Mitochondrial Replacement Therapy: Are Mito-nuclear Interactions Likely To Be a Problem?

Abstract

It has been suggested that deleterious interactions between the mitochondrial and nuclear genomes could pose a problem for mitochondrial replacement therapy (MRT). This is because the mitochondrial genome is placed in a novel nuclear environment using this technique. In contrast, it is inherited with half the mother’s genome during normal reproduction, a genome that it is relatively compatible with, since the mother is alive. Here, I review the evidence of whether mito-nuclear interactions are likely to pose a problem for MRT. The majority of the available experimental evidence, both in humans and other species, suggests that MRT is not harmful. These results are consistent with population genetic theory, which predicts that deleterious mito-nuclear interactions are unlikely to be much more prevalent in individuals born to MRT than normal reproduction, particularly in a species such as humans with low population differentiation. This is because selection is unlikely to be strong enough to establish significant linkage disequilibrium between the mitochondrial and nuclear genomes. These results are supported by a meta-analysis of 231 cases, from a variety of animals, in which the mitochondrial DNA (mtDNA) from one strain has been introgressed into the nuclear background of another strain of the same species. Overall, there is little tendency for introgression of mtDNA to be harmful.

A number of diseases, such as Leber’s Hereditary Neuropathy, Leigh’s syndrome, and Myoclonic Epilepsy with Ragged Red Fibers, are known to be caused by mutations in mtDNA (Tuppen et al. 2010). These diseases are debilitating and have no cure. Furthermore, due to the maternal inheritance of mitochondrial DNA (mtDNA), a woman with one of these diseases has a high probability of transmitting the disease to her offspring; the probability is not 100% because she may be heteroplasmic for the disease-causing mutation; therefore, by chance, she may have offspring with a low proportion of her mitochondria with the pathogenic mutation (Tuppen et al. 2010). However, it is now possible to transfer her nucleus to the enucleated egg of a donor, effectively replacing her mitochondria with those from a donor who does not carry the pathogenic mitochondrial mutation, and hence eliminate the chance that her offspring will be affected by the disease (Wolf et al. 2015). This technique is known as mitochondrial replacement (MR) or mitochondrial replacement therapy (MRT) (Wolf et al. 2015). The procedure was approved in 2015 by the British government for use in the clinic, and the first individual conceived by this technique has been reported to have been born in Mexico (Zhang et al. 2016). However, concerns have been raised about the potential deleterious genetic effects of mixing the nuclear genome of the recipient with mitochondrial genome of the donor due to potential incompatibilities between the mitochondrial and nuclear genomes (Reinhardt et al. 2013; Gemmell and Wolff 2015; Hamilton 2015; Morrow et al. 2015). Others have argued that MRT is not very dissimilar to normal sex and that such incompatibilities are unlikely to be an issue (Chinnery et al. 2014) (http://www.hfea.gov.uk/8178.html). Here, I discuss whether such incompatibilities are likely to be a problem in humans from a population and evolutionary genetic viewpoint.

In thinking about MRT, it is important to differentiate between two possible sources of deleterious genetic effects associated with the process that have been conflated, to some degree, by previous commentators (Reinhardt et al. 2013). First, there are effects that are independent of nuclear background, e.g., mtDNA X is always more fit than mtDNA Y in all nuclear backgrounds, and to the same degree. It seems unlikely that these effects will be a problem for MR since it is simple to avoid them; choose a donor who is healthy, and if you are concerned that the mtDNA might perform differently in a male, check that she has had a healthy male child. Second, there is potential for the mtDNA to interact with the nuclear genome such that mtDNA X is fitter than Y in nuclear background A but (relatively) worse in nuclear background B, i.e., there is epistasis. These interactions are more problematic because they are essentially unpredictable, and it is these latter effects that chiefly concern us here.

