210.13935v2 [math.AP] 31 Oct 2023

Contents

1 Introduction 4

2 Methodology 6

2.1 Models............................................... 6

2.2 Settingoftheproblem...................................... 7

2.3 Glossary.............................................. 8

3 Results 10

3.1 Model with perfect conversion in the zygote . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.1.1 Preliminary statements on the model . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.1.2 r= +∞.......................................... 10

3.1.3 r= 0 ............................................ 11

3.1.4 Comparison between the outcomes when r= +∞and r= 0 ............. 12

3.2 Models with partial conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2.1 Conversion occurring in the zygote . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2.1.1 Preliminary statements on the model . . . . . . . . . . . . . . . . . . . . 13

3.2.1.2 r= +∞..................................... 14

3.2.1.3 r= 0 ...................................... 15

3.2.1.4 Numerical illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2.2 Conversion occurring in the germline . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2.2.1 Preliminary statements on the model . . . . . . . . . . . . . . . . . . . . 20

3.2.2.2 r= +∞..................................... 20

3.2.2.3 r= 0 ...................................... 21

3.2.2.4 Numerical illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2.3 Conclusion ........................................ 22

4 Discussion 22

5 Acknowledgements 23

Appendix

A Model with partial conversion: growth term details 24

A.1 Conversion occurring in the zygote . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

A.2 Conversion occurring in the germline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

B System rewritten with variables (n, pD)26

B.1 Modelwithperfectconversion.................................. 26

B.2 Modelwithpartialconversion.................................. 27

B.2.1 Conversioninthezygote................................. 27

B.2.2 Conversion in the germline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

C Proofs for model (12) with perfect conversion in the zygote 29

C.1 Numerical evidence for the continuity when r→0...................... 29

C.2 Proof of the statements in Tables 3 and 4 when perfect conversion occurs in the zygote . . 30

C.2.1 Gene drive clearance for s∈(0.5,1) when r= 0 .................... 31

C.2.2 Numerical approximation of sthreshold value for the pulled/pushed wave when

r= +∞.......................................... 31

D Critical traveling wave for an SI similar model. 33

D.1 Existence of critical traveling wave solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 33

D.2 Proofofthetheorem....................................... 33

D.2.1 Construction of sub- and super-solutions . . . . . . . . . . . . . . . . . . . . . . . . 34

D.2.2 Existence and positivity of a critical traveling wave solution . . . . . . . . . . . . . 37

E Study of the reaction term when r= +∞in section 3.2 38

E.1 Conversion occurring in the zygote . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

E.1.1 Monostable drive invasion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

E.1.2 Monostable wild-type invasion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

E.1.3 Monostable coexistence state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

E.2 Conversion occurring in the germline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

E.2.1 Monostable drive invasion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

E.2.2 Monostable wild-type invasion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

E.2.3 Monostable coexistence state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

F Heatmap supplementary materials 46

F.1 Eﬀect of ﬁtness disadvantage (s) and dominance coeﬃcient (h) on drive dynamics, for

r= +∞............................................... 46

F.2 Heatmaplines........................................... 46

F.2.1 Puredriveline ...................................... 46

F.2.2 Composite persistence line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

1 Introduction

A highly accurate, cost-eﬀective and easy-to-use technology, the CRISPR-Cas genome editing system has

been favoring the development of promising innovations [25]. Among them, CRISPR-Cas9 gene drive

[1], which aims to spread a trait of interest in a wild type population in a relatively short number of

generations [26]. Application ﬁelds are numerous, and include i) the eradication of insect-borne diseases

[10, 18, 26]; ii) the elimination of herbicide and pesticide resistance in pest populations [31]; iii) the

control of destructive invasive species [19, 21]; iv) the conservation of biodiversity by spreading beneﬁcial

traits in endangered species [17, 35].

