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Proc. NatI. Acad. Sci. USA Vol. 81, pp. 1764-1767, March 1984

Mitochondrial flow ( genetics/interspedfic gene flow/mitochondrial DNA) NAOYUKI TAKAHATA*t AND MONTGOMERY SLATKIN Department of , NJ-15, University of Washington, Seattle, WA 98195 Communicated by W. T. Edmondson, November 28, 1983

ABSTRACT To account for the transmission of mitochon- drial because of the linkage disequilibrium between drial DNA between Nonspecific Drosophila pseudoob- them. This could be important for interspecific gene flow scura and D. pprsimilis in reported by J. R. Powell especially, because it is reasonable to expect some selection [Powell, J. R. (1983) Proc. Nail. Acpd. Sci. USA 80, 492-495], against hybrids. Powell (1) proposes selection against het- a simple model of gene flow and selection in infinite popula- erozygotes as one mechanism for preventing the interchange tions is analyzed. The model assumes two at each of two of nuclear among the two Drosophila species. Another loci, one of which is coded by an autosome and the other' by mechanism Powell suggests is . If the hy- mitochondrial DNA. Viability selection is'presumed to be un- brids are able to mate with only one of the parental species derdominant-i.e., heterozygous inferiority to the homozy- and the backcrosses tend to mate with the same parental spe- gotes- at an autosomal , and neutral or deleterious at a cies, then nuclear and mitochondrial gene flow between spe- mitochondrial locus, with'the combined action being multipli- cies could be lower than expected on the basis of'the number cative. Extremely strong selection against heterozygotes may of hybrids observed. We will not consider this possibility prevent the transmission of mitochondrial DNA between two here, but we will show that the available data are consistent species, but otlerwise the transmission can easily occur over with a model that does not incorporate this effect. species bptindaries. The rate of opproach to equilibrium is de- termined by the level of gene flow and is not affected much by Model and Analysis selection against an autosomal locus. The divergence'of the nu- We consider a simple model of two loci. One locus is carried clear genomes of thp two species is reexamined. Based on pub- lished data on enzyme loci, we conclude that there has been by an autosomal' and the other is carried by an mitochondrial gene flow between these species for a long extranuclear (mitochondrial or chloroplast) chromosome. enough time that several nuclear loci examined could diverge We assume two alleles at each locus and denote them A and because of accumulation of neutral . a for autosomal alleles and M and m for extranuclear alleles. Assuming the completely maternal inheritance of extranu- clear (8), we can treat the extranuclear locus Three recent studies have shown that the geographic pat- as haploid and represent the types of female gametes by AM, terns exhibited by mitochondrial DNA do not necessarily co- Am, aM, and am and the male gametes by A and a. The incide with the geographic patterns of anatomical traits and of the individuals are, therefore, represented by of nuclear genes. Powell (1) has suggested the interspecific the pairs of these gametes as given in Table 1. transfer of mitochondrial DNA between Drosophila pseu- We consider a randomly mating population that is of effec- doobscura and D. persimilis in their region of sympatry, tively infinite size but that exchanges only females with a even though the two species are distinguishable on cytologi- population (representing the other species) consisting of aam cal and behavioral grounds (2) and have differentiated at sev- individuals. Immigration of males does not alter the follow- eral nuclear loci (3, 4). Ferris et al. (5) have found a similar ing treatment by much, but we assume that males are pattern in two of Mus musculus: M. m. musculus sterile, as is usually observed in conspecific matings (see ref. in Scandinavia contains mitochondrial DNA of the species 1 and refs. therein). We assume that selection occurs only M. m. domesticus. Yonekawa'et al. (6) showed there is mito- through the difference in viability. For the autosomal locus, chondrial gene flow between two subspecies of M. muscu- we assume underdominance (the heterozygote inferior to the lus. In Japan, only one subspecies is found, M. m. molos- homozygotes) to model a locus assumed'to be responsible sinus, which is homogeneous in and in electro- for the between the two in phoretically detectable nuclear loci. In the northern part of question. Thus, we consider only the case s > 0, but we do Honshu and in Hokkaido, however, the mitochondrial DNA not exclude the possibility of a large s, which may be as large is apparently the same as that in another subspecies, M. m. as 0.5 or more because of lowered viability of F1 hybrids and castaneus, which occurs in southern China and the Philip- because of backcross progeny. For the extranuclear locus, pines. In light of these findings, it is appropriate to investi- on the other hand, we assume neutrality or selective disad- gate the theory of. mitochondrial gene flow with particular vantage of the m (t . 0). The selection regime is also reference to the conditions under which the geographic pat- given in Table 1. Immigration of females, random mating, terns exhibited by mitochondrial and nuclear genes differ. and selection are assumed to occur in this order, and the In this paper we will analyze a simple model of gene flow generations are nonoverlapping and discrete. and selection affecting both mitochondrial and nuclear Let g be the immigration rate of females per generation, genes, and we use this model to discuss the data of Powell and P1, P2, P3, and P4 be the gamete frequencies ofAM, Am, (1). We will consider, in particular, the association between aM, and am in females, and Pt and PI be the frequencies of mitochondrial and autosomal loci generated by gene flow A and a in males. (7). It is possible, at least, that selection at an autosomal lo- cus could the establishment of a neutral mitochon- prevent *Permanent address: National Institute of Genetics, Mishima, 411 Shizuoka-ken, Japan. The publication costs of this article were defrayed in part by page charge tPresent address (until Aug. 31, 1984): Center for Demographic & payment. This article must therefore be hereby marked "advertisement" , University of Texas at Houston, Houston, in accordance with 18 U.S.C. §1734 solely to indicate this fact. TX 77225. 1764 Downloaded by guest on September 24, 2021 Genetics: Takahata and Slatkin Proc. Natl. Acad. Sci. USA 81 (1984) 1765

