Kin Selection–Mutation Balance: A Model for the Origin, Maintenance, and Consequences ofSocial CheatingAuthor(s): J. David Van Dyken, Timothy A. Linksvayer, Michael J. WadeSource: The American Naturalist, Vol. 177, No. 3 (March 2011), pp. 288-300Published by: The University of Chicago Press for The American Society of NaturalistsStable URL: http://www.jstor.org/stable/10.1086/658365 .Accessed: 21/02/2011 14:54

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vol. 177, no. 3 the american naturalist march 2011

Kin Selection–Mutation Balance: A Model for the Origin,

Maintenance, and Consequences of Social Cheating

J. David Van Dyken,1,* Timothy A. Linksvayer,2,† and Michael J. Wade1

1. Department of Biology, Indiana University, Bloomington, Indiana 47405; 2. Centre for Social Evolution, Department of Biology,University of Copenhagen, Universitetsparken 15, DK-2100 Copenhagen, Denmark

Submitted July 28, 2010; Accepted November 15, 2010; Electronically published February 9, 2011

abstract: Social conflict, in the form of intraspecific selfish “cheat-ing,” has been observed in a number of natural systems. However,a formal, evolutionary genetic theory of social cheating that providesan explanatory, predictive framework for these observations is lack-ing. Here we derive the kin selection–mutation balance, which pro-vides an evolutionary null hypothesis for the statics and dynamicsof cheating. When social interactions have linear fitness effects andHamilton’s rule is satisfied, selection is never strong enough to elim-inate recurrent cheater mutants from a population, but cheater lin-eages are transient and do not invade. Instead, cheating lineages areeliminated by kin selection but are constantly reintroduced by mu-tation, maintaining a stable equilibrium frequency of cheaters. Thepresence of cheaters at equilibrium creates a “cheater load” thatselects for mechanisms of cheater control, such as policing. We findthat increasing relatedness reduces the cheater load more efficientlythan does policing the costs and benefits of cooperation. Our resultsprovide new insight into the effects of genetic systems, mating sys-tems, ecology, and patterns of sex-limited expression on social evo-lution. We offer an explanation for the widespread cheater/altruistpolymorphism found in nature and suggest that the common fearof conflict-induced social collapse is unwarranted.

Keywords: kin selection, cheating, conflict, mutation, load, policing.

Introduction

Intraspecific cooperation and altruism are prevalent in na-ture, with examples spanning the continuum of biologicalcomplexity from unicellular prokaryotes and eukaryotesto plants and animals. Altruism is believed to play anessential role in the evolutionary transitions that broughtabout the origin of the genome, multicellularity, and so-cieties (Buss 1987; Maynard Smith and Szathmary 1998;Michod 1999), making altruism a fundamental and ubiq-uitous feature of life. Yet explaining the existence of al-

* Corresponding author; e-mail: jdvandyk@indiana.edu.†

Present address: Department of Biology, University of Pennsylvania, Phil-

adelphia, Pennsylvania 19104.

Am. Nat. 2011. Vol. 177, pp. 288–300. � 2011 by The University of Chicago.

0003-0147/2011/17703-52339$15.00. All rights reserved.

DOI: 10.1086/658365

truism has posed a challenge to evolutionary theory. Theproblem is that altruists always decline in frequency withingroups as they are outcompeted by selfish “cheaters,” whoreap the benefits of altruism from others but do not paythe costs (Haldane 1932; Wright 1945). A solution to thisproblem is formulated by kin selection theory, whichshows that selection within groups favoring cheaters canbe counteracted by selection among groups favoring al-truists (Darwin 1859; Hamilton 1975; Wade 1980, 1985;Queller 1992). The response to among-group selectionagainst cheaters exceeds the opposing response to within-group selection favoring cheaters when Hamilton’s rule,

, is satisfied (Hamilton 1975; Wade 1980), wherebr � c 1 0b is the benefit to recipients of altruism, r is the relatednessof altruists to recipients, and c is the fitness cost of actingaltruistically (Hamilton 1963, 1964). The condition for theinvasion and spread of cheaters, , is the inversebr � c ! 0of Hamilton’s rule (Wade and Breden 1980). BecauseHamilton’s additive linear model does not permit internalequilibria (Michod 1982), a polymorphism with coexis-tence of cheaters and cooperators is not possible.

Contrary to these theoretical predictions, natural popu-lations are commonly polymorphic for cooperative andcheating phenotypes. For example, cheater/cooperator poly-morphisms have been observed in nature in the slime moldDictyostelium discoideum (Buss 1982; Strassmann et al.2000), quorum-sensing systems of pathogenic bacteria (Dig-gle et al. 2007; Sandoz et al. 2007), nitrogen-fixing bacteriaof the genus Bradyrhizobium (Sachs et al. 2010), endomy-corrhizal fungi (Denison et al. 2003), clinical isolates of theopportunistic pathogen Pseudomonas aerugenosa (Lee et al.2005), cancerous cells in multicellular organisms (Buss1987), and egg-laying workers and queen-biased develop-ment (i.e., “royal cheats”; Hughes and Boomsma 2008) inthe eusocial Hymenoptera (Starr 1984; Ratnieks andVisscher 1989; Wenseleers and Ratnieks 2006).

The effort to bridge the gap between theory and datahas been hampered by the absence of a rigorous, formalpopulation genetic model of cheating. In classical popu-lation genetics, there are two primary mechanisms that

Kin Selection–Mutation Balance 289

maintain polymorphism in natural populations (Lewontin1974): frequency-dependent selection and recurrent mu-tation. By analogy, there are two approaches to reconcilingtheory with data in social evolution. In the first, one couldalter the underlying model of social interactions substan-tially by making fitness nonlinear, thus abandoning thestandard form of Hamilton’s rule (smith et al. 2010). Likefrequency dependence in classical population genetics,nonlinear social interactions can maintain a stable poly-morphism of cooperators and cheaters (Doebeli et al. 2004;smith et al. 2010). Nonlinear frequency-dependent socialinteractions arise when the social benefit contributed byan altruist depends on the number of altruists in a group.For example, if doubling the number of altruists in a groupalways exactly doubles the total benefit of altruism, thensocial interactions are said to be linear. However, if thetotal benefit is more than doubled (or less than doubled)or if it is doubled only when altruists are at a specificfrequency within the group, then social interactions arenonlinear. A disadvantage of this approach is that the num-ber of possible nonlinear fitness functions is vast, makingit difficult to extract general predictions. Further, thismodel does not provide a satisfactory evolutionary nullhypothesis (sensu Lynch 2007), because selection is re-quired to actively maintain the cheater/altruist poly-morphism.

