The Population Genetics of Using Homing Endonuclease Genes in … · The Population Genetics of...

Copyright � 2008 by the Genetics Society of AmericaDOI: 10.1534/genetics.108.089037

The Population Genetics of Using Homing Endonuclease Genes inVector and Pest Management

Anne Deredec,* Austin Burt* and H. C. J. Godfray†,1

*NERC Centre for Population Biology, Department of Biology, Imperial College London, Ascot, Berks SL5 7PY,United Kingdom and †Department of Zoology, University of Oxford, Oxford OX1 3PS, United Kingdom

Manuscript received March 7, 2008Accepted for publication April 28, 2008

ABSTRACT

Homing endonuclease genes (HEGs) encode proteins that in the heterozygous state cause double-strand breaks in the homologous chromosome at the precise position opposite the HEG. If the double-strand break is repaired using the homologous chromosome, the HEG becomes homozygous, and thisrepresents a powerful genetic drive mechanism that might be used as a tool in managing vector or pestpopulations. HEGs may be used to decrease population fitness to drive down population densities(possibly causing local extinction) or, in disease vectors, to knock out a gene required for pathogentransmission. The relative advantages of HEGs that target viability or fecundity, that are active in one sexor both, and whose target is expressed before or after homing are explored. The conditions under whichescape mutants arise are also analyzed. A different strategy is to place HEGs on the Y chromosome thatcause one or more breaks on the X chromosome and so disrupt sex ratio. This strategy can cause severesex-ratio biases with efficiencies that depend on the details of sperm competition and zygote mortality.This strategy is probably less susceptible to escape mutants, especially when multiple X shredders are used.

THE possibility of controlling man’s major pests,pathogens, and disease vectors using genetic ma-

nipulation has long been discussed (Hamilton 1967;Curtis 1968) and is of great current interest (Turelli

and Hoffmann 1999; Alphey et al. 2002; James 2005;Sinkins and Gould 2006). A broad spectrum ofpossible strategies has been explored. Organisms canbe manipulated to be conditionally sterile or lethal andreleased into the environment to disrupt mating or toreduce the fecundity of the wild population (Thomas

et al. 2000; Atkinson et al. 2007; Phuc et al. 2007). Withthese inundative techniques the manipulated constructis not required to persist in the environment. Adifferent approach is to introduce a beneficial geneticconstruct into a wild population with a drive mecha-nism that causes it to increase in frequency. The con-struct might impose a fitness load on the population,reducing its density or causing it to go extinct. Al-ternatively, it may alter the phenotype of the organismwith no or minor changes to its fitness. The latter is ofparticular relevance to disease vectors where it may bepossible to reduce or eliminate transmission. Recentadvances in molecular genetics have demonstrated thatknocking out certain Anopheles mosquito genes, orinserting new constructs, prevents the insect fromtransmitting Plasmodium, the malaria pathogen (Ito

et al. 2002; Moreira et al. 2002), while RNAi techniqueshave been used to prevent Aedes mosquitoes fromtransmitting the dengue virus (Franz et al. 2006).Enthusiasm for these control strategies is tempered bythe realization that any method involving geneticmanipulation will require the highest scrutiny andinvestigation prior to implementation and that supportfrom the public will be essential for any project to goahead (Alphey et al. 2002; James 2005; Knols et al.2006).

A variety of different mechanisms for driving genesthrough a population have been considered, most ofthem based on elements with non-Mendelian heritancethat have been discovered in nature (Burt and Trivers

2006). Some genes cause the chromosomes on whichthey reside to be overrepresented in the gamete pooland thus could be used to increase the frequency of anintroduced linked gene (Burt and Trivers 2006).Genetic constructs can be designed that show under-dominance—heterozygote inferiority—and hence willincrease in frequency once their abundance passes acertain threshold (Davis et al. 2001; Magori andGould 2006). Elements that jump between chromo-somes can be used as vectors for beneficial constructs,and transposable elements in particular have received alot of attention (Coates et al. 1998). Heterozygotefemales carrying medea elements modify their eggs suchthat they survive only if they carry the medea gene or arefertilized by sperm that carry the element. This dis-advantages wild-type alleles and allows medea to spread

1Corresponding author: Department of Zoology, University of Oxford,S. Parks Rd., Oxford OX1 3PS, United Kingdom. E-mail: [email protected]

Genetics 179: 2013–2026 (August 2008)

(Wade and Beeman 1994). Artificially engineered medeaelements have recently been developed and offer animportant new potential drive mechanism (Chen et al.2007). Certain symbiotic microorganisms with verticalinheritance spread by manipulating host reproductionsuch that infected individuals produce more daughtersthan uninfected individuals. Introducing the beneficialgene into the symbiont could then lead to its spread,though with the disadvantage that the gene may not beexpressed in the correct tissue. The intracellular bacte-rium Wolbachia that is present in a very large fraction ofinsects and that spreads through cytoplasmic incom-patibility (noninfected females are at a disadvantagebecause they cannot use the sperm of infected males) isthe most important candidate drive mechanism of thistype (Werren 1997; Turelli and Hoffmann 1999).Finally, a variant of these techniques is to use the drivemechanism to impose a fitness cost on the organism andthen to link the beneficial gene to a construct thatmitigates the cost and hence is selected to spread(Sinkins and Godfray 2004). In comparing drivemechanisms the most important factors likely to in-fluence success or acceptability include the evolutionarystability of the construct, the degree to which the within-and between-individual spread of the element can bepredicted, whether the construct can increase in fre-quency from rare or if a threshold frequency must beexceeded before spread occurs, and whether it is pos-sible to reverse the manipulation.

An exciting potential drive strategy is to use site-specific selfish genes such as homing endonucleasegenes (HEGs) (Burt 2003). A HEG codes for a proteinthat recognizes and cuts DNA containing a specific 20-to 30-bp sequence (Stoddard 2005). Critically thissequence is found only on chromosomes not containingthe HEG and at the precise location where the HEGoccurs. After a double-strand break in a heterozygote,the cell’s recombinational repair mechanism uses thechromosome carrying the HEG as a template and theHEG is thus copied from one chromosome to the other,converting a heterozygote to a homozygote. If there areno fitness costs to the HEG, it spreads until it reachesfixation. Other elements such as group II introns andcertain LINE-like transposable elements have similarstrategies for spread, though with a more complicatedmechanism involving RNA intermediaries (Burt andTrivers 2006). Below we concentrate just on HEGs thatoffer the most straightforward site-specific selfish genesfor exploitation.

HEGs are found in nature in single-celled fungi,plants, protists, and bacteria, but not in higher animals.They tend to reside in noncoding regions (especiallyintrons) and so have little effect on fitness because theyare spliced out prior to translation into protein. Due totheir low fitness costs they are expected to spread tofixation, but then decay because once they are fixedthere is no selection for their maintenance. Compara-

tive studies have shown that HEGs probably survive byjumping from species to species and that maintenancerequires that the rate of species jumps must exceed HEG‘‘death’’ in a lineage (Goddard and Burt 1999; Burt

and Koufopanou 2004). It is likely that it is easier forHEGs to jump among single-cell organisms than amonganimals with segregated germlines, which may explaintheir absence from the latter.

