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Spatial gene drives and pushed genetic wavesHidenori Tanakaa,b,1, Howard A. Stonec, and David R. Nelsona,d,e,1

aSchool of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138; bKavli Institute for Bionano Science and Technology, HarvardUniversity, Cambridge, MA 02138; cDepartment of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544; dDepartment ofPhysics, Harvard University, Cambridge, MA 02138; and eDepartment of Molecular and Cellular Biology, Harvard University, Cambridge, MA 02138

Edited by Andrea J. Liu, University of Pennsylvania, Philadelphia, PA, and approved June 20, 2017 (received for review April 9, 2017)

Gene drives have the potential to rapidly replace a harmful wild-type allele with a gene drive allele engineered to have desiredfunctionalities. However, an accidental or premature release ofa gene drive construct to the natural environment could dam-age an ecosystem irreversibly. Thus, it is important to understandthe spatiotemporal consequences of the super-Mendelian pop-ulation genetics before potential applications. Here, we use areaction–diffusion model for sexually reproducing diploid organ-isms to study how a locally introduced gene drive allele spreadsto replace the wild-type allele, although it possesses a selectivedisadvantage s > 0. Using methods developed by Barton and col-laborators, we show that socially responsible gene drives require0.5 < s < 0.697, a rather narrow range. In this “pushed wave”regime, the spatial spreading of gene drives will be initiated onlywhen the initial frequency distribution is above a threshold pro-file called “critical propagule,” which acts as a safeguard againstaccidental release. We also study how the spatial spread of thepushed wave can be stopped by making gene drives uniquely vul-nerable (“sensitizing drive”) in a way that is harmless for a wild-type allele. Finally, we show that appropriately sensitized drivesin two dimensions can be stopped, even by imperfect barriers per-forated by a series of gaps.

gene drive | Fisher wave | bistable wave

The development of the CRISPR-Cas9 system (1–4), derivedfrom an adaptive immune system in prokaryotes (5), has

received much recent attention, in part because of its excep-tional versatility as a gene editor in sexually reproducingorganisms compared with similar exploitations of homologousrecombination, such as zinc-finger nucleases and the TALENSsystem (4, 6). Part of the appeal is the potential for introducing anovel gene into a population, allowing control of highly pesticide-resistant crop pests and disease vectors, such as mosquitoes(7–10). Although the genetic modifications typically introduce afitness cost or a “selective disadvantage,” the enhanced inher-itance rate embodied in CRISPR-Cas9 gene drives neverthe-less allows edited genes to spread, even when the fitness costof the inserted gene is large. The idea of using constructs thatbias gene transmission rates to rapidly introduce novel genes intoecosystems has been discussed for many decades (11–16). Simi-lar “homing endonuclease genes” (in the case of CRISPR-Cas9,the homing ability is provided by a guide RNA) were consid-ered earlier by ecologists in the context of control of malaria inAfrica (17, 18).

As a hypothetical example of a gene drive applied to apathogen vector requiring both a vertebrate and an insect host,consider plasmodium, carried by mosquitoes and injected withits saliva into humans (Fig. 1). Female mosquitoes typicallyhatch from eggs in small standing pools of water and after mat-ing, search for a human to feed on. They then lay their eggsand repeat the process, thus spreading the infection over a fewgonotrophic cycles. A gene drive could alter the function of aprotein manufactured in the salivary gland of female mosquitoesfrom, say, type a , anesthetizing nerve cells when it bites humans,to instead type A, clogging up essential chemoreceptors in plas-modium and thus killing these eukaryotes. In the absence of agene drive, there would be a selective disadvantage or fitness cost

s to losing this protein. Even if the fitness cost s was zero, it isunlikely that this new trait would be able to escape genetic drift inlarge populations. However, as we describe below, the trait couldspread easily if linked to a gene drive that converts heterozygotesto homozygotes with efficiency c close to one (Fig. 1A). Remark-ably, high conversion rates have already been achieved with themutagenic chain reaction (MCR) realized by the CRISPR-Cas9system (1–3) for yeast (cyeast > 0.995) (19), fruit flies (cflies = 0.97)(20), and the malaria vector mosquito, Anopheles stephensi, withengineered malaria resistance (cmosquito≥ 0.98) (21).