Here we suggest a simple but overlooked alternativeexplanation for the statics and dynamics of social conflictin nature. Specifically, we propose taking the canonicalmodel of social evolution introduced by Hamilton (1964)and adding a previously neglected but fundamental evo-lutionary force: mutation. In this case, polymorphism ismaintained by recurrent mutation despite linear, purifyingkin selection. Mutation is a necessary, ubiquitous, andinexorable evolutionary force that has been widely ne-glected in social-evolution theory (although see Michod1996, 1997, 1999; Michod and Roze 2000, 2001). Mutationis the ultimate source of genetic variation and serves as acentral parameter in the modern study of genetic variationat the molecular level, where sequencing technology isproviding an ever-increasing supply of data. Thus, a mu-tational theory of social evolution provides both a system-atic theory of cheating and allows theoretical extension toaddress patterns of gene sequence variation (Cruickshankand Wade 2008; Linskvayer and Wade 2009; Van Dykenand Wade 2010), bringing the population genetics of socialevolution into the postgenome era.

The use of Hamilton’s additive linear model with re-current mutation as an evolutionary null model for themaintenance of polymorphism in social traits parallels theuse of purifying selection with recurrent mutation as anevolutionary null model for nonsocial traits (Kimura 1983;Lynch 2007). By “evolutionary null,” we mean a hypothesis

whereby adaptive evolution is not responsible for activelymaintaining cheaters or creating the patterns of social con-flict observed in nature. This is not necessarily a claimabout nature but rather an empirical tool for distinguish-ing among alternative hypotheses, whereby the most par-simonious (i.e., nonadaptive) explanation is taken as thenull and can be rejected by data.

We show that the steady influx of cheater mutants intopopulations is balanced at equilibrium by their removalby kin selection in a process that we call the “kin selection–mutation balance.” We analyze this process for differentgenetic systems, mating systems, patterns of sex-limitedexpression, and scales of density regulation, and we in-vestigate its long-term evolutionary consequences.

Results

The Model

Consider a population with discrete, nonoverlapping gen-erations where two alleles, A and a, segregate at a locusthat controls a cooperative trait. The population is sub-divided into cooperative groups, wherein interactions oc-cur at random with respect to genotype (i.e., indiscrimi-nate, whole-group sociality). These groups are equivalentto “trait groups” (Wilson 1975). We consider haploid, dip-loid, and haplodiploid populations and assume that thereis random mating (nonrandom mating and dominancewill be considered elsewhere [J. D. Van Dyken and M. J.Wade, unpublished manuscript]). In diploids and haplo-diploids, homozygous AA individuals are “altruists,” aahomozygotes are socially defective “cheaters,” and Aa het-erozygotes express the altruist and cheater phenotypes withprobability 1/2 each (i.e., the alleles act additively). Cheat-ers are socially defective individuals that gain the benefitsof altruism received from altruists within their group butdo not suffer a cost to individual fitness because they donot express the social trait. In haploids, A individuals arealtruists and a individuals are cheaters.

Altruists suffer a relative decrease in fitness by anamount c, but they increase the fitness of each individualk with which they interact by an amount bk. When altruistsform a sterile caste, as in the eusocial Hymenoptera, thedirect fitness cost c can be interpreted as the probabilitythat an individual possessing an altruist genotype becomesa sterile worker (which then has a direct fitness of 0). Inthis case, a colony fixed for an altruism allele will be com-posed of some proportion c of sterile workers. The totalgross benefit of altruism received by an individual in agroup is , where N is the number of individualsNb p bk

in a group, unless expression of altruism is sex limited, inwhich case the total benefit is proportional to the numberof the altruistic sex in a group. (Here, we are assuming

290 The American Naturalist

that benefit accrues to the performer as well as to othergroup members. If it is assumed that it does not, then theequations below are only modestly altered; see Van Dyken2010.) Costs and benefits are measured with respect to areference allele, in this case the a allele, such that the fitnesscosts and benefits of the AA homozygotes are equal to thedifference between its absolute costs and benefits and thoseof aa homozygotes ( ; ). Thisc p c � c b p b � bAA aa k k, AA k, aa

allows us to investigate mutant alleles expressing an ar-bitrary degree of social defect rather than simply purealtruists and pure cheaters. For example, we can investigatecheaters that provide less benefit to the group ( ) orb 1 0k

cheaters that incur lower costs ( ). Both cases are par-c 1 0ticularly important in tests of the theory using DNA se-quence data, where mutational effect sizes may vary amongnucleotides within a sequence.

Allele Frequency Recursion

We use the hierarchical Price equation (Price 1972; Ham-ilton 1975; Wade 1985; Rice 2004) and Hamilton’s linearfitness model (Hamilton 1964; Wilson 1975; Wade 1978,1979; Wade and McCauley 1980) to derive an expressionfor the change in frequency of a cheater allele under kinselection (Van Dyken 2010). In the simplest case of equalploidy and expression in both sexes (which we relax be-low), it can be shown that

bV � cVb tDq p � (1)

W

(app. A), where q is the global frequency of the a “cheater”allele, Vb is the variance of mean frequency of the a alleleamong groups, Vt is the variance of allele a among indi-viduals over the global population, and is the meanWfitness of the metapopulation.

Note that cheaters are selected against ( ) whenDq ! 0, which is a restatement of Hamilton’s rule withbV 1 cVb t

r equal to , the fraction of the total genetic variationV /Vb t

that is among groups (Wade 1980). When Hamilton’s ruleis satisfied, cheaters are equivalent to deleterious mutants,even though they are favored within groups. If ,bV ! cVb t

then cheating is net advantageous (after summing selectionwithin and among groups) and will spread throughout thepopulation.

