NatI. Acad. Sci.USA. Vol. 81, pp. 1764-1767, March 1984. Genetics. Mitochondrial gene flow. (population genetics/interspedfic gene flow/mitochondrial DNA).
Proc. NatI. Acad. Sci. USA Vol. 81, pp. 1764-1767, March 1984 Genetics
Mitochondrial
gene flow (population genetics/interspedfic gene flow/mitochondrial DNA) NAOYUKI TAKAHATA*t AND MONTGOMERY SLATKIN
Department of Zoology, NJ-15, University of Washington, Seattle, WA 98195
Communicated by W. T. Edmondson, November 28, 1983
To account for the transmission of mitochonABSTRACT drial DNA between Nonspecific species Drosophila pseudoobscura and D. pprsimilis in sympatry reported by J. R. Powell [Powell, J. R. (1983) Proc. Nail. Acpd. Sci. USA 80, 492-495], a simple model of gene flow and selection in infinite populations is analyzed. The model assumes two alleles at each of two loci, one of which is coded by an autosome and the other' by mitochondrial DNA. Viability selection is'presumed to be underdominant-i.e., heterozygous inferiority to the homozygotes- at an autosomal locus, and neutral or deleterious at a mitochondrial locus, with'the combined action being multiplicative. Extremely strong selection against heterozygotes may prevent the transmission of mitochondrial DNA between two species, but otlerwise the transmission can easily occur over species bptindaries. The rate of opproach to equilibrium is determined by the level of gene flow and is not affected much by selection against an autosomal locus. The divergence'of the nuclear genomes of thp two species is reexamined. Based on published data on enzyme loci, we conclude that there has been mitochondrial gene flow between these species for a long enough time that several nuclear loci examined could diverge because of accumulation of neutral mutations.
drial genotype because of the linkage disequilibrium between
them. This could be important for interspecific gene flow especially, because it is reasonable to expect some selection against hybrids. Powell (1) proposes selection against heterozygotes as one mechanism for preventing the interchange of nuclear genes among the two Drosophila species. Another mechanism Powell suggests is assortative mating. If the hybrids are able to mate with only one of the parental species and the backcrosses tend to mate with the same parental species, then nuclear and mitochondrial gene flow between species could be lower than expected on the basis of'the number of hybrids observed. We will not consider this possibility here, but we will show that the available data are consistent with a model that does not incorporate this effect. Model and Analysis We consider a simple model of two loci. One locus is carried by an autosomal' chromosome and the other is carried by an extranuclear (mitochondrial or chloroplast) chromosome. We assume two alleles at each locus and denote them A and a for autosomal alleles and M and m for extranuclear alleles. Assuming the completely maternal inheritance of extranuclear chromosomes (8), we can treat the extranuclear locus as haploid and represent the types of female gametes by AM, Am, aM, and am and the male gametes by A and a. The genotypes of the individuals are, therefore, represented by the pairs of these gametes as given in Table 1. We consider a randomly mating population that is of effectively infinite size but that exchanges only females with a population (representing the other species) consisting of aam individuals. Immigration of males does not alter the following treatment by much, but we assume that hybrid males are sterile, as is usually observed in conspecific matings (see ref. 1 and refs. therein). We assume that selection occurs only through the difference in viability. For the autosomal locus, we assume underdominance (the heterozygote inferior to the homozygotes) to model a locus assumed'to be responsible for the reproductive isolation between the two populations in question. Thus, we consider only the case s > 0, but we do not exclude the possibility of a large s, which may be as large as 0.5 or more because of lowered viability of F1 hybrids and because of backcross progeny. For the extranuclear locus, on the other hand, we assume neutrality or selective disadvantage of the m Allele (t . 0). The selection regime is also given in Table 1. Immigration of females, random mating, and selection are assumed to occur in this order, and the generations are nonoverlapping and discrete. Let g be the immigration rate of females per generation, and P1, P2, P3, and P4 be the gamete frequencies of AM, Am, aM, and am in females, and Pt and PI be the frequencies of A and a in males.