However, the gene drives’ intrinsic nature of irreversibly alter-ing wild-type (WT) populations raises biosafety concerns (9) andcalls for confinement strategies to prevent unintentional escapeand spread of the gene drive constructs (22). Although variousgenetic design or containment strategies have been discussed (9,20, 23, 24) and a few computational simulations were conducted(17, 18, 25), the spatial spreading of the gene drive alleles hasreceived less attention.

To understand such phenomena in a spatial context, we willexploit a methodology developed by Barton and coworkers (26–28) originally in an effort to understand adaptation and specia-tion of diploid sexually reproducing organisms in genetic hybridzones. We apply these techniques to a spatial generalization ofa model of diploid CRISPR-Cas9 population genetics proposedby Unckless et al. (29) and highlight two distinct ways in whichgene drive alleles can spread spatially. The non-Mendelian [or“super-Mendelian” (30)] population genetics of gene drives areremarkable, because individuals homozygous for a gene drivecan, in fact, spread into WT populations, even if they carry a pos-itive selective disadvantage s (Fig. 1B). For small selective disad-vantages (0< s < 0.5 in our case), the spatial spreading proceedsvia a well-known Fisher–Kolmogorov–Petrovsky–Piskunov wave(31, 32). Such pulled genetic waves (33–35) are driven by growth

Significance

Gene constructs introduced into natural environments havebeen proposed to solve various ecological problems. TheCRISPR-Cas9 technology greatly facilitates construction ofgene drives that allow desired traits to rapidly replace wildtypes, even if these convey a selective growth rate disad-vantage s > 0. However, accidental release of a gene drivecould damage ecosystems irreversibly. We have modeled thespatial spread of gene drives and find a preferred range ofselective disadvantages, 0.5 < s < 0.697. In this regime, genedrives spread but only when a nucleus exceeds a critical sizeand intensity. By making gene drives uniquely susceptible toa compound, their advance can be stopped in two dimensionsby finite-width barriers, even when interrupted by gaps.

Author contributions: H.T., H.A.S., and D.R.N. designed research, performed research,contributed new reagents/analytic tools, analyzed data, and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.1To whom correspondence may be addressed. Email: tanaka@g.harvard.edu ordrnelson@fas.harvard.edu.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1705868114/-/DCSupplemental.

www.pnas.org/cgi/doi/10.1073/pnas.1705868114 PNAS Early Edition | 1 of 6

Wild typeGene driveEgg

MCR

A B

Fig. 1. Schematics of the gene drive machinery with a perfect conver-sion efficiency c = 1. (A) Every time an individual homozygous for the driveconstruct and a WT mate, heterozygotes in the embryo are converted tohomozygotes by the MCR. (B) Gene drives enhance their inheritance ratebeyond that of the conventional Mendelian population genetics and canspread even with a selective disadvantage.

and diffusive dispersal at the leading edge, and they are difficultto slow down and stop.

However, for somewhat larger selective disadvantages (0.5<s < 0.697), we find that propagation proceeds instead via apushed genetic wave (33–35), where the genetic wave advancesvia accentuated growth from populations somewhat behind thefront that spill over the leading edge. These waves, characterizedby a strong Allee effect (36, 37), are more socially responsiblethan the pulled Fisher waves, because (i) only inoculations withspatial size and density that exceed a critical nucleus or “criti-cal propagule” (28) are able to spread spatially, thus providingprotection against a premature or accidental release of a genedrive; (ii) the gene drive pushed waves can be stopped by makingthem uniquely vulnerable to a specific compound [“sensitizingdrive” (9)], which is harmless for a WT allele; and (iii) appro-priately sensitized gene drives can be stopped even by barrierspunctuated by defects, analogous to regularly spaced fire breaksused to contain forest fires. Similar pushed or “excitable” wavesalso arise, for example, in neuroscience in simplified versions ofthe Hodgkin–Huxley model of action potentials (38). When theselective disadvantage associated with the gene drive is too large(s > 0.697 in our model), the excitable wave reverses direction,and the region occupied by the gene drive homozygotes collapsesto zero.