Equation (1) applies equally to haploids and diploidsunder the current assumptions, although the value of thevariance components depends on the genetic system(Wright 1951, 1965). For diploids,

V p pqf , (2a)b st

pq(1 � f )itV p , (2b)t 2

where is the frequency of the altruism allelep p (1 � q)A; fit is the total inbreeding coefficient, which accounts forboth population subdivision (fst) and nonrandom matingwithin groups (fis). These components are related by theequation (Wright 1951). While1 � f p (1 � f )(1 � f )it st is

we assume throughout that there is random mating( , so that ), this general form is preferablef p 0 f p fis it st

because it allows us to link the variance components ofhaploids and diploids. This is done by noting that f pis

in haploids (Hamilton 1972). Substituting intof p 1it

equations (2a) and (2b), we obtain the haploid variances,

V p pqf , (2c)b st

V p pq. (2d)t

For all genetic systems, the average relatedness withingroups (i.e., the regression coefficient of relatedness) isexpressed as

Vbr p (3)Vt

(Crow and Kimura 1970; Hamilton 1975; Michod andHamilton 1980). In haplodiploids (see below and app. B),the diploid sex obeys equations (2a) and (2b) while thehaploid sex obeys equations (2c) and (2d).

Importantly, identity coefficients (fst and r) are not in-fluenced by allele frequency (Wright 1951, 1965; Rousset2004). This allows us to solve for the equilibrium allelefrequency across many different metapopulation geneticstructures. In addition, identity coefficients will typicallybe independent of mutation rate, as long as the mutationrate is small relative to the rate of migration (Wright 1951,1965; Rousset 2004). When mutation is sufficiently high,it can contribute to an increase in variance within groups,which drives a decline in values of fst and r. Here we limitour explicit attention to the common case in which dis-persal is much higher than mutation, although our modelis general enough to apply to situations of low dispersaland high mutation.

With recurrent mutation, the A allele is mutated to aat a rate m, and a is mutated to A at a rate u. It is reasonableto assume that spontaneous mutations are more likely toknock out or reduce cooperative allele function than toimprove it, such that the likelihood of creating an altruisticallele by mutation, u, is negligible. This is justified by nat-ural and laboratory populations, where cheaters typicallyarise when cooperation alleles sustain loss-of-function ordiminution-of-function mutations (Foster et al. 2007). In-deed, deleting or otherwise impairing “cooperation” genesis a common method for creating cheater strains for usein social-evolution experiments (reviewed in Foster et al.2007). Adding mutation to equation (1) gives us the ex-pression for change in frequency of a cheater allele under

Kin Selection–Mutation Balance 291

selection and mutation for the additive, linear model withequal ploidy and expression in both sexes (these latter twoassumptions will be relaxed for haplodiploidy below):

bV � cVb tDq p � � pm. (4)

W

Kin Selection–Mutation Balance

Haploid Populations. From equations (2c), (2d), and (3),we find that for haploids, . Substituting into equa-r p fst

tion (4) and rearranging, we find the change in frequencyof the cheater allele under kin selection and mutation tobe

pq(br � c)Dq p � � pm. (5)

W

Setting equation (5) equal to 0 and assuming weak selec-tion, such that , we can solve for the equilibriumW ∼ 1frequency of cheaters at the kin selection–mutation bal-ance for haploids as

mq � . (6)

br � c

Diplodiploid Populations. Substituting equations (2a) and(2b) into equation (4), we find that

pq[2bf � c(1 � f )]st stDq p � � pm. (7)

2W

From equations (2c), (2d), and (3), the relatedness in dip-loid populations with no inbreeding ( ) isf p 0 r pis

. Substituting r into equation (7) and re-2f /(1 � f )st st

arranging, we find that

pq(1 � f )(br � c)stDq p � � pm. (8)

2W

Assuming that , setting in equation (8), andW ∼ 1 Dq p 0solving for q, we find the equilibrium frequency of cheateralleles to be

2mq � . (9)

(1 � f )(br � c)st

This is the kin selection–mutation balance for diploids.The factor 2 in the numerator reflects the fact that thecheating mutation has been assumed to act additively( ). When there is no population structure (h p 1/2 f pst

), equation (9) retrieves the mutation-selection bal-r p 0ance for an additively acting locus on a direct-effect mu-tation (Muller 1950), and when , it retrieves theb p 0mutation-selection balance for direct-effect loci in struc-

tured populations under hard selection (Whitlock 2002).Note, too, that for , the equilibrium frequency off ! 1st

cheaters is always greater for diploids than for haploidsbecause of the factor in equation (9).2/(1 � f )st

Haplodiploid Populations. In haplodiploid species, socialtraits are commonly expressed by only a single sex, typi-cally females (reviewed in Linksvayer and Wade 2005). Inthe eusocial Hymenoptera (including ants and some beesand wasps), there is a division of labor between fully fertile(i.e., “queen”) and nonreproductive helper (i.e., “worker”)females, while all males are reproductive (Anderson et al.2008; Schwander et al. 2010). We focus especially on thiscase of female-limited expression of sib care because theeusocial Hymenoptera are well-established model systemsfor studying social evolution. In this case, the cost of help-ing is experienced only by worker females, and the benefitaccrues to fully fertile sibling males and females. Mutationsoccur in both sexes but are expressed only in females.

