Quick viewing(Text Mode)

Waiting Time to Parapatric Speciation

Waiting Time to Parapatric Speciation

doi 10.1098/rspb.2000.1309

Waiting time to parapatric Sergey Gavrilets Departments of and Evolutionary and Mathematics, University of Tennessee, Knoxville,TN 37996, USA ([email protected]) Using a weak migration and weak approximation, I studied the average waiting time to . The description of used is based on the classical Dobzhansky model and its recently proposed multilocus generalizations. The dynamics of parapatric speciation are modelled as a biased random walk performed by the average genetic distance between the residents and immigrants. If a small number of genetic changes is su¤cient for complete reproductive isolation, mutation and random alone can cause speciation on the time-scale of ten to 1000 times the inverse of the mutation rate over a set of loci underlying reproductive isolation. Even relatively weak selection for local can dramatically decrease the waiting time to speciation. The actual duration of the parapatric speciation process (that is the duration of intermediate forms in the actual transition to a state of complete reproductive isolation) is shorter by orders of magnitude than the overall waiting time to speciation. For a wide range of parameter values, the actual duration of parapatric speciation is of the order of one over the mutation rate. In general, parapatric speciation is expected to be triggered by changes in the environment. Keywords: ; ; parapatric speciation; waiting time; mathematical models; Dobzhansky model

1996; Orr & Orr 1996; Gavrilets & Boake 1998; Gavrilets 1. INTRODUCTION 1999; Johnson & Porter 2000), parapatric (Gavrilets Parapatric speciation is usually de¢ned as the process of 1999; Gavrilets et al. 1998, 2000a; Johnson et al. 2000) formation in the presence of some gene £ow and sympatric (e.g. Turner & Burrows 1995; Gavrilets & between diverging . From a theoretical point Boake 1998; Van Doorn et al. 1998; Dieckmann & Doebeli of view, parapatric speciation represents the most general 1999) scenarios. Here, I develop a new stochastic scenario of speciation which includes both allopatric and approach to modelling speciation as a biased random as extreme cases (of zero gene £ow walk with absorption. I use this framework to ¢nd the and a very large gene £ow, respectively). The geogra- average waiting time to parapatric speciation. I also phical structure of most species, which are usually study the average actual duration of parapatric speciation composed of many local populations experiencing little which is de¢ned as the duration of intermediate forms in genetic contact for long periods of time (Avise 2000), ¢ts the actual transition to a state of complete reproductive the one implied in the parapatric speciation scenario. In isolation. My results provide insights into a number of spite of this, parapatric speciation has received relatively important evolutionary questions about the role of little attention compared to a large number of empirical di¡erent factors (such as mutation, migration, random and theoretical studies devoted to allopatric and sympa- genetic drift, selection for and the tric modes (but see Ripley & Beehler 1990; Burger 1995; genetic architecture of reproductive isolation) in control- Friesen & Anderson 1997; Rola¨ n-Alvarez et al. 1997; Frias ling the time-scale of parapatric speciation. & Atria 1998; Macnair & Gardner 1998). Traditionally, The method for modelling reproductive isolation studies of parapatric speciation have emphasized the adapted below is based on the classical Dobzhansky importance of strong selection for local adaptation in (1937) model which has been discussed in detail in a overcoming the homogenizing e¡ects of migration (e.g. number of recent publications (e.g. Orr 1995; Gavrilets & Endler 1977; Slatkin 1982). It has recently been theoreti- Hastings 1996; Orr & Orr 1996; Gavrilets 1997). The cally shown that rapid parapatric speciation is possible Dobzhansky model, as originally described, has two even without selection for local adaptation if there are important and somewhat independent features (Orr many loci a¡ecting reproductive isolation and mutation is 1995). First, the Dobzhansky model suggests that, in some not too small relative to migration (Gavrilets et al. 1998, cases, reproductive isolation can be reduced to interac- 2000a; Gavrilets 1999). tions of `complementary’ genes (that is, genes that Earlier theoretical studies of speciation have mostly decrease ¢tness when present simultaneously in an concentrated on the accumulation of genetic di¡erences ). Second, it postulates the existence of a `ridge’ that could eventually lead to complete reproductive of well-¢t that connects two reproductively isolation. However, within the modelling frameworks isolated genotypes in space. This `ridge’ makes it previously used complete reproductive isolation was not possible for a to evolve from one state to a possible (but see Nei et al. 1983; Wu 1985). Recently, new reproductively isolated state without passing through any approaches describing the whole process of speciation maladaptive states (`adaptive valleys’). The original from origination to completion have been developed and Dobzhansky model was formulated for the two- applied to allopatric (Orr 1995; Gavrilets & Hastings case. The development of multilocus generalizations has