Table 1. Combinations of female gametes (AM, Am, aM, and W am) and male gametes (A and a) _ q" = (1 - t)(2 (1) ) )+ sq q AM Am aM am [6b] t A 1 1 - t 1 - S (1 - S)(1 - t) + -f(1 - s - 2q)p + (1 - s)qr] a 1 - S (1- S)(1 - t) 1 1 - t 2 Relative viabilities of the six genotypes are shown, assuming mul- tiplicative action of nuclear and extranuclear genes and underdomi- 1 g r" = s(2q - 1)p + (1 - sq)r, [6c] nance at the nuclear locus; 1 - s is the relative viability ofautosomal heterozygotes and 1 - t is the relative viability of individuals carry- ing the m extranuclear allele. where After immigration, the gamete frequencies in females change from Pi (i = 1, 2, 3, 4) to W = A0(q) + Al(q)p + A2(q)r, AO(q) = (1 - t)[1 - s(q + q' - 2qq')], [q' = (1 - g)q] [7] P! = (1 - g)Pi (i = 1, 2, 3) Al(q) = st(1 - g)(2q - 1), [1] 1~~~~~~~= A2(q) = t(l - g)(1 - sq). P4 + ( - 9)P4 I pi= l ,

Neutral Mitochondrial Genes (t = 0) but no change occurs in males-i.e., P*' = P* (i = 1, 2), where the primes denote the frequencies after immigration. We study first the case in which the extranuclear genes are Random mating and the subsequent viability selection selectively neutral to find the effect of an underdominant nu- change the gamete frequencies in the next generation. The clear gene on an extranuclear one. In this case, Eq. 6b be- mean of the population is given by comes W = Pt'[P' + (1 - t)P2 + (1 - s)P3 + (1 - s)(1 - t)P4] wo q,, = [(2 g)(1 s) + sq1q + P2*'[(1 - S)PW + (1 - s)(1 - t)P2 + P [2] [8] 1 g 2( 1- g) + (1 - t4], in which and the gamete frequencies in females become WO = 1 - s[q + q' - 2qq']. WP'X = PiPt' + (1 - s)(PiPt' + P'Pt')/2 [3a] On the other hand, Eqs. 6a and 6c do not change except for the replacement of W by W0. WP2 = (1 - t)[P2Pt' + (1 - S)(P2PI' + P'Pt')/2] [3b] We first examine their equilibrium properties. From Eq. 8, WP'3 = P3PS' + (1 - S)(PiPl' + P3Pt')/2 [3C] it is readily seen that there are three equilibria for q, denoted qo, q_, and q+. Clearly, one equilibrium is WP'4 = (1 - t)[P4Pl' + (1 - S)(P2Pt' + P4Pt')/2]. [3d] Ao = 0 In a similar way, we can calculate the gamete frequencies in males and obtain the following relationship. and the other two are the solutions of