With haplodiploidy, allele frequency change is measuredseparately in males and females and then averaged, weight-ing males by a factor of 1/3 and females by 2/3, which isthe “reproductive value” of each sex, defined as the as-ymptotic proportion of genes in future generations con-tributed by each sex (Crow and Kimura 1970; Taylor 1988).In appendix B, we show that the allele frequency recursionunder the present assumptions is given by

pq(1 � f )[bs r � (2/3)c]st, f fDq p � � pm, (10)

2W

where fst, f is the value of fst taken among females in themetapopulation and , wherer p (1/3)r � (2/3)r r pfm ff fm

is the regression coefficient of related-Cov (p , p )/V(p )b f m t f

ness of females to males and is the re-r p V (p )/V(p )ff b f t f

gression coefficient of relatedness among females. Thevalue sf is the total proportion of females in a group( ). This includes both fully fertile and steriles p N /Nf f

females (i.e., both queens and workers) and is not to beconfused with the operational sex ratio, which generatesFisherian sex ratio selection (Fisher 1958), opposing a de-viation from the optimal sex ratio. For simplicity, ourmodel does not correct for deviations from an equal re-productive sex ratio, which can be done with methodsdiscussed by Taylor (1988) and Grafen (1986). We assumethat the sex ratio is at its equilibrium and is determinedby factors extrinsic to our model. Equation (10) showsthat the inclusive fitness effect is greater for higher sf (afemale-biased sex ratio), which simply means that the col-ony has higher viability when there are more female al-truists contributing to group benefit.

From equation (10), we find the kin selection–mutationbalance to be

292 The American Naturalist

Figure 1: Kin selection–mutation balance as a function of cheaterbenefit (altruist cost) c for different genetic systems. Parameter values:

and . In A, social groups are families with single-�3b p 0.1 m p 10mated foundresses, while in B, foundresses are multiply mated. Whilenot shown here, haploidy has a vertical asymptote at (i.e.,c p 0.1where ). Vertical asymptotes indicate where the inclusive fitnessc p beffect is equal to 0.

2mq � . (11)

(1 � f )[bs r � (2/3)c]st, f f

Note the similarity to equation (9), with fst, f here insteadof fst and the factors sf modifying the total benefit and2/3 modifying the cost; all three differences reflect theeffect of limiting the expression of social behavior to fe-males, which weakens selection (Wade 1998) and inflatesthe frequency of cheaters relative to expression in bothsexes.

Comparison of Genetic Systems atKin Selection–Mutation Balance

Ever since Hamilton (1964) suggested that haplodiploidypromotes the evolution of altruism, there has been intenseinterest in determining how genetic system affects socialevolution (e.g., Reeve 1993; Reeve and Shellman-Reeve1997; Wade 2001; Linksvayer and Wade 2005). In orderto compare the equilibrium frequency of cheaters amongthe three genetic systems modeled above, we first comparethe case where social groups are composed of siblings inmonogamous, single-mother (queen) families (fig. 1A). Inthis case, relatedness differs between genetic systems. Inhaplodiploids, following Grafen (1986) and Bourke andFranks (1995), the two regression coefficients of related-ness for full sibs are and , so thatr p 1/2 r p 3/4fm ff

for haplodiploids under the current assumptions.r p 2/3Relatedness is more straightforward in haploids and dip-loids, with (we assume asexual haploids) andr p 1 r p

, respectively. Also, with these assumptions and apply-1/2ing equations (2a), (2b), and (3), we find that f p 1/3st

in diploids, and in haplodiploids among females we findthat . In order to compare diploidy and haplo-f p 3/5st, f

diploidy on an equal footing (i.e., to not confound geneticsystem with sex-limited expression), we also consider dip-loidy with female-limited expression of altruism. Analyt-ical results for this case are given in appendix B and areplotted in figure 1.

Figure 1A shows that the expected frequency of cheatersis lowest for haploids and highest for diploids with female-limited expression of altruism. Interestingly, haplodiploidywith female-limited expression, as found in the eusocialHymenoptera, leads to a greater frequency of cheaters thandiploidy with expression in both sexes. This is a conse-quence of relaxed selection on cheating mutations due tofemale-limited social expression (Wade 1998; Linksvayerand Wade 2009; Van Dyken and Wade 2010). This hasimportant implications, as most cases of extreme diploidsociality (e.g., naked mole rats and termites) have bothmale and female expressions of altruism (Reeve and Shell-man-Reeve 1997), while in haplodiploids altruism is typ-ically female limited (Linksvayer and Wade 2005). This

clearly demonstrates how an advantage of haplodiploidyfor the evolution of altruism is largely negated by sex-limited gene expression.

Polyandry (i.e., multiple mating by foundresses) greatlyincreases the frequency of cheaters at equilibrium (fig. 1B).When hymenopteran queens mate multiply, relatedness offemales within a colony decreases rapidly with the numberof sires, from to , while relatedness ofr p 3/4 r p 1/4ff ff

females to males remains unchanged at a value of r pfm

. The total relatedness in our model for haplodiploids1/2under multiple mating, then, is . The relatednessr p 1/3in a diploid family with a single, multiply mated foundressis . Likewise, fst values are altered accordingly forr p 1/4each of these cases.

The kin selection–mutation balance draws an analogybetween cheating and deleterious mutation at nonsocialloci. Without mutation, the condition for an increase in

Kin Selection–Mutation Balance 293

frequency of the altruism allele in haploids and diploidsis , the familiar Hamilton’s rule (Hamilton 1964).br � c 1 0In haplodiploids with female-limited expression of altruism,this condition becomes . If these conditionsbs r � (2/3)c 1 0f

hold, then there will be no cheaters in a population whenmutation is absent. If , then the right-hand sidebr � c ! 0of equations (5), (8), and (10) will be positive and cheatingwill be favored by natural selection ( ). By analogyDq 1 0with the classic recursion of allele frequency for alleles withdirect effects on viability (Haldane 1932), we see that

or , also called the “inclusive fitness ef-br � c bs r � (2/3)cf

fect” of the altruism allele (Hamilton 1964; Michod 1982),is equivalent to the net selection coefficient (summed acrossboth levels of selection) favoring the altruism allele or,equivalently, to that opposing the cheating allele. Thus, ifHamilton’s rule is satisfied, then cheaters are simply dele-terious mutants opposed by net selection (see also Wadeand Breden 1980).