The same mathematical analyses apply to spatial evolution-ary games of two competing species in one dimension, whichare governed by a class of reaction–diffusion equations thatresemble the gene drive system. The fitness levels of the twointeracting red and green species (wR, wG) are related totheir frequencies [f (x , t), 1 − f (x , t)] by wR(x , t) = g + α[1 −f (x , t)],wG(x , t) = g +βf (x , t), where g is a background fitness,assumed identical for the two alleles for simplicity. The mutual-istic regime α> 0, β > 0 in the first quadrant of Fig. 2 has beenstudied already (40), including the effect of genetic drift, withtwo lines of directed “percolation” transitions out of a mutual-istic phase. Here, we apply the methods from ref. 28 to studythe evolutionary dynamics near the line of first-order transitionsthat characterize the competitive exclusion regime in the thirdquadrant of Fig. 2. Because the mathematics parallels the anal-ysis inspired by gene drive systems in the text, we relegate dis-cussion of this topic to SI Appendix, which also has discussion ofconversion efficiencies c < 1, an analogy with nucleation theory,and other matters.

Mathematical Model of the CRISPR Gene DrivesWe start with a Hardy–Weinberg model (42) and incorporatean MCR with 100% conversion rate to construct a model for awell-mixed system. This model is the limiting case of “c = 1” inthe work by Unckless et al. (29). Conversion efficiencies c< 1can be handled by similar techniques. We consider a well-mixed

diploid system with a WT allele a and a gene drive allele A withfrequencies p = p(t) and q = q(t), respectively, at time t , withp(t) + q(t) = 1. Within a random mating model, the allele fre-quencies after one generation time τg are given by

(pa + qA)2 = p2(a, a) + 2pq(a,A) + q2(A,A), [1]

and the ratios of fertilized eggs with diploid types (a, a), (a,A),and (A,A) are p2 : 2pq : q2. In a heterozygous (a,A) egg,the CRISPR-Cas9 machinery encoded on a gene drive alleleA converts the WT allele a into a gene drive allele A. Here,we assume a perfect conversion rate (a,A)

c=1−−−→MCR

(A,A) in the

embryo, which has been approximated already for yeast (19) andfruit flies (20). Genetic engineering will typically reduce the fit-ness of individuals carrying the gene drive alleles compared withWT organisms, which have already gone through natural evolu-tion and may be near a fitness maximum.

The selective disadvantage of a gene drive allele s is defined bythe ratio of the fitness wwild of WT organisms (a, a) to the fitnesswdrive of (A,A) individuals carrying the gene drive:

wdrive

wwild≡ 1− s, 0 ≤ s. [2]

[In the limit c → 1, no heterozygous (a,A) individuals are born(29).] Taking the fitness into account, the allele frequencies afterone generation time τg are

p′ : q ′ = wwildp2 : wwild(1− s)(q2 + 2pq), [3]

where p′ ≡ p(t + τg) and q ′ ≡ q(t + τg). On approximatingq ′ − q = q(t + τg) − q(t) by τgdq/dt , we obtain a differentialequation

Mutualis

m

DP

DP

Pushed

Waves

Limit of metastability

Lim

it of

met

asta

bilit

y

Pushed

Waves

Pulled Waves

Pulled Waves

1st PT

Fig. 2. Schematic phase diagram of the spatial evolutionary games in onedimension (39–41). The parametersα andβ control interactions between redand green haploid organisms. Positiveαmeans that the presence of the greenallele favors the red allele, and positive β enhances the green allele whenred is present, etc. (SI Appendix has a detailed description of the model.)Pulled Fisher wave regimes (controlling, for example, the dynamics of selec-tive dominance in the second and fourth quadrants) and pushed excitablewave regimes (third quadrant, competitive exclusion dynamics) are boundedby the black dashed spinodal lines α= 0, β < 0 and α< 0, β= 0. These twobistable regimes are separated by the first-order phase transition (PT) lineα= β < 0, which is drawn as a black solid line. DP, directed percolation.