Proc. R. Soc. Lond. B (2000) 267, 2483^2492 2483 © 2000 The Royal Society Received 7 June 2000 Accepted 14 September 2000 2484 S. Gavrilets Waiting time toparapatric speciation proceeded in two directions. A mathematical theory of each new potentially has a selective advantage s the build up of incompatible genes leading to steri- (50) over the corresponding ancestral allele in the local lity or inviability was developed by Orr (1995; Orr & Orr environment. 1996) who applied it to allopatric speciation. A comple- I will use a weak mutation and weak migration mentary approach placing the most emphasis on `ridges’ approximation (e.g. Slatkin 1976, 1981; Lande 1979, 1985a; rather than on `incompatibilities’ was advanced by Gavri- Tachida & Iizuka 1991; Barton 1993) neglecting within- lets (1997, 1999; Gavrilets & Gravner 1997; Gavrilets et al. population variation. Under this approximation the only 1998, 2000a,b). This approach makes use of a recent role of mutation and migration is to introduce new discovery that the existence of `ridges’ is a general feature which quickly get ¢xed or lost. I will assume that the of multi-dimensional adaptive landscapes rather than a processes of ¢xation and loss of alleles at di¡erent loci are property of a speci¢c genetic architecture (Gavrilets 1997, independent. Within this approximation, the relevant 2000; Gavrilets & Gravner 1997). Here, I will use the dynamic variable is the number of loci Db at which a `ridges-based’ approach assuming that mating and the typical individual in the population is di¡erent from the development of viable and fertile o¡spring is possible only immigrants. Variable Db is the average genetic distance between that are not too di¡erent over a between residents and immigrants computed over the loci speci¢c set of loci responsible for reproductive isolation. underlying reproductive isolation. The dynamics of The adaptive landscape arising in this model is an speciation will be modelled as a random walk performed example of `holey adaptive landscapes’ (Gavrilets 1997, by D on a set of integers 0, 1, : : :, K, K 1. In what b ‡ 2000; Gavrilets & Gravner 1997) of which the original follows I will use li and ·i for the probabilities that Db two-locus, two-allele, Dobzhansky model is the simplest changes from i to i 1 or i 1 in one time-step (genera- ‡ partial case. My general results are directly applicable to tion). The former outcome occurs¡ if a new allele supplied the original Dobzhansky model. by mutation gets ¢xed in the population. The latter outcome occurs if an ancestral allele brought by immi- grants replaces a new, previously ¢xed allele. I disregard 2. THE MODEL the possibility of more than one substitution in one time- I consider a ¢nite population of sexual diploid - step. Probabilities li and ·i are small and depend on the isms with discrete non-overlapping generations. The probability of migration per generation m, the probability population is subject to immigration from another popu- of mutation per gamete per generation v, the strength of lation. For example, one can think of a peripheral popula- selection for local adaptation s and the population size N. tion (or an island) receiving immigrants from a central Speciation occurs when d hits the (absorbing) boundary population (or the mainland). All immigrants are homo- K 1. If this happens, the population is completely ‡ zygous and have a ¢xed `ancestral’ genotype. Mutation reproductively isolated from the ancestral genotypes. I do supplies new genes in the population, some of which may not consider the possibility of backward mutation towards be ¢xed by random genetic drift and/or selection for local an ancestral state. Fixing new alleles at K 1 loci ‡ adaptation. Migration brings ancestral genes which, if completes the process of speciation. ¢xed, will decrease genetic di¡erentiation of the popula- tion from its ancestral state. 3. RESULTS In this paper, I consider only the loci potentially a¡ecting reproductive isolation. The degree of reproduc- I will compute two important characteristics of the tive isolation depends on the extent of speciation process. The ¢rst is the average waiting time to at these loci. Let d be the number of loci at which two speciation t0, which is de¢ned as the average time to reach individuals di¡er. I posit that the probability w that two the state of complete reproductive isolation (D K 1) b ˆ ‡ individuals are able to mate and produce viable and starting at the ancestral state (D 0). In general, during b ˆ fertile o¡spring is a non-increasing function of d such that the interval from t 0 to the time of speciation the popu- ˆ w(0) 1, w(d)40 for d4 K and w(d) 0 for all d4K, lation will repeatedly accumulate a few substitutions only ˆ ˆ where K is a parameter of the model specifying the to lose them and return to the ancestral state at D 0. b ˆ genetic architecture of reproductive isolation. This implies The second characteristic is the average actual duration of that individuals with identical genotypes at the loci under speciation T0, which is de¢ned as the time that it takes to consideration are completely compatible whereas indivi- get from the ancestral state (D 0) to the state of b ˆ duals that di¡er in more than K loci are completely complete reproductive isolation (D K 1) without b ˆ ‡ reproductively isolated. A small K means that a small returning to the ancestral state. The actual duration of number of genetic changes are su¤cient for complete speciation is similar to the conditional time that a new reproductive isolation. A large K means that signi¢cant allele destined to be ¢xed segregates before ¢xation. It genetic divergence is necessary for completely reproduc- characterizes the length of the time-interval during which tive isolation. If K is equal to the overall number of loci, intermediate forms are present. complete reproductive isolation is impossible. This simple model is appropriate for a variety of isolating barriers (a) Allopatric speciation including pre-mating, post-mating pre-zygotic and post- It is illuminating to start with the case of no immigra- zygotic (Gavrilets et al. 1998, 2000a,b; Gavrilets 1999). I tion (cf. Orr 1985; Orr & Orr 1996; Gavrilets 1999, will allow the loci responsible for reproductive isolation to pp. 6^8). In this case, the process of accumulation of new have pleiotropic e¡ects on the degree of adaptation to the is irreversible and the average duration of local environment (Gavrilets 1999; cf. Slatkin 1981; Rice speciation T0 is equal to the average waiting time to 1984; Rice & Salt 1988). Speci¢cally, I will assume that speciation t0.

Proc. R. Soc. Lond. B (2000) Waiting time toparapatric speciation S. Gavrilets 2485

* Table 1. Exact expressions for the average waiting time to speciation t 0¤ and the average duration of speciation T0 for small K with no selection for local adaptation and a threshold function of reproductive compatibility (R m/v) ˆ

parapatric case

* K allopatric case (t T ) t ¤ T 0 ˆ 0 0 0

2 R 1 1 2/v (2 R)(1/v) ‡ ‡ 1 R v ‡ 3 4R 2R2 1 2 3/v (3 3R 2R2)(1/v) ‡ ‡ ‡ ‡ 1 R 2R2 v ‡ ‡ 4 8R 13R2 6R3 1 3 4/v (4 6R 8R2 6R3)(1/v) ‡ ‡ ‡ ‡ ‡ ‡ 1 R 2R2 6R3 v ‡ ‡ ‡

(i) No selection for local adaptation unless the genetic distance Db exceeds K. I start with the With no or very little within-population genetic worst-case scenario for speciation when not only immi- variation the process of accumulation of substitutions grants can easily mate with residents but also selection for leading to reproductive isolation is e¡ectively neutral (cf. local adaptation is absent (cf. Gavrilets et al. 1998, 2000a; Orr 1995; Orr & Orr 1996). The average number of Gavrilets 1999). neutral mutations ¢xed per generation equals the muta- tion rate v (Kimura 1983). Thus, the average time to ¢x K 1 mutations is No selection for local adaptation ‡ With no selection for local adaptation and neglecting K 1 t0 ‡ . (1) within-population , the process of ¢xa- ˆ v tion is approximately neutral. The probability of ¢xation of an allele is equal to its initial frequency. The average (ii) Selection for local adaptation frequency of new alleles per generation is approximately In a diploid population of size N, the number of muta- the mutation rate v. If the immigrants di¡er from the residents at D i loci, there are i loci that can ¢x ances- tions per generation is 2Nv. The probability of a mutant b ˆ allele with a small selective advantage s being ¢xed is ca. tral alleles brought by migration. The average frequency 2s/(1 exp( 4Ns)) (Kimura 1983). Thus, the average of such alleles per generation is im. Thus, the probabilities ¡ ¡ time to ¢x K 1 mutations is of stochastic transitions increasing and decreasing Db by ‡ one are ca.