Pt" = P'1 + P'2 (P2*" = P'3 + Pf4)- [4] g 4(1 g) q2 - 2(3 - 2g)q + + 2 - g = 0 [9] S Therefore, after one generation the frequencies of the auto- somal genes become the same in both sexes. Although only when s > 4g(1 - g). females immigrate, so that Pt' + P' + P', the frequency of The equilibrium points of qO and the larger root q+ are A on an autosomal locus becomes identical in both sexes stable, while q_ is unstable. When s is much larger than g, after random mating. To simplify the subsequent analysis, we assume that Pt = P1 + P2 in the initial population. For convenience, we transform the variables Pi as follows - 1 g(1 s) q+ 1 2s [10] - p = Pi q_ (1+ g/s)/2 q =P1 + P2 [5]

r = P1 + P3, to a good approximation. The model of the autosomal locus is similar to that of Slatkin (9). Using the above result, we can show that the equilibrium which denote the frequencies of AM, A, and M, respective- point of r and, therefore, ofp is 0 (p r). Thus, if s > 4g( 1 - ly. From Eqs. 1 and 3, we have g), the equilibrium states (p5, 4, r) = (0, 0, 0) and (0, 4+, 0) are stable, while if s < 4g(1 - g), the only stable equilibrium is given by (0, 0, 0). 2W For the = p" = (1 - s + 2sq)p + (1 - s)qr [6a] equilibrium (pO, q, r) (0, 4+, 0), it is of interest to 1-ge know the rate at which the equilibrium state is attained. We Downloaded by guest on September 24, 2021 1766 Genetics: Takahata and Slatkin Proc. NatL Acad Sci. USA 81 (1984) linearize the equations around this equilibrium state (0, q+, q, which we must determine. Letting q" = q = q in Eq. 6b 0). Letting p - j + Ep, q = q + Eq, and r = P + Er, we have and substituting 1 - E for 4, we find 1 - s - g(1 - 3s)/2 [11a] - - Eq 11-sg q g(l t)(1 S) 2(s - t + st) [17] and t (:/)Ep 8 1__- g and, thus

A 1 9 = .. - - - - ) r 1 (1 S) [18] (1 s(l 21), (1 s) EP lb] 1 g-t 2s(2q - 1), 2(1 - s ') / Er s a (1 )[ sgs(1-t)1+ [19] where W = 1 - 2q [g + 2(1 - g)(1 - 4)]. The larger eigenval- 2(s' t + St)j. ue of the matrix in Eq. lib is approximately The equilibrium value of f-i.e., the equilibrium frequency 1 - (1 - s)g. [idc] of M, decreases almost linearly to zero as g increases to t/(1 - s), as shown in Fig. 1. It is also important to note in Eq. 18 Thus, and when g << s, we can see from Eqs. Ha and lic that r depends on the , s, against a het- that the frequency of q approaches to 4+ at the rate of s, erozygous autosomal locus. As shown in Eq. 14, the quasi- p r O at rate of (1 - while the frequencies of and approach the linkage equilibrium between autosomal and extranuclear loci speaking, the fixation of the m allele in a popu- s)g. Roughly is attained at equilibrium, but the selection does affect the lation takes place on the time scale set by the immigration equilibrium frequency of the extranuclear gene. If s = r = rate of females. This process is very slow compared with the - - but r increases as s increases: in rate that A attains its equilibrium frequency. (1 g/t)/(1 g) exactly, fact, is close to 1 if the hybrid is lethal (s = 1). In other selection, but only very strong selection, against an Genes (t > 0) words, Deleterious Mitochondrial autosomal locus obstructs extranuclear gene flow. In this case, we look for the equilibrium values of (p, r) other Non-Zero Paterhal Contribution than (0, 0). We note that letting x = p6/, we have The above results are derived under the assumption of com- [1 - s(1 - 24)]x + (1 - s)q [12] pletely maternal inheritance. For the robustness, however, a of ex- 2[1 -sq - s(l - 2q)x] model incorporating the possibility of paternal leakage tranuclear genomes is more desirable. We will not give the full analysis of such a model here, but we present a few re- from Eqs. 6a and 6c. Eq. 12 has one and only one real root in neutral mutations. which is by sults for (0, 1), given We considered a model that allows the paternal leakage and that guarantees the complete fixation of extranuclear genes in a cell during one generation. The fixation within a

