Popul Ecol (2001) 43:179–188
© The Society of Population Ecology and Springer-Verlag Tokyo 2001
ORIGINAL ARTICLE
Takehiko I. Hayashi · Masakado Kawata
Evolution of postmating isolation: comparison of three models based on possible genetic mechanisms
Received: January 29, 2001 / Accepted: July 4, 2001
Abstract In this study, we simulated the process of the evolution of postmating isolation using three models in which postmating isolation is caused by (1) genetic divergence through collaborative coevolution, (2) genetic divergence through antagonistic coevolution resulting from sexual conflict, and (3) genetic divergence through combinational incompatibility. The collaborative coevolution model and the combinational incompatibility model showed a similar decreasing pattern of hybrid compatibility over generations depending on population size and mutation rates. The antagonistic coevolution model showed that reproductive isolation can evolve rapidly depending on the intensity of selection. In the combinational incompatibility model, the increasing number of loci that interact and result in incompatibility would have both promoting and inhibiting effects on the formation of hybrid incompatibility in the earlier stage of isolation. Mutation rates for genes causing incompatibility significantly affect the number of generations required for postmating isolation, which indicates that models assuming high mutation rates (e.g., µ 104) might predict much faster evolution for reproductive isolation than those observed in real populations. Key words Speciation · Reproductive isolation · Postzygotic isolation · Coevolution
T.I. Hayashi (*) Department of Ecology and Evolutionary Biology, Biological Institute, Graduate School of Science, Tohoku University, Aoba-ku, Sendai 980-8578, Japan Tel. 81-22-217-6689; Fax 81-22-217-6689 e-mail:
[email protected] M. Kawata Division of Ecology and Evolutionary Biology, Graduate School of Life Sciences, Tohoku University, Sendai, Japan
Introduction The tempo and mode of the formation of postmating isolation depends on its genetic basis. The traits for postmating isolation are maladaptive, because these traits cause inviability or sterility in hybrid offspring. Thus, selection can prevent the traits from spreading within a population if the traits reduce the fitness of individuals within the population. Therefore, many models for the evolution of postmating isolation have assumed genetic mechanisms that do not reduce the fitness of offspring parents from the same population but reduce the fitness of those from different populations (review in Orr 1996; Gavrilets 1997). However, several different genetic mechanisms for postmating isolation can be considered and the predictions by the models depend on which genetic mechanism is assumed. Thus, knowledge of the genetic basis of postmating isolation in real organisms is necessary to examine how reproductive isolation has evolved. Recent studies have revealed several aspects of the genetic basis of postmating isolation. First, large numbers of loci are responsible for genetic incompatibility, and there are strong interactions among them. Using high-resolution mapping, a series of studies in the Drosophila simulans clade showed that a genetic incompatibility is caused by the strong interaction among alleles and that each of these alleles has only a slight deleterious effect alone (for review, see Wu and Hollocher 1998). Furthermore, these studies also suggest that a large number of loci (e.g., 120 loci) are responsible for genetic incompatibility even when closely related species hybridize. The number and interactions of loci responsible for genetic incompatibility are important in modeling a genetic system. Second, several studies have suggested that there is positive selection for genes affecting reproductive characters (Lee et al. 1995; Metz and Palumbi 1996; Tsaur et al. 1998; Ting et al. 1998; Nurminsky et al. 1998). Positive selection is often attributed to antagonistic coevolution between genes for male and female reproductive traits. Such antagonistic coevolution can produce rapid divergence of a reproductive character, and this divergence
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would result in genetic incompatibility between populations. Therefore, antagonistic coevolution has been receiving increasing attention as an important factor causing reproductive isolation (Rice 1998; Palumbi 1998; Partridge and Parker 1999). Third, detailed characterization of genetic interactions among incompatibility genes has suggested that a gene regulatory system is involved in postmating isolation (Sawamura 1999). Several studies have shown that deleterious effects in hybrids are caused by genes producing a transcriptional factor (e.g., Sry; recently reviewed in Capel 2000). Furthermore, Ludwig et al. (2000) have suggested that a series of compensatory mutations in regulatory