K 1 1 exp( S) li v, ·i im. (4) t ‡ ¡ ¡ , (2) ˆ ˆ 0 ˆ v S With small K the exact expressions for t0 and T0 found in where S 4Ns. With S increasing from zero to, for Appendix A are relatively compact (see table 1). With ˆ larger K, the approximate equations are more example, ten the time to speciation t0 decreases to approximately one-tenth of that in the case of no selection illuminating. The average waiting time to speciation is ca. for local adaptation. 1 m K t¤ K!. (5) 0 º v v (b) Parapatric speciation With immigration, the dynamics of Db are controlled The average duration of speciation is ca. by two opposing types of forces. Mutation and selection 1 C(K 1) ® act to increase Db whereas migration acts to decrease Db. T * 1 ‡ ‡ , (6) 0 ‡ Appendix A presents exact formulae for t0 and T0 in the º v m/v case of parapatric speciation. Below I give some simple where ® 0:577 is Euler’s constant and C( ) is the psi approximations that are valid if (m/v)exp( S) is not too º ¢ (digamma) function (Gradshteyn & Ryzhik 1994). small. ¡ (Function C(K 1) ® slowly increases with K and is ‡ ‡ equal to 1 at K 1, to 2.93 at K 10 and to 5.19 at (i) Threshold function of reproductive compatibility ˆ ˆ K 100.) For example, if m 0:01; v 0:001 and Here, I assume that the function w(d), which speci¢es ˆ ˆ ˆ K 5, then the waiting time to speciation is very long the probability that two individuals are not repro- ˆ 10 (t ¤ 1:35 10 ) generations, but if speciation does ductively isolated, has a threshold form of 0 happºen its£ duration is relatively short (T * 1236 0 º 1 for d4 K generations). Figure 1 illustrates the dependence of t 0¤ and w(d) , (3) * ˆ 0 for d4K T0 on the model parameters. (Using composite variables for the y-axis in this and other ¢gures allows one to (Gavrilets et al. 1998, 2000a,b; Gavrilets 1999; cf. Higgs represent relevant dependencies in two dimensions.) * & Derrida 1992). This function implies that immigrants Notice that T0 is order 1=v across a wide range of have absolutely no problems mating with the residents parameter values.

Proc. R. Soc. Lond. B (2000) 2486 S. Gavrilets Waiting time toparapatric speciation

4Ns 4Ns (a) l v , · im . (7) 16 i ˆ 1 exp( 4Ns) i ˆ exp(4Ns) 1 K = 10 ¡ ¡ ¡ 14 K = 8 The waiting time to speciation is ca. 12 1 exp( S) t0 t 0¤ exp( KS) ¡ ¡ , (8) ) º ¡ S 0 10 *

t K = 6 v (

where t 0¤ is given by equation (5). The average duration of 0

1 8

g speciation is ca. o l 6 K = 4 1 C(K 1) ® 1 exp( S) T ‡ ‡ ¡ ¡ : 0 1 S (9) º v ‡ (m/v)e¡ S 4 K = 2 For example, if m 0:01, v 0:001, K 5 and S 2, ˆ ˆ ˆ ˆ 2 then t 2:74 104 generations and T 2170 genera- K = 1 0 0 ˆ tions. Thus,º selection£ for local adaptation dramatically 0 decreases t (by a factor of ca. 50 000 in the numerical 0 2 4 6 8 10 0 T m/v example) and somewhat increases 0 relative to the case of speciation driven by mutation and genetic drift. Intui- 5 (b) tively, with selection for local adaptation counteracting the e¡ects of migration, the population can `a¡ord’ more backward steps on its route to a state of complete repro- ductive isolation than when such selection is absent. These 4 additional backward steps and the steps necessary for `compensating’ for them increase the duration of specia- tion. Figure 2a illustrates the e¡ect of selection for local adaptation on t0 in more detail. 0 *

T 3 v (ii) Linear function of reproductive compatibility Here, I assume that the probability of no reproductive isolation decreases linearly with genetic distance d from 2 one at d 0 to zero at d K 1: ˆ ˆ ‡ 1 i/=(K 1) for d4 K w ¡ ‡ . (10) ˆ 0 for d4K 1 1 2 3 4 5 6 7 8 9 10 Now, immigrants experience problems in ¢nding compa- tible mates even when the genetic distance is below K 1. m/v ‡

Figure 1. The average waiting time to speciation t 0¤ and * the average duration of speciation T 0. The x-axes give No selection for local adaptation the ratio of migration and mutation rates. (a) The y-axis With no selection for local adaptation, the probabilities gives the product of t 0¤ and the mutation rate v on the of stochastic transitions li and ·i are given by equation (4) * logarithmic scale. (b) The y-axis gives the product of T0 with m substituted for an `e¡ective’ migration rate: and the mutation rate v on the linear scale. Solid lines correspond to the exact values (found in Appendix A) and i m m 1 . (11) dashed lines correspond to the approximate equations (5) i ˆ ¡ K 1 and (6). (b) The lines correspond to K 1, 2, 4, 6, 8 and ‡ ˆ 10 (from bottom to top). The waiting time to speciation is ca.