Interactions between the nuclear and mitochondrial genomes are expected because the genes involved in mitochondrial function are found in both the nucleus and the mitochondria. In humans, mtDNA contains 36 genes whereas the nuclear DNA (nuDNA) encodes >1158 genes that have been shown to localize to the mitochondrion (Calvo et al. 2016), and among those genes that encode for the electron transport system, 13 are found in mtDNA and ∼70 in nuDNA (Shoubridge 2001). Indeed, there are a number of cases in which it has been demonstrated that the nuclear background affects the phenotype of a mitochondrial mutation, or vice versa, [for example Bykhovskaya et al. (2004); Ballana et al. (2007); Bonaiti et al. (2010)].

Deleterious interactions between the nuclear and mitochondrial genomes are expected to be, on average, more prevalent during MR because the mitochondria will experience a completely novel nuclear background in MR, whereas in normal sex the mitochondria will be inherited with half the mother’s nuclear genome, a genome with which the mitochondria have shown themselves to be relatively compatible, because the mother is alive. Nevertheless, during normal sex, the mitochondria will experience a seminovel nuclear environment because they will not have previously experienced the half of the nuclear genome from the father (unless there is a high degree of inbreeding). So, the critical question is not whether deleterious mito-nuclear interactions (DMNI) are more likely under MR, but how much more likely they are. If DMNIs are common, strong, and MR substantially increases their frequency, then we should be concerned about them; but if they are rare, weak, or MR has little effect on their frequency, then they are not something that should concern us.

Within Populations

Both sexes

The association between alleles that are relatively compatible within the mother can be established by two processes. First, within-a-population selection against DMNIs establishes linkage disequilibrium (LD) between the interacting mutations; for example, if there is a dominant embryonically lethal interaction between a nuclear and mitochondrial mutation, then the mother will never carry the two variants that interact. However, this LD is broken down each generation by segregation. The rate of recombination is 0.5 between a nuclear and mitochondrial locus and only selection of a similar magnitude ($\mathrm{\gg }0.1$) is likely to maintain substantial LD between the nuclear and mitochondrial genomes. Analysis of simple models suggests that DMNIs will be, at most, only twice as common in individuals born through MR compared to individuals born through normal sex (Appendix); this occurs when the nuclear variant is completely dominant and the interaction is lethal. If the interaction is not strongly selected or the nuclear mutation is only partially dominant or recessive then there will not be strong LD between the nuclear and mitochondrial genomes, and even if DMNIs are common, they will only be a slightly more frequent in MR relative to normal sex.

Such strong selection is thought to be rare in humans and most multi-cellular eukaryotes (Eyre-Walker and Keightley 2007). This is evident in humans from the fact that only 2–3% of human babies are born with visible birth defects (http://www.cdc.gov/ncbddd/birthdefects/data.html and http://www.eurocat-network.eu/accessprevalencedata/prevalencetables), and only 2% of couples are primary infertile (i.e., cannot have at least one child within 5 years of trying) (Mascarenhas et al. 2012). So, unless most of these problems are due to genetics and in particular DMNIs, which seems unlikely and for which there is no evidence, strongly selected DMNIs must be rare. The limited available evidence from Drosophila melanogaster suggests that interactions between the mitochondrial and nuclear genomes involve many mutations of small effect (Camus et al. 2012). Furthermore, most mutations of large effect tend to be recessive (Simmons and Crow 1977), which will greatly reduce the level of LD between the mitochondrial and nuclear genomes (Appendix).

There is one form of selection that can increase the likelihood of DMNIs substantially in MR relative to normal individuals. As Morrow et al. (2015) point out, if there is selection on the oocyte before fertilization, then some DMNIs might be removed during normal sex that would not be removed by MRT. A variant of this model could allow the frequency of DMNIs to be zero under normal sex but nonzero under MRT; if a mutation is expressed in both the haploid and diploid stages, is lethal at both stages but recessive in the diploid stage, then a mother will never pass the interacting alleles on to her offspring and her offspring will never suffer from the DMNI, because the nuclear allele is recessive. However, under MR it is possible to introduce the mitochondrial allele into an individual who is homozygous for the interacting nuclear mutation.