elements would cause an incompatibility between populations. In light of recent genetic findings, previous theoretical studies of postmating isolation often assumed inadequate genetic mechanisms. For instance, several studies assumed that genetic incompatibility depends on the number of heterozygotic loci (Higgs and Derrida 1992; Manzo and Peliti 1994; Gavrilets et al. 1998; Gavrilets 1999). Under this assumption, underdominance (innerlocus interaction) rather than epistasis (interlocus interaction) is likely to cause incompatibility. In addition, these studies also assumed that the interactions among a large number of loci cause genetic incompatibilities (a group of genes that interact with each other will be called a gene set). For example, Gavrilets et al. (1998) assumed that incompatibility arises when the number of heterozygotic loci is larger than a certain threshold value. However, it is well known that postmating isolation is usually caused by epistasis rather than by the degree of heterozygosity (reviewed by Wu and Palopoli 1994). Furthermore, empirical studies have shown that incompatibilities occur due to a relatively small number of loci that are restricted to a particular site of the chromosome (Wu and Hollocher 1998), whereas the total number of loci responsible for incompatibility is large and the loci are distributed over the whole genome. Thus, a valid assumption is that there are many interaction sets, within which interactions among a small number of loci lead to the incompatibility of hybrids. In this study, we examined what factors promote the evolution of postmating isolation using three models based on recent findings on the genetics of reproductive isolation. In all three models, we assumed that (a) interactions among loci cause incompatibility and that (b) there are many genetic interaction sets within which a small number of loci interact with each other and cause incompatibilities. In the first model, genetic divergence occurs as a result of collaborative coevolution, which can be applied when genes for postmating isolation are related to gene regulatory systems. In the second model, genetic divergence occurs as a result of antagonistic coevolution by sexual conflict, which can be applied to evolutionary conflict between sperm and eggs. In the third model, postmating isolation evolves because of the combination of different loci (combinational incompatibility). We compare the predictions from these three models and discuss what factors are important when considering genetic mechanisms of postmating isolation.
Models Three models for postmating isolation were constructed based on three different genetic mechanisms for hybrid incompatibility. In all the models, loci for incompatibility were assumed and the allele frequencies at these loci change over generations within populations. After several generations, the degree of the fitness reduction of hybrid offspring between two independently evolved populations (the degree of postmating isolation) and the degree of population differentiation were calculated.
Collaborative coevolution model There are two loci that interact with each other and affect genetic incompatibility. Thus, genes at these loci coevolve within a genome in a population. For instance, in a gene regulatory system, one locus codes transcription factors such as binding proteins and another locus codes the binding sites. When a mutation causes a change in the sequence of the binding protein, compensatory mutations for the binding site will be favored. Accordingly, the binding protein and the binding site will coevolve within a population. Sawamura (1999) has suggested that genetic incompatibilities of hybrids occur because of inconsistency between the transcription factors and their biding sites and that coevolution in the regulatory system would thus be important for postmating isolation. We used a model assuming two loci with multiple alleles and stepwise mutations to simulate coevolution of a transcription factor and its binding site in a gene regulatory system (Fig. 1). The model used was based on Nei et al. (1983). Ai and Bl represent an allele at loci A and B, respectively. The subscript i and l represent the values of the two quantitative traits, transcriptional factor and its binding site, respectively. The mutation increases or decreases these trait values by 1 with a probability of 0.5. Initially, all the alleles at two loci have the same trait values (i l). The gene regulatory system can function depending on the difference between the values of the two traits. In truncation selection, the value between the two loci (dil) and the compatibility is dil |i l|
(1)
Fig. 1. Multiallele stepwise mutation in coevolution models. Each allele indicates the corresponding phenotypic value of the trait. The value increases or decreases by 1 with a probability of 0.5 when the mutation occurs