t0 t0¤ p2º exp( K), (12) Selection for local adaptation º ¡ Assume that `new’ alleles improve adaptation to the where t 0¤ is given by equation (5). The average duration of local conditions. Let s be the average selective advantage speciation is ca. of a new allele over the corresponding ancestral allele. 1 C(K 1) ® T 1 2 ‡ ‡ , (13) Each generation, there are 2Nv such alleles supplied by 0 º v ‡ m/v mutation. The probability of ¢xation of an advantageous allele is ca. 2s/ 1 exp( 4Ns) . Migration brings ca. where ® is Euler’s constant and C( ) is the psi (digamma) ‰ ¡ ¡ Š ¢ 2Nmi ancestral alleles at the loci that have previously function. The last equation di¡ers from equation (6) only ¢xed new alleles. These alleles are deleterious in the new by the factor two inside the parentheses. For example, environment. The probability of ¢xation of a deleterious with the same parameter values as above t 2:36 108 0 º £ allele is ca. 2s/ exp(4Ns) 1 (Kimura 1983). Thus, the generations and T 1470 generations. Thus, t is signif- ‰ ¡ Š 0 ˆ 0 probabilities of stochastic transitions increasing and icantly reduced (by a factor of 57) whereas T0 is some- decreasing Db by one are ca. what larger than in the case of the threshold function of

Proc. R. Soc. Lond. B (2000) Waiting time toparapatric speciation S. Gavrilets 2487

0 K = 1 K = 2

K = 4 s s

- 5 e n t i f )

1 s K = 6 ’ l F ( a

u 0 1 d i g v o i l

K = 8 d n i - 10 AA K = 10 A b fir a b st loc Bb us aa cus B nd lo B seco (a) - 15 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Figure 3. Adaptive landscape in the two-locus, two-allele S Dobzhansky model. Alleles a and B are incompatible. The 0 arrows specify the chain of gene substitutions leading to complete reproductive isolation.

- 1 local adaptation substantially decreases t0 and slightly increases T0. ) 2

F 4. DISCUSSION (

0 - 2 1

g The results presented in ½ 3 allow one to obtain insights o l about the time-scale of parapatric speciation driven by mutation, random genetic drift and/or selection for local - 3 adaptation. I start the discussion of these results by considering the original Dobzhansky model.

(a) Two-locus, two-allele Dobzhansky model (b) - 4 Dobzhansky’s original model describes a two-locus, 1 2 3 4 5 6 7 8 9 10 two-allele system where a speci¢c pair of alleles is incom- K patible in the sense that the interaction of these alleles `produces one of the physiological isolating mechanisms’ Figure 2. E¡ects of selection for local adaptation and of linear (p. 282). Let us assume that the immigrants have function of reproductive compatibility on the waiting time to ancestral haplotyp e ab and that the derived allele B is speciation. (a) The proportion (on the logarithm scale) by incompatible with the ancestral allele a (see ¢gure 3). In which selection for local adaptation decreases t0 (F exp( KS)(1 exp( S)/S)); (see equation (8)). (b) The this case, the population can evolve to a state repro- 1 ˆ ¡ ¡ ¡ proportion (on the logarithm scale) by which t0 is reduced ductively isolated from the ancestral state via a state with relative to t if the function of reproductive compatibility is haplotype Ab ¢xed: ab Ab AB. Let ¸ be the 0¤ ! ! linear (F2 p2º exp( K)) (see equation (12)). probability of mutation from an ancestral allele (a and b) ˆ ¡ to the corresponding derived allele (A or B). The average waiting time to and the average duration of parapatric reproductive compatibility. Figure 2b illustrates the e¡ect of linear function of reproductive compatibility on t in speciation in this system are given by our general 0 equations with K 1 and v ¸. Allowing for equal more detail. ˆ ˆ selective advantage s of derived alleles over the ancestral alleles, Selection for local adaptation S S S (2¸ me¡ )(1 e¡ ) m 1 e¡ S With selection for local adaptation, the probabilities of t ‡ ¡ ¡ e¡ (15) 0 ˆ 2 2 stochastic transitions li and ·i are given by equation (7) ¸ S º ¸ S with m substituted for an `e¡ective’ migration rate and (equation (11)). The average time to speciation is ca. S S S 1 exp( S) (2¸ me¡ )(1 e¡ ) 1 1 e¡ p T0 ‡ ¡ ¡ , (16) t0 t 0¤ 2º exp( K) exp( KS) ¡ ¡ . (14) ˆ ¸S(¸ me S) º ¸ S º ¡ ¡ S ‡ ¡ The average duration of speciation T is given by where S 4Ns and the approximations are good if 0 ˆ equation (9) with an additional factor two placed in front (m=¸) exp( S)441. With no selection for local adapta- ¡ of the ratio in the parentheses. As before, selection for tion (that is if S 0 ), t m/¸2 and T 1/¸. ˆ 0 º 0 º Proc. R. Soc. Lond. B (2000) 2488 S. Gavrilets Waiting time toparapatric speciation

5 Let m 0:01 and ¸ 10¡ . Then, with no selection for ˆ ˆ local adaptation, the average waiting time to speciation is 16 (a) very long: t 108 generations and T 105 genera- 0 º 0 º tions. However, even with relatively weak selection for 14 local adaptation, t0 can decrease by one to two orders of magnitude. For example, with S 1, t 2:34 107 and 12 ˆ 0 º £ T 6:34 104, with S 2, t 5:94 106 and 0 0 ) º £ 4 ˆ º £ 6 0 10 * T0 4:36 10 and with S 3, t0 1:64 10 and t ˆ v º £ 4 º £ ( T0 3:23 10 . Because the waiting time to speciation 0 1 8

º £ g

in the two-locus Dobzhansky model scales as one over the o l mutation rate per locus squared, this time is rather long. 6 However, the overall number of loci involved in the initial stages of reproductive isolation is at least in the 4 order of tens to hundreds (e.g. Singh 1990; Wu & Palopoli 1994; Coyne & Orr 1998; Naveira & Masida 1998). This 2 increases the overall mutation rate and can make speciation much more rapid. 0 (b) (b) Average waiting time to parapatric speciation 16 In the models studied here, reproductive isolation is a consequence of cumulative genetic divergence over a set 14 of loci potentially a¡ecting mating behaviour, fertilization processes and/or o¡spring viability and fertility. The 12 underlying biological intuition is that organisms that are )