Hamilton’s Rule with Mutation

Equations (6), (9), and (11) demonstrate that despite sat-isfying Hamilton’s rule (i.e., ), cooperative allelesbr � c 1 0may nevertheless be rare in the population (see also fig.1). As the inclusive fitness effect ( or )br � c bs r � (2/3)cf

approaches 0, the cheater allele approaches fixation in thepopulation because of recurrent mutation pressure. Ham-ilton’s rule, then, does not preclude the existence of cheat-ers in an otherwise cooperative population, nor does itpredict the frequency of cooperators or cheaters in a pop-ulation. When cheaters arise at a constant rate via recur-rent mutation, the inclusive fitness effect must be suffi-ciently greater than 0 for cooperation to be common atequilibrium.

We suggest a corollary to Hamilton’s rule that accountsfor mutation. Cooperation will be “common” in a pop-ulation when , where � is an arbitrary cutoffq ! 1 � �frequency of cooperators such that the frequency of co-operators exceeds this cutoff when

mbr � c 1 , (12a)

1 � �

2mbr � c 1 , (12b)

(1 � f )(1 � �)st

2 2mbs r � c 1 , (12c)f 3 (1 � f )(1 � �)st, f

for haploids, diploids, and haplodiploids, respectively.These conditions predict the evolutionary stability of

cooperation in the face of recurrent cheating mutations.Note that these conditions are always more stringent thanHamilton’s rule. If explaining the prevalence or stability

of cooperation/altruism within a species is desirable, equa-tions (12) provide the appropriate reformulation of Ham-ilton’s rule.

Note that equations (12) cannot be satisfied if .� p 1This demonstrates that at equilibrium, kin selection isnever strong enough to eliminate all selfish cheaters froma population. Yet the introduction of cheaters into a pop-ulation by mutation implies neither the destabilization ofcooperation nor the spread of cheaters throughout theglobal population, as is frequently assumed (see Haldane1932; Maynard Smith 1964; Hamilton 1971; West et al.2006a, 2006b).

The Cheater Load

The cheater load is the reduction in mean individual (orequivalently, mean group) fitness due to the presence ofcheating mutants (Travisano and Velicer 2004). Eventhough cheaters may not spread to fixation, their ineluc-table presence via recurrent mutation generates importantlong-term evolutionary consequences.

The mean fitness in a population in the current modelat equilibrium is for haploids andˆW p 1 � (b � c)(1 � q)diploids and for haplodip-ˆW p 1 � [bs � (2/3)c](1 � q)f

loids. As long as , which is required for the cooper-b 1 cative trait to persist, the mean fitness is maximal when thecooperative allele is fixed ( ), demonstrating that theq p 0presence of cheaters causes a cheater load that reducesmean fitness by the amount (or byˆ(b � c)q [bs �f

for haplodiploids).ˆ(2/3)c]qThe cheater load is a function of both the reduction in

mean cost and the loss of benefits, because cheaters neitherbear the costs nor contribute benefits. The mean fitnessin the population is not explicitly dependent on populationstructure (fst and r do not appear in the expression for

), although it is implicitly dependent through the effectWof population structure on equilibrium cheater frequency,

(eqq. [6], [9], [11]).qThe cheater load can be quantified, by analogy to other

types of genetic load (Crow 1970; Charlesworth andCharlesworth 1998), as the difference between the maxi-mum attainable mean fitness and the observed mean fit-ness, . Substituting and equations (6),L p W � W Wmax

(9), and (11) into L, we find

m(b � c)L p , (13a)haploid br � c

2m(b � c)L p , (13b)diploid (1 � f )(br � c)st

2m[bs � (2/3)c]fL p . (13c)haplodiploid (1 � f )[bs r � (2/3)c]st, f f

294 The American Naturalist

Figure 2: Cheater load as a function of relatedness for diploids, withparameter values or 0.5, where is the cost/benefitl p 0.1 l p c/bratio, and . The curves for other genetic systems are omitted�3m p 10for clarity, but the relationship between genetic systems in terms ofrelative magnitude of the cheater load follows that for the equilibriumcheater frequency in figure 1. Note that the cheater load decreasesas relatedness increases.

When , equations (13) retrieve the genetic loadb p 0for direct-effect genes in structured populations underhard selection (Whitlock 2002). When ,b p f p r p 0st

these equations reduce to the classic result for genetic loadon direct-effect genes in unstructured populations (Muller1950; Werren 1993), as expected (although our haplodip-loid model accounts for female-limited expression, whilethe previous results did not).

The ratio in equations (13) is the ratio(b � c)/(br � c)of the group-selection gradient to the total-selection gra-dient. Whenever , this ratio is greater than 1, increas-r ! 1ing the load. This can be viewed as a consequence of theadditional mean fitness–reducing effect of selection withingroups that generates a conflict between levels of selection(Michod 1999). The apparent dependence of the load onabsolute fitness can be removed by defining the parameter

, which then leads tol p c/b

m(1 � l)L p , (14a)haploid r � l

2m(1 � l)L p , (14b)diploid (1 � f )(r � l)st

2m[s � (2/3)l]fL p , (14c)haplodiploid (1 � f )[s r � (2/3)l]st, f f

under the constraint that , which is the condition forr 1 l

Hamilton’s rule to hold (Hamilton 1963, 1964).Like the mutation load on direct-effect genes (Muller

1950), the cheater load does not depend on the value ofthe fitness effect of mutations. However, unlike the classicresult, the mutation load on social loci depends on therelative magnitudes of direct and indirect effects, throughl, and on the population structure, through fst and r (fig.2). When , there is no levels-of-selection conflictf p r p 1st

to further reduce mean fitness. Mutations that arise arelocalized within clonal groups of mutants, so that selectionis equivalent to selection among “individuals,” which is whyequations (14) retrieve the classic results for individual-leveltraits when . Whenever , the mutation load isr p 1 r ! 1always greater for social loci than for nonsocial loci becauseof the conflict between the levels of selection.

The Targets of Load Modification

Because the cheater load reduces fitness, we might expectmodifiers of the cheater load to arise and be favored byselection. Mutual policing is one such mechanism ofcheater control (Ratnieks 1988; Frank 1995; Travisano andVelicer 2004; Brandvain and Wade 2007). The canonicalexample of policing is the enforcement of worker sterilityin the eusocial Hymenoptera (Ratnieks 1988), where un-fertilized “sterile” workers can cheat by laying eggs that

develop into sons (arrhenotoky) or daughters (thelytoky).This is thought to reduce the colony-beneficial produc-tivity of the cheater, as such cheaters have been found tospend less time foraging or in colony defense, both ofwhich cause a reduction in colony fitness (West-Eberhard1975; Cole 1986; Hartmann et al. 2003; Dampney et al.2004; Dijkstra and Boomsma 2006).