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τgdq

dt=

(1− s)(q2 + 2pq)

p2 + (1− s)(q2 + 2pq)− q

=sq(1− q)(q − q∗)

1− sq(2− q), where q∗ =

2s − 1

s,

[4]

which governs population dynamics of the MCR with 100% con-version efficiency in a well-mixed system. To take spatial dynam-ics into account, we add a diffusion term (28) and obtain a deter-ministic reaction–diffusion equation for the MCR model, namely

τg∂q

∂t= τgD

∂2q

∂x2+

sq(1− q)(q − q∗)

1− sq(2− q), [5]

which will be the main focus of this article. For later discussions,we name the reaction term of the reaction–diffusion equation

fMCR(q , s) =sq(1− q)(q − q∗)

1− sq(2− q). [6]

The reaction term reduces to a simpler cubic expression

fcubic(q , s) = sq(1− q)(q − q∗) [7]

by ignoring −sq(2 − q) in the denominator, which is a reason-able approximation if the selective disadvantage s is small. Thisform of the reaction–diffusion equation has been well-studied, asreviewed in ref. 28.

Although population genetics is often studied in the limitof small s , s is, in fact, fairly large in the regime of pushedexcitable waves of most interest to us here, 0.5< s < 1.0. Hence,we will keep the denominator of the reaction term, which wasalso done in ref. 28 with a different reaction term. Compari-son of results for the full nonlinear reaction term with thosefor the cubic approximation will give us a sense of the robust-ness of the cubic approximation. Although it might also be ofinterest to study corrections to the continuous time approxima-tion arising from higher-order time derivatives in (q ′ − q)/τg =∂q/∂t + 1/2τg∂

2q/∂t2 + ... (contributions from τg∂2q/∂t2 are

formally of order s2), this complicated problem will be neglectedhere; however, ref. 43 is a study of the robustness of the con-tinuous time approximation motivated by a model of dengue-suppressing Wolbachia in mosquitoes.

Initiation of the Pushed WavesThe reaction terms fMCR(q , s) and fcubic(q , s) have three identi-cal fixed points, q = 0, 1 and q∗

(= 2s − 1/s

). As discussed in SI

Appendix in connection with classical nucleation theory in physicsand following ref. 26, we can define the potential energy function

U (q) = − 1

τg

q∫0

sq ′(1− q ′)(q ′ − q∗)

1− sq ′(2− q ′)dq ′ [8]

to identify qualitatively different parameter regimes. In a well-mixed system, without spatial structure, the gene drive frequencyq(t) obeys Eq. 4 and evolves in time, so that it arrives at a localminimum of U (q). For the spatial model of interest here, q(x , t)shows qualitatively distinct behaviors in three parameter regimesdepending on the selective disadvantage s (Fig. 3A). We plot thepotential energy functions U (q) in these parameter regimes inFig. 3B.

i) First, when s < smin = 0.5, fixation of a gene drive alleleq(x ) = 1 for all x is the unique stable state, and there is noenergy barrier to reach the ground state starting from q ≈ 0.In this regime, any finite frequency of gene drive allele locallyintroduced in space (provided it overcomes genetic drift) willspread and replace the WT allele. The frequency profile willevolve as a pulled traveling wave q(x , t) =Q(x − vt) withwave velocity v . Such a wave was first found by Fisher (31)and Kolmogorov et al. (32) in the 1930s in studies of howlocally introduced organisms with advantageous genes spa-

tially spread and replace inferior genes. However, the thresh-oldless initiation of population waves of engineered genedrives with relatively small selective disadvantages seemshighly undesirable, because the accidental escape of a sin-gle gene drive construct can establish a population wave thatspreads freely into the extended environment.

ii) There is a second regime for 0.5< s < 0.697, in which thepotential energy function U (q) exhibits an energy barrierbetween q = 0 and q = 1. In this regime, a pushed travelingwave can be excited only when a threshold gene drive allelefrequency is introduced over a sufficiently broad region of

B

A

iii) Wave reverses direction

i) Pulled wave regime

s: selective disadvantagewdrive = (1 s ) wwild

ii) Pushed wave regime

s=0.0

s=0.5

s=0.697

s=1.0

Frequency q

U(q

)τg

s = 0.8s = 0.7s = 0.6s = 0.5s = 0.4

0 0.2 0.4 0.6 0.8 1.