0 10 *

reproductively compatible should not be too di¡erent t v (

genetically. Most species consist of geographically struc- 0 1 8 g

tured populations, some of which experience little genetic o l contact for long periods of time (Avise 2000). Di¡erent 6 mutations are expected to appear ¢rst and increase in frequency in di¡erent populations necessarily resulting in 4 some geographical di¡erentiation even without any varia- tion in local selection regimes. An interesting question is 2 whether mutation and drift alone are su¤cient to result in parapatric speciation. This question is particularly 0 0 1 2 3 4 5 6 7 8 9 10 important given a growing amount of data suggesting m/v that rapid evolution of reproductive isolation is possible without selection for local adaptation involved (e.g. Figure 4. Waiting time to speciation with selection for local adaptation for K 10. Di¡erent lines correspond to Palumbi 1998; Vacquier 1998; Howard 1999). Our results ˆ S 0:0, 0:5, 1:0, 1:5, 2:0, 2:5 and 3:0 (from top to bottom). provide an a¤rmative answer to this question (see also ˆ Gavrilets et al. 1998, 2000a; Gavrilets 1999). However, (a) Threshold function of reproductive compatibility here the waiting time to speciation is relatively short if (equation (3)). (b) Linear function of reproductive compatibility (equation (10)). only a very small number of genetic changes is su¤cient for complete reproductive isolation. For example, t0 is in the order of ten to 1000 times the inverse of the mutation Another is that the expected distribution of allele rate if K 1 or 2 with a threshold function of repro- frequency in the island model changes from U-shaped ˆ ductive compatibility and if K 1, 2 or 3 with a linear (which implies at least some genetic di¡erentiation) to ˆ function of reproductive compatibility. It is well bell-shaped (which implies no genetic di¡erentiation on recognized that selection for local adaptation can result average) as the average number of migrants become in speciation in the presence of some gene £ow (e.g. larger than one per generation (e.g. Crow & Kimura Slatkin 1981; Rice 1984; Rice & Salt 1988; Rice & Hostert 1970). However, the equilibrium expectations derived 1993; Schluter 1998). Our results show that even relatively under neutrality theory can be rather misleading if there weak selection can dramatically reduce the waiting time is a possibility of evolving complete reproductive isolation. to speciation by orders of magnitude (see ¢gure 2a). For example, in the model with no selection for local adaptation considered above the expected change per (c) How much migration prevents speciation? generation in the genetic distance Db between the immi- In general, evolutionary biologists accept that very grants and residents is small levels of migration are su¤cient for preventing any ¢Db v mDb, (17) signi¢cant genetic di¡erentiation of the populations not ˆ ¡

to mention speciation (e.g. Slatkin 1987; but see Wade & where the ¢rst term describes an expected increase in Db McCauley 1984). To a large degree, this belief appears to because of new mutations and the second term describes be based on two observations. One is that the expected an expected decrease in Db because of the in£ux of ances- value of the ¢xation index FST is small even with a single tral genotypes. This equation predicts that Db will reach migrant per generation (e.g. Hartl & Clark 1997). an equilibrium value of v/m. From this one can be

Proc. R. Soc. Lond. B (2000) Waiting time toparapatric speciation S. Gavrilets 2489 tempted to conclude that, unless the migration rate is (d) Average duration of parapatric speciation smaller than that of mutation (v4m), Db cannot be larger In our model, the average waiting time to and the than one and, thus, no speciation is possible. However, average duration of allopatric speciation are identical. this argument is £awed. Because of the inherent stochasti- Lande (1985b) and Newman et al. (1985) have previously city of the system there is always a non-zero probability of studied how an isolated population can move from one Db moving any pre-speci¢ed distance from zero which adaptive peak to another by random genetic drift. They will lead to reproductive isolation. showed that the average duration of stochastic transitions Strictly speaking, in the models studied here migration between the peaks is much shorter than the time that the does not prevent but rather delays speciation. (The population spends in a neighbourhood of the initial peak resulting delay can be substantial and, for all practical before the transition. Within the framework used by these reasons, in¢nite.) For de¢niteness, I will say that specia- authors stochastic transitions are possible in a reasonable tion is e¡ectively prevented if the average waiting time to time only if the adaptive valley separating the peaks is speciation is larger than 1000 times the inverse of the shallow. This implies that reproductive isolation resulting mutation rate (that is, if log10(vt0)43 ). If the number of from a single transition is very small. Potentially, strong or genetic substitutions necessary for speciation is small (for even complete reproductive isolation (that is, speciation) example, K 1, as in the original Dobzhansky model or can result from a series of peak shifts along a chain of ˆ K 2), then migration rates higher than 10v will `intermediate’ adaptive peaks such that each individual ˆ e¡ectively prevent speciation in the absence of selection transition is across a shallow valley but the cumulative 3 for local adaptation. For example, if v 10¡ , then e¡ect of many peak shifts is large (Walsh 1982). In this case, ˆ speciation is possible with m as high as 0.01. However, if the results of Lande (1985b) and Newman et al. (1985) actu- 5 v 10¡ , then any migration rate higher than 0.0001 will ally imply that the population will spend a very long time ˆ e¡ectively prevent speciation. If the number of genetic at each of the intermediate adaptive peaks. This would lead changes required for speciation is relatively large, for to a very long duration of allopatric speciation that is in fact example if K 10, then without selection for local adap- comparable to the overall waiting time to speciation. ˆ tation speciation is e¡ectively prevented (see ¢gure 4a). For parapatric speciation, the predictions are very However, relatively weak selection, for example with di¡erent. Our results on the duration of speciation lead to S 2:5, would overcome migration rates as high as 10v if three important generalizations. The ¢rst is that the ˆ the strength of reproductive isolation increases linearly average duration of parapatric speciation T0 is much with genetic distance (see ¢gure 4b). smaller than the average waiting time to speciation t0. Within the modelling framework used, all immigrants This feature of the models studied here is compatible with had a ¢xed genetic composition which did not change in the patterns observed in the fossil record which form the time. Alternatively, one can imagine two populations empirical basis of the theory of exchanging migrants assuming that both populations can (Eldredge 1971; Eldredge & Gould 1972). The second evolve. If there is no selection for local adaptation, this generalization concerns the absolute value of T0. The case is mathematically equivalent to that studied above but waiting time to speciation changes dramatically with with the mutation rate being twice as large as in the case slight changes in parameter values. In contrast, the dura- of a single evolving population. Therefore, the waiting tion of speciation is of the order of one over the mutation time to speciation in the two-population case will dramati- rate over a subset of the loci a¡ecting reproductive isola- cally decrease relative to that in the single-population tion for a wide range of migration rates, population sizes, case. The maximum migration rates compatible with intensities of selection for local adaptation and the speciation will be twice as large as before. number of genetic changes required for reproductive isolation. Given a `typical’ mutation rate in the order of 5 6 (c) The role of environment 10¡ to 10¡ per locus per generation (e.g. Gri¤ths et al.