4sq - 1 - s + [(4sq - 1 - S)2 + 8s(1 - s)4(24 - 1)]1/2 1.0 4s(2q - 1) [13]

Because selection against the m allele is expected not to de- crease q+ by much and because we are interested in the case s g, we assume = 1 - where £ <« 1. with >> that q E 0 Substituting q = 1 - E into Eq. 13 and, ignoring the higher order terms of E, we have

X = 1 - E [14]

or x = q. In other words, = approximately holds, and the linkage disequilibrium d = p - qr is very small at equilib- rium. Using this fact and Eq. 6c, we have b.~ .0

- -1) A t sg(2q = [15] r t(l g) t(i g) [1 2sq'(1 @)]

For r to be in the interval (0, 1) we must have 0 t/(1 -s) 9 ga < t < a [16] FIG. 1. Equilibrium frequency of M, the mitochondrial allele, as - < where a = (1 - sq)/[l - 2sq(1 - q)]. a of g, the immigration rate, under the condition g(1 s) The above results are given in terms of the unknown value t < (1 - s). Downloaded by guest on September 24, 2021 Genetics: Takahata and Slatkin Proc. Natl. Acad. Sci. USA 81 (1984) 1767

cell is caused by random transmission of extranuclear genes Table 2. Genetic distances D (ref. 10) between D. pseudoobscura at mitosis. The fixation probability, u, of extranuclear genes and D. persimilis for each of the loci found to be polymorphic that are transmitted from a female gamete is an important Locus D Locus D quantity. If u = 1 either because of complete maternal inheri- tance or because of preferential replication of maternal ge- Pt-7 0.0014 G6PDH 0.0341 nomes, the present model reduces to the one described earli- Pt-12 0.2562 LAP 0.0320 er. But if u < 1, we have Pt-13 0.0020 Pt-10 0.1617 MDH 0.0005 XDH 1.2239* ODH 0.0005 Amy 0.0877 p" = 1/2[(1 - ug)p + (1 - g)qr] AO-2 0.0020 Est-5 3.6282 Pt-8 0.2790 q" = (1 - )q [20] Data are from table 38 in ref. 3. Except as noted, the allele fre- quencies were obtained using a single electrophoretic technique. Eleven other loci were monomorphic for the same allele in both spe- r" = (1 - ug)r cies. *From the data of Coyne (4), who used several techniques to reveal hidden variability, D = 1.8597. instead of Eq. 6 with t = s = 0. From Eq. 20 we can conclude that the incomplete fixation (u < 1) possibly retards the trophoretic study of the XDH locus, found additional alleles, process, but the effect is small unless u is small. From this thereby increasing the genetic distance at that locus, but the result and from the study of the case of t > 0 and s > 0, we XDH locus was already markedly different between the spe- can show that our conclusions are not affected by the pater- cies. Comparable results have not been found for other loci. nal leakage. Our conclusion is that the divergence of the nuclear genomes of these two species is not as extensive as was indicated by Discussion and Conclusions Powell. Nei and Feldman (12), in a model of gene flow and genetic We conclude from the preceding analysis that a very small drift, found that the expected value of genetic distance be- amount of immigration is sufficient to establish a neutral tween two partially isolated populations is approximately mitochondrial genotype in a population unless hybrids are ,u/g, if ,u << g << 1 where u is the mutation rate and g is lethal or nearly so. The potential association with an under- the migration rate. If a value of 10-4 is accurate for g, then dominant autosomal locus hardly affects the progress of the the data for most loci in Table 2 are consistent with the as- immigrant mitochondrial genotype. The time required for the sumption that the different alleles are equally fit