–0.08

–0.04

0.

0.04

0.08

Fig. 3. (A) Spatial dynamics of gene drives can be determined by both theselective disadvantage s and (when 0.5< s< 0.697), the size and intensityof the initial condition. (B) The energy landscapes U(q) with various selectivedisadvantages s. (i) Pulled Fisher wave regime. When s is small, s≤ smin = 0.5(red and yellow curves), fixation of the gene drive allele (q = 1) is the uniquestable state, and there is no energy barrier between q = 0 and 1. Any finiteintroduction of a gene drive allele is sufficient to initiate a pulled Fisher pop-ulation wave that spreads through space to saturate the system. (ii) Pushedexcitable wave regime. When s is slightly larger (green curve) and satisfiessmin = 0.5< s< smax = 0.697, q = 1 is still the preferred stable state, but anenergy barrier at q = q∗ appears between q = 0 and 1. In this regime, theintroduction of the gene drive allele at sufficient concentration and over asufficiently large spatial extent is required for a pushed wave to spread toglobal fixation. (iii) Wave reverses direction. When s is large, s> smax = 0.697(blue and purple curves), q = 0 is the unique ground state, and the genedrive species cannot establish a traveling population wave and thereforedies out.

Tanaka et al. PNAS Early Edition | 3 of 6

space that exceeds the size of a critical nucleus, which weinvestigate in the next section. The existence of this thresholdacts as a safeguard against accidental release. In addition, suchexcitable waves are easier to stop, which we will discuss later.It seems that gene drives in this relatively narrow intermediateregime are the most desirable from a biosafety perspective.

iii) When s > smax = 0.697, the fixation of a gene drive allelethroughout space is no longer absolutely stable (Fig. 3B), anda gene drive population wave cannot be established. Indeed,the excitable wave reverses direction for s > smax. An implicitequation for smax results from equating U (0) =U (1) = 0,which yields

0 =

1∫0

sq(1− q)(q − q∗)

1− sq(2− q)dq ,

or 0 =−2 + smax + 2

√−1 + 1

smaxarcsin(

√smax)

2smax[9]

⇒ smax ≈ 0.697,

where we used q∗= (2s − 1)/s . When s > smax, the locally intro-duced gene drive allele contracts rather than expands relative tothe WT allele and simply dies out. SI Appendix has the analogousresults with an arbitrary conversion rate (0< c< 1).

Critical Nucleus in the Pushed Wave RegimeWhen the selective disadvantage s is in the intermediate regime,smin = 1/2< s < smax = 0.697, we can control initiation of the

Distance x/ τgD

t

0 5 10 15 20 250.00.20.40.60.81.0

0 5 10 15 20 250.00.20.40.60.81.0

t

Freq

uenc

y q(

x,t)

B

A

Freq

uenc

y q(

x,t)

Fig. 4. The excitable population wave carrying a gene drive can be estab-lished only when the initial concentration is above a threshold distribu-tion and over a region of sufficient spatial extent [the critical nucleus orcritical propagule (28)]. Numerical solutions of τg∂q/∂t = τgD∂2q/∂x2 +

sq(1− q)(q− q∗)/1− sq(2− q) with q∗ = 2s− 1/s are plotted with timeincrement ∆t = 2.5τg. The early time response is shown in red, with latertimes in blue. Selective disadvantage of the gene drive allele relative tothe WT allele is set to s = 0.58. In the case illustrated here, the genedrive allele can either die out or saturate the entire system, depending on

the width of the initial Gaussian population profile of q(x, 0) = ae−(x/B)2 .(A) With a narrow distribution of the initially introduced gene drivespecies (a = 0.5, B = 3.0

√τgD), the population quickly fizzles out. (B) With a

broader distribution of the initial gene drive allele (a = 0.5, B = 6.0√τgD),

the gene drive allele successfully establishes a pushed population wave,leading to q(x) = 1 over the entire system.

s = 0.66s = 0.58s = 0.51cubicMCR

0 5 10 15 200

0.2

0.4

0.6

0.8

1.