The waiting time to speciation t0 is extremely sensitive 1996; Futuyma 1997) and assuming that there are at least to parameters: changing a parameter by a small factor, in the order of ten to 100 genes involved in the initial for example two or three, can increase or decrease t0 by stages of the evolution of reproductive isolation (e.g. several orders of magnitude. Looking across a range of Singh 1990; Wu & Palopoli 1994; Coyne & Orr 1998; parameter values, t0 is either relatively short (if the para- Naveira & Masida 1998), the duration of speciation is meters are right) or e¡ectively in¢nite. Most of the para- predicted to range between 103 and 105 generations with meters of the model (such as the migration rate, intensity the average in the order of 104 generations. The third of selection for local adaptation, the population size and, generalization is about the likelihood of situations where probably, the mutation rate) depend directly on the state strong but not complete reproductive isolation between of the environment (biotic and abiotic) the population populations is maintained for an extended period of time experiences. This suggests that speciation can be triggered (much longer than the inverse of the mutation rate) in the by changes in the environment (cf. Eldredge 2000). Note presence of small migration without the populations that the time-lag between an environmental change initi- becoming completely isolated or completely compatible. ating speciation and an actual attainment of reproductive Judging from our theoretical results, such situations isolation can be quite substantial as our model shows. If it appear to be extremely improbable. is an environmental change that initiates speciation, the populations of di¡erent species inhabiting the same (e) Validity of the approximations used geographical area should all be a¡ected. In this case, one The results presented here are based on a number of expects more or less synchronized bursts of speciation in a approximations the most important of which is the geographical area, that is a `turnover pulse’ (Vrba 1985). assumption that within-population genetic variation in

Proc. R. Soc. Lond. B (2000) 2490 S. Gavrilets Waiting time toparapatric speciation

the loci underlying reproductive isolation can be for i 0, 1, : : :, K with tK 1 0 (e.g. Norris 1997). I ˆ ‡ ˆ neglected. A biological scenario to which this assumption assume that the transition probabilities are pi,i 1 li , ‡ ˆ is most applicable is that of a small (peripheral) popula- pi,i 1 ·i and pij 0 if i j 41 with ·K 1 0. In this ¡ ˆ ˆ ¡ ‡ ˆ tions with not much genetic variation maintained and case, the system of linearj equationj (A1) can be solved by with an occasional in£ux of immigrants from the main standard methods (e.g. Karlin & Taylor 1975). population. (Note that within-population genetic varia- Let zi ti ti 1. From equation (A1) with i 0 one ˆ ¡ ‡ ˆ tion in the loci underlying reproductive isolation has to be ¢nds an equality t 1 l t (1 l )t which can be 0 ˆ ‡ 0 1 ‡ ¡ 0 0 manifested in reproductive incompatibilities between rewritten as some members of the population. However, the overall z 1/l : (A2) proportion of incompatible mating pairs within the popu- 0 ˆ 0 lation is not expected to be large (e.g. Wills 1977; Nei et al. In a similar way, for i40 one ¢nds an equality t 1 1983; Gavrilets 1999).) Intuitively, one might expect that i ˆ liti 1 ·iti 1 (1 li ·i)ti which can be rewritten increasing within-population variation would substan- ‡ ‡ ‡ ¡ ‡ ¡ ¡ as tially increase the rate of substitutions by random genetic drift and make speciation easier. However, in poly- morphic populations the alleles a¡ecting the degree of ·i 1 zi zi 1 : (A3) reproductive isolation cannot be treated as neutral ˆ li ¡ ‡ li because they are weakly selected against than rare (Gavrilets et al. 1998, 2000a; Gavrilets 1999). In the The solution of the system of linear recurrence equations absence of selection for local adaptation this might make (A2) and (A3) is speciation somewhat more di¤cult. Allowing for genetic « i « variation among immigrants can increase the plausibility z i i , (A4) i ˆ ‡ of speciation. For example, if new alleles are deleterious l0 j 1 lj«j in the ancestral environment and are maintained there by ˆ mutation, their equilibrium frequency will be order v/s¤, where s¤ where is the selection coe¤cient against new alleles in · · : : : · the ancestral environment. Thus, the overall frequency of « 1 2 j (A5) j ˆ new alleles in the population per generation will increase l1l2 : : : lj

from v to ca. v mv/s¤. Intuitively, this can result in a K ‡ with «0 1. One can also see that §i 0 zi substantial reduction in the waiting time to speciation. ˆ ˆ (t0 t1) (t1 t2) : : : (tK tK 1) t0. Thus, t0 can The overall e¡ect of genetic variation (both within ˆ ‡ ‡ ‡ ˆ be found¡ by summing¡ up equation¡ (A4) to obtain population and among immigrants) on the waiting time to parapatric speciation has to be explored in a systematic K K i i 0 «i «i way. This is particularly important given that the t ˆ . (A6) 0 ˆ ‡ l0 i 1 j 1 lj«j individual-based simulations reported in Gavrilets et al. ˆ ˆ (1998, 2000a) show that rapid speciation is possible well beyond the domain of parameter values identi¢ed here as The absorption times ti corresponding to i40 can be conducive to speciation. As for the duration of speciation, found recursively using equation (A3). I expect it to have an order of one over the level of genetic With a threshold function of reproductive compatibility variation maintained in the loci underlying reproductive (equation (3)), isolation. As such, with genetic variation, the duration of « R jj!, (A7) speciation is expected to be (much) shorter than 1=v. j ˆ where R (m/v) exp( S). With a linear function of ˆ ¡ I am grateful to Janko Gravner for helpful suggestions, to Chris reproductive compatibility (equation (10)), Boake and two reviewers for comments on the manuscript and to the members of a working group organized by Niles Eldredge K! and John Thompson at the National Center for Ecological « R jj! . (A8) j ˆ (K 1)j(K j)! Analysis and Synthesis (NCEES), University of California, ‡ ¡ Santa Barbara, for illuminating discussions. This work was supported by National Institutes of Health grant GM56693 and by an Exhibit, Performance and Publication Expense fund (b) Average duration of speciation (University of Tennessee, Knoxville, TN, USA). The average duration of speciation T0 can be de¢ned as the average time that it takes to walk from state 0 to APPENDIX A state K 1 without returning to state 0. Ewens (1979, ‡ ½ 2.11) provided formulae that can be used to ¢nd T . (a) Average waiting time to speciation 0 These formulae are summarized below. I consider a Markov chain with K 1 states ‡ The probability of entering state K 1 before state 0 0, 1, : : :, K, K 1. Let p be the corresponding tran- ‡ ‡ ij starting from i is sition probabilities. I assume that the state K 1 is ‡ i 1 K absorbing but the state 0 is not. Let ti be the average time ¡ to absorption starting from i. The mean absorption times º « « . (A9) i ˆ j j j 0 j 0 satisfy to a general system of linear equations ˆ ˆ t 1 p t , (A1) i ˆ ‡ ij j Starting from state i, the mean time spent in state j before j entering state 0 or state K 1 is ‡