The aim of this article is to describe the different waysin which HEGs might be used as part of a genetic controlstrategy and to develop and analyze the population ge-netic models that will be required to assess their relativeadvantages and disadvantages. It builds on the analysesof Burt (2003), who derived the equilibrium frequencyand genetic load of HEGs with different homing fre-quencies that were either lethal or sterile to one or bothsexes. He also discussed alternative strategies such as theuse of multiple HEGs and their use as ‘‘X chromosomeshredders’’ that are analyzed formally here for the firsttime. We also study the population genetics of muta-tions that might nullify the action of the HEG.

We first treat ‘‘classical’’ HEGs that spread by copyingthemselves into homologous chromosomes after dou-ble-strand breaks. We derive equations for (i) the spreadand equilibria of HEGs that are active after gene ex-pression and (ii) HEGs that are active before. We thenexplore (iii) the possible advantages of sex-specificexpression and (iv) the risk of mutations arising thatprevent HEG spread. Second, we study HEGs on the Ychromosome that cause X chromosome breaks—Xshredders. We (i) derive equilibrium sex ratios for dif-ferent numbers of shredders, (ii) analyze the effects ofreduced sperm number and competition for zygotes,and (iii) study the evolution of escape mutants.

THEORETICAL RESULTS: DRIVING HEGs

HEG active after gene expression: Consider anengineered HEG that is introduced into a chromosomeopposite a functional gene. Let the homing rate (theprobability of a successful gene conversion) be e andthe fitness costs of disrupting gene function be s for thehomozygote and sh for the heterozygote. We begin byassuming fitness costs are equal in males and femalesand that homing occurs at meiosis after gene expression(so any costs of being homozygote are not experiencedby the individual in which homing occurs). If q and p arethe gametic frequencies of the HEG and wild-typealleles, respectively, the recurrence for q is

q9 ¼ ð1� sÞq2 1 ð1� shÞpqð1 1 eÞ1� sq2 � 2shpq

: ð1Þ

The equilibrium frequency for the HEG, q*, is

q* ¼ 0; s .e

1� h 1 ehand s .

e

hð1 1 eÞ ; ð2Þ

2014 A. Deredec, A. Burt and H. J. C. Godfray

q* ¼ 1; s ,e

1� h 1 ehand s ,

e

hð1 1 eÞ : ð3Þ

When these inequalities do not hold there is an interiorequilibrium

q* ¼ e � ð1 1 eÞhs

sð1� 2hÞ ; ð4Þ

which is stable if

e

1� h 1 eh, s ,

e

hð1 1 eÞ ð5Þ

and unstable if

e

1� h 1 eh. s .

e

hð1 1 eÞ : ð6Þ

The latter implies low heterozygote fitness, h . 12 , in which

case the HEG either goes extinct or reaches fixationdepending on whether the initial frequency is less than orgreater than the unstable equilibrium (Figure 1).

We define the ‘‘HEG load’’ (L) to be the relativereduction in the growth rate of the population in thepresence of the HEG. Assume a population with discretegenerations and let l0 be the rate at which thepopulation increases in the absence of any density-dependent effects (equivalent here to per capita femalefecundity). Then define L ¼ 1� l90=l0, where l90 is thepopulation growth rate when the HEG is present. HEGload is thus a quantity very similar to genetic load asusually interpreted in classical population genetics,except that it does not include effects on males andcan include processes that bias the sex ratio (see below).In using HEGs to drive down vector and pest densities itis important to note that a HEG load of L does notnecessarily mean that the population density is reducedby a factor L. The observed reduction will depend onthe precise form of density dependence operating in the

population, as well as on the relative ordering of hom-ing, target gene expression, and density dependence inthe life cycle. This article is concerned only with geneticdynamics and so we cannot predict absolute populationreductions. Nevertheless, calculating L provides a usefulcomparative measure of potential population reduc-tions. In this case, the HEG load experienced by thepopulation at equilibrium is

L ¼ e2ð1� hsÞ2 � s2h2

sð1� 2hÞ ð7Þ

for 0 , q* ,1, no load when q*¼ 0, and L¼ s for q*¼ 1.Consider first the special case in which HEGs are

employed to knock out an essential gene with the aim ofmaximizing HEG load and driving a population toextinction. For a fully recessive homozygote lethal (s ¼1, h ¼ 0) the equilibrium HEG frequency is e and theload e2. Thus very substantial loads, and hence potentialreductions in population numbers, are possible ashoming frequencies approach unity. However, the high-est equilibrium HEG loads do not occur when the HEGis invariably lethal in the homozygote. For a given valueof the homing rate (e), the greatest load occurs whenthe fitness costs are at the maximum that still allows theHEG to become fixed (s ¼ e), in which case the loadequals the selection pressure (L ¼ s) (Figure 1).

If we assume that the heterozygote also has reducedfitness (h . 0), then a greater range of equilibriumbehaviors may be observed (Figure 1). The HEG isalways fixed when fitness costs (s) are low and homingrates (e) are high, but away from this region of pa-rameter space decreasing heterozygote fitness first seesa reduction in the parameter combinations where theHEG equilibrium frequency is less than one but greaterthan zero (an internal equilibrium). Then, whenheterozygote fitness is closer to the homozygote HEGthan to the wild type (h . 1

2 ), a region of bistability appears(the HEG goes to extinction or fixation depending on

Figure 1.—Equilibrium frequency andload of a HEG that is active (homes) aftergene expression. Top row: equilibriumHEG frequency as a function of homing rateand fitness costs. The gene is fixed in thesolid region and lost in the open region.Where an interior equilibrium is possiblethe darkness of the shading is proportionalto the equilibrium frequency. In the stripedregion the gene is either fixed or lost de-pending on its initial frequency. Bottomrow: load imposed by the HEG in the sameregions of parameter space as in the toprow. The darkness of the shading is propor-tional to the HEG load.

Population Genetics of Homing Endonucleases 2015

its initial frequency) that increases as the HEG becomesfully dominant. In the absence of stochastic effects, formost parameter combinations a HEG with h . 1

2 will notspread from rare. As before, for fixed e, load is generallymaximized at the highest value of s that allows fixation.

A slightly different strategy is to engineer a HEG totarget a gene that is required for reproduction ratherthan survival. In the simplest case, if the knockout pre-vents the individual from participating in mating, thenthe dynamics and HEG load are exactly the same asdescribed above. But suppose the knockout acts later,such that mating occurs normally but is less productive(for example, the male makes defective sperm thatfertilize the eggs normally but result in inviable prog-eny), so that any matings involving either a male or afemale carry the HEG lead to fewer offspring (a post-mating fertility effect). Then while the dynamics ofspread and equilibrium would still be as describedabove, the genetic load would be greater. Only thosematings not involving an infertile carrier of the HEG, afraction (1� q 2s� 2pqhs)2, would produce offspring andhence the genetic load would be 1 � (1 � q 2s � 2pqhs)2.The load is thus always greater than or equal to theequivalent load (q2s 1 2pqhs) for a HEG targeting sur-vival. If the knockout mutation is recessive and abolishesreproductive success completely (s ¼ 1, h ¼ 0), theHEG equilibrium frequency is q ¼ e and the load isL ¼ e2ð2� e2Þ. e2 (Figure 2).

