Monogamy promotes worker sterility in insect societies

Inclusive-fitness theory highlights monogamy as a key driver of altruistic sib-rearing. Accordingly, monogamy should promote the evolution of worker sterility in social insects when sterile workers make for better helpers. However, a recent population-genetics analysis (Olejarz et al. 2015) found no clear effect of monogamy on worker sterility. Here, we revisit this analysis. First, we relax genetic assumptions, considering not only alleles of extreme effect—encoding either no sterility or complete sterility—but also alleles with intermediate worker-sterility effects. Second, we broaden the stability analysis—which focused on the invasibility of populations where either all workers are fully-sterile or all workers are fully-reproductive—to identify where intermediate pure or mixed evolutionarily-stable states may occur. Finally, we consider additional, demographically-explicit ecological scenarios relevant to worker non-reproduction. This extended analysis demonstrates that an exact population-genetics approach strongly supports the prediction of inclusive-fitness theory that monogamy promotes sib-directed altruism in social insects.

If we assume that only full-sterility alleles can arise, double mating sometimes promotes the invasion of sterility over single mating. But (b) if we assume that alleles encoding intermediate worker sterility may arise, double mating never promotes the invasion of sterility over single mating, depending on the colony efficiency values r 0 = 1, r 1/4 , and r 1/2 . This is because (c) for a rare allele encoding full sterility, mutant colonies have the phenotype z = 1/2 under single mating and z = 1/4 under double mating. Therefore, sterility may invade more easily under double mating if colony efficiency is relatively peaked near z = 1/4. But (d) for a rare allele encoding intermediate sterility, mutant colonies may express any phenotype 0 < z ≤ 1/2 under single mating and 0 < z ≤ 1/4 under double mating, depending on the allele's effect, and so mutant phenotypes are less constrained by the population's mating number. In order to facilitate comparison with

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iii.  . (a) There are many possible ways to construct the colony efficiency function r z based on picking random numbers from a normal distribution. Five alternatives are shown here, including the two procedures used by Olejarz et al. ("Random noise", their Procedure 1, and "Plateau", their Procedure 2). For testing whether sterility invades, only two points are needed (solid lines), but this can be extended to four points (dashed lines) for measuring sterility at equilibrium. (b) We record the frequency of invasion of a full-sterility allele under single (n = 1) versus double mating (n = 2), running 10 million experiments for each scenario. Percentages beneath the bar chart show that an initially-decelerating r z is required for sterility to invade under double mating only (see Methods). (c) We record the average worker sterility at equilibrium over 5000 experiments for each scenario. Except when r z is constructed using the "random noise" or "plateau" procedure and the magnitude of efficiency effects is small (asterisks), single mating tends to promote average worker sterility at equilibrium over double mating (the 0/0 denotes no worker sterility under either single or double mating). This can happen even if sterility is more likely to invade under double mating (for example, compare results of procedures i-iii in panel (b) versus panel (c)). Arrowheads beneath the x-axis show where parameters coincide with those used in panel (b). The "magnitude of colony efficiency effects" is the standard deviation of normally-distributed variates used for constructing r z . For panels (b) and (c), we assume p z = 0.2 + 0.8z. See Methods for details.  ), matching well with the predicted evolutionarily-stable levels of worker sterility. To illustrate a scenario where constraints on heritable variation may lead to promiscuity promoting worker sterility over monogamy, we use the colony efficiency function r z = 1 + bz − z 2 , with a "benefit of worker sterility" term bz and a "decelerating" term -z 2 . For the proportion of male eggs laid by the queen, we again use p z = 0.2 + 0.8z.
Alternative ecological scenarios: monogamy promotes worker sterility 155 Finally, we consider some alternative scenarios for the evolution of worker non-reproduction, us-156 ing a demographically-explicit model of queen-worker competition over egg-laying. Whether we we have shown that the long-term evolutionary outcome is readily described, conceptualised, and 171 explained by standard inclusive-fitness theory. In sum, a more comprehensive analysis based on 172 Olejarz et al.'s (2015) exact population-genetics approach supports inclusive-fitness theory and its 173 prediction that monogamy promotes the evolution of worker sterility.