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for dil K Ï1, Cil Ì Ó1 P , for dil K
(2)
where (Cil) is compatibility between loci Ai and Bl, P is the degree of reduction for compatibility, and K is the threshold value for compatibility. In linear selection, Cil 1, 0.5 or 0 if dil 1, dil 2, or dil 3, respectively. We show results with truncation selection in which both P and K were set at 1, unless specified. The fitness of diploid genotype (WAiAjBkBl) was calculated as WAiAjBkBl Cik Cil Cjk Cjl. The gene frequency change for each locus was predicted by computer simulations (see the next section). In this model, mutation and drift are divergent forces against stabilizing selection. The model is essentially similar to the premating isolation model by Nei et al. (1983). However, to apply this model to postmating isolation, we assume no sexspecific expression. Antagonistic coevolution model Similar to the collaborative coevolution model, there are two loci that interact with each other and affect genetic incompatibility. Contrary to the collaborative coevolution model, the benefits of the loci conflict with each other, and selection may lead to rapid divergence. There has been some evidence that antagonistic coevolution by sexual conflict is important in reproductive isolation (for review, see Rice 1998; Palumbi 1998; Partridge and Parker 1999). The present model assumes occurrence of sexual conflict because the optimal rates of sperm transgression into the egg for males are different from those for females. The benefits of eggs and sperm often contradict each other when more than one sperm penetrates into one egg (Rice 1998; Palumbi 1998; Partridge and Parker 1999). Faster rates of penetration are advantageous for a sperm to fertilize an egg, whereas it might be disadvantageous for an egg because the egg may suffer the risk of polyspermy if the penetration rates of sperm are too fast. Thus, it is beneficial for an egg to suppress the rate of penetration by the sperm. We used a model assuming two loci with multiple alleles and stepwise mutations to simulate antagonistic coevolution of male and female traits. Sis, Sek, Esj, and Eel represent alleles at loci coding sperm trait in the sperm, unexpressed sperm trait in the egg, unexpressed egg trait in the sperm, and egg trait in the egg, respectively. The subscripts i (or k) and l (or j) represent the rates of the sperm and the suppression rates of sperm penetration by the egg, respectively. The superscripts s and e indicate that the allele is involved in the sperm and egg, respectively. Increasing values of i (or k) and l (or j) indicate an increase in the rate of penetration and suppression, respectively. Initially, all the alleles at the two loci have the same trait values (i k l j). Random mating is assumed so that gametes are randomly combined. The compatibility of a zygote (Zil) was calculated according to the difference in trait values (hil) between Si in the sperm and El in the egg. hil i l
(3)
for K hil K Ï1, Ô Zil Ì1 Qe , for hil K Ô0, for hil K or hil K Ó
(4)
where Qe represents the intensity of egg–sperm conflict, and K is the threshold values for compatibility. When hil K, the rate of sperm penetration is too fast for the eggs to suppress the penetration, which consequently reduces the fitness of the zygote. This condition assumes that evolution of male traits due to sperm competition can cause partial reduction of zygote fitness (Rice 1998). When hil K or hil K, the trait values of the egg and sperm are too different for fertilization to take place. K was set at 1, and Qe was set at 0.001 unless specified. The relative fertilizing ability of a sperm might depend on the relative rates of penetration of other sperm because the sperm with the faster rate of penetration would be more likely to fertilize an egg. Thus, the fertilizing ability of the sperm (MSisSos ) with trait i against other sperms with trait q q within the same population is Ï1 Qs , for i q Ô MSis Sqos Ì1, for i q Ô1 Q , for i q s Ó
(5)
where superscript os indicates an allele in other sperms in the same population and Qs represents the intensity of sperm competition. Qs was set at 0.01 unless specified. The relative number of offspring expected from the combination of SisEsj sperm and SekEel egg is WSisE js SkeEle Zil
n
 p(S )M os q
q0
Sis Sqos
(6)
os where p(Sos q ) is the frequency of allele Sq and n is the number of alleles at locus S in the population. Equation 6 means that sperm competition always exists between two males. The trait of sperm may evolve so as to have larger values because a large trait value of sperm is advantageous for sperm competition. On the other hand, if the traits of sperm evolved to the extent that they have larger values than those of the trait of eggs, the decrease in the compatibility Z reduces the fitness of the sperm. Thus, the evolution of these traits depends on the parameters Qe and Qs.