Freq

uenc

y q(

x,t)

Distance x/ τgD

Fig. 5. Initial critical frequency profiles of the MCR allele qc(x) just suffi-cient to excite a pushed genetic wave in 1D (critical propagule). Numeri-cally calculated critical propagules for the MCR model of Eq. 5 (solid lines)are compared with analytical results available for the cubic model of Eq. 7(dashed lines) (28). When s = 0.51, the two equations gives almost identicalresults, but as s increases, the critical propagule shape of the MCR modeldeviates significantly from that of the cubic model. The critical propag-ule of the cubic equation consistently overestimates the height of qc(x),because the sq(2−q)> 0 term in the denominator of the MCR model alwaysincreases the growth rate.

pushed excitable wave by the initial frequency profile of the genedrive allele q(x , 0) as shown in Fig. 4. For example, in Fig. 4A,an initially introduced gene drive allele (in the form of a Gaus-sian) diminishes and dies out, because the width of the initialfrequency distribution q(x , 0) is not sufficient to excite the pop-ulation wave. In contrast, the results in Fig. 4B show the suc-cessful establishment of the excitable wave starting from a suf-ficiently broad (Gaussian) initial distribution of a gene driveallele. Roughly speaking (provided 1/2< s < smax), two condi-tions must be satisfied to obtain a critical propagule. (i) The ini-tial condition q(0, 0) at the center of the inoculant must exceedq∗= 2s − 1/s , the local maximum of the function U (q) plottedin Fig. 3. (ii) The spatial spread ∆x of the inoculant q(x , t = 0)

must satisfy ∆x & const√

Dτg , where the dimensionless constantdepends on s . Thus, the initial width should exceed the width ofthe pushed wave that is being launched.

We show the spatial concentration profile qc(x ) that consti-tutes that (Gaussian) critical nucleus is just sufficient to initiate anexcitable wave in Fig. 5. The solid lines in Fig. 5 represent numer-ically obtained critical nuclei of the MCR model. Note the con-sistency for s = 0.58 with the pushed excitable waves shown inFig. 4. The dashed lines in Fig. 5 represent analytically derivedcritical propagules of the cubic model as a reference (details arein SI Appendix). Fig. 5 shows that the cubic model overestimatesthe height of critical propagule, particularly for larger s . The dif-ference between the reaction terms of the MCR model fMCR(q)(Eq. 6) and those of its cubic approximation fcubic(q) (Eq. 7) arisesfrom the term −sq(2 − q) in the denominator of Eq. 5. In thebiologically relevant regime (0< s < 1, 0< q < 1), sq(2 − q) isalways positive, and fMCR(q)> fcubic(q) is satisfied, which explainswhy there is a larger critical propagule in the cubic approxima-tion and why the discrepancy is larger for larger s . The criticalnucleus with a step function-like circular boundary is studied bothnumerically and analytically in two dimensions in SI Appendix.

Stopping of Pushed, Excitable Waves by a SelectiveDisadvantage BarrierThus far, we have found that (i) we can control initiation ofthe spatial spread of a gene drive provided smin = 0.5< s <smax = 0.697 and that (ii) the pushed population waves in this

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regime slow down and eventually stop (and reverse direction)when s > smax (SI Appendix). In this section, we examine alterna-tive ways to confine an excitable gene drive wave to attain greatercontrol over its spread in this regime.

Imagine exploiting the CRISPR-Cas9 system to encode mul-tiple functionalities into the gene drive machinery (1–3, 20). Forexample, one could produce genetically engineered mosquitoesthat are not only resistant to malaria but also, specifically vulner-able to an insecticide that is harmless for the WT alleles. Such agene drive, which is uniquely vulnerable to an otherwise harmlesscompound, is a sensitizing drive (9). The effect of laying downinsecticide in a prescribed spatial pattern on a sensitizing drivecan be incorporated in our model by increasing the selective dis-advantage to a value sb(> s) within a “selective disadvantage bar-rier” region.