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Acknowledgements 175 We thank Kevin Foster for helpful comments.  Here, we determine the stable level of worker sterility under four demographically-explicit models of worker sterility; see Methods for full details. (a) One possible assumption is that worker-laid males only compete with the queen's sons (cf. Olejarz et al. 2015). In this case, monogamy promotes worker sterility over promiscuity. (b) It is also possible to assume that worker-laid males compete with the queen's offspring of both sexes, and not just with the queen's sons. In this case, monogamy promotes worker sterility over promiscuity. (c) In the gall-forming thrips, the foundress produces an initial brood of female and male soldiers, who may produce part of the next brood by inbreeding amongst themselves (Chapman et al. 2002). Female soldiers can sacrifice part of their reproductive potential to invest more in defending their nestmates. In this case, monogamy promotes worker sterility over promiscuity. (d) A possible model for the evolution of eusociality involves dispersing, fully-reproductive females evolving into sterile workers, who stay in the nest to help, producing no offspring (Boomsma 2007(Boomsma , 2009(Boomsma , 2013. In this case, monogamy promotes worker sterility over promiscuity. We show results for k = 4 in (a) and k = 2 in (b) and (c) (see Methods for details).
worker sterility in the social Hymenoptera. Adv. Stud. Behav., 48, 251-317.   225 Throughout the main text, our focus is on helpful worker sterility, where giving up some or all 226 of her reproductive potential allows a worker to provide more help within her colony, as this 227 biological assumption underpins most work on altruistic sib-rearing in social insects. However,  In this model, harmful worker sterility may occur via two routes-one operating through 234 colony efficiency, r z , and one operating through the queen's production of males, p z . The first 235 case occurs when an increase in average worker sterility decreases colony efficiency-for exam-236 ple, if the sterility allele has a pleiotropic effect on worker condition which results in less-efficient 237 work. In such a case, monogamy will inhibit the evolution of worker sterility relative to promis-238 cuity, since promiscuity decreases relatedness between relatives, thereby lessening the harmful 239 impact of sterility upon a worker's inclusive fitness via colony efficiency.

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The second case occurs when an an increase in a focal worker's sterility harms the reproduc-241 tive success of other workers. In the main text, we assume that when a worker becomes sterile, 242 her forfeited sons are replaced partly by the queen's sons and partly by her sisters' sons, such that 243 by forfeiting sons she gains both nephews and brothers. But if, due to the shape of the p z function, 244 the queen gains a larger proportion of sons than the worker forfeits (that is, when p z > 1−p z 1−z ), this 245 "outsized gain" by the queen must be balanced by decreased male production by other workers, ( Similarly, we find that a dominant allele encoding worker sterility v can invade a population 268 monomorphic for sterility u when

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In order to find when natural selection will favour a small increase in sterility δz, we make the Linearizing r z and p z around z = u, we replace r u+ δz 2n with r + δz 2n r , where r = r u and r = dr dz | z=u .

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Similarly, we replace p u+ δz 2n with p + δz 2n p , where p = p u and p = dp dz | z=u . This yields Eliminating the fractions on both sides, discarding terms of order δz 2 or higher, substituting z for 285 u and simplifying yields which is condition 1 of the main text.

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Similarly, for a dominant sterility allele, substituting v = u + δz into condition 4 yields By linearizing r z and p z around z = u as above, we obtain Expanding all terms, discarding terms of order δz 2 or higher, substituting z for u and simplifying which, again, is condition 1 of the main text.

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In the main text, we discuss why promiscuity can sometimes favour the invasion of a recessive  , we assume p z = 0.2 + 0.8z, and for r z we use the unique quadratic curve passing through the points specified by r 0 = 1, r 1/2 , and r 1 . mean 0 and standard deviation σ. In all cases, we assume r 0 = 1, and use the random variates to 317 generate r 1/4 , r 1/2 , r 3/4 , and r 1 , which suffice to numerically integrate the evolutionary dynamics 318 of worker sterility using the system of ODEs described by Olejarz et al. (2015). We restrict our 319 attention here to the invasion of an allele encoding full sterility in its carriers, under either recessive 320 or dominant genetics.