Combinational incompatibility model In this model, the combinations of deleterious alleles among several loci cause incompatibility. Within populations, particular combinations of alleles do not arise because mutations creating the combinations are deleterious. Such deleterious combinations, however, may arise when two independent populations hybridize. Thus, incompatibility due to the combination of genes can readily evolve without inhibition by selection acting in each population. The importance of epistasis in evolution of reproductive isolation was first suggested by Dobzhansky (1937) and
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Muller (1942). Experimental studies have also shown that loci that have little or no deleterious effect per se cause incompatibility in a certain combination of alleles at the different loci (for review, see Wu and Palopoli 1994). This incompatibility can be explained either by coevolution among loci (i.e., collaborative coevolution and antagonistic coevolution) or by the particular combinations of deleterious alleles at different loci. The latter case is treated in the combinational incompatibility model. The combinational incompatibility model assumes two, three, or six loci, each of which has two alleles (0 or 1). Initially, all the alleles at all loci are assumed to be 0 state. Mutations always change the state of the allele from 0-allele to 1-allele. If all the loci show either heterozygous (1 and 0) or a homozygous (1 and 1) state, the fitness of zygotes is reduced. In the two-locus model, Ai (or Aj) and Bk (or Bl) represent alleles at loci A and B, respectively. The fitness of the genotype AiAjBkBl is
Ï1 R, for Ai Aj 0 « Bk Bl 0 WAi Aj Bk Bl Ì for Ai Aj 0 » Bk Bl 0 Ó1,
(7)
where R is the degree of fitness reduction due to the combination of genes. For the three-locus and six-locus model, the fitness reduces by R when all the loci are in either a heterozygous (1 and 0) or a homozygous (1 and 1) state. R was set at 1 unless indicated. Nei (1976) first studied postmating isolation caused by combinational incompatibility between two loci with simple formal analysis. He only discussed the condition in which the allele causing incompatibility can increase. Orr (1995) and Orr and Orr (1996) studied postmating isolation assuming that the number of novel gene combinations can cause incompatibility when isolated populations hybridize. Their model, however, ignored the process of spreading and fixation of new alleles within populations. Gavrilets and Hastings (1996) studied postmating isolation with combinational incompatibility of one, two, or three loci as well as of polygenic traits, but they took only founder effect speciation into consideration. Some of the studies on the lose of duplicated loci have also examined two-locus models in which the combinations of the loci cause genetic incompatibility (e.g., Takahata and Maruyama 1979; Li 1980; Watterson 1983). These studies assumed mainly two-locus systems and mainly focused on the loss of alleles and did not examine the evolution of incompatibility.
were calculated generation to generation. We assumed free recombination and linkage equilibrium in the models because a mathematical treatment for gamete frequencies becomes difficult when the number of alleles (coevolution model) and loci (combination model) increase. In our model, however, linkage equilibrium may not hold even under free recombination as a particular combination among alleles was assumed to be advantageous or disadvantageous. Thus, we conducted individual-based simulations using these three genetic models but allowing linkage disequilibrium. The simulations showed that linkage disequilibrium had little or no effect on the results in the present study, at least for N 5000. Thus, the effect of linkage disequibrium is relatively small in our models. Gene frequency change by genetic drift at each locus is calculated by an improved pseudosampling method (Kimura and Takahata 1982). Following Kimura and Takahata (1982), sampling of alleles was calculated sequentially from A0 to An. Let p(Ai) be an allele frequency of Ai. An allele frequency after sampling was calculated as
Ê p¢( Ai ) ci 2 N yi¢xi 2 N yi¢ Á 1 Ë