Males and postreproductive characters

It has been suggested that DMNIs might be particularly prevalent in males born to MRT (Reinhardt et al. 2013; Morrow et al. 2015). Mitochondrial mutations are expected to have more deleterious effects in males, because mtDNA is not (usually) inherited from males and hence selection in males can have no effect on the frequency of mitochondrial mutations. Therefore, there is an expectation that mutations that are advantageous or neutral in females, but deleterious in males, can accumulate; this has been termed the “Mother’s Curse.” Indeed, it has been demonstrated that genetic variation in mtDNA affects several traits more in males than in females, such as gene expression (Innocenti et al. 2011) and ageing (Camus et al. 2012), although a recent meta-analysis found no evidence for this (Dobler et al. 2014). However, MRT will not exacerbate these effects if the mitochondrial and nuclear genomes are taken from the same population, for the simple reason that selection in males cannot generate the LD between the mitochondrial and nuclear genomes that make DMNIs less likely in normal sex. For similar reasons, DMNIs might affect postreproductive characters, but these will not be more common among MR individuals because selection cannot generate LD in individuals who have finished reproducing.

Between Populations

The second process that can lead to an association between mitochondrial and nuclear variants in the mother is population substructure. This can cause there to be incompatibilities between individuals that are sampled from different populations if there is limited migration. For example, let us imagine that we have a nuclear locus with two alleles N and n, and a mitochondrial locus with alleles M and m; let us assume that the ancestral population has the genotype of NN M, that the n allele is either advantageous or neutral if the mitochondrial locus is M, and that the m allele is advantageous or neutral if the nuclear locus is NN, but that the genotypes Nn m and nn m are deleterious (i.e., there is a negative epistatic interaction between the n and m alleles). Under these conditions, a single panmictic population is very unlikely to evolve from NN M to nn m, unless the population size is small and genetic drift is effective. However, if the population splits into two or more subpopulations then the first could evolve to be nn M and the second to NN m (or vice versa). If we cross individuals from these populations, we will generate individuals who are Nn m who suffer from the negative epistatic interaction between alleles n and m. With limited migration, and as time progresses, these incompatibilities can accumulate to an extent that hybrids between populations are extremely unfit; the accumulation of these incompatibilities can eventually lead to speciation, at which point they are referred to as Dobzhansky–Muller incompatabilities.

Human populations are well-known to show little evidence of population differentiation (Rosenberg et al. 2002), although there are clearly genetic differences, since there is regional variation in morphology and physiology. However, it seems unlikely that many strongly DMNIs have accumulated between human populations because there is no evidence that interbreeding between populations leads to a decrease in fitness. There is also no evidence of significantly more LD between human populations between mitochondrial and nuclear genes than one would expect given the limited levels of population substructure that exist in humans (Sloan et al. 2015), although it should be appreciated that LD would have had to be very strong between substantial numbers of mitochondrial and nuclear polymorphisms for this test to detect an effect.

