Effective population size in social Hymenoptera with worker-produced males

Heredity 63 (1989) 59-65 The Genetical Society of Great Britain Received 22 December 1988 Effective population size in social Hymenoptera with worker-produced males Robin E. Owen* and A. R. G. Owent * Department of Biological Sciences, University of Calgary, Alberta, Canada T2N 1N4. t Department of Zoology University of Toronto, Toronto, Ontario, Canada M5S 1AI. In the social Hymenoptera, which have haplodiploid inheritance, a proportion, ii of the (haploid) males can be produced by the workers. It is shown that, for the special case where each laying worker produces exactly one male that survives to maturity and mates, the variance effective population size Ne(v) = (3— il,)2 FMI(2F + (2— il,)2M), where F and M are, respectively, the number of queens and males in the population. If the sex ratio is unity or female biased then Ne is reduced if there are worker-produced males, however with male biased sex ratios Ne is increased compared to its value with il,=0. An alternative situation, in which laying workers can each produce more than one male offspring, was investigated using computer simulations. In this case worker-produced males reduce N) regardless of sex ratio, although the effect is relatively the most weak with male biased sex ratios. A reduction in effective population size due to worker-produced males may contribute to the generally low levels of genetic variation found in the Hymenoptera. INTRODUCTION The insect order Hymenoptera is characterized by a haplodiploid genetic system in which, at the phenomenological level, diploid females are derived from fertilized eggs while haploid males arise from unfertilized eggs. In actual fact sex is determined by an underlying genic mechanism (Crozier 1971). Inheritance in haplodiploid species is equivalent to sex-linked inheritance in diploid species. The Hymenoptera are also characterized by the occurence of eusociality in which there is a reproductive division of labour. Colonies are founded by mated females (queens) who produce daughters (workers and young queens) and males. Workers 1984; Bourke, 1988). Similarly the effect of worker- produced males on deterministic aspects of hymenopteran population genetics has been investigated (Owen, 1980, 1985a, 1986). In this paper we examine the influence that worker-produced males have on random genetic drift through effective population size, Ne. There are two contrasting views in the literature on how worker-produced males might affect Ne. Kerr (1975) suggests that given a constant total number of males, then Ne will increase as the proportion of males produced by workers increases. The reasoning followed is that the number of laying work- ers should be added to the number of queens, to give the total number of "genetically active" do not mate, yet in many species retain their females (Contel and Kerr, 1976) in the population, which is then just used in Wright's (1933) standard ovaries, and under certain conditions can undergo sex-linked formula for Ne. On the other hand, ovarian development and lay unfertilized eggs which develop into males. This production of males by workers is now recognized to be wide- Crozier (1979) argues that worker-produced males as stingless bees, honey bees, bumble bees, vespine will inevitably reduce effective population size, because an extra round of gametic sampling is introduced each generation and therefore the rate of random genetic drift is increased. optimum sex ratios has received attention (Oster vives to maturity and mates. We derive an expression for effective population size and and Wilson, 1978; Aoki and Moody, 1981; Pamilo, confirm the results using computer simulations. In spread in the higher eusocial Hymenoptera, such wasps and higher ants (for a recent review see Bourke, 1988). The effect that worker-produced males have on the evolution of sociality and We first examine the simplest case where each laying worker produces exactly one male that sur- R. E. OWEN AND A. R. G. OWEN 60 this case it turns out that whether worker-produced expected values of p and p,. are both equal to males increase or decrease N depends on the sex-ratio. If the sex-ratio is unity or female biased Ep = Ep = PfPm + (p-q + qp,,,) then Ne is reduced by the presence of workerproduced males, however with male biased sex =(pr+pm). ratios N is increased, Next we consider one alternative scenario in which laying workers produce different numbers of males that survive to maturity and mate. In this particular case we find using computer simulations, that worker-produced males reduce N regardless of sex ratio, although the effect is weakest with male biased sex ratios. (1) The conditional variance of p is V(p), where FV(p) = PfPrn + ()2(pfq, + qfpfl) — E(p)2 =PfPm(Pf+Pm2PfPn)(Pf+Prn)2 = (pq+p,q), i.e., V(p) = (pfqf+pq)/4F. (2) ONE OFFSPRING PER LAYING WORKER For queen-produced males Ep1 =p, and the variance V(p1) is just that of the proportion of The case considered here includes that of ordinary A1 gametes in a total of (1 — sex-linkage and of haplodiploid inheritance. Assume that in one generation F queens mate probabilities p. of being A1 and qf of being A2. at random with M (haploid) males and their offspring consist of F queens, (1 — q)M males and a number of workers. Furthermore it is supposed that l/JM of these workers each produce exactly one male by parthenogenesis that survives to maturity and mates. The quantity ii is the propor- tion of males produced by the workers which can vary from zero to one (Owen, 1980). Thus the mating population in the next generation still con- sists of F queens and M males. A population of fixed size is postulated with numbers F and M constant from generation to generation. Consider a single locus with alleles A1 and A2 at frequencies in the first generation denoted as follows: Queens Workers Queen-produced males Worker-produced males All males V(p1)=p1q/(1—i)M, The actual gene frequency of A1 among worker- produced males equals (number of A1 workerproduced males)/4'M. This is the proportion of A1 genes in a sample of /iM genes obtained by randomly drawing one gene from each of the workers who each give rise to a male. Each such gene has independently of the others in the sample, a probability of being A1 equal to the expected gene frequency of A1 among the qM workers. This is therefore (p+p,) as in equation (1). Moreover the number of I/fM genes in the I/iM males produced by workers is binomially distributed. There- Pr q1 fore the variance for P2 is Pmi qmi V(p2) pm qm Pm2 m2 The actual gene frequency of A1 among queens Fp=(number of A1A1 queens) +(number of A1A2 queens), similarly the actual gene frequency of A1 among queen-produced males given by In the next generation a queen or a worker belongs to the genotypes A1A1 and A1A2 with respective probabilities PiPr and (ptq,. + q pm), thus the (Pf+Pm)[l (pr+prn)]/çIjM = 2p1q1+ 2pq + (pr— pm)21/ (...truncated)