The second special case is when a HEG is employed toknock out a gene required for an insect to vector apathogen. Ideally the gene would have no fitness costs tothe host (s ¼ 0), in which case it would always spread andcause no HEG load. A HEG whose fitness effects are

manifest only in homozygotes becomes fixed provided e .

s and causes a load of L ¼ s when it affects survival orfecundity. When fertility is affected and determined aftermating by the genotype of both partners, then L¼ s(2� s).

Were a HEG to be used in a vector or pest controlprogram, not only the ultimate outcome but also therate at which it is attained would be significant. In Figure3 we plot the number of generations that it takes for aHEG to increase in frequency from 0.05 to 0.9. For thoseHEGs that can reach fixation, spread is faster for highhoming rates and for genes with recessive fitness costs.For much of this parameter space rapid spread occurswithin 10–15 generations, which for many insect speciesis just a couple of years and so is highly relevant to pestand vector control on relatively short timescales.

HEG active before gene expression: We now assumethat homing and gene conversion occur prior to theexpression of the gene containing the HEG recognition

Figure 2.—HEG load (L) as a function of homing fre-quency (e) when the target gene is essential and the wild typeis fully dominant (s¼ 1, h¼ 0). We plot three cases: in the firsttwo the effects of the HEG are experienced equally by bothsexes, the homozygote either being lethal (solid line) or hav-ing no postmating fertility through either sperm or ova (dot-ted-and-dashed line). The third case is a female-specific HEGwhere effects on either viability or postmating fertility lead tothe same load (dotted line).

Figure 3.—The number of generations taken for a HEG toincrease in frequency from 0.05 to 0.9 as a function of fitnesscosts (s), homing frequency (e), and dominance (h).


sequence. Any fitness consequences of disrupting thegene are now experienced both by homozygotes and bythe ‘‘transformed’’ heterozygotes. The recurrence forHEG frequency is now

q9 ¼ ð1� sÞðq2 1 2pqeÞ1 ð1� shÞpqð1� eÞ1� sðq2 1 2pqeÞ � 2pqshð1� eÞ : ð8Þ

The equilibrium frequency for the HEG, q*, is

q* ¼ 0; s .e

1� h 1 ehand s .

e

2e 1 h � eh; ð9Þ

q* ¼ 1; s ,e

1� h 1 ehand s ,

e

2e 1 h � eh: ð10Þ

When these inequalities do not hold, there is an interiorequilibrium

q* ¼ eð1� 2sÞ � hsð1� eÞsð1� 2e � 2hð1� eÞÞ ; ð11Þ

which is stable if

e

1� h 1 eh, s ,

e

2e 1 h � ehð12Þ

and unstable if

e

1� h 1 eh. s .

e

2e 1 h � eh: ð13Þ

In the last case the HEG either goes extinct or reachesfixation depending on whether the initial frequency is lessthan or greater than the unstable equilibrium (Figure 4).

The HEG load is s when the HEG becomes fixed and

L ¼ ðe � eð2� hÞs � hsÞðe 1 ð1� eÞhsÞsð1� 2eð1� hÞ � 2hÞ ð14Þ

for the interior equilibrium.

A HEG targeting a fully recessive gene with a sig-nificant effect on fitness (h¼ 0, s . 1

2 ) can invade only ife . s and if the initial HEG frequency exceeds athreshold of eð2s � 1Þ=sð2e � 1Þ. Thus the strategy ofusing HEGs to create a recessive lethal (s¼ 1, h¼ 0) willnot work if homing occurs prior to gene expression. Thebenefits of gene conversion in the heterozygote arenullified by the fitness costs of creating a homozygote.

In the limit, when there are no costs to carrying aHEG, it makes no difference when homing occurs andthe HEG will always spread. For moderate fitness costs,the condition for the fixation of the HEG is the sameirrespective of the order of homing and expression.However, in the regions where both fixation and ex-tinction of the HEG can occur depending on initial con-ditions, fixation now requires higher gene frequenciescompared to the case where the HEG is active after geneexpression. Where the HEG is not fixed, its equilibriumfrequency and load are always lower when homing occursprior to expression, and the rate of spread of the gene isalso relatively slower (data not shown).

Sex-specific expression: Returning to our originalmodel of the HEG being active after gene expression,we now assume that it targets a gene that has differenteffects on males and females and/or that the HEG hasdifferent rates of homing in the two sexes. Let qx, ex, sx,and hx have the same meanings as before except now weassume their values may be different in males (x¼m) orfemales (x ¼ f). The dynamics are given by the coupledrecurrence equations

q9x ¼ð1� sxÞqxqx9 1 ð1� sxhxÞð1=2Þðpxqx9 1 px9qxÞð1 1 exÞ

1� sxqxqx9 � sxhxðpxqx9 1 px9qxÞ;

fx; x9g ¼ fm; f g and ff ;mg: ð15Þ

The general solution to these equations is too com-plex to be helpful and we focus on a few specific cases.First, assume the knockouts are recessive and affect onlyfemales (h f ¼ 0, s f . 0, sm ¼ 0). Then

Figure 4.—Equilibrium frequency and loadof a HEG that is active (homes) before geneexpression. Drawing conventions and parame-ters are the same as in Figure 1.


q*m ¼

ðef 1 emÞð1 1 emÞsfð1� e2

mÞ1 2emðef 1 emÞ; q*

f ¼ef 1 em

sfð1 1 emÞ;

ef 1 em

1 1 em, sf ; ð16Þ

q*m ¼ q*

f ¼ 1;ef 1 em

1 1 em. sf : ð17Þ

The HEG load is sf qm qf or

L ¼ ðef 1 emÞ2sfð1� e2

mÞ1 2emðef 1 emÞð18Þ

for q*m; q

*f , 1 and L¼ sf otherwise. If we assume homing

frequency is the same in the two sexes (ef¼ em¼ e) and theknockout is a female-specific recessive lethal or sterile (sf

¼ 1), then the load is L ¼ 4e2=ð3e2 1 1Þ (Figure 2). Thisload is always greater than that when both sexes are killedor removed from the mating pool, for which the load is L¼e2. Killing males is counterproductive because it reducesthe frequency of the HEG without reducing populationproductivity. We can also compare the load caused by afemale-specific lethal or sterile with that of a HEG thatdisrupts both male and female fertility so that only zygotesproduced by parents neither of whom are homozygousHEG carriers survive (L¼ e2(2� e2)). The female-specificHEG is superior unless homing rates (e) are large (Figure2). As before, the maximum HEG load, for a given homingrate, occurs for the highest homozygote fitness cost atwhich the HEG can become fixed. Here this is found whensf ¼ 2e=ð1 1 eÞ in which case L ¼ sf. Larger loads, for thesame homing rate, are thus possible for sex-specific fitnesseffects. The rate of spread of sex-specific HEGs is similar tothat of nonspecific genes.