Experimental Evidence

MR

MR has been performed in three species: humans, macaques, and mice. In humans, the experiments have generally only been allowed to proceed to the blastocyst stage of development, but in one case the pregnancy was allowed to go to term (Zhang et al. 2016). In several of these experiments, MR zygotes were significantly less likely to develop to the blastocyst stage than unmanipulated controls (Tachibana et al. 2009; Craven et al. 2010; Hyslop et al. 2016). This could be due to the manipulation of the oocyte or the presence of DMNIs. The evidence is inconclusive. In one study, there was evidence that the problems were at least in part due to the experimental manipulation of the oocytes (Tachibana et al. 2009). However, in the most informative experiment to date, Hyslop et al. (2016) performed two forms of MR; they reinjected the nucleus back into the oocyte it had been removed from [autologous pronuclear transfer (PNT)] and they performed reciprocal swaps between eggs (heterologous PNT). They found that zygotes derived by autologous PNT were as likely to develop to the blastocyst stage as unmanipulated controls. However, the products of heterologous PNT were almost 50% less likely to develop to the blastocyst stage than either the unmanipulated controls or the autologous PNT-derived embryos (one-tailed Fisher’s exact test P = 0.009 and P = 0.027 for unmanipulated and autologous controls, respectively). Hyslop et al. (2016) suggest that the difference might arise because one of the eggs involved in the heterologous transfer had been previously frozen, whereas all other eggs used in the experiment were fresh. Unfortunately, they did not track the survival of the two products of heterologous PNT separately (i.e., frozen vs. fresh eggs used as the donor). Subsequent analysis showed that both autologous and heterologous PNT-derived blastocysts were of similar quality to unmanipulated controls and that gene expression profiles did not differ (Hyslop et al. 2016). Analysis of mitochondrial function in other experiments of MR-derived stem cells and fibroblasts showed either no significant difference to non-MR controls (Paull et al. 2013; Yamada et al. 2016), or to significant differences in which the MR cells could be significantly better or worse (Kang et al. 2016). Significantly, in these latter experiments, there was no correlation between mitochondrial performance and the divergence between the donor and recipient mtDNAs (Kang et al. 2016). Overall, there seems little evidence that MRT is harmful in humans, but the experiment performed by Hyslop et al. (2016) suggests that further experiments are warranted.

MR has also been performed in macaques, with four individuals born to this technique (Tachibana et al. 2009). At 3 years of age, these individuals appear to have developed normally with normal mitochondrial function (Tachibana et al. 2013) despite the macaques apparently coming from different subspecies (http://www.hfea.gov.uk/8178.html). In mice, it has been shown that MRT can be used successfully to replace mitochondria with respiratory defects (Sato et al. 2005) and that MRT yields healthy offspring, with normal respiration, as frequently as in vitro fertilization (Wang et al. 2014).

Conplastic strains

Although, MR has only been performed in three species and very few mito-nuclear combinations have been studied, mtDNA has been introduced into various nuclear backgrounds, by repeated backcrossing, in a number of other animal species; these strains are known as conplastic. This effectively performs MR, although it should be appreciated that very strong DMNIs are likely to be removed by selection during backcrossing.

The effects of introgressing mtDNA from one strain in another are mixed. In some cases, the introgression of mtDNA into a strain leads to a loss of fitness; for example, in D. melanogaster, the introgression of a mtDNA from Brownsville, Texas, onto a standard lab background led to male sterility (Clancy 2008). These deleterious effects have led to the suggestion that there is coadaptation between the mitochondrial and nuclear genomes, and that MR could be harmful (Reinhardt et al. 2013; Gemmell and Wolff 2015; Hamilton 2015; Morrow et al. 2015). However, there are also cases in which fitness and health are enhanced in conplastic strains; for example, the introgression of the mtDNA from the mouse strain NZB/OlaHsd into C57BL/6 led to an increase in longevity relative to the C57BL/6 strain with its own mtDNA (Latorre-Pellicer et al. 2016).

To investigate whether this form of MR tends to be deleterious on average, estimates of the effect of replacing the “native” with “foreign” mitochondria were compiled from the literature, where native mtDNA is isolated with the nuclear genome, and foreign mtDNA is sampled from the same species, but alongside a different nuclear genome. Note that, in this case, we are not differentiating between additive and epistatic effects; hence, low fitness in the conplastic strain might be because the mtDNA reduces fitness in all strains that it is in, or it might be because it has an interaction with a specific nuclear background. The data set comprised experiments in animals in which an organismal-level trait had been measured; I ignored studies in which only mitochondrial function had been studied, since it is organismal-level traits that we are ultimately interested in and there is a poor correlation between mitochondrial function and organismal fitness [e.g., see Latorre-Pellicer et al. (2016)]. The magnitude of the effect was quantified as the proportional difference between the lines with the foreign and native mtDNAs (i.e., the difference divided by the mean, calculated such that positive values represent the case when the line with the foreign mtDNA was better than the line with the native mtDNA); as such, only studies using an untransformed scale were included (e.g., residual sperm length was not included). Many of the traits that have been studied are probably under stabilizing selection in natural populations and hence it can be difficult to judge whether an increase or decrease in the trait is beneficial; therefore, we assigned the beneficial direction based on what we might expect if mitochondrial function was seriously compromised; i.e., individuals with very poorly functioning mitochondria might be expected to have a lower chance of survival, develop more slowly, be smaller, and have lower fertility. For some traits, we took the direction as that which would be desirable to humans—e.g., increased life span and ability to learn—but for some traits, such as exploratory behavior in mice, it was difficult to assign a beneficial direction; these studies were ignored. Where the same trait had been assayed multiple times or at different temperatures, we took the middle assay time and temperature to reduce nonindependence.