Three recent studies have shown that the geographic patterns exhibited by mitochondrial DNA do not necessarily coincide with the geographic patterns of anatomical traits and of nuclear genes. Powell (1) has suggested the interspecific transfer of mitochondrial DNA between Drosophila pseudoobscura and D. persimilis in their region of sympatry, even though the two species are distinguishable on cytological and behavioral grounds (2) and have differentiated at several nuclear loci (3, 4). Ferris et al. (5) have found a similar pattern in two subspecies of Mus musculus: M. m. musculus in Scandinavia contains mitochondrial DNA of the species M. m. domesticus. Yonekawa'et al. (6) showed there is mitochondrial gene flow between two subspecies of M. musculus. In Japan, only one subspecies is found, M. m. molossinus, which is homogeneous in morphology and in electrophoretically detectable nuclear loci. In the northern part of Honshu and in Hokkaido, however, the mitochondrial DNA is apparently the same as that in another subspecies, M. m. castaneus, which occurs in southern China and the Philippines. In light of these findings, it is appropriate to investigate the theory of. mitochondrial gene flow with particular reference to the conditions under which the geographic patterns exhibited by mitochondrial and nuclear genes differ. In this paper we will analyze a simple model of gene flow and selection affecting both mitochondrial and nuclear genes, and we use this model to discuss the data of Powell (1). We will consider, in particular, the association between mitochondrial and autosomal loci generated by gene flow (7). It is possible, at least, that selection at an autosomal locus
could prevent the establishment of a neutral mitochon-
*Permanent address: National Institute of Genetics, Mishima, 411 Shizuoka-ken, Japan. tPresent address (until Aug. 31, 1984): Center for Demographic & Population Genetics, University of Texas at Houston, Houston, TX 77225.
The publication costs of this article were defrayed in part by page charge payment. This article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. §1734 solely to indicate this fact.
1764
Genetics: Takahata and Slatkin
Proc. Natl. Acad. Sci. USA 81 (1984)
Table 1. Combinations of female gametes (AM, Am, aM, and am) and male gametes (A and a) AM Am aM am A 1 1-t 1-S (1 - S)(1 - t) a 1- S 1 1- t (1- S)(1 - t) Relative viabilities of the six genotypes are shown, assuming multiplicative action of nuclear and extranuclear genes and underdominance at the nuclear locus; 1 - s is the relative viability of autosomal heterozygotes and 1 - t is the relative viability of individuals carrying the m extranuclear allele.
W
_
q" =
(1
-
t)(2 (1)
1765
) )+ sq q
[6b]
t
+ -f(1 - s - 2q)p + (1 - s)qr] 2
1
[6c]
g r" = s(2q - 1)p + (1 - sq)r,
where
After immigration, the gamete frequencies in females change from Pi (i = 1, 2, 3, 4) to P!
(1
=
-
g)Pi
(i
=
1, 2, 3) [1]
+ ( - 9)P4 P41~~~~~~~
I pi= l
=
,
but no change occurs in males-i.e., P*' = P* (i = 1, 2), where the primes denote the frequencies after immigration. Random mating and the subsequent viability selection change the gamete frequencies in the next generation. The mean fitness of the population is given by W=
Pt'[P' +
(1 - t)P2 + (1 - s)P3 + (1 - s)(1 P2*'[(1 S)PW + (1 s)(1 t)P2 + P +
-
+ (1
-
-
-
-
W = A0(q) + Al(q)p + A2(q)r, AO(q) = (1 - t)[1 - s(q + q' - 2qq')], [q' = (1 Al(q) = st(1 - g)(2q - 1), A2(q) = t(l - g)(1 - sq).
-
g)q] [7]
Neutral Mitochondrial Genes (t = 0) We study first the case in which the extranuclear genes are selectively neutral to find the effect of an underdominant nuclear gene on an extranuclear one. In this case, Eq. 6b becomes
t)P4]
q,, = [(2
wo
[2]
g)(1
1 g
2(
1-
s) + sq1q
g)
[8]
t4], in which
and the gamete frequencies in females become
WO WP'X
=
PiPt' + (1
WP2
=
(1
WP'3
=
P3PS'
WP'4
=
(1 -
-
-
t)[P2Pt' +
(1
-
t)[P4Pl'
s)(PiPt' + P'Pt')/2
[3a]
S)(P2PI' + P'Pt')/2] S)(PiPl' + P3Pt')/2
[3b]
+
+
(1
(1
-
-
S)(P2Pt'
+
P4Pt')/2].
