Adaptive co‐evolution of mitochondria and the Y‐chromosome: A resolution to conflict between evolutionary opponents

Abstract In most species with motile sperm, male fertility depends upon genes located on the Y‐chromosome and in the mitochondrial genome. Coordinated adaptive evolution for the function of male fertility between genes on the Y and the mitochondrion is hampered by their uniparental inheritance in opposing sexes: The Y‐chromosome is inherited uniparentally, father to son, and the mitochondrion is inherited maternally, mother to offspring. Preserving male fertility is problematic, because maternal inheritance permits mitochondrial mutations advantageous to females, but deleterious to male fertility, to accumulate in a population. Although uniparental inheritance with sex‐restricted adaptation also affects genes on the Y‐chromosome, females lack a Y‐chromosome and escape the potential maladaptive consequences of male‐limited selection. Evolutionary models have shown that mitochondrial mutations deleterious to male fertility can be countered by compensatory evolution of Y‐linked mutations that restore it. However, direct adaptive coevolution of Y‐ and mitochondrial gene combinations has not yet been mathematically characterized. We use population genetic models to show that adaptive coevolution of Y and mitochondrial genes are possible when Y‐mt gene combinations have positive effects on male fertility and populations are inbred.

mtDNA mutations, having little or no effect on female fitness, in male fertility and sperm function (Ruiz-Pesini et al., 2000) points to the existence and maintenance of sexually antagonistic cytoplasmic interactions. Further, uniparental inheritance with sex-restricted adaptation also affects genes on the Y-chromosome, but females escape the potential maladaptive consequences since they do not have a Y-chromosome. Theory suggests that it is possible for Ylinked genes that enhance male fitness to diminish female fitness indirectly via Y-autosome gene interactions (Ågren et al., 2019a). The opposing modes of uniparental inheritance for Y-linked and mitochondria genes could engender perpetual evolutionary genetic conflict because the maternally transmitted mitochondria are essential to male gamete function (Meiklejohn & Tao, 2010;Zeh & Zeh, 2005).
In nonrandomly mating populations, the evolution of mitochondrial genes with effects on male fertility can be constrained (Unckless & Herren, 2009;Wade & Brandvain, 2009), because inbreeding creates an association between the deleterious mitochondrial effects on male fertility or viability and the fitness of the female mitochondrial lineages causing those effects. These single gene cytoplasmic models did not examine Y-mitochondrial gene interactions or any kind of nuclear-mitochondrial epistasis for fitness. Empirically, however, studies by Dean et al. (2015), Yee et al. (2015), and Ågren et al. (2019b) in Drosophila melanogaster discovered abundant mitochondrial-Y chromosome epistasis. The findings of Ågren et al. (2019b, p. 1) are that, "In particular, genes involved in male reproduction appear to be especially sensitive to such interactions." In this article, we examine theoretically the potential for coevolution between Ylinked and mitochondrial genes.
The efficacy of epistatic selection acting on gene combinations depends upon the relative strengths of selection, s, and recombination, r. Positive epistatic selection creates linkage disequilibrium (LD) between genes at different loci, while subsequent recombination reduces the LD created by selection. Relative to selection on single genes, this tends to reduce the heritability of the effects of a gene combination on fitness. Positive epistatic selection acting on Y-mt gene combinations would appear to be particularly ineffective because Y-linked and mitochondrial genes are not only unlinked, but they are necessarily independently inherited from different sex parents. Because epistatically fertile fathers cannot pass Y-mt gene combinations directly to their sons, it would appear to be impossible for epistatic selection to sustain Y-mt gene combinations in affecting male fertility in LD. Theoretical studies by Connallon et al. (2018) and Ågren et al. (2019a) investigated the extent to which nuclear gene evolution could compensate for the deleterious effects of mitochondrial genes on male fitness. Such compensatory evolution is inherently epistatic. These authors investigated only one type of epistasis, compensation for a mitochondrial male fitness reduction, and did so only in randomly mating populations. Both studies investigated invasion conditions following fixation of a deleterious mitochondrial allele. When one background is fixed (here the mitochondrial haplotype), epistatic variation is converted into additive variation for the remaining segregating gene. Thus, Ågren et al. (2019a, p. 12) found that the conditions for invasion of a compensatory y allele are independent of mitochondrial allele frequency and that an invading y allele fixes whenever the compensatory effect exceeds the deleterious mitochondrial effect on males; that is, whenever the effect of the y on male fitness is positive, it fixes in the population. Neither study calculated LD between Y-linked and mitochondrial alleles.
We use population genetic models to show how these potential evolutionary opponents, the mitochondrion and the Y-chromosome, might adaptively coevolve and discuss the empirical evidence in support of our theory. We further show how our coadaptive process depends upon epistasis and inbreeding to create and sustain LD between Y-linked and mitochondrial genes.

| THE MODEL
We assume there are two Y-linked alleles, Y 1 and Y 2 , and two mitochondrial C alleles, C 1 and C 2 . We let allele Y 1 have a direct effect, s, on male fertility and an additional epistatic effect, e, when paired with C 1 . We let the eight matings (or families) in Table 2 column 10 have frequencies, F i (i = 1, 2, … 8) under random mating which change with inbreeding, k, to the values in column 11. Here, k determines the amount of "within family mating" or the excess frequency with which individuals with the same mitochondrial types mate. We define the frequency of Y 1 C 1 males as G 1 ; that of Y 1 C 2 males as G 2 ; Y 2 C 1 males as G 3 ; and Y 2 C 2 males as G 4 where genotype frequencies can be expressed as functions of the family frequencies, for example, G 1 = F 1 + F 5 . Similarly, the allele frequencies can be expressed as functions of the genotype frequencies. Here, the frequency of Y 1 Table 1 for a summary of symbols used.