Proc. R. Soc. Lond. B (2000) Waiting time toparapatric speciation S. Gavrilets 2491

j 1 Futuyma, D. J. 1997 . Sunderland, MA: ¡ t (1 º ) « /(« l ) for j 1, : : :, i, (A10) Sinauer. i j ˆ i k j j ˆ ¡ k 0 Gavrilets, S. 1997 Evolution and speciation on holey adaptive ˆ landscapes.Trends Ecol. Evol. 12, 307^312. and Gavrilets, S. 1999 A dynamical theory of speciation on holey

K adaptive landscapes. Am. Nat. 154, 1^22. Gavrilets, S. 2000 Evolution and speciation in a hyperspace: the t º « /(« l ) for j i 1, : : :, K. (A11) i j ˆ i k j j ˆ ‡ roles of neutrality, selection, mutation and random drift. In k j ˆ Towards a comprehensive dynamics of evolutionöexploring the Starting from state i, the conditional mean time spent in interplay of selection, neutrality, accident, and function (ed. J. state j for those cases for which the state K 1 is entered Crutch¢eld and P. Schuster). New York: Oxford University ‡ before state 0 is Press. (In the press.) Gavrilets, S. & Boake, C. R. B. 1998 On the evolution of tij¤ tijºj/ºi. (A12) premating isolation after a founder event. Am. Nat. 152, ˆ 706^716. The condition mean time till absorption in K 1 is Gavrilets, S. & Gravner, J. 1997 Percolation on the ¢tness hyper- ‡ cube and the evolution of reproductive isolation. J.Theor. Biol. K 184, 51^64. t ¤ t ¤ . (A13) i ˆ ij Gavrilets, S. & Hastings, A. 1996 Founder e¡ect speciation: a j 1 ˆ theoretical reassessment. Am. Nat. 147, 466^491. The average duration of speciation is the sum of the Gavrilets, S., Li, H. & Vose, M. D. 1998 Rapid parapatric average time spent in state 0 before moving to state 1, speciation on holey adaptive landscapes. Proc. R. Soc. Lond. B 265, 1483^1489. which is 1/l , plus the conditional mean time till absorp- 0 Gavrilets, S., Li, H. & Vose, M. D. 2000a Patterns of parapatric tion in K 1 starting from state 1, which is t ¤, ‡ 1 speciation. Evolution 54, 1126^1134. Gavrilets, S., Acton, R. & Gravner, J. 2000b Dynamics of T0 1/l0 t 1¤. (A14) ˆ ‡ speciation and diversi¢cation in a metapopulation. Evolution 54, 1493^1501. Gradshteyn, I. S. & Ryzhik, I. M. 1994 Tables of integrals, series, REFERENCES and products. San Diego, CA: Academic Press. Avise, J. C. 2000 . Cambridge, MA: Harvard Gri¤ths, A. J. F., Miller, J. H., Suzuki, D. T., Lewontin, R. C. University Press. & Gelbart, W. M. 1996 An introduction to genetic analysis, 6th Barton, N. H. 1993 The probability of ¢xation of a favoured edn. New York: W. H. Freeman. allele in a subdivided population. Genet. Res. 62, 149^157. Hartl, D. L. & Clark, A. G. 1997 Principles of population . Burger, W. 1995 Montane species-limits in Costa Rica and Sunderland, MA: Sinauer Associates. evidence for local speciation on attitudinal gradients. In Higgs, P. G. & Derrida, B. 1992 Genetic distance and species and conservation of neotropical montane forests (ed. S. P. formation in evolving populations. J. Mol. Evol. 35, 454^465. Churchill), pp. 127^133. New York: The New York Botanical Howard, D. J. 1999 Conspeci¢c sperm and pollen precedence Garden. and speciation. A. Rev. Ecol. Syst. 30, 109^132. Coyne, J. A. & Orr, H. A. 1998 The evolutionary genetics of Johnson, K. P., Adler, F. R. & Cherry, J. L. 2000 Genetic and speciation. Phil.Trans. R. Soc. Lond. B 353, 287^305. phylogenetic consequences of island . Evolution Crow, J. F. & Kimura, M. 1970 An introduction to 54, 387^396. theory. Minneapolis, MN: Burgess Publishing Company. Johnson, N. A. & Porter, A. H. 2000 Rapid speciation via Dieckmann, U. & Doebeli, M. 1999 by parallel, on regulatory genetic pathways. sympatric speciation. Nature, 400, 354^357. J.Theor. Biol. 205, 527^542. Dobzhansky, T. G. 1937 Genetics and the origin of species. New York: Karlin, S. & Taylor, H. M. 1975 A ¢rst course in stochastic processes, Columbia University Press. 2nd edn. San Diego, CA: Academic Press. Eldredge, N. 1971 The allopatric model and phylogeny in Kimura, M. 1983 The neutral theory of . New Paleozoic invertebrates. Evolution 25, 156^167. York: Cambridge University Press. Eldredge, N. 2000 The sloshing bucket: how the physical realm Lande, R. 1979 E¡ective size during long-term evolution controls evolution. In Towards a comprehensive dynamics of evolu- estimated from rates of chromosomal rearrangement. Evolution tionöexploring the interplay of selection, neutrality, accident, and 33, 234^251. function (ed. J. Crutch¢eld and P. Schuster). New York: Oxford Lande, R. 1985a The ¢xation of chromosomal rearrangements University Press. (In the press.) in a subdivided population with local and coloniza- Eldredge, N. & Gould, S. J. 1972 Punctuated equilibria: an tion. Heredity 54, 323^332. alternative to . In Models in paleobiology Lande, R. 1985b Expected time for random genetic drift of a (ed. T. J. Schopf ), pp. 82^115. San Francisco, CA: Freeman, population between stable phenotypic states. Proc. Natl Acad. Cooper. Sci. USA 82, 7641^7645. Endler, J. A. 1977 Geographic variation, speciation and clines. Macnair, M. R. & Gardner, M. 1998 The evolution of edaphic Princeton University Press. endemics. In Endless forms: species and speciation (ed. D. J. Ewens,W. J. 1979 Mathematicalpopulation genetics. Berlin: Springer. Howard & S. H. Berlocher), pp. 157^171. New York: Oxford Frias, D. & Atria, J. 1998 Chromosomal variation, macro- University Press. evolution and possible parapatric speciation in Mepraia Naveira, H. F. & Masida, X. R. 1998 The genetics of hybrid spinolai (Porter) (Hemiptera: Reduviidae). Gen. Mol. Evol. 21, male sterility in . In Endless forms: species and specia- 179^184. tion (ed. D. J. Howard & S. H. Berlocher), pp. 330^338. New Friesen,V. L. & Anderson, D. J. 1997 Phylogeny and evolution of York: Oxford University Press. the Sulidae (Aves: Pelecaniformes): a test of alternative modes Nei, M., Maruyama, T. & Wu, C.-I. 1983 Models of evolution of of speciation. Mol. Phylogenet. Evol. 7, 252^260. reproductive isolation. Genetics 103, 557^579.