Second, assume that the fitness effects of the HEG arethe same for both males and females, but that homingrates are different (sf ¼ sm, hf ¼ hm ¼ 0, ef 6¼ em). Recallthat when the homing rate is the same in the two sexes,fixation requires s , e : in the present case s must beless than the average rate of homing in the two sexes. Ifsf ¼ sm ¼ 1, then fixation cannot occur and the load is

L ¼ 12

�1 1 ef em �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� e2

fð Þ 1� e2m

� �q �. If the average

homing rate is kept constant, the load is at a minimumwhen rates are the same in the two sexes and increases asthe difference gets larger.

Third, consider the case where both homing andcosts are sex specific. If the HEG homes only in femalesand is a female-specific lethal or sterile (em ¼ 0, ef . 0,sm ¼ 0, sf¼ 1), then the loads produced are identical tothe non-sex-specific case. But if homing is restricted tomales rather than females, then a female lethal or sterileHEG (em . 0, ef ¼ 0, sm ¼ 0, sf ¼ 1) causes substantiallylower loads to occur (L ¼ e2

m=ð1 1 e2mÞ), with the maxi-

mum obtainable load (as homing frequency approachesone) being L ¼ 1

2 rather than 1. The reason for this is

that when homozygous females are rendered dead orsterile, then heterozygous females make a relativelylarger contribution to the next generation, and hencethe spread of the HEG is particularly influenced by thehoming that occurs in these heterozygotes. The case ofnon-sex-specific lethality but homing only in a single sexprovides an even worse outcome in terms of load thanfemale lethality and male homing.

Consider now using HEGs to knock out a geneessential for vector transmission. Suppose there aremild costs (s > e) to the knockout that may beexperienced by males, females, or both sexes equally.The HEG will always go to fixation and there are onlyminor differences in the speed at which this happens(fastest when only one sex suffers the fitness cost).

We have also explored sex-specific fitness costs whenthe HEG is active prior to gene expression. Qualitativelythe conclusions are very similar to the comparison of thetwo situations in the non-sex-specific case: HEG activityprior to expression always tends to reduce the rate ofspread, genetic load, and equilibrium frequency.

Dynamics of HEG-escape mutants: When a homingendonuclease cuts a chromosome it is normally re-paired using the second chromosome as a template, themechanism through which the HEG increases in fre-quency. But it is possible that the chromosome isrejoined in a different way. Possibly an incomplete copyof the HEG is transferred, or alternatively the ends ofthe chromosome may be ligated without the use ofa template. If the wild-type chromosome is preciselyreconstituted, then the initial cut leaves no trace and theonly dynamic effect is a reduction in the efficiency ofhoming, a lower value of the parameter e. But if therepair destroys the HEG recognition site, then non-standard repair can be very important. A critical issue iswhether the repaired chromosome, without a HEGrecognition site, contains a functioning gene.

To explore this assume there are three classes ofallele, the wild type (1), the HEG (H), and a misre-paired allele (M) at frequencies of p, qH, and qM,respectively. In reality there are likely to be severalclasses of misrepaired allele, although we simplify thesituation by allowing just a single type. The genotypefrequencies and fitnesses are given in the table,

where the subscripted s and h parameters describe thefitness costs of the different alleles and their pattern ofdominance. Population fitness, �w, is the average geno-type fitness weighted by frequency (and so the HEG loadis 1� �w) and g is the frequency of misrepair [and hence(1 � g) is the frequency of repair resulting in afunctional HEG].

Genotypes 11 1H 1M HH HM MM

Frequency p2 2pqH 2pqM qH2 2qMqH qM

2

Fitness 1 1 � hHsH 1 � hMsM 1 � sH 1 � sI 1 � sM,


With these assumptions we can write recurrences forqH and qM,

q9H ¼1�Wðq2

Hð1� sHÞ1 pqHð1 1 eð1� gÞÞð1� hHsHÞ

1 qMqHð1� sIÞÞ; ð19Þ

q9M ¼1�Wðq2

Mð1� sMÞ1 pqMð1� hMsMÞ

1 pqHð1� hHsHÞeg 1 qMqHð1� sIÞÞ: ð20Þ

The general solution of these equations is too complexto be useful so we explore some special cases.

Assume first that the wild-type allele is fully dominant(hH¼ hM¼ 0) and that both the HEG and the misrepairallele cause nonfunctional gene products leading to deathin the absence of a wild-type allele (sH ¼ sM ¼ sI ¼ 1).The equilibrium frequency of the two alleles is

q*H ¼ eð1� gÞ2; q*

M ¼ egð1� gÞ; ð21Þ

and the HEG load is e2(1 � g)2. Thus if the aim of theprogram is to reduce population densities by targeting alethal gene, the effect of misrepair leading to non-functional alleles is simply to make homing less efficient.

Now assume that the misrepair allele produces a geneproduct that is at least partially functional. Specificallylet sM ¼ sI ¼ s while the HEG remains homozygouslethal (sH¼ 1) and all other parameters are as above. Inthe case of s ¼ 0 the misrepaired allele is completelyfunctional and it can be shown formally that there is nostable equilibrium with the HEG present, no matter howsmall is the rate with which escape arises (g). For s . 0 anequilibrium exists with both the HEG and the escapemutant present. As the costs of the escape mutant rise,the equilibrium frequency of the HEG also increases.Higher homing rates (e) and higher probabilities oflegitimate repair (1 � g) lead to both greater frequen-cies of the HEG and greater load (Figure 5).

Recall that in the absence of misrepair a HEG that hasno fitness costs (sH¼ 0) inexorably becomes fixed. If lessthan half the time (g , 1

2 ) misrepair generates non-functional or partially functional alleles (sM ¼ sI . 0),then fixation of a cost-free HEG still occurs, though itmay happen more slowly. If functional alleles are pro-duced that have no fitness costs (sM¼ sI¼ 0), then boththe HEG and the misrepair allele increase in frequencyuntil the wild-type allele disappears. The equilibriumfrequency of the HEG (qH

* ¼ 1 � g (1 � qH0)) dependson the frequency with which these mutant alleles arise(g) as well as the initial HEG frequency qH0. After thisthe HEG and misrepair alleles will show neutral dynam-ics affected only by drift.

Complex dynamics may occur if the HEG targets agene whose knockout is neither lethal nor of no fitnessconsequence to the organism (0 , sH , 1). We have notperformed a full analysis of all possible scenarios but

HEGs are more likely to become fixed if they have highfitness in the homozygote and lost if the homozygote iscostly. High homing rates (e) tend to favor HEG fixationand low rates loss; high misrepair rates (g) also increasethe chance of loss, while low relative escape mutantfitness (sI . sH) tends to favor fixation. For intermediatevalues of sH polymorphisms may occur with the HEGfrequency depending on the relative costs of the HEGand escape mutant, as well as the frequency with whichthe latter arises. Simulations of the dynamics showcomplex behaviors including dependence on initialconditions and cycles where the frequency of the HEGat times gets so close to zero that in a real population itcould be lost due to stochastic effects. We note thatsimilar, complex dynamics have been observed in othermeiotic drive systems (Charlesworth and Hartl

1978; Nauta and Hoekstra 1993).