In general, for a given mutation rate, increasing relat-edness (r) and reducing the ratio of costs and benefits (l)are the only ways to reduce the cheater load. The effecton the cheater load of a small change in r and l is foundby taking the partial derivative of load with respect to rand l, respectively. For haploids, we have

�L m(1 � l)p � , (15a)

2�r (1 � r)

�L mp . (15b)

�l 1 � r

Equations (15) show that the load is reduced by increasingr and/or by lowering l and that increasing relatedness hasa much larger effect on load reduction than does modi-fication of costs and benefits, which is true for all valuesof r and l that satisfy Hamilton’s rule.

Modifiers of relatedness might act to affect the numberof individuals founding a new group, the number of malesinseminating a female that founds a colony, the migrationrate among groups, or kin recognition. Alternatively, mi-crobes may secrete anticompetitor toxins that prevent theinvasion of a clonal group by outsiders (“xenophobia”; seeTravisano and Velicer 2004). Modifiers of l include po-

Kin Selection–Mutation Balance 295

Figure 3: Effect of kin competition (i.e., soft selection) on the equi-librium frequency of cheaters at kin selection mutation balance. Pa-rameter values are , , , and . This�3b p 0.4 c p 0.1 r p 0.5 m p 10curve represents the effect for diploids, which is qualitatively thesame for all other genetic systems.

licing behavior, which diminishes the detrimental effectthat cheaters have on group fitness (Ratnieks 1988; Frank1995; Travisano and Velicer 2004). According to equations(15), policing should evolve only when there are con-straints (genetic, developmental, ecological, or phyloge-netic) that prevent modification of relatedness.

Kin Competition

Kin competition (i.e., soft selection) occurs when localdensity regulation diminishes the expression of differentialgroup productivity, resulting in a diminished response togroup selection (Van Dyken 2010; see app. A). Kin com-petition has been shown to be a potentially severe con-straint on the evolution of altruism (Grafen 1984; Wade1985; Kelly 1992, 1994; Taylor 1992; Wilson et al. 1992;Frank 1998). We would like to know how kin competitionaffects the kin selection–mutation balance and the cheaterload. Following Van Dyken (2010; see app. A), we findthat the equilibrium frequency of cheaters under an ar-bitrary intensity of kin competition a is

mq p , (16a)haploid br � c � ar(b � c)

2mq p , (16b)diploid (1 � f )[br � c � ar(b � c)]st

q phaplodiploid

2m. (16c)

(1 � f ){bs r � (2/3)c � ar[bs � (2/3)c]}st, f f f

When , there is no kin competition (the efficacy ofa p 0kin selection is not diminished by local density regulation;i.e., there is “hard selection”), and when , there isa p 1strict kin competition (“soft selection”; Wallace 1968; Wade1985; Whitlock 2002; Van Dyken 2010). Figure 3 shows thatincreasing the degree of kin competition inflates the equi-librium frequency of cheaters in a population. Importantly,for all parameter values, there exists some degree of kincompetition above which cheaters will fix in the population.

While greater kin competition monotonically increasescheater frequency, its affect on the cheater load is morecomplicated, revealing some interesting properties of kincompetition. The mean fitness with an arbitrary degree ofkin competition a is forˆW p 1 � (1 � a)(b � c)(1 � q)haploids and diploids and W p 1 � (1 � a)[bs �f

for haplodiploids with female-limited altru-ˆ(2/3)c](1 � q)ism expression. The maximum fitness under kin compe-tition occurs when , givingq p 0 W p 1 � (1 �max

for haploids and diploids anda)(b � c) W p 1 �max

for haplodiploids with female-limited(1 � a)[bs � (2/3)c]f

altruism. Applying and a (eqq. [16]) to the mean fitnessqand solving for , we findL p W � Wmax

m(1 � a)(b � c)L p , (17a)haploid br � c � ar(b � c)

2m(1 � a)(b � c)L p , (17b)diploid (1 � f )[br � c � ar(b � c)]st

L phaplodiploid

2m(1 � a)[bv � (2/3)c]f . (17c)(1 � f ){bs r � (2/3)c � ar[bs � (2/3)c]}st, f f f

It is seen immediately from equations (17) that withstrict kin competition ( ), the cheater load will be 0.a p 1The load is experienced as an increased death rate relativeto birth rate in a population, but by the definition of strictkin competition, all local groups are maintained at a con-stant carrying capacity, so that cheaters (or any other del-eterious mutations) do not cause a net increase in selectivedeaths (Wallace 1968; Whitlock 2002; Van Dyken 2010).Although cheaters are more abundant, their effect on meanfitness is diminished. Differently put, with strict kin com-petition, selection is blind to the social effects of traits(Van Dyken 2010), so that the loss of group benefits owingto cheaters has no (negative) effect on mean fitness.

Figure 4 shows a unimodal relationship between cheaterload and degree of soft selection. The cheater load increaseswith increasing soft selection until the cheating allele fixesin the population, at which point the load decreases as

. Figure 4 compares the cheater load under(1 � a)(b � c)kin competition with the cheater load under strict hardselection with an equivalent frequency of cheaters (fig. 4,

296 The American Naturalist

Figure 4: Cheater load under kin competition (i.e., soft selection)is shown by the solid line. The maximum occurs at the value of a

where cheaters fix because of soft selection. The dashed line repre-sents the predicted cheater load if the population represented by thesolid line were experiencing hard selection. While the cheater loadincreases initially with soft selection, it does not increase as muchas would be predicted by its effect on cheater frequency (dashed line).With strict kin competition, there is no load. Parameter values are

, , , and .�3b p 0.4 c p 0.1 r p 0.5 m p 10

dashed line). While kin competition increases the cheaterload by increasing the frequency of cheaters, the resultingload is not as great as it would be with the same cheaterfrequency under hard selection.