We calculated change in gene frequency using Monte Carlo simulations. The same method, except for fitness calculation, which was described in the previous section, was used for all three models of genetic incompatibility. For all the models, ten populations were simulated for the same parameter settings. Discrete generation, random mating, free recombination, and no gene flow were assumed. Changes in gene frequency by drift, selection, and mutation
ˆ
j
(8)
j0
where ci (i 0,1,2, . . . , n) is the number of Ai-bearing gametes in the population. xi is the remainder of the gamete after Ai is removed; that is, xi 2N c0 c1 . . . ci1 and x0 2N. y i was calculated as follows. If xi¢ 20, y i ζi/xi. ζi is a binominal random number following B(xi,yi), where yi is defined as yi p(Ai)/(1 p(A0) p(A1) . . . p(Ai1)) and y0 p(A0). If xi 20 and xiyi 3, y i ηi/xi. If xi 20 and xi(1 yi) 3, then y i 1 ri/xi. ηi and ri are Poisson random numbers with the mean xiyi and xi(1 yi), respectively. When xi 20 and none of xiyi and xi(1 yi) is less ———— than 3, y i yi Ui÷(1 yi)/xi. Uis are mutually independent arbitrary random variables, each with mean 0 and variance 1. The change in allele frequency by selection for collaborative coevolution models was calculated as
(
∆ p( Ai ) p( Ai ) WAi W WAi
)
W
(9)
 p(B ) p( A ) p(B )W
(10)
n
j
k
l
j , k ,l 0
W
Ai Bj Ak Bl
n
 p( A ) p(B ) p( A ) p(B )W i
j
k
l
i , j , k ,l 0
Gene frequency change
i1
 p¢( A )˜¯
Ai Bj Ak Bl
(11)
where WAiAjBkBl is the fitness of genotype AiAjBkBl, and W is the average fitness. The change in gene frequency for the combinational incompatibility model was calculated in the same way as that for the collaborative coevolution model. Egg- or sperm-specific expression was assumed in the antagonistic coevolution model. Thus, change in allele frequency by selection, the average fitness of Si (WSi) or Ei (WEi), and the average fitness of population ( W ) were ∆ p(Si )
1 p(Si ) WSi W 2
(
)
W
(12)
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∆ p(Ei )
1 p(Ei ) WEi W 2
(
)
(13)
W
n
W
 p(S ) p(E ) p(S ) p(E )W a i
s j
e k
e l
a , j , k ,l 0
WSi WEi
Sas E js Ske Ele
(14)
n
 p(E ) p(S ) p(E )W s j
e k
e l
j , k ,l 0
Sis E js Ske Ele
(15)
n
 p(S ) p(E ) p(S )W s a
a , j , k0
s j
e k
Sas E js Ske Eie
(16)
The number of new mutations per allele per generation in a population was assumed to be a Poisson distribution with mean 2Nµ, where µ is the mutation rate per allele per generation. The mutation increases or decreases the i value of the traits by 1 with a probability of 0.5 in both the coevolution models and changes allele value from 0 to 1 in the combinational model. The initial frequency of the mutant allele was set at 1/2N. Allele Ai was considered to disappear within the population when p(Ai) 0. CI (compatibility index) was used as a value indicating the degree of postmating isolation between populations and was calculated as follows. The average fitness of hybrids was calculated assuming random mating between two populations. The fitness is calculated in the same way as that within populations. Then, these fitness values between pairs of populations were averaged over all possible paired populations. These average values were regarded as CI. We assumed that considerable incompatibility evolved when CI 0.9 because CI 0.9 for one gene set would cause a significant decrease in CI for a large number of gene sets in one genome (see Discussion). In the collaborative model and in the antagonistic coevolution model, the number of mutational steps is also shown to indicate the degree of population differentiation. The number of mutational steps is calculated as the average difference of phenotypic values (i.e., subscript of alleles) on the A (collaborative model) locus or the S (antagonistic model) locus between the populations at the 500 000th (collaborative model) or the 50 000th (antagonistic model) generation from the initial (0) generation. The number of mutational steps is used in the coevolution model to show the relation of the degree of differentiation and incompatibility between populations. Population size N was set at 500, 5000, 10 000, 100 000, and 500 000; the mutation rate µ was set at 104, 105, and 106.