Proc. R. Soc. Lond. B (2000) 2492 S. Gavrilets Waiting time toparapatric speciation

Newman, C. M., Cohen, J. E. & Kipnis, C. 1985 Neo- Slatkin, M. 1976 The rate of spread of an advantageous allele in a Darwinian evolution implies punctuated equilibria. Nature subdivided population. In Population genetics and ecology (ed. S. 315, 400^401. Karlin & E. Nevo), pp. 767^780. NewYork: Academic Press. Norris, J. R. 1997 Markov chains. Cambridge University Press. Slatkin, M. 1981 Fixation probabilities and ¢xation times in a Orr, H. A. 1995 The population genetics of speciation: subdivided population. Evolution 35, 477^488. the evolution of hybrid in compatibilities. Genetics 139, Slatkin, M. 1982 and parapatric speciation. Evolution 1803^1813. 36, 263^270. Orr, H. A. & Orr, L. H. 1996 Waiting for speciation: the e¡ect Slatkin, M. 1987 Gene £ow and the geographic structure of of population subdivision on the waiting time to speciation. natural populations. Science 236, 787^792. Evolution 50, 1742^1749. Tachida, H. & Iizuka, M. 1991 Fixation probability in spatially Palumbi, S. R. 1998 Species formation and the evolution of varying environments. Genet. Res. 58, 243^251. gamete recognition loci. In Endless forms: species and speciation Turner, G. F. & Burrows, M. T. 1995 A model of sympatric (ed. D. J. Howard & S. H. Berlocher), pp. 271^278. New York: speciation by . Proc. R. Soc. Lond. B 260, Oxford University Press. 287^292. Rice, W. R. 1984 Disruptive selection on preferences and Vacquier, V. D. 1998 Evolution of gamete recognition proteins. the evolution of reproductive isolation: a simulation study. Science 281, 1995^1998. Evolution 38, 1251^1260. Van Doorn, G. S., Noest, A. J. & Hogeweg, P. 1998 Sympatric Rice, W. R. & Hostert, E. E. 1993 Laboratory experiments on speciation and extinction driven by environment dependent speciation: what have we learned in 40 years? Evolution 47, sexual selection. Proc. R. Soc. Lond. B 265, 1915^1919. 1637^1653. Vrba, E. S. 1985 Environment and evolution: alternative causes Rice, W. R. & Salt, G. B. 1988 Speciation via disruptive selec- of the temporal distribution of evolutionary events. S. Afr. J. tion on habitat preference: experimental evidence. Am. Nat. Sci. 81, 229^236. 131, 911^917. Wade, M. J. & McCauley, D. E. 1984 Group selection: the inter- Ripley, S. D. & Beehler, B. M. 1990. Patterns of speciation in action of local deme size and migration in the di¡erentiation Indian birds. J. Biogeog. 17, 639^648. of small populations. Evolution 38, 1047^1058. Rola¨ n-Alvarez, E., Johannesson, K. & Erlandson, J. 1997 The Walsh, J. B. 1982 Rate of accumulation of reproductive isolation maintenance of a in the marine snail Littorina saxatilis: by rearrangements. Am. Nat. 120, 510^532. the role of home site advantage and hybrid ¢tness. Evolution Wills, C. J. 1977 A mechanism for rapid allopatric speciation. 51, 1838^1847. Am. Nat. 111, 603^605. Schluter, D. 1998 Ecological causes of speciation. In Endless Wu, C.-I. 1985 A stochastic simulation study of speciation by forms: species and speciation (ed. D. J. Howard & S. H. sexual selection. Evolution 39, 66^82. Berlocher), pp. 114^129. New York: Oxford University Press. Wu, C.-I. & Palopoli, M. F. 1994 Genetics of postmating repro- Singh, R. S. 1990 Patterns of species divergence and genetic ductive isolation in animals. A. Rev. Genet. 27, 283^308. theories of speciation. In Population biology. Ecological and evolutionary viewpoints (ed. K. Wo« hrmann & C. K. Jain), As this paper exceeds the maximum length normally permitted, pp. 231^265. Berlin: Springer. the author has agreed to contribute to production costs.

Proc. R. Soc. Lond. B (2000)

Top View