Discussion

The current perception of cheaters as a perpetual threat tocomplex social systems has a long history in evolutionarystudies. In the earliest genetic model of the evolution ofaltruism, Haldane (1932, p. 207) showed how alleles for“socially valuable but individually disadvantageous char-acters” (i.e., altruistic traits) could increase in frequency tofixation when a population was genetically structured intogroups of relatives. He argued, however, that selfish cheatersor “reverse mutations” would occur and spread, concludingthat, because a system of altruism was invasible by selfishalleles, “it [is] difficult to suppose that many genes for ab-solute altruism are common in man” (p. 210). Similarly,Maynard Smith (1964, p. 1146) argued that all social systemsare invasible by selfish or cheater alleles and that “any plau-sible model of group selection must explain why they donot spread.” More recently, West et al. (2006a, p. R482)summarize the common narrative that because altruisticpopulations are vulnerable to invasion by cheaters, “wewould not expect altruistic behaviours to be maintained ina population—put formally, altruism should not be evo-lutionarily stable.” Consideration of mutation has generatedfurther confusion. Using inclusive fitness reasoning, Westet al. (2006b, p. 606) argue that “mutation can produce

‘unrelated’ cheater lineages, which could spread through apopulation.”

We have demonstrated that these fears of social collapsein the face of recurrent cheater mutations are typicallyunwarranted (but for counterexamples, see Michod 1996,1997, 1999; Michod and Roze 2000, 2001). Specifically,cooperation is evolutionarily stable, despite recurrentcheating mutations, when the condition of a modifiedform of Hamilton’s rule is satisfied (eqq. [12]). The per-ceived threat posed by cheaters arises only when selectionis viewed as a process occurring at a single level of or-ganization (specifically, within groups). By explicitly mod-eling selection of recurrent cheating mutations within andamong groups, we have shown that the within-group ad-vantage of cheating, which is the basis of most argumentsidentifying cheating as a problem (e.g., Haldane 1932;Maynard Smith 1964; West et al. 2006a, 2006b), can beovercome by selection among groups opposing cheating.The strength of among-group selection is weakened byfactors that reduce relatedness (e.g., high dispersal rate,multiple foundresses, polyandry) and/or increase kin com-petition. Proper conceptualization of the levels of selectioninfluencing cheating evolution prevents the view thatcheaters are a general problem to the stability of coop-erative groups and generates predictions for when cheatingwill and will not be a legitimate threat to social cohesion.

While cheaters do not necessarily undermine the evo-lutionary persistence of cooperation, the maintenance ofcheating mutants at equilibrium nonetheless has long-termevolutionary consequences. By generating a cheater load(i.e., a reduction in mean fitness due to the presence ofcheaters), recurrent cheating mutation creates selection forload modifiers, such as worker policing or modificationof relatedness through monogamy or limited dispersal.Theories of policing have been developed without explic-itly considering the mechanism by which cheaters originateor are maintained in the population (Frank 1995; Brand-vain and Wade 2007; Wenseleers et al. 2004). Without thisinformation, it is impossible to quantify the strength ofselection favoring policing, which is central to understand-ing when policing will or will not evolve. The kin selection–mutation balance provides a quantitative prediction forthe strength of selection favoring policing that may beuseful in explaining or predicting patterns of policing innature. It also provides a mechanism whereby cheaters aremaintained in the population over a long enough evolu-tionary time, without spreading to fixation, to allow se-lection for policing.

How many cheaters should we expect to find in nature?For the simplest single-locus case, as modeled here, wecan take the rate of deleterious mutation to be on theorder of 10�6 to 10�7 per gene per generation (Kibota andLynch 1996; Haag-Liautard et al. 2007). There are few

Kin Selection–Mutation Balance 297

estimates of the value of the inclusive fitness effect, butan average estimate of 0.01 has been calculated for de-velopmental cheaters in the social bacterium Myxococcusxanthus (smith et al. 2010). This fits with the magnitudeof beneficial effects found for nonsocial loci (Perfeito etal. 2007). Given these rough estimates, we would expectto find one cheater for every 10,000–100,000 individuals.This amounts to a large number of cheaters at equilibriumin microbial populations, where densities can be of order108 cells mL�1, and in eusocial insects, where colonies cancontain millions of individuals (Holldobler and Wilson1990). Of course, population structure prevents mutantsfrom being uniformly distributed among groups, so theactual cheater count in any given group in nature will behigher or lower than this value. With multiple loci (higherm) and/or weaker selection, the equilibrium frequency ofcheaters will be much higher. Clearly, more estimates ofthe mutation rate m, selection coefficients (e.g., ),br � cand local population regulation a are needed for socialtraits before we can forecast the expected number of cheat-ers in any system. Our theory shows that cheaters will bepresent in all social systems and that their frequency de-pends on the strength of selection, the population geneticstructure, and the ecology of local density regulation.

The mathematical models developed here are simple butnonetheless incorporate a great deal of biologically realisticcomplexity. We have accounted for multiple genetic sys-tems (haploidy, diploidy, haplodiploidy), patterns of sex-limited expression (female-limited expression), matingsystems (monogamy, polyandry), and the effects of ecologythrough the spatial scale of density regulation. Futuremodels should add overlapping generations, multiple loci,dominance, and frequency-dependent selection in orderto encompass a more complete range of biological com-plexity. While we have contrasted frequency dependenceand recurrent mutation as processes that maintain cheatersin populations, these phenomena are not mutually exclu-sive: mutation will always introduce a constant stream ofcheaters into populations, which will contribute to vari-ation even when cheaters are maintained by nonlinearfrequency dependence. In general, more theoretical workis necessary for the full development of a mutational the-ory of kin selection, which will be important in generatingnovel explanatory and predictive theories for social evo-lution, particularly at the molecular level.

Acknowledgments

We would like to thank j. smith and G. Velicer for helpfuldiscussions and two anonymous reviewers for suggestionson improving the manuscript. J.D.V. and M.J.W. weresupported by National Institutes of Health grant

R01GM084238, and T.A.L. was supported by a EuropeanUnion Marie Curie postdoctoral fellowship.