and the mu- mitochondrial genotype to become common is of the order of tation rates are of the order of 10-6 or 10-7. The few loci that the inverse of the immigration rate, 1/g in our notation, if have differentiated more extensively require the assumption hybrids are viable. Also, we found that, if the immigration of either higher rates, some form of selection (pos- rate is small, as must be the case for different species, then sibly on closely linked loci), or both. Finally, the extent of only a small amount of selection against immigrant mito- replacement of one mitochondrial genotype by the other sug- chondrial genotype is sufficient to prevent its establishment. gests that there has been gene flow between the two species We can apply these results to find if there is a relatively for a long enough time that the nuclear genomes could have simple explanation for the data of Powell (1). D. persimilis diverged through the action of and mutation and D. pseudoobscura are nearly reproductively isolated but alone, if gene flow were absent. the F1 hybrid females are fertile. Naturally occurring hybrids have been found in an approximate frequency of 1 in 10,000 We thank J. F. Crow and J. R. Powell for numerous helpful com- (1). According to our model, this small amount of gene flow ments on the earlier version of this paper. This research has been between the two species would be sufficient to lead to the supported in part by National Science Foundation Grant DEB- replacement of the mitochondrial genotype of one species by 8120580 and in part by a Grant-in-Aid from the Ministry of Educa- that of the other in =10,000 generations as long as the two tion, Science and Culture, Japan. mitochondrial genotypes are equally fit. Association with underdominant nuclear loci would not significantly affect 1. Powell, J. R. (1983) Proc. Natl. Acad. Sci USA 80, 492-495. that time scale of replacement. 2. Dobzhansky, Th. (1951) Genetics and the Origin of Species (Columbia Univ. Press, New York), 3rd Ed. The question posed by Powell (1) in light of his finding is 3. Lewontin, R. C. (1974) Genetic Basis ofEvolutionary Change how this gene flow could not also prevent the divergence of (Columbia Univ. Press, New York). the nuclear genomes of the two species in sympatry. Powell 4. Coyne, J. A. (1976) Genetics 84, 593-607. characterizes the nuclear genomes as "completely diver- 5. Ferris, S. D., Sage, R. D., Huang, C.-M., Nielsen, J. T., gent" (p. 494 of ref. 1). As shown in Table 2, however, the Ritte, U. & Wilson, A. C. (1983) Proc. Nati. Acad. Sci. USA available data do not support that characterization. Of the 24 80, 2290-2294. loci examined by Prakash and his co-worker (data reported 6. Yonekawa, H., Tagashira, S. & Moriwaki, K. (1983) Iden 37, in ref. 3, table 38), 11 are monomorphic for the same allele. 29-37 (in Japanese). For the 13 polymorphic loci, the genetic distance (10) ex- 7. Prout, T. (1973) Genetics 73, 493-496. 8. Wallace, D. C. (1982) Microbiol. Rev. 46, 208-240. ceeds 0.1 for only 5 of the loci. In fact at only 3 loci are the 9. Slatkin, M. (1982) 36, 263-270. most common alleles different. Lewontin (ref. 3, p. 174) con- 10. Nei, M. (1972) Am. Nat. 106, 283-292. cluded that, "there is very little differentiation in gene fre- 11. Prakash, S. (1976) Evolution 31, 14-23. quencies." The more extensive study by Prakash (11) rein- 12. Nei, M. & Feldman, M. W. (1972) Theor. Popul. Biol. 3, 460- forces this conclusion. Coyne (4), in a more exhaustive elec- 465. Downloaded by guest on September 24, 2021