THEORETICAL RESULTS: X CHROMOSOMESHREDDERS

Assume that in a species with heterogametic males kHEGs are inserted into the Y chromosome and eachrecognizes a specific, different site on the X chromo-some causing a break with probability e. The chance thata particular X chromosome survives this assault is thus(1 � e)k and the fraction of gametes carrying the Y

Figure 5.—The equilibrium HEG frequency (top) andHEG load (bottom) when the repair of the cut chromosomecan produce mutant alleles with intermediate fitness costs,0 , sM , 1. A homing rate of e ¼ 0.9 is assumed and the threelines represent different probabilities of misrepair: g ¼ 0.5(solid line), 0.1 (dotted line), and 0.01 (dashed line).


chromosome increases from 12 to 1/[1 1 (1 � e)k]. If

there are no fitness costs to reduced sperm volume andthe frequency of HEG-bearing Y chromosomes (as afraction of all Y chromosomes) in the current genera-tion is q, then in the next generation it will be

q9 ¼ qð1=ð1 1 ð1� eÞkÞÞqð1=ð1 1 ð1� eÞkÞÞ1 ð1� qÞð1=2Þ : ð22Þ

The HEG spreads to fixation on the Y chromosome(Figure 6) and hence the sex ratio (proportion ofmales) is equal to the fraction of Y-bearing gametes1/[1 1 (1� e)k]. Note that in the special case of a singleHEG the equilibrium sex ratio is 1/(2 � e). We definethe HEG load when there is a biased sex ratio to beL ¼ 1� 2ð1� r Þðl9f=lfÞ, where r is the sex ratio and l9fand lf are the average population growth rates (exclud-ing the sex-ratio effect) in the presence and the absenceof the HEG, respectively. In this case, once the HEG hasbecome fixed, L ¼ ð1� ð1� eÞkÞ=ð1 1 ð1� eÞkÞ.

The equilibrium sex ratio as a function of thechromosome break frequency (e) and the number ofHEGs are shown in Figure 7. If breaks occur with a highprobability, then the insertion of a single HEG can skewthe sex ratio so strongly to males that the population isunlikely to persist. If breaks occur with lower probability,then high skew and population extinction can still beachieved using multiple HEGs. The spread of the geneis relatively fast. For example, a single X shredder with acutting rate of e ¼ 0.9 can increase in frequency from0.01 to 0.99 in �15 generations. The same speed ofspread for lower cutting rates can be achieved withmultiple X shredders (Figure 6).

Note that the mathematics is unaltered if the param-eter k refers not to the number of unique target sites ofmultiple HEGs, but to the multiple targets of a singleHEG. Theoretically, very high values of k might thusbe achieved if the HEG is engineered to recognizecommon sequences of a tandem array of genes orother multiple-gene families. Note that at some pointthe concentration of the homing endonuclease mightbecome limiting, depending upon how strongly it isexpressed.

Costs due to lower sperm numbers: The argumentsabove assume that males with reduced sperm produc-tion suffer no fitness penalties. The simplest way to relaxthis assumption is to let the relative cost paid byengineered males be a function, f (x), of relative spermvolume (x ¼ 1/2[1 1 (1 � e)k]), where f (x) decreasesfrom one to zero as x varies over the same range.Intuitively, males are penalized by producing fewersperm. If the costs are sufficiently high that HEG-bearing sperm fertilize fewer eggs than they would havein the absence of X shredding, then the spread of theHEG can be prevented ( f (x) . 1 � x). Below thisthreshold the HEG still spreads to fixation and the finalsex ratio is the same, but the costs can considerably slowthe rate at which this is attained.

A more mechanistic model of the costs of reducedsperm production can also be constructed. Multiplemating will reduce the advantage of the Y chromosomebearing the HEG as some of the benefits of the smallernumber of X gametes will be shared with wild-type Ychromosomes from other males. To model this, assumethat when a female is mated by x males carrying the HEGand y carrying the wild-type allele, her eggs are fertilizedrandomly by the pool of sperm produced by the x 1 ymales. The recurrence for the frequency of the HEGbecomes

Figure 6.—The spread of an X-shredding HEG (solidlines) and its consequences for the population sex ratio(dashed lines). For each pair of lines the bottom (solid) curverepresents the case of a single HEG recognition site on the Xchromosome (k ¼ 1) and the top (shaded) curve representsfive recognition sites (k ¼ 5). A cutting frequency of e ¼ 0.8 isassumed. At equilibrium the HEG loads are 0.67 and 0.99 fork ¼ 1 and 5, respectively.

Figure 7.—Equilibrium sex ratio in the presence of X-shredding HEGs as a function of the chromosome break fre-quency (e) and the number of HEGs or HEG recognition sites(k). In all cases the HEG-bearing Y chromosome goes to fix-ation at equilibrium.


q9 ¼P

x;y pðx; yÞðx=ðð1 1 ð1� eÞkÞx 1 2yÞPx;y pðx; yÞððx 1 yÞ=ðð1 1 ð1� eÞkÞx 1 2yÞÞ ; ð23Þ

where p(x, y) is the probability of the combination [x, y].If females mate only once at random with the two typesof males [so p(1, 0)¼ q, p(0, 1)¼ 1� q, and all other p(x,y) ¼ 0], then Equation 23 simplifies to Equation 22.Another limit is the case when all females are mated by alarge number of males (x 1 y / ‘), in which case q9 ¼ q :the HEG allele has exactly the same fitness as the wild-type allele because it gets no special benefit from thereduction in X gametes it brings about.

Figure 8 explores intermediate cases assuming p(x, y)is a compound distribution with the total number ofmatings (x 1 y) determined by a geometric distribution[the probability that a female is mated x 1 y times is(1 � u)ux1y�1 with u , 1] and the numbers of x and yconditional on their sum described by the binomialdistribution. Increased frequencies of multiple matingreduce the advantages of X shredding and slow thespread of the HEG, without affecting the final outcome.

Costs due to lower zygote numbers: A different typeof cost occurs when sperm that carry shredded andhence inviable X chromosomes can pseudofertilizezygotes. The latter die and are not available forfertilization by the HEG-carrying Y. Let u ¼ (1 � e)k bethe fraction of X chromosomes that avoid shredding.Assume that a fraction z of the remaining (1 � u)shredded X chromosomes pseudofertilize a zygote. Inthe absence of X shredding the Y chromosome couldexpect to fertilize 1

2 the zygotes, and with X shreddingbut not pseudofertilization this fraction goes up to 1/(1 1 u). However, if pseudofertilization occurs thefraction drops to 1=ð1 1 u 1 ð1� uÞzÞ. Clearly the ad-

vantage of X shredding disappears if z¼ 1 while for z , 1the HEG still spreads to fixation with the final sex ratiounaffected, though the speed of spread declines as z in-creases (Figure 9).

X-shredder escape mutants: Breaks in the X chromo-some may be repaired by ligation of the two ends (repairusing the homologous chromosome as a template is notof course possible in the heterogametic sex). If therepair regenerates the HEG recognition site, then this issimply equivalent to a reduction in the efficiency ofshredding. But if the recognition site is lost, and if therepaired chromosome suffers no or mild fitness costsafter repair, then an X-shredder escape mutant will havebeen generated.