Results In the collaborative coevolution model, compatibility (CI) decreased linearly as generations increased (Fig. 2). CI decreased more rapidly with decreasing population size (Figs. 2 and 3a) and increasing mutation rates (Fig. 3a). CI decreased to less than 0.9 within 300 000 generations for µ 104. CI did not decrease greatly within 1 000 000 generations when µ 105 and 106 for N 100 000. A tenfold difference in mutation rates resulted in a severalfold differ-
Fig. 2. Decreases in compatibility index (CI ) during 1 000 000 generations for different population sizes in the collaborative coevolution model. µ 105
ence in decreasing rates of CI, except for N 500. There was little difference in the rates of decrease in CI between K 1, K 3, and linear selection, where K is the threshold value of phenotype distance that causes incompatibility. ‘Linear selection’ assumes that the compatibility decreased linearly with increasing phenotype difference (Fig. 3b). The average number of mutational steps during 500 000 generations (distance of phenotype value from the initial value at A locus) increased with decreasing population size and increasing mutation rates (Fig. 4a). The increase in number of mutational steps represents the increase in the phenotypic differentiation within each population. The dependence of population size and mutation rates on the phenotypic differentiation was consistent with that of the decrease in CI. In the antagonistic coevolution model, however, the effects of population size and mutation rates on the decrease of CI were not consistent with those on the increase in the average number of mutational steps. This inconsistency may be an artifact caused by the similar divergence rates among populations caused by the strong and constant selection under the assumption of a one-dimensional trait (see Discussion). Thus, in the antagonistic coevolution model, only the result of the number of mutational steps is shown. The number of mutational steps during 50 000 generations increased with increasing population size and mutation rates (Fig. 4b). When the mutation rate was low, the effects of population size on the number of mutational steps became large. A 10-fold or 100-fold difference in mutation rates resulted in a severalfold difference in the number of mutational steps. The effects of intensity of sperm competition (Qs) and egg–sperm conflict (Qe) are shown in Fig. 5. The number of mutational steps generally decreased with increased Qe during 101–103, whereas the number of mutational steps for Qe 104 was smaller than that for Qe 103. In the combinational incompatibility model, CI generally decreased to constant values (Fig. 6a–c). These constant values were defined by the number of possible fixation patterns of mutations. For instance, in the two-locus model,
184 Fig. 3. Relationships between the numbers of generations required for CI 0.9 and the population size N in the collaborative coevolution model. a The mutation rates (µ) were varied. K 1. b The fitness functions were varied. µ 105. In the linear fitness function, dij 1, dij 2, and dij 3 when C 1, C 0.5, and C 0, respectively, where dij is the distance of phenotypic values between alleles at each loci. C is the compatibility between these pairs of alleles
Fig. 4. Relationships between population size (N) and the number of mutational steps (mean SD) for different mutation rates µ in the collaborative coevolution model a and the antagonistic coevolution model b. The number of mutational steps was calculated as the average distance in phenotypic values of alleles at A a or S b loci at the 500 000th a or the 50 000th b generation from the initial value. The average number of 10 populations is shown
Fig. 5. Relationships between the number of mutational steps (mean SD) and the intensity of sexual conflict in the antagonistic coevolution model. Qs and Qe are the intensity of sperm competition and egg– sperm conflict, respectively. The number of mutational steps was calculated as the average distance of phenotypic values at the S loci at the 50 000th generation from the initial values. The average number of 10 populations is shown
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because a pair of 1-alleles (mutational allele) on each loci causes incompatibility, only two fixation patterns of mutation are possible [1-allele fixed on locus A and 0-allele (wild-type allele) fixed on locus B, or the converse]. Thus, even in different populations, the same fixation pattern of mutation was finally fixed in half of the cases in the two-locus model. Because matings among populations having same fixation pattern of mutation yield compatible offspring, CI will finally decrease to 0.5. Similarly, in the three-locus model, two-thirds of the population mates to produce incompatible offspring and thus CI decreased to one-third, and in the six-locus model, CI decreased to onesixth. CI mostly decreased linearly until reaching constant values (Fig. 6a–c). In a large population, however, CI did not decrease during the first 150 000 to 300 000 generations. The decreasing rates of CI were not greatly affected by the number of loci involved in the compatibility (Fig. 7a–c). Like the collaborative coevolution model, decreasing rates of CI increased with decreasing population size and increasing mutation rates. CI decreased to less than 0.9 within 100 000 generations for µ 104, and within 700 000 generations for µ 105. CI did not greatly decrease within 1 000 000 generations for µ 106.