The data set comprised 231 estimates, the vast majority of which came from either Drosophila species or the beetle Callosobruchus maculatus (Supplemental Material, Table S1). There were also a moderate number of estimates from the house mouse Mus mus domesticus. It should be appreciated that, while there are a substantial number of estimates, they come from a limited set of lines; for example, the 92 estimates from C. maculatus come from 25 lines that have been repeatedly used. Therefore, there is substantial nonindependence in the data and, as a consequence, no formal statistical analysis was performed. It should also be noted that the vast majority of estimates come from inbred lines; this may exacerbate any interactions between the nuclear and mitochondrial genomes, since mutations with large effects tend to be recessive (Simmons and Crow 1977). The extent of inbreeding depression in some of these data are evident in the work of Clancy on female longevity in D. melanogaster (Clancy 2008). He backcrossed mtDNAs from different localities on to a standard homozygous nuclear background; the mean longevity for mtDNAs from Alstonville, Dahomey, and Japan was 27.4 days. However, when those lines were made heterozygous by crossing to other strains of flies, but not the line the mtDNA was sampled from, the mean longevity increased to 38.9 days, a 42% increase.

Taking all the data together, there is a slight tendency for conplastic strains to be less fit than native strains (Figure 1); 54% (125) of the comparisons show a negative effect of introgression, 44% (102) show a positive effect, and 2% (4) show no discernible effect (Figure 1). However, the average effect is very close to zero at −3.4% and the majority of effects are small; 56% of the estimates are <10%.

Figure 1.

The distribution of proportional effects for cases in which the mitochondrial DNA (mtDNA) from one strain has been introgressed onto the nuclear background of another strain. The statistic is calculated such that positive values indicate that the line with “foreign” mtDNA is fitter than that with the “native” mtDNA.

If we consider each species individually, for which we have >10 estimates, we observe qualitatively similar patterns (Table 1). There is one exception, the copepod Tigriopus californicus. In this species, the strain with the native mtDNA is always fitter than the strain with the foreign mtDNA, and the average effect size is >10%. It is notable that this species shows extremely high levels of population differentiation (Edmands and Harrison 2003); F_ST between the Southern populations of this species, from where the lines were sampled, is 0.81 (Edmands and Harrison 2003). However, high levels of population differentiation do not always result in consistently negative consequences of introgressing mtDNA from one population into another, as illustrated by C. maculatus, which shows high levels of population differentiation (Tuda et al. 2014) but no overall trend toward negative effects (Table 1). It is worth noting that F_ST in humans is very low at ∼0.05 (Rosenberg et al. 2002), similar to the levels observed in D. melanogaster (Verspoor and Haddrill 2011), which also show no consistent pattern in the effects of introgressing mtDNA into different nuclear backgrounds (Table 1).

Table 1. The proportion of cases in which the strain with the native mtDNA is better than the strain with the foreign mtDNA, and the average proportional effect (negative values indicate the foreign strain has lower fitness).