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The first procedure, "random noise", is equivalent to Procedure 1 in Olejarz et al. (2015). Here, 322 we set r 1/4 = r 0 + a, r 1/2 = r 0 + b, r 3/4 = r 0 + c, and r 1 = r 0 + d. Note that the four values are where ρ is the desired correlation between each variate. By multiplying the vector of uncorre-333 lated variates by the Cholesky decomposition of this matrix, one obtains four correlated variates Now, we set r 1/4 = r0 + a , r 1/2 = r0 + b , r 3/4 = r0 + c , and r 1 = r 0 + d . Note that, because 335 the variables are correlated, the first "step" (from r 0 to r 1/4 ) tends to be larger in magnitude than 336 subsequent "steps" (i.e., from r 1/4 to r 1/2 , r 1/2 to r 3/4 , or r 3/4 to r 1 ), which is why we have named 337 this procedure "plateau". This procedure might generate plausible r z functions for a population in 338 which worker sterility brings diminishing returns to colony productivity, where these diminishing 339 returns happen to set in near z = 1 /4.

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Note that both the "random noise" and "plateau" procedures tend to produce r z functions that 341 disadvantage single mating relative to double mating. For the "random noise" procedure, this 342 is because although the procedure is just as likely to produce a peak at z = 1 /2 (which would 343 favour single mating) as at z = 1 /4 (which would favour double mating), workers at z = 1 /2 are 344 typically "trading away" more male production than workers at z = 1 /4 (since p 1/2 ≥ p 1/4 ), yet, 345 on average, they are receiving the same expected increase in productivity; hence, single mating 346 is relatively disfavoured. And since the "plateau" procedure tends to produce colony efficiency 347 functions with diminishing returns on worker sterility for colonies with z > 1 /4, it is much more 348 likely to produce an r z function with a relative peak at z = 1 /4 rather than a relative peak at z = 1 /2.

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This procedure might generate plausible r z functions if each increase in worker sterility had a 352 random increasing or decreasing effect on colony productivity. The fourth procedure, "increasing 353 steps", is similar, except steps are constrained to be positive: r 1/4 = r 0 + |a|, r 1/2 = r 1/4 + |b|, 354 r 3/4 = r 1/2 + |c|, and r 1 = r3/4 + |d|. This procedure might generate plausible r z functions 355 if each increase in worker sterility added a random increase to colony productivity. The fifth 356 procedure, "linear", uses a single normal variate to establish a constant step size for r z : r 1/4 = 357 r 0 + a, r 1/2 = r 1/4 + a, r 3/4 = r 1/2 + a, and r 1 = r 3/4 + a. This procedure might generate plausible 358 r z functions if each increase in worker sterility had a consistent increasing or decreasing effect on 359 colony productivity. For each of these new procedures, later points in r z depend on earlier points, but there is no tendency for "steps" between points in r z to change in average magnitude.

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In Fig. 2, we test each of these 5 procedures to see whether single or double mating promotes 362 the invasion (Fig. 2b) or equilibrium level of sterility ( Fig. 2c)  will favour an increase to worker sterility, z, when where R son = 1 2 , R neph = 2+n 8n , R sis = (1 + p z ) 2+n 8n , and R bro =  Similarly, natural selection favours an increase to the queen's sex allocation, x (her proportion 407 of resources allocated to daughters), when That is, natural selection favours an increased investment into daughters when x < 1 /2, and a 409 decreased investment into daughters when x > 1 /2, such that an even sex ratio is favoured overall, 410 regardless of worker sterility. own; alternatively, it may apply if rather than replacing the queen's eggs, the workers simply lay 416 their eggs in the communal nest, and all queen-produced and worker-produced offspring have the 417 same expected survival. Following these assumptions, we find that natural selection will favour 418 an increase to worker sterility, z, when where p z is the proportion of all juveniles on the patch that are produced by the queen, R son = 1 2 , In this scenario, queen sex allocation is not independent of worker sterility. We find that natural 434 selection favours an increase to the queen's investment in daughters, x, when hence, when all colony offspring are queen-laid (p z = 1), the queen favours an even sex ratio 436 (x = 1 /2), but as the proportion of colony offspring laid by workers increases, the queen favours 437 an increasingly female-biased sex ratio. Specifically, the equilibrium sex ratio is x * = 1+p z 1+3p z .

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Scenario C. Worker sterility among claustral inbreeders 439 Here, we assume that the queen produces a first brood of female and male soldiers, who mate 440 amongst themselves; the second brood of female and male dispersers is partly produced by the 441 queen and partly produced by the soldiers, as in the gall-forming social thrips (Chapman 2002).

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For simplicity, we assume here that queens and soldiers produce an even sex ratio for the second 443 brood, but allowing sex ratio evolution does not change the results qualitatively (not shown).