In Fig. 6, we numerically simulate the MCR model definedby Eq. 5 in one dimension with a barrier of strength sb = 0.958

placed in a region 25√τgD < x < 27

√τgD . When the selective

disadvantage outside the barrier is small (s < 0.5) and the pop-ulation wave travels as the pulled Fisher wave, even a tiny frac-tion of the MCR allele diffusing through the insecticide regioncan easily reestablish the population wave, which is shown inFig. 6A. However, when the system is in the pushed wave regime0.5< s < 0.697, the wave can be stopped provided that the spa-tial profile of the gene drive allele that leaks through does notconstitute a critical nucleus, which is illustrated in Fig. 6B. SIAppendix has numerically calculated plots of the critical widthand barrier selective disadvantage needed to stop pushed wavesfor various values of s .

Distance x/ τgD

0 10 20 30 40 50 60 70 800.00.20.40.60.81.0

Freq

uenc

y q(

x,t)

Pulled wave

0 10 20 30 40 50 60 70 800.00.20.40.60.81.0

Pushed wave

t

wild-type gene drive

B

A

Freq

uenc

y q(

x,t)

Fig. 6. Numerical simulations of pushed, excitable waves generated by Eq.5 with barriers in one dimension, with time increments ∆t = 5.0τg. As thewaves advance from left to right, the early time response is shown in red,with later times shown in blue. The fitness disadvantage inside the bar-rier is set to sb = 0.958 within a region 25

√τgD< x< 27

√τgD (shown as

purple bars). The initial conditions are step function-like, q(x, 0) = q0/(1 +

e10(x−x0)/√τgD), with q0 = 1.0 and x0 = 5.0

√τgD, similar to the initial con-

dition SI Appendix, Eq. S30 that we used in two dimensions (SI Appendix).(A) In the case of a Fisher wave with s = 0.479< smin = 0.5, a small numberof individuals diffuse through the barrier, which is sufficient to reestablish arobust traveling wave. (B) In the case of the excitable wave s = 0.542 >

smin = 0.5, a small number of individuals also diffuse through the bar-rier. However, because the tail of the penetrating wave front is insuffi-cient to create a critical nucleus, the barrier causes the excitable wave todie out.

1.01.0 1.0

1.0 1.0s=0.62

s=0.48

gene drive

gene drive

Pushed wave

Pulled waveA

B

t=0 20 40 60 80

wild type

wild type

Fig. 7. Population waves impeded by a selective disadvantage barrierof strength sb = 1.0 (purple) with a gap. This imperfect barrier has aregion without insecticide in the middle of width 6

√τgD. (A) The pulled

Fisher wave with s = 0.48< 0.5 always leaks through the gap and reestab-lishes the gene drive wave (red and yellow). (B) The pushed wave thatarises when s = 0.62> 0.5 is deexcited by a gapped barrier, provided thatthe gap width is comparable to or smaller than the width of the genedrive wave.

Excitable Wave Dynamics with Gapped Barriers in TwoDimensionsIn the previous section, we showed that pushed excitable wavescan be stopped by a selective disadvantage barrier in one dimen-sion. However, in two dimensions, it may be difficult to makebarriers without defects. Hence, we have also studied the effectof a gap in a 2D selective disadvantage barrier. We find that,although the gene drive population wave in the Fisher waveregime s < 0.5 always leaks through the gaps, the excitable wavewith 0.5< s < 0.697 can be stopped, provided that the gap iscomparable with or smaller than the width of the traveling wavefront. In Fig. 7, we illustrate the gene drive dynamics for two dif-ferent parameter choices. In Fig. 7, the strength of the selectivedisadvantage barrier is set to be sb = 1.0, and the width of thegap in the barrier is set to be 6

√τgD . The engineered selec-

tive disadvantage in the nonbarrier region s differs in the twoplots. In Fig. 7A, s = 0.48< 0.5; therefore, the gene drive wavepropagates as a pulled Fisher wave, and the wave easily leaksthrough the gap. If genetic drift can be neglected, we expectthat Fisher wave excitations will leak through any gap, regard-less of how small. However, when the selective disadvantage bar-rier is in the pushed wave regime 0.5< s < 0.697, the populationwave can be stopped by a gapped selective disadvantage barrieras shown in Fig. 7B. To stop a pushed excitable wave, the gapdimensions must be smaller than the front width; alternatively,we can say that the gap must be smaller than size of the criticalnucleus.