Species	Number of Comparisons	Proportion of Cases in which Native Is Better	Average Proportional Effect (%)
Callosobruchus maculatus	92	0.41	−1.5
Drosophila melanogaster	66	0.59	−3.4
Drosophila subobscura	32	0.44	−0.1
Mus musculus domesticus	14	0.64	−5.0
Tigriopus californicus	18	1.0	−14

It has been hypothesized that the effects of MR might be more apparent in males (Reinhardt et al. 2013; Morrow et al. 2015) because mtDNA is not inherited from this sex, and hence mutations that are deleterious in males, but neutral or beneficial in females, can accumulate. There is little evidence that effects of introgression are worse in males; the mean effect in males is −3.3% (86 estimates) and in females it is −2.6% (93 estimates) (Figure 2). There is also little evidence that sampling the mtDNAs from different populations is more deleterious than taking them from the same population; the mean effect for between populations is −3.7% (181 estimates) and within it is −0.0% (34 estimates).

Figure 2.

The distribution of proportional effects among 212 estimates in males and females separately. Positive values indicate that replacing the mitochondrial DNA (mtDNA) in a strain with a “foreign” mtDNA leads to an increase in fitness.

The presence of a deleterious interaction between the nuclear and mitochondrial genomes is often interpreted as evidence of coadaptation, i.e., that the two genomes have coevolved together. However, it is possible that such cases simply represent segregating mutations that have a deleterious interaction. For example, the mtDNA in D. melanogaster from Brownsville, Texas, might cause male sterility on the w1118 nuclear background, because the nuclear and mitochondrial genomes have coevolved and diverged from the population that w1118 was sampled from; or it might simply be that there is a mitochondrial mutation segregating in Texas that is incompatible with the single w1118 genome. To establish coevolution and coadaptation it is necessary, but not sufficient, to show that all or most mtDNAs taken from Texas are compatible with all or most nuclear genomes taken from Texas, and incompatible with all or most genomes sampled from the w1118 population. This type of experiment is rarely done. The above meta-analysis suggests that there is little evidence for coadaptation between the mitochondrial and nuclear genomes because there is little tendency for the introgression of a foreign mtDNA onto a nuclear background to be deleterious.

Haplotype Matching

It is potentially possible to completely eliminate the prospect of DMNIs by using donor mtDNA that is identical to the recipient’s in all but the pathogenic mutation, by sampling the donor from a close maternal relative. More generally, it has been suggested that sampling the donor mtDNA from the same haplogroup as the recipient might be a worthwhile strategy (Vogel 2014; Wolf et al. 2015). The fact that there is little evidence that transferring a mtDNA from one nuclear background to another, either by MR or introgression, is deleterious, suggests that this is not necessary in terms of DMNIs. However, there is evidence from human MR experiments that the maternal mtDNA can outcompete the donor mtDNA leading to loss of the donor mtDNA (Kang et al. 2016). In some cases, this reversion appears to be due to a specific polymorphism in the control region of the mtDNA (Kang et al. 2016).

Summary

It has been suggested that placing mtDNA into a completely novel genetic background, as will happen in MR, may lead to harmful effects because of deleterious interactions between the mitochondrial and nuclear genomes. There is one experiment that suggests that this might be the case; Hyslop et al. (2016) found that oocytes that had received their nucleus from a donor were less likely to develop into a blastocyst than oocytes who had had their own nucleus injected back into them. This might be due to DMNIs, but it also could be due to the fact that the one of the oocytes had been frozen in the heterologous treatment. Otherwise, there is little evidence that MR is harmful because of DMNIs. MR in humans leads to blastocysts with normal gene expression (Hyslop et al. 2016) and OXPHOS function (Paull et al. 2013; Kang et al. 2016; Yamada et al. 2016). MRT also appears to have no deleterious effects in macaques (Tachibana et al. 2009, 2013) and mice (Wang et al. 2014). Furthermore, the introgression of mtDNA into a new “foreign” nuclear background in other species does not appear to be deleterious on average. These results are consistent with population genetic theory. Only if selection against DMNIs is very strong will they be more frequent in individuals born using MR than normal reproduction if the species is relatively panmictic, and even then, the effects are at most twice as likely. Finally, it is worth noting that the risk of DMNIs from MRT must be much smaller than the risks of having an affected child through normal reproduction, if the mother has a pathogenic mitochondrial mutation.

Supplementary Material

Supplemental material is available online at www.genetics.org/lookup/suppl/doi:10.1534/genetics.116.196436/-/DC1.