Targeting sexually reproducing species, CRISPR-Cas9 gene drive biases the transmission of an al-

lele from a parent to its oﬀspring. This biased inheritance occurs through gene conversion (also called

“homing” [15]): in a heterozygous cell, the gene drive cassette present on one chromosome induces a

double-strand break at a speciﬁc target site on the homologous chromosome, and the repair process du-

plicates the cassette. Overall, this process increases the chances of transmitting the gene drive cassette

compared to its wild-part counterpart, and the mechanism repeats through the generations. Gene con-

version can potentially take place at diﬀerent timings of the life cycle: from very early on, in the zygote,

meaning that potentially every single cell of the individual could become homozygous for the gene drive,

to, in the germline, where only the gametes are converted.

Gene drives can be classiﬁed into two main categories depending on the purpose of their use [16, 20].

A “replacement drive” is aimed at spreading a genetic modiﬁcation in order to introduce an important and

durable feature in the natural population. Population size is then not signiﬁcantly aﬀected and the drive

construct may in principle persist indeﬁnitely in the environment. A “suppression drive” on the other

hand is meant to reduce population size by spreading a detrimental trait, such as a sex ratio distorter

[29] or by altering fertility [26], for example. The term “eradication drive” can be used for the extreme

case where population extinction is the aim.

As with any new tool, it is essential to balance risks (safety) and beneﬁts (eﬃcacy) of the technique

before running any ﬁeld trials. Experiments currently conducted in laboratories provide small- to medium-

scale information; mathematical models can help to extend these empirical results and identify the features

that are the most important in determining the dynamics at larger scales [13].

Early gene drive models [11, 15, 40] used classical population genetics frameworks, and considered

discrete non-overlapping generations in a well-mixed population. These simpliﬁcations helped to draw

general conclusions, but it is important to challenge them. First of all, most of the species targeted

in the context of gene drive do not have synchronous generations (for instance mosquitoes [18, 23, 10,

26], ﬂies [19], mice [21]). Secondly, the assumption of a single well-mixed collection of individuals living

across a uniform space is usually not realistic. In fact, most of the natural landscapes are heterogeneous.

Individuals are also more likely to interact with others that are in closer proximity, which might result in

local genetic variations. Finally, releases of transgenic individuals are limited in range, which is another

factor of spatial heterogeneity.

Taking into account spatio-temporal dynamics of the population size is another key step towards

more realistic models. For the sake of simplicity, most early models focused on allele frequencies and

considered a constant population density. However in the context of gene drive, the introduction of

maladapted transgenic individuals can lead to the reduction (or even extinction) of the population [16].

When considering a spatially structured population, variations in population density naturally generate a

demographic ﬂux from denser to less dense areas. This demographic ﬂux is directed in opposition to the

spread of the drive allele. It was previously shown [20] that the advantage conferred by gene conversion

may nevertheless counteract the demographic eﬀect linked to the ﬁtness cost.

The main goal of this paper is to clarify the impact of variations in population density over the course

of drive propagation over space.

We study partial diﬀerential equations which follow the propagation of the drive in space and time.

We explore numerically and analytically two models: a ﬁrst model based on perfect conversion in the

zygote, already introduced in [20] in a spatially structured population, corresponding to an idealized

case where gene conversion always succeeds; second, a more realistic model with partial conversion and

presence of heterozygous individuals, already studied in [35] in a well-mixed, non spatial population. In

order to investigate the possible spreading of gene drives through space after local introduction, we focus

on the description of traveling waves solutions, that is, particular solutions which are stationary in a

frame moving at constant speed. Our analysis goes beyond [20] by several means: we extend it to the

case of partial conversion, and we systematically analyze the case where the demographic eﬀects are the

strongest, in the regime of vanishing growth rate. The latter is possible through an insightful connection

with an epidemiological SI model.

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Pulled,pushedorfailed:thedemographicimpactofagenedrivecanchangethenatureofitsspatialspreadLénaKläy∗,1,LéoGirardin2,VincentCalvez2,andFlorenceDébarre11InstituteofEcologyandEnvironmentalSciencesParis(IEESParis),SorbonneUniversité,CNRS,IRD,INRAE,UniversitéParisEstCreteil,UniversitédeParis,ParisCedex5,F...

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Pulled pushed or failed the demographic impact of a gene drive can change the nature of its spatial spread Léna Kläy1 Léo Girardin2 Vincent Calvez2 and Florence Débarre1

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