Discussion Significance and problems of these models In this article, we chose three models for postmating isolation because recent genetic and molecular studies have suggested that these three mechanisms are possible causes of postmating isolation. There might be other mechanisms causing postmating isolation, but the present three models provide the basic framework of the mechanisms for evolution of postmating isolation. In the present models, we assumed a single gene set within which genetic compatibility is caused by interactions among two or more loci. We also assumed that there are many of these gene sets in a genome. In real organisms such as Drosophila, a large number of loci would be potentially related to genetic incompatibility, and some of these loci actually would induce observed incompatibility of hybrids between populations. Wu and Hollocher (1998) estimated that the number of loci causing hybrid male sterility was at least 120 per genome. The number of loci that are potentially related to incompatibility may be larger than these estimated values. Detailed studies in the Drosophila simulans clade have shown that although a large number of loci are involved in observed hybrid sterility, even a small portion in a genome (e.g., 1%–5%) can cause incompatibility (for review, see Wu and Hollocher 1998). These studies have shown that within a small portion of a genome there might be several loci (e.g., 2–6 loci) involved in genetic compatibility. However, these loci are very close to each other, at least in these Drosophila studies, so that they may behave as a single or a few loci. The facts revealed by these Drosophila studies
Fig. 6. Decrease in compatibility index (CI) during 1 000 000 generations in the combinational incompatibility model. Mutation rates were set at 105. a Two-locus model; b three-locus model; c six-locus model
support our assumption that there are a large number of gene sets within which the interaction among a few loci causes genetic compatibility. We assumed that postmating isolation is achieved when, in a gene set, genetic compatibility index CI is less than 0.9. The value 0.9 was arbitrary and somewhat high. Our model simulated evolution of incompatibility that is caused by a single gene set. In real organisms, there would be a large number of these gene sets, as already discussed. If there are n gene sets acting independently to cause incompatibility in one genome, the incompatibility per genome will be ap-
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Fig. 7. Relationships between the number of generations required for CI 0.9 and the population size (N) in the combinational incompatibility model; a two-locus model; b three-locus model; c six-locus model
n proximately CIave , where CIave is the average of CI among n gene sets. Thus, if there are many of these gene sets, CIave 0.9 is sufficient to cause almost complete incompatibility between populations, e.g., if n 25, CIave 0.9, 0.925 ⯝ 0.07. It is well known that the X chromosome often has a major effect on postmating isolation (Wu et al. 1996). For loci located in the sex chromosome, the effective population size is reduced, and the dominance may enhance its expression in hemizygote hybrids. Thus, reproductive isolation might be promoted. We did not assume the effect of the sex chromosome on incompatibilities. However, in all our models, alleles were assumed to exhibit a dominant-like expression such as with the X chromosome because incompatibility arises in F1 hybrids. Thus, including the effect of the X chromosome on the evolution of incompatibility might not greatly change the present results. We found that the decreasing rates of CI were not consistent with the number of mutational steps in the antagonistic coevolution model; this was because different populations show similar rates of phenotype evolution so that the difference in the value of the phenotype among different populations remained small, especially due to strong and constant selection with regard to the one-dimensional traits. If different populations evolve the same traits independently (similar protein in eggs evolves in different populations in egg–sperm coevolution) with the same temporal pattern, the decreasing rates of CI will be not consistent with the degree of divergence. However, such parallel evolution among different populations might be rare, and so the decreasing rates of CI in the present results will be inadequate and artifacts of model assumptions. Thus, we show only the results of the average number of mutational steps. These numbers reflect the rates of divergence of different popula-
tions and become a good qualitative index of the degree of postmating isolation.