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Following these assumptions, we find that natural selection favours an increase to the sterility of where, under haplodiploidy, , and R bro = 1 3 . Because this scenario does not require arrhenotokous partheno-448 genesis of males, it also applies to diploid populations. Under diploidy, R dau = R son = 11+p z 16 449 and R niece = R neph = R sis = R bro = 1+n 4n (Fig. 6a). Similarly to condition 7, the left-hand side of 450 condition 9 can be interpreted as the inclusive-fitness effect experienced by a worker who stops 451 laying male eggs; but in condition 9, the female worker's "sacrifice effect" involves giving up 452 both daughters and sons; the "efficiency effect" involves an increase in both niece and nephew production as well as sister and brother production; and the "juvenile production effect" involves 454 the focal worker gaining both sisters and brothers, while her gain or loss of nieces and nephews 455 balances her forfeited offspring and her gained siblings.

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Scenario D. The evolution of eusociality 457 Here, we assume that the queen produces and provisions a first brood of females, and then pro- and help to raise the queen's second-brood offspring without producing any offspring of her own. 461 We assume that each worker can raise b siblings, on average, in her natal nest, and that each dis- assumptions, we find that natural selection will favour an increase to worker sterility, z, when where R dau = R son = 1 2 , R sis = 2+n 4n , and R bro = 1 4 . As with scenario C, this scenario also applies 469 to diploid populations; under diploidy, R dau = R son = 1 2 and R sis = R bro = 1+n 4n (Fig. 6b). When z = 0, this condition reduces to 2n is required for natural selection to favour the invasion of sterile workers ( Fig. 4d; Fig. 6b). In The r z function above has three components: a baseline efficiency of 1; bz, representing a linear 482 fitness benefit for each sterile worker; and sz 2 , representing an "interaction effect" of worker steril-483 ity. We use the parameter s to examine scenarios where multiple sterile workers results in either 484 synergy (s > 0) or diminishing returns (s < 0) to colony productivity.

485
The p z function given above corresponds to a model in which the queen and k(1 − z) repro-486 ductive workers each take an equal share of offspring production. Alternatively, k can capture not 487 only the total number of workers but also their ability to control offspring production relative to 488 the queen; for example, halving k could represent either a halving in the number of workers or 489 a halving of their relative ability to control offspring production, keeping the number of workers 490 constant.

491
A function of this form can also model more complicated demographic processes: for example, 492 if we assume that there are N workers, each of whom replaces a random egg with their own at rate 493 W, while the queen can replace a workers' egg with her own at rate Q, then the form above gives 494 the proportion of eggs produced by the queen at equilibrium when k = NW Q . In models where worker-laid and queen-laid individuals compete equally, regardless of their sex, production of eggs and replacement of eggs will often be equivalent processes: that is, the form given above 497 for p z also holds if workers, rather than replacing the queen's eggs, simply lay their own eggs in 498 the communal nest without replacement. In that case, the r z function would capture the overall 499 production and survival of eggs. value Z = γ, while a diploid individual with genotype γ 1 , γ 2 has breeding value Z = (γ 1 + γ 2 )/2.

513
At the beginning of each generation, M mated females each produce K female workers on 514 their home patch. Each worker has a probability Z of being sterile. The patch average sterility 515 z determines the colony productivity r z and the proportion of males produced by the queen p z .

516
The next generation of breeders is then produced: first, a patch is randomly selected from the 517 population with probability proportional to its colony efficiency, r z , and a female is produced by 518 the queen on that patch; then, another n patches are randomly selected with replacement, with 519 probability proportional to their colony efficiency, and each of these n patches produces a male 520 (from the queen with probability p z , or from a random reproductive worker on that patch with The death rate of existing colonies, φ, is defined as in order to enforce a density constraint, namely: These equations can be understood as follows. First, note that in an AA, m colony, a fraction Aa females, and m 2n aa females; and aa, m colonies produce n−m n Aa females and m n aa females.