[3C]
= 1
-
s[q
-
2qq'].
On the other hand, Eqs. 6a and 6c do not change except for the replacement of W by W0. We first examine their equilibrium properties. From Eq. 8, it is readily seen that there are three equilibria for q, denoted qo, q_, and q+. Clearly, one equilibrium is
Ao
(P2*" = P'3 + Pf4)-
q'
[3d]
In a similar way, we can calculate the gamete frequencies in males and obtain the following relationship. Pt" = P'1 + P'2
+
[4]
and the other two 4(1
are
g) q2
-
=
0
the solutions of 2(3
-
2g)q + g + 2 S
Therefore, after one generation the frequencies of the autosomal genes become the same in both sexes. Although only females immigrate, so that Pt' + P' + P', the frequency of A on an autosomal locus becomes identical in both sexes 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
-
g = 0
[9]
when s > 4g(1 - g). The equilibrium points of qO and the larger root q+ are stable, while q_ is unstable. When s is much larger than g, 1
q+
1
g(1
-
s)
2s [10]
p
q_
= Pi
=P1 + P2 r = P1 + P3,
to a good approximation. The model of the autosomal locus is similar to that of Slatkin (9).
which denote the frequencies of AM, A, and M, respectively. From Eqs. 1 and 3, we have 2W
p"
=
(1
-
s +
2sq)p + (1
(1+ g/s)/2
[5]
q
1-ge
-
-
s)qr
[6a]
Using the above result, we can show that the equilibrium point of r and, therefore, of p is 0 (p r). Thus, if s > 4g( 1 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). For the equilibrium (pO, q, r) = (0, 4+, 0), it is of interest to know the rate at which the equilibrium state is attained. We -
Genetics: Takahata and Slatkin
1766
Proc. NatL Acad Sci. USA 81
linearize the equations around this equilibrium state (0, q+, 0). Letting p - j + Ep, q = q + Eq, and r = P + Er, we have 1 - s - g(1 - 3s)/2 11-sg
Eq
q, which we must determine. Letting q" = q = q in Eq. 6b and substituting 1 - E for 4, we find
[11a]
q
g(l 2(s
and t
1
Ep 8 (:/)
__-
g
(1
-
s(l -21), (1 -
1),
-
2(1
s)
-
EP )
s ') /
A
lb]
r
1
-
(1
-
[idc]
s)g.
Thus, and when g 0)
In this case, we look for the equilibrium values of (p, r) other than (0, 0). We note that letting x = p6/, we have s(1
-
2[1 -sq
(1 s)q s(l - 2q)x]
24)]x -
+
[12]
-
from Eqs. 6a and 6c. Eq. 12 has one and only one real root in (0, 1), which is given by
4sq
-
1
-
s
[(4sq
+
S)2 + 8s(1 4s(2q 1)
-
1
-
-
1 =
..
1
Er
where W = 1 - 2q [g + 2(1 - g)(1 - 4)]. The larger eigenvalue of the matrix in Eq. lib is approximately
-
-
t)(1 S) -
-
t + st)
[17]
S)
[18]
and, thus
2s(2q
[1
(1984)
s)4(24
-
a
(1
1
9
-
g-t )[
(1
sgs(1-t)1 +s St)j.
2(s'
[19]
t +
The equilibrium value of f-i.e., the equilibrium frequency 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 that r depends on the selection coefficient, s, against a heterozygous autosomal locus. As shown in Eq. 14, the quasilinkage equilibrium between autosomal and extranuclear loci is attained at equilibrium, but the selection does affect the equilibrium frequency of the extranuclear gene. If s = r = (1 g/t)/(1 g) exactly, but r increases as s increases: in fact, is close to 1 if the hybrid is lethal (s = 1). In other words, selection, but only very strong selection, against an autosomal locus obstructs extranuclear gene flow. -
-
Non-Zero Paterhal Contribution The above results are derived under the assumption of completely maternal inheritance. For the robustness, however, a model incorporating the possibility of paternal leakage of extranuclear genomes is more desirable. We will not give the full analysis of such a model here, but we present a few results for neutral mutations. 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
1)]1/2
1.0
-
[13]
Because selection against the m allele is expected not to deq+ by much and because we are interested in the case with s >> g, we assume that q = 1 - E where £