APPENDIX A

Derivation of Allele Frequency Recursion

Following the procedure in appendix A of Van Dyken(2010), equation (1) can be derived by use of a standardn-ploid population genetic model with Hamilton’s linearfitness function. We present the derivation for diploids,which is readily extended to haploids. Consider a diploidpopulation with alleles A and a in frequency p and q,respectively, segregating at a single biallelic locus control-ling altruism (see “The Model”). Let pij be the frequencyof allele A in individual j in group i, with the values (0,1/2, 1); pi be the mean frequency of allele A in group i,and p be the mean frequency of A in the global population.Expressing the altruistic trait decreases the fitness of thealtruist j by an amount cj but increases the fitness of eachgroup member by an average amount bk. Assuming thatHardy-Weinberg proportions obtain, on average, withingroups, the mean and genotypic fitnesses in group i are

2W p p W � 2p (1 � p )Wi AA, i i i Aa, ii

2� (1 � p ) W , (A1a)i aa, i

W p 1 � b N � c, (A1b)i, AA k alt, i

W p 1 � b N � hc, (A1c)i, Aa k alt, i

W p 1 � b N , (A1d)i, aa k alt, i

where is the number of al-2N p N [p � 2p (1 � p )h]alt, i i i i i

truists in group i (Wilson 1975; Wade 1978, 1979, 1980).This fitness model is for “whole-group” altruism (Pepper2000), where altruists receive the benefit of their altruisticactions. “Others-group” altruism can be modeled by mak-ing a simple substitution to this fitness model (see app. Aof Van Dyken 2010).

The multilevel Price equation (Price 1972; Hamilton1975; Wade 1985; Rice 2004) can be used to generate anallele frequency recursion for our model. We can write thePrice equation as

WDp p Cov (w , p ) � E(Cov (w , p )), (A2)i i ij ij

where the first term is the change in allele frequency dueto selection among groups and the second term is changedue to selection within groups. We can rewrite this equa-tion into its regression coefficient form (Wade 1985),which, assuming additivity ( ), givesh p 1/2

298 The American Naturalist

WDp p b V � b V , (A3)w , p b w , p wi i ij ij

where and are the partial regression coefficientsb bw , p w , pi i ij ij

of group and individual allele frequency on group andindividual fitness, respectively (Frank 1998; Rice 2004), Vb

is the variance in allele frequency among all groups, andVw is the average within-group variance in individual allelefrequency (i.e., . Values for these vari-V p E(Var (p ))w w ij

ance components are given for neutral alleles by Wright(1951, 1965). The among-group partial regression coef-ficient can be found by taking the derivative of equa-bw , pi i

tion (A1a) with respect to pi. The within-group partialregression coefficient is the difference in the marginalbw , pij ij

within-group fitness of each allele (Rice 2004). This canbe written as , where∗ ∗ ∗b p (W � W ) W pw , p A, i a, i A, iij ij

and are∗pW � (1 � p )W W p (1 � p )W � pWi AA, i i Aa, i a, i i aa, i i Aa, i

the marginal fitnesses of the A and a alleles, respectively,within group i. Thus, we find that andb p (b � c)w , pi i

. Substituting into equation (A3), further sub-b p �cw , pij ij

stituting for p, and then simplifying gives equation1 � q(1).

Kin competition can be modeled by modifying the mul-tilevel Price equation as follows:

WDp p (1 � a) Cov (w , p ) � E(Cov (w , p )) (A4)i i ij ij

(Van Dyken 2010). Here, a is the intensity of kin com-petition. Following the same procedure as above but usingequation (A4) instead of equation (A2) leads to equations(16) and (17).

APPENDIX B

Haplodiploid and Diplodiploid Models withFemale-Limited Altruism Expression

If a population is structured into different classes of in-dividuals, such as different sexes, and if classes differ inploidy, then allele frequency change must be computedseparately for each class. In haplodiploids, each sex com-poses a class, where one sex (typically males) is haploidand the other (typically females) is diploid. The allele fre-quency change in each sex is weighted by the reproductivevalue of the sex. In haplodiploids, males are weighted by1/3 and females by 2/3 (Taylor 1988). Thus, the total allelefrequency change is given by

1 2Dp p Dp � Dp , (B1)m f3 3

where for each sex l the change is

W Dp p Cov (w , p ) � E(Cov (w , p )). (B2)l li li lij lijl

Assuming that only females (the diploid sex) express al-

truism, we can write the following genotypic fitnesses forfemales and males:

W p 1 � b N � c,fi, AA k alt, i

W p 1 � b N � hc,fi, Aa k alt, i

W p 1 � b N , (B3)fi, aa k alt, i

W p 1 � b N ,mi, A k alt, i

W p 1 � b N .mi, a k alt, i

Assuming that selection is weak, Hardy-Weinberg pro-portions hold approximately and allele frequency is ap-proximately equivalent in males and females after repro-duction. With these assumptions, combining equations(B1)–(B3) and doing some algebra gives

2 1 2WDp p bs V � Cov (p , p ) � c V , (B4)f b, f fi mi t, f( )3 3 3

where sf is the proportion of females in a group (s pf

), which appears because only females are altruists.N /Nf

Using the relations of equations (2a), (2b), and (3) andnoting that (the regression relatedness amongV /V p rb, f t, f ff

females) and (the regression re-(Cov (p , p ))/V p rfi mi t, f fm

latedness of females to males), we obtain equation (10).To find the recursion for female-limited expression of

altruism in diploids, we follow the same procedure asabove but let both sexes be diploid and weight the con-tribution of each sex by 1/2 (i.e., the reproductive valueof each sex in diploids). After some algebra, we find that

pq(1 � f )[bs r � (1/2)c]st, f fDp p . (B5)

2W

This requires the fact that there are no relatedness asym-metries in diplodiploids, so that females are equally relatedto sisters and brothers.

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Associate Editor: Sean H. RiceEditor: Ruth G. Shaw