568
Male production is more complicated, since both queens and workers produce males, but the 569 principle is the same. We will take the first term in curly braces in the y A line, These equations can be understood similarly to equation 14; in fact, they are identical, except 591 for two general changes. First, the subscripts to r z and p z are different, because the mutant allele is and Aa, 0 colonies. Colony types with more copies of the mutant allele are rarer, and so will have 602 a negligible effect on invasion. Therefore, from equation 11, we need only consider: We start with a wild-type population (X AA,0 = 1) and introduce a small perturbation of magni- which implies that Substituting 17 into 14, and keeping terms only up to order , gives Finally, substituting 12, 18, and 19 into 16 and discarding powers of 2 or higher gives If the dominant eigenvalue of the above matrix is greater than zero, then a dominant sterility 611 allele with penetrance v can invade a population monomorphic for sterility with penetrance u. AA,0 = x AA y n A − φX AA,0 X AA,1 = nx AA y n−1 A y a − φX AA,1 X Aa,0 = x Aa y n A − φX Aa,0 X AA,2 = n(n − 1) 2 x AA y n−2 A y 2 a − φX AA,2 X Aa,1 = nx Aa y n−1 A y a − φX Aa,1 X aa,0 = x aa y n A − φX aa,0 .

655
In this model, we denote a focal worker's sterility by Z, the average sterility on a focal patch by 656 z, and the average sterility in the population byz. A focal queen's sex ratio strategy for her second 657 brood is denoted by x, and the average sex ratio strategy among all queens in the population is 658 denoted byx. The production of queen-laid second-brood females on a focal patch is f = f (z, x); 659 the production of queen-laid second-brood males on a focal patch is m = m(z, x); the production 660 of worker-laid females by a focal worker is φ = φ(Z, z, x); and the production of worker-laid males 661 by a focal worker is µ = µ(Z, z, x). We denote byf = f (z,x),m = m(z,x),φ = φ(z,z,x), and 662μ = µ(z,z,x) the population-average production of each of these four classes, respectively, and by f = f /f ,m = m/m,φ = φ/φ, andμ = µ/μ the relative production of each of these four classes.
For a gene increasing worker sterility to spread, its carriers, on average, should leave more 665 descendants than other members of the population. Accordingly, natural selection will favour an 666 increase in worker sterility, z, when  gives up reproduction to become sterile. These interpretations are mathematically equivalent, but 677 we focus on the inclusive-fitness interpretation here, as it is conceptually simpler.

678
Similarly, natural selection will favour an increase in the queen's sex allocation strategy (her 679 investment in daughters), x, when Above, R dau|Q is the relatedness between a focal queen and her daughter, R son|Q is the relat- gives up one of her sons to raise an extra daughter.

693
(1 − y)(1 − Z)(1 − c). Substituting these definitions into conditions 32 and 33 recovers conditions  Accordingly, consanguinities needed for the conditions above can be found in Table 1. The con-703 sanguinities for a female worker under claustral inbreeding are obtained by first calculating the 704 coefficient of inbreeding for a foundress in this mating system (the probability that her two genes 705 at a given locus are identical by descent). Suppose that a juvenile is foundress-laid with probabil-706 ity Q, and soldier-laid with probability 1 − Q. If foundress-laid, her coefficient of consanguinity 707 is zero, because patch founders are unrelated. If worker-laid, then her paternally-inherited gene 708 comes from her grandmother, and her maternally-inherited gene comes, with equal probability, 709 either from her grandfather-who is unrelated to her grandmother-or from her grandmother; in 710 the latter case, her two genes are either copies of the "same" gene in her grandmother, in which 711 case they are identical by descent with probability 1, or are copies of "different" genes from her 712 grandmother, in which case they are identical by descent with probability G, where G is the ju-  venile's grandmother's coefficient of inbreeding. That is, overall, the probability that these two Defining Q =f f +φ as the probability that a random female is 721 queen-laid, and P =m m+μ as the probability that a random male is queen-laid, note that a random When all second-brood juveniles are queen-laid (P = Q = 1), this yields the expected result 731 that c f = 2 /3, c m = 1 /3, c φ = 0, and c µ = 0; when all second-brood juveniles are worker-laid It is illustrative to examine a special case. When all second-brood females are queen-laid (Q = 735 1), this reduces to In this case, when P = 1, we have the expected result that the total value of juvenile females is 737 2 /3 and the total value of juvenile males is 1 /3, because of the usual asymmetries of haplodiploidy.

738
But when P = 0, the total value of juvenile females is 1 /2 and the total value of juvenile males is 739 1 /2. This is because new juvenile females get half their genes from their mother and half from their 740 father, while new juvenile males are parthenogenetically produced by worker females, and so ul-741 timately get half their genes from their mother's mother and half their genes from their mother's 742 father. In this way, juvenile females and males have an equal share in producing the next genera-