Appendix

The Frequency of DMNIs in MR and Normal Reproduction

It would seem likely that the biggest difference in the frequency of DMNIs between MR and normal reproduction will occur when the mother cannot pass on the alleles during normal reproduction that interact negatively. For this to happen the interaction must be lethal. Let us consider two scenarios that fulfill this condition. In the first, we imagine there is a mitochondrial mutation m, which is neutral unless it is in a nuclear background with the nuclear allele n, in which case the allele is lethal, i.e. there is dominant lethal interaction (model I in Table A1). Consider an individual m allele. Under normal sex, the chance that an m allele will encounter an n allele in the next generation is approximately p, the frequency of the n allele in the population, because we know the mother must have been NNm, otherwise she would have been dead. This is only an approximation, because we have ignored the n alleles that have been removed because they found themselves in a zygote that also had the m allele; this is a reasonable approximation, because the m and/or n alleles are likely to be at low frequency since the interaction is lethal. In contrast, under MR the chance that the m allele encounters an n allele is ∼2p(1 − p) + p². If the n allele is rare this is ∼2p and it is <2p if the allele is common. Hence, at most there is twice the chance of generating a genotype with the lethal combination during MR compared to normal reproduction.

Table A1. The fitness of the six genotypes under several models of selection.

Nuclear Genotype	Mitochondrial Genotype	Fitness Model I	Fitness Model II	Fitness Model III
NN	M	1	1	1
Nn	M	1	0	1
nn	M	1	0	1
NN	m	1	0	1
Nn	m	0	0	1
nn	m	0	1	0

In the second scenario, let us imagine that the two genotypes NNM and nnm are viable, and that all other genotypes are inviable (model II in Table A1). We might think of this as a “rescue” model in which the n and m mutations are lethal relative to the wild-type genotype NNM unless they cooccur, in which case they rescue each other. Under this model all females (and males) are either NNM or nnm. Consider an M allele under this model. The chance that this allele encounters an n allele during normal reproduction is approximately p, the frequency of the n allele, and the chance that it encounters it during MR is ∼2p(1 − p) + p², which is 2p or less. Now consider an m allele; the chance that it encounters an N allele is ∼(1 − p) during normal reproduction and 2p(1 − p) + (1 − p)² during MR. Overall, the chance that there will be a lethal interaction during normal reproduction is fp + (1 − f)(1 − p) and during MR it is f(2p(1 − p) + p²) + (1 − f)(2p(1 − p) + (1 − p)²), where f is the frequency of M allele. Logic dictates that the chance of a lethal interaction during MR must be less than twice the chance under normal sex, since both 2p(1 − p) + p² < 2p and 2p(1 − p) + (1 − p)² < 2(1 − p) (as above). However, over much of the parameter range (i.e. values of p and f) the effect is attenuated, particularly when the mitochondrial variant segregates at a higher frequency than the nuclear variant.

In both of these models, we have assumed that the interaction is dominant and lethal; if the mutation is either partially dominant or recessive, or more weakly selected, then the LD between the nuclear and mitochondrial variants will be less strong and the difference in the frequency of DMNIs between MR and normal reproduction will be less, because the mother will have some chance of passing on the interacting alleles to her offspring. It is instructive, for example, to consider the case where the interaction is lethal but recessive; for example, the m allele is only lethal on an nn background (model III in Table A1). It is evident, in this case, that some females that carry the m allele will also carry the n allele and hence pass on the n allele to her offspring. Consider an m allele and assume that both the n and m alleles are rare. The chance that the m allele is in a female who is heterozygous for the n allele is ∼2p(1 − p) and the chance that the m allele finds itself in an oocyte with the n allele is p(1 − p), hence the probability of generating a zygote that is nnm is p²(1 − p). The probability that we form an nnm genotype given an m allele using MR is simply p². Hence, to a first approximation, the probability of a DMNI during MR relative to normal sex is 1/(1 − p). If p is 1% the increase in the frequency of DMNIs due to MR would be ∼1%.