DiscussionThe CRISPR-Cas9 system has greatly expanded the designspace for genome editing and construction of MCRs with non-Mendelian inheritance. We analyzed the spatial spreading ofgene drive constructs, applying reaction–diffusion formulationsthat have been developed to understand spatial genetic waveswith bistable dynamics (26–28). For a continuous time and spaceversion of the model by Unckless et al. (29), in the limit of 100%conversion efficiency, we found that a critical nucleus or propag-ule is required to establish a gene drive population wave whenthe selective disadvantage satisfies 0.5< s < 0.697. This range iseven narrower when we relax the assumption of 100% conver-sion efficiency, so that c< 1 (SI Appendix). Our model led usto study termination of pushed gene drive waves using a bar-rier that acts only on gene drive homozygotes, correspondingto an insecticide in the case of mosquitoes. In this parameter

Tanaka et al. PNAS Early Edition | 5 of 6

regime, the properties of pushed waves allow safeguards againstthe accidental release and spreading of the gene drives. Onecan, in effect, construct switches that initiate and terminate thegene drive wave. In the future, it would be interesting to studythe stochasticity caused by finite population size (genetic drift),which is known to play a role in the first quadrant in Fig. 2 (40,41). We expect that genetic drift can be neglected provided thatNeff� 1, where Neff is an effective population size, say, the num-ber of organisms in a well-mixed critical propagule. SI Appendixhas a brief discussion of genetic drift. It could also be impor-tant to study the effect of additional mutations on an excitablegene drive wave, particularly those that move the organism out-side the preferred range 0.5< s < 0.697. Finally, we address pos-sible experimental tests of the theoretical predictions. Because itseems inadvisable to conduct field tests without thorough under-standing of the system, laboratory experiments with microbeswould be a good starting point. Recently, the transition frompulled to pushed waves was qualitatively investigated with hap-loid microbial populations (35). Because the MCR has alreadybeen realized in Saccharomyces cerevisiae (19), it may also bepossible to test the theory in the context of range expansions ona petri dish, which has already been done for haploid mutualis-tic yeast strains in ref. 44. Here, the frontier approximates a 1D

stepping stone model, and jostling of daughter cells at the fron-tier leads to an effective diffusion constant in one dimension (45,46). Finally, as illustrated in SI Appendix, Fig. S2, the mathemat-ics of the spatial evolutionary games in one dimension parallelsthe dynamics of diploid gene drives in the pushed wave regime,providing another arena for experimental tests, including theeffects of genetic drift.

Materials and MethodsTo simulate the dynamics governed by Eq. 5 in Figs. 4, 6, and 7 and SIAppendix, Fig. S6, we used the method of lines and discretized spatial vari-ables to map the partial differential equation to a system of coupled ordi-nary equations (ODEs). Then, we solved the coupled ODEs with a standardODE solver. The widths of the spatial grids were varied from 1/200

√τgD to

1/20√τgD, always making sure that the mesh size was much smaller than

the width of the fronts of the pushed and pulled genetic waves that westudied.

ACKNOWLEDGMENTS. We thank N. Barton, S. Block, S. Sawyer, T. Stearns,and M. Turelli for helpful discussions and two anonymous reviewers for use-ful suggestions. N. Barton also provided a critical reading of our manuscript.Work by H.T. and D.R.N. was supported by National Science FoundationGrant DMR1608501 and Harvard Materials Science Research and Engineer-ing Center Grant DMR1435999. H.A.S. acknowledges support from NationalScience Foundation Grants MCB1344191 and DMS1614907.

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