To explore this for the case of an X shredder with asingle recognition site (k¼ 1) we model the dynamics offour kinds of chromosome: the wild-type sex chromo-somes (Y and X), Y chromosomes bearing a HEG (Yh),and X chromosomes bearing the escape mutant (Xm).Among Y chromosomes the frequency of Yh is q whilethe frequency of Xm among X chromosomes is jm insperm and jf in eggs. The genotype frequencies andfitnesses are given in the following tables:

Figure 8.—The effect of multiple mating and sperm com-petition on the rate at which an X-shredder HEG spreadsthrough the population. The precise assumptions madeabout the distribution of mating frequencies are describedin the text and the average number of matings per femaleis m. We assumed a cutting frequency (e) of 0.9, a HEG withone recognition site (k ¼ 1), and a HEG initial frequency of0.01.

Figure 9.—The effect of ‘‘pseudofertilization’’ on the rateat which an X-shredder HEG spreads through the population.It is assumed that a fraction z of sperm with cut X chromo-somes fertilize ova that subsequently die. We assumed a cut-ting frequency (e) of 0.7, a HEG with two recognition sites(k ¼ 2), and a HEG initial frequency of 0.01.

Malegenotypes XY XmY XYh XmYh

Frequency (1 � q)(1 � jf) (1 � q)jf q(1 � jf) qjf

Fitness 1 1 � sm 1 � sH 1 � sm

Femalegenotypes XX XXm XmXm

Frequency (1 � jm)(1 � jf)

jm(1 � jf) 1

(1 � jm)jf

jmjf

Fitness 1 1 � sf hf 1 � sf.


The subscripted s and h parameters describe the fitnesscosts of the different alleles and their pattern ofdominance. We assume that the costs of bearing aHEG arise directly from the X-shredding process.

Assuming that escape mutants arise with frequency g,the recurrences for gamete frequencies are

q9 ¼ qð2� 2sHð1� jf Þ � 2smjf � eð1� smÞð1� gÞjf Þ2� 2qsHð1� jf Þ � 2smjf � eð1� gÞð1� qð1� jf Þ � smjf Þ

j9m ¼2ð1� smÞjf 1 eð2qgð1� sHÞð1� jf Þ � ð1� smÞð1� gÞjf Þ

2� 2qsHð1� jf Þ � 2smjf � eð1� gÞð1 1 qð1� 2sHÞð1� jf Þ � smjf Þ

j9f ¼ð1� hf sf Þðjf 1 jmÞ � 2ð1� hf Þsf jf jm

2ð1� sf jf jm � sf hf ðjm 1 jf � 2jmjf ÞÞ: ð24Þ

The general solution is very complex and we focus onsome special cases. First, assume that there are no coststo HEG carriage (sH ¼ 0). If the escape mutant also hasno costs (sm ¼ sf ¼ 0), then it will always spread tofixation. If an escape mutant suffers fitness costs sM inthe homozygote and the hemizygote (sm ¼ sf ¼ sM) andhMsM in the heterozygote, then fixation of the escapemutant generally occurs below a critical threshold givenby the two inequalities,

sM ,eð1 1 gÞ

2� e 1 egand

eð1 1 gÞ � sMð3� 2sMÞð2� e 1 egÞsMðeð1 1 gÞ � ð2� sMÞð2� e 1 egÞÞ , hM:

ð25Þ

Fixation of the mutant restores equal sex ratios and theremaining HEG alleles in the population will then haveneutral dynamics because they have identical fitness tothe wild type. If the costs are above this threshold, thenthe HEG will spread to fixation with the populationremaining polymorphic for the escape mutant (Figure10). Thus a single HEG is least affected by resistantescape mutants when its target sequence is an essentialgene whose function is lost after repair (sM¼ 1, hM¼ 0).Similar dynamics are observed when the gene targetedby the HEG is essential only in males or in females.

Now suppose that there is a cost to X shredding(sH . 0), perhaps because sperm volume is reduced orthrough pseudofertilization (see above), and focus oncases where the target gene cut on the X chromosomemay be essential. If HEG costs exceed a threshold(sH . e(1 � g)/2) that depends on the frequency withwhich escape mutants arise, then the HEG always goesextinct and the sex ratio returns to equality. If the cost ofthe HEG is below this threshold, then the HEG becomesfixed and the predicted sex ratio depends on the fitnessof the escape mutant. If the escape mutant has wild-typefitness it spreads to fixation and equal sex ratio isrestored. But if repair gives rise to an escape mutantthat at least in some circumstances is lethal, then apolymorphism may result where the sex ratio shows anintermediate bias toward males. For instance, if themutant is dominant and lethal in females (sf¼ 1, hf¼ 1,

sm¼ 0), the sex ratio evolves toward 1/(2� e 1 eg), andat each generation a proportion eg/(1 � e 1 eg) of thefemales die before reproducing.

This analysis suggests that single X shredders, or Xshredders with single recognition sites (k¼ 1), are likelyto fail if cost-free escape mutants occur. However, theprobability of escape mutants of this type arising can bemarkedly reduced by using multiple HEGs or HEGs thatcut at multiple targets on the X chromosome. An escapemutant requires that the X chromosome be rejoinedwith the loss of the HEG recognition site at all k sitesbefore it can avoid further attack. We have not modeledthis scenario in detail but the main effect of using a HEGwith multiple shredding sites can be captured by re-ducing g, the rate at which escape mutants are gener-ated. Though formally for all g . 0 long-termpersistence of the HEG is not possible when escapemutants have no costs, for small g this may take a verylong time. A more detailed model of escape mutantevolution would need to take into account the fre-quency of X chromosomes that had lost the HEGrecognition site at some but not all sites. Also, for HEGstargeting recognition sites in multicopy genes, the pos-sibility of gene conversion and molecular drive wouldneed to be considered.

DISCUSSION

Parasitic or selfish genetic elements have clear attrac-tions as potential tools for population genetic engineer-ing and control since they are able to spread through apopulation even if they do not increase the fitness oforganisms carrying them—indeed, they may spreadeven if they cause some harm. Moreover, our ever-

Figure 10.—Examples of the spread of an X-shredder es-cape mutant. Three different assumptions about the fitnessof the escape mutant are made (solid lines, sM ¼ 0; dashedlines, sM ¼ 0.5; dotted-and-dashed lines, sM ¼ 1), and thesex ratio (shaded lines) and escape mutant frequency (amongmales, solid lines) are plotted. We assumed a cutting fre-quency (e) of 0.9, a HEG with a single recognition site (k ¼1), that the rate at which escape mutants are generated (g)is 0.01, and a HEG initial frequency of 0.01.


increasing understanding of the molecular basis of drivemeans that it is becoming feasible to synthesize artificialelements ourselves (Han and Boeke 2004; Adelman

et al. 2007; Chen et al. 2007). This article is a theoreticalinvestigation of the conditions under which one class ofparasitic genetic element, HEGs, might be useful forpopulation genetic engineering and in particular forapplications to control disease vectors or pests. The aimof the work presented here is to provide guidance indesigning efficient constructs with the lowest likelihoodof resistance arising.