Comparison among three models Incompatibility of hybrids in both the collaborative coevolution model and the combinational incompatibility model showed similar decreasing patterns depending on population size and mutation rates. This similarity indicates that it is difficult to discriminate whether epistasis of deleterious mutations at different loci or coevolution between loci causes reproductive isolation from the empirical/observational relationships between population size and estimated time since divergence. In the combination incompatibility model, CI decreased by half in the two-locus model and by one-third and onesixth in the three- and six-locus models, respectively (see Fig. 6). In the two-locus model, half the population remained compatible with the other half when CI reached the minimum, which suggests that larger numbers of loci in a gene set may more readily cause incompatibility, as suggested theoretically in Orr (1995) and Cabot et al. (1994), and thus the increasing number of loci has increasing effects on the rate of decreasing incompatibility. On the other hand, the difference in the number of loci did not greatly affect the decreasing rates of CI in the earlier stage of isolation (see Fig. 7). When m mutational alleles are required to cause incompatibility and the probability of fixation for each allele is p, the probability that incompatibility will arise will be pm. pm decreases with increasing m, and the decreasing effects become remarkable when p approach 0. Thus, in the earlier stage of isolation, increasing m
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has both increasing and decreasing effects on the rates of decreasing compatibility; this may be the reason for the small effect of the number of loci on decreasing rates of CI in the earlier stage of isolation. In the combinational incompatibility model with large population size, a certain number of generations seems to be required before CI begins to decrease because the probability of the fixation of the mutational alleles by drift is low in large population sizes; this might cause simultaneous speciation among geographically isolated populations (Gavrilets et al. 1998). In the antagonistic coevolution model, mutational steps rapidly increased (see Fig. 4b). The rates of the increase of the number of mutational steps depended mainly on the selection intensity of sexual conflict (see Fig. 5). Increase in the values of N and µ increased the rates of the increase of the number of mutational steps, but the effects of N and µ were smaller than those of the selection intensity. The models in the present study cannot predict absolute rates of increasing incompatibility because the rates depend on several assumptions and parameter values. The threshold value for incompatibility in the present models is rather severe because incompatibility is assumed to occur when the distance between phenotype values is a minimum value (e.g., one). Thus, the predicted rates by the present models might be the minimum estimates. Coyne and Orr (1989, 1997) suggested that 1.5–5 million years was required to achieve initial postmating isolation in Drosophila. Compared with this, isolation as shown by the antagonistic coevolution model evolved much faster than that in Drosophila data as the model showed much differentiation in phenotypic values among populations within only 50 000 generations (see Fig. 4b). Thus, antagonistic coevolution with strong selection by sexual conflict cannot explain the pattern of increasing rates of incompatibilities observed in Drosophila. This result indicates that antagonistic coevolution might not be a major cause of postmating isolation in Drosophila. If antagonistic coevolution producing postmating isolation is common, one possibility explaining this inconsistency is that the assumption of a constant strong selection causing faster evolution is invalid or unrealistic in nature. In natural populations, the intensity of sexual conflict may depend on ecological factors such as population density (Gavrilets 2000). Another possibility is that mutations causing genetic variations in the characters for antagonistic coevolution in Drosophila are so rare that the rate of evolution is effectively suppressed. However, there may be much genetic variation as a source of evolution because an experiment of antagonistic coevolution has shown considerable selection response (Rice 1996).
The effects of mutation rates on postmating isolation In all our models, µ 104 led to CI 0.9 within 300 000 generations. Therefore, this mutation rate might be higher than those in real organisms. Gavrilets et al. (1998) and Gavrilets (1999) applied the concept of holey adaptive landscapes to explain postmating reproductive isolation. Their
simulation showed that reproductive isolation can occur with a gene flow in several hundred generations (Gavrilets et al. 1998). Their model used 104 for mutation rates and several hundred individuals for local population size. Thus, it is uncertain whether the rapid speciation rates of the model were the result of the high mutation rates or the assumption underling holey adaptive landscapes. If the much larger population size with lower mutation rates still results in rapid speciation in the Gavrilets model, the assumption involved in holey adaptive landscapes might cause much faster postmating isolation than those observed in Drosophila. Our model showed that mutation rates significantly affect the number of generations required for postmating isolation and its population size dependency (see Figs. 3 and 7). This finding suggests that mutation rates significantly affect the prediction of the models. Thus, models for postmating isolation should use a wide range of mutation rates. Acknowledgments We thank K. Sawamura and A. Sasaki for helpful suggestions.
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