Two logically different uses of HEGs are considered.First, we investigated how their intrinsic homing abilitymight be put to use. A HEG could be designed to targetan essential gene and hence to impose a genetic load ona population or in a vector to knock out a gene that isessential for disease transmission but nonessential forthe host. Second, we consider a HEG placed on the Ychromosome that recognizes and cuts one or more siteson the X chromosome. By reducing competition fromX-carrying sperm the driving Y chromosome spreads,causing a male-biased sex ratio that can severely reducepopulation growth rate.

Our models show that a HEG targeting an essentialgene can potentially cause a substantial reduction inpopulation fitness, while one that targets a gene whoseknockout has minor consequences for host fitnessquickly goes to fixation. These strategies rely on thebreak in the chromosome caused by the HEG beingrepaired by gene conversion with the homologous chro-mosome as the template. In our models the frequencyof homing by this mechanism is described by theparameter e that typically should be as high as possible.But repair by gene conversion is not the only possiblepathway, and experiments have shown that the fre-quency of different types of repair can depend upon thegenomic context. For example, if the cleavage site isflanked by direct repeats, then the single-strand anneal-ing (SSA) pathway usually seems to predominate, atleast in the premeiotic male germline. In this process allDNA sequence between the repeats is removed: thus theHEG site is lost but a functional gene is unlikely to bereconstituted. Thus SSA reduces the efficiency of hom-ing rather than giving rise to resistant alleles. In one setof experiments with Drosophila the SSA pathway wasused two-thirds of the time in the premeiotic malegermline with conversion accounting for only 10–20%of repairs (Preston et al. 2006). In other experiments,without direct repeats flanking the cleavage site, con-version repair was observed �85% of the time [exclud-ing repair of the sister chromatid, which preciselyregenerates the target site, producing a chromosomethat can simply be cut again (Rong and Golic 2003)].These experiments indicate that it will be important toavoid targets that are flanked by direct repeats. Thetiming of HEG expression can also be important: ifcleavage is late in spermatogenesis, then nonhomolo-

gous end joining (NHEJ) can predominate (Preston

et al. 2006). Here the ends of the chromosome aredirectly ligated without involvement of the homolog(and, as discussed below, with a high probability of theloss of the HEG recognition sequence), again notthe desired result. There may also be differences inthe frequency of conversion repair (e), depending uponwhere the target is located in the genome.

If the aim of the intervention is to impose a load onthe population, the models suggest that it is better totarget a gene that affects fertility rather than viabilityand that it is in general better for the target to berecessive (though in some cases maximum load isobtained when heterozygote fitness is somewhat re-duced, 0 , h , 0.5). In Drosophila there are thought tobe �3000 genes essential for viability and 50–100needed for female fertility, most of which are recessive(Ashburner et al. 1999, 2005). It is also possible toimpose a load if one targets male fertility. In Drosophilathere are two genes (misfire and sneaky) that, whendisrupted, have the effect of impairing embryo de-velopment after fertilization (Ohsako et al. 2003;Wilson et al. 2006; Smith and Wakimoto 2007). Misfireis needed for correct sperm head decondensation andsneaky is required for the proper breakdown of thesperm plasma membrane. Homologs of these genes inpest and vector species are potential targets. In princi-ple, if a gene were to affect both male fertility (in thisway) and female fertility, then targeting it could be moreefficient than knocking out an essential gene, but it isnot clear if any such genes exist in insects.

It may also be possible to target genes that have fitnessconsequences in only one sex or that have differenthoming frequencies in the two sexes. Targeting onlyfemales is advantageous as the reduced overall costsallow the HEG to reach higher equilibrium frequencies(and spread faster) and then impose a greater load onthe population. For most human disease vectors it is thefemale that requires a blood meal prior to ovipositionand transmits the pathogen, another advantage forpreferentially targeting this sex.

Different control sequences can be used to determinethe relative timing of homing and the expression of thegene targeted by the HEG. We find constructs are moreinvasive if HEG expression is after that of the targetgene, so that heterozygotes have more-or-less normalfitness. In many cases HEGs cannot spread if the fitnessconsequences of gene conversion are experienced bythe individuals in which it occurs, and the speed ofspread and eventual load of any HEG that has a sig-nificant fitness cost is lower if homing occurs beforeexpression of the target gene.

There is one circumstance in which the inability of anearly-acting HEG to spread may be a positive advantage.Successful control of several pests has been obtained bythe mass release of males that have been sterilizedusing chemicals or radiation (sterile insect techniques,


SIT). A development of this is to engineer condition-dependent dominant lethals that are mass reared in thepresence of an artificial repressor that inactivates thisgene (Thomas et al. 2000; Atkinson et al. 2007; Phuc

et al. 2007). If released into the environment in suf-ficient numbers they bring about population collapse bycompeting with wild-type males for mates that then failto produce offspring (in the absence of the artificialrepressor in the field). If the gene is only lethal forfemales (Thomas et al. 2000), this release of insects witha dominant lethal (RIDL) technique has the additionaladvantage that the male progeny of released insects cantransmit the female lethality to the next generation.Although in the absence of further releases the traitquickly disappears, its temporary persistence in theenvironment is an advantage of RIDL over SIT. A HEGthat could not spread through a population could becombined with a condition-dependent female-lethalconstruct in a ‘‘RIDL-with-drive’’ strategy, which wouldbe more effective than RIDL by itself (Thomas et al.2000), though again without long-term persistence.

The models show that for broad regions of parameterspace, typically involving HEGs with substantial fitnesscosts in the heterozygote state, the fate of the constructdepends on its initial frequency. Although these HEGscannot spread from rare, were they to be released insufficient numbers, the constructs would spread tofixation. The ability of many HEG constructs to spreadfrom rare, even in the presence of costs, distinguishesthem from a number of potential drive elements—forexample, medea elements, Wolbachia, and underdomi-nant chromosomes—that in the presence of costsrequire frequencies to exceed a threshold before spreadoccurs (Sinkins and Gould 2006). If a HEG constructwas developed and then found to spread only above amodest threshold frequency, it might still be useful as adrive mechanism. Currently, we see no particular advan-tages to such constructs and efforts should be directedat developing HEGs that can spread from rare.

For insect-transmitted diseases a different strategyinvolving homing is to target a gene whose disablementhas no or relatively low costs to the vector but that isessential for successful parasite transmission. SuchHEGs rapidly rise in frequency to fixation in the popu-lation and considerations such as whether the HEG actsbefore or after target gene expression have little or noeffect on whether it spreads. Currently, there is intensivestudy of mosquito gene products required by Plasmo-dium and other disease agents (e.g., Dinglasan et al.2007; Ecker et al. 2007) and measures of the costs to thevector of their knockout would be very interesting.

Any strategy that aims to impose a fitness load on apopulation inevitably leads to strong selection forresistance. For HEGs, the easiest way for resistance tooccur is for an allele to arise that codes for a functioninggene but that does not possess the HEG recognition site.This could arise through point mutation or, most likely,

through the chromosome break caused by the HEG notbeing rejoined by homologous repair but by NHEJ.Here we show that if an escape mutant has normal fit-ness, then it will spread and the HEG will eventuallydisappear. If the HEG has low fitness costs, then thespread of the escape mutant will occur slowly andsignificant short-term benefits in population manage-ment may still occur. But for HEGs whose primepurpose is imposing load any escape mutant that ariseswill quickly destroy any benefit.

It is thus critical to design a HEG where loss of therecognition site in the target gene also implies loss ofgene function. Some HEGs that target protein-codinggenes have evolved to ‘‘ignore’’ the third base silent sitesin the target sequence, so as to broaden their ownspecificity (Koufopanou et al. 2002; Kurokawa et al.2005). If this feature can be maintained in the engi-neered HEGs, then the most obvious sort of resistantmutation can be nullified. Loss of the recognition sitethat arises from NHEJ normally involves the deletion ofa short stretch of DNA, and hence engineering a HEG torecognize the DNA coding for an active site of anenzyme, or equivalent conserved motif, should ensureits loss causes nonfunctionality. Further strategies thatmight be explored include targeting multiple siteswithin the same gene simultaneously and targeting mul-tiple genes simultaneously. Akin to multiple-drug re-sistance, these measures might substantially delay theevolution of resistance, possibly allowing enough timefor local extinction to occur.

The second distinct way of using a HEG is to destroy Xchromosomes at male meiosis and so bias the sex ratio(Burt 2003), a strategy formally modeled here for thefirst time. This mechanism does not rely on homing perse but on the HEG providing an extra-Mendelianadvantage to its Y chromosome carrier. Naturally occur-ring driving Y chromosomes are known from twomedically important genera of mosquito, Aedes andCulex, which is encouraging for this strategy (Newton

et al. 1976; Sweeny and Barr 1978; Wood and Newton

1991; Cha et al. 2006). Nothing is known at the mo-lecular level of how they work, but cytologically drive isassociated with X chromosome breaks at male meiosis(Cazemajor et al. 2000; Windbichler et al. 2007). Thedegree to which breaking the X chromosome favors theY chromosome depends quite critically on the organ-ism’s biology. We show that if there are costs to reducedsperm production, or if multiple mating occurs andfertilization is a lottery, then the spread of the HEG-bearing Y may be slowed. Similarly, if damaged Xchromosomes ‘‘pseudofertilize’’ eggs, that is, if they com-pete with other sperm and create zygotes that then die,the HEG-bearing Y spread may be slowed and in the limiteven totally prevented. We lack information on theseissues for major vectors and pests and studies on thereproductive biology of a candidate species would needto be done in advance to assess the likely outcome of


adopting this approach. For example, if irradiation ofsperm (which causes chromosome breaks) leads to deathof zygotes and not sperm, as occurs in some species, thenthis would suggest pseudofertilization might occur.

An escape mutant on the X chromosome that de-stroys the HEG recognition site and does not dramati-cally reduce fitness would increase in frequency andultimately return the sex ratio to equality. We would alsoexpect a suppressor mutant on an autosome to spread ina similar manner and destroy the sex-ratio bias, thoughwe have not modeled this scenario. Again, as withclassical HEGs, the probability of resistance occurringcan be reduced by designing the HEG to cut multiplesites on the X chromosome. This is also advantageous asfor cutting frequencies (e) less than one multiple siteincreases the equilibrium sex-ratio bias. An example ofhow this might be implemented is provided by Anophelesgambiae, the most important vector of malaria in sub-Saharan Africa. In this species rDNA repeats are re-stricted to the X chromosome, and the I-PpoI homingendonuclease of Physarum slime molds recognizes andcuts a sequence within the gene (Windbichler et al.2007). Given that there are .100 copies of the rDNAgene (Collins and Paskewitz 1996), the evolution ofresistance will be a very rare event. Note that if theX-shredding HEG was placed on an autosome, it wouldstill disrupt X sperm but not spread. This strategy couldbe employed in a noninvasive pest or vector manage-ment campaign, although it may be necessary to makeHEG expression condition dependent (using a RIDL-like strategy) to enable efficient mass rearing.

Hybridsbetweenthe twomainstrategies—classicalhom-ing and biasing the sex ratio—might also be devised. Inthe major agricultural pest, the medfly (Ceratitis capitata),XX individuals require at least one functional copy of theautosomal gene tra (transformer) to be phenotypicfemales (Pane et al. 2005). If a HEG was engineered toknock out the tra gene, it would drive the Y chromosometo extinction so that all males would be XX tra� tra� andthe sex ratio would be (1 � e)/2, where e is the homingefficiency (detailed modeling not shown).

Throughout this article we have measured the effectof HEGs designed to reduce pest or vector populationdensities by what we have called ‘‘HEG load,’’ thedecrease in population fitness caused by reductions inviability and fertility. The effect of sex-ratio biases onpopulation fitness can be measured using the samemetric. It is critically important to stress that how HEGload translates into real population reductions dependson the particular demography and population dynamicsof the species involved and also the context in whichpest or vector management is sought. For example, ifpopulation control is attempted for a seasonally invasivespecies or one with boom-and-bust population dynam-ics, then the most important concern may be to reducemaximum population growth rate. In these circum-stances HEG load will map directly onto population

reduction and be a simple means of comparing differ-ent strategies. However, if a population is subject todensity-dependent mortality, then it is possible for asubstantial HEG load to be present but only modest oreven no reductions in pest or vector numbers to be ob-served. In theses circumstances comparing HEG loadprovides a ranking of the effectiveness of differentstrategies, though not a quantitative metric. In this articlewe have deliberately not extended the models developedhere beyond frequency into the density domain as webelieve analyses based on particular systems and specificbiologies are the most effective means of exploring theseissues. We are also aware that our models have beenpurely deterministic and have assumed discrete gener-ations: stochastic models with overlapping generationswill be valuable in providing guidance about the num-bers of HEG-bearing insects that should be released in anactual management program.

The aim of the work described here is primarily tohelp in the design of HEGs that might be used in pestand vector management, especially involving insects.The success of a HEG-based strategy will depend on (i)the discovery of suitable recognition sites or the success-ful engineering of HEGs to recognize new sites, (ii) theability of HEGs to cut and home in insects, (iii) thedesign of constructs and implementation strategies toavoid or delay resistance, and (iv) the regulatory ac-ceptance of a pest or vector management strategy thatinvolves genetic manipulation. We have mentionedabove one significant recognition site in a malaria vectorthat can be cut by a naturally occurring HEG, whilethere has been substantial recent progress on reengin-eering HEGS to recognize novel sites (Ashworth et al.2006; Arnould et al. 2007). Preliminary work suggestsartificially introduced HEGs can function in mosquitocells (Windbichler et al. 2007). Much remains to bedone but we believe HEGs offer exciting prospects fornovel approaches to pest and vector management andthat consideration of their population genetics will beimportant in guiding this development.

We are grateful to Michael Ashburner, Andrea Crisanti, Fred Gould,Penny Hancock, Ray Monnat, Samantha O’Loughlin, Steve Russell,Barry Stoddard, and an anonymous referee for helpful discussions.This work was funded by a grant from the Foundation for the NationalInstitutes of Health through the Grand Challenges in Global Health.

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