| Random mating
We begin by considering the case of random mating (k = 0) and proceed to build a model that includes the effects of inbreeding and epistatis. In the random case, the mating frequencies can be found in Table 2 column 10. Here the mean fitness is given by We first consider the evolution of males and the change in frequency of the Y 1 allele (u) over a single generation. Because,

after fertility selection (indicated by a prime ′) we have
We can expand the expressions for Combing these and simplifying we find that the change in u (Δu) is given by , Y 2 is rare and initially, LD = 0), then, we get for the change in frequency of Y 1 . Note that the selective effect of epistasis, e, depends upon p, the frequency of the mitochondrial background with which Y 2 interacts. Finally, we consider evolution in the female population by calculating the change in the frequency of C 1 .
Following the same method as above, we get that the frequency of C 1 after selection is And so, the change in p is This implies that in the case of random mating, Mother's Curse is still in effect despite mito-nuclear epistasis, because evolution of the male Y does not result in evolution of the mitochondrial C. Here, fertility evolution on the Y-chromosome could compensate for male deleterious but female beneficial mitochondrial genes, one aspect of genomic conflict typically hypothesized to characterize these uniparentally inherited genes.

| Non-random mating
We now consider how the results above change when inbreeding structures the mating system and allows the eight possible matings (or families) to have frequencies F i with i = {1, …, 8} shown in Table 2, modified by k as shown in Table 2 column 11. We can show that the mean fitness is now given by W = 1 + us + H 1 e, where k). First, we calculate the change in u, the frequency of the Y 1 allele. As before, we have that Expanding the expressions for G ′ 1 and G ′ 2 we get the following: Similarly, for G ′ 2 we get Thus, the change in the frequency of Y 1 is where Again, following the same method as above, we can calculate the frequency of C 1 after selection. This is given by And so, the overall change in the frequency of C 1 after selection is Note that, F 1 + F 2 = up, F 5 + F 6 = uq, F 1 = G 1 p, and F 5 = G 1 q. Thus, we can simplify giving This implies that, with inbreeding, evolution of the male Y affects evolution of female C in proportion to the epistatic effect, e, and the degree of inbreeding, k. Therefore, fertility evolution involving genes on the Y could affect mitochondrial evolution.

| LD in males
Finally, we are interested in calculating the LD or the covariance of Y 1 and C 1 in males. This is defined as LD = G 1 − up. If LD is initially 0, after selection (indicated by a prime ′) it equals LD′ or (G � 1 − u � p � ), where G � 1 = G 1 + ΔG 1 , u′ = u + Δu, and p′ = p + Δp. Noting that, if the population were initially in random mating proportions, so that F 4 = F 7 and F 2 = F 5 , then G � 4 = F 7 + F 8 ∕W = G 4 ∕W and G � 3 = F 3 + F 4 ∕W = G 3 ∕W Continuing, we find that, Using the expressions above and rearranging, we get since, F 5 = qG 1 (see under Equation 22).
The LD′ reduces to 0 when either e or k = 0, as above. We illustrate the behavior of the LD as the mating system changes and the amount of inbreeding increases from 0 in Figure 1.

| SUPP ORTING E VIDEN CE AND DISCUSS ION
Many evolutionary biologists believe that mitochondrial genes and Y-linked genes are "inexorably at odds" owing to their opposing modes of uniparental inheritance (Zeh & Zeh, 2005). Enigmatically, they share common adaptive functions essential for sperm motility and male fertility. Genes on both the Y-chromosome and mitochondrial genes are necessary for sperm development and differentiation; deletions or mutations in either group of genes can have serious effects on adult male fertility (Dhanoa et al., 2016;Gemmell & Allendorf, 2001;Pal et al., 2017). Mitochondrial effects on male sperm performance are well known in birds (Froman & Kirby, 2005), insects (Rand et al., 2001;Yee et al., 2013), and mammals (Cardullo & Baltz, 1991), including humans (Kao et al., 1995;Moore & Reijo-Pera, 2000;Ruiz-Pesini et al., 2000). Because the mitochondrial genome is exclusively maternally transmitted (Frank & Hurst, 1996;Pominankowski, 1999), a selection response against maledetrimental mitochondrial genes or for male-beneficial genes is not possible in a randomly mating population. In human fertility studies, mitochondria are "the most important organelles for the evaluation of sperm quality" (Luo et al., 2013;Nakada et al., 2006) and Y chromosome microdeletions are the most common cause of human male infertility (Bansal et al., 2017).
Contrary to established evolutionary theory, our model shows that direct co-adaptation between a Y-linked gene and a mitochondrial gene is possible when there is Y-mt epistasis for male fertility There is empirical evidence to support our model assumptions.
First, in D. melanogaster, mitochondrial and Y-linked genes interact to affect male fertility, meaning that e ≠ 0. Y chromosome variation (Branco et al., 2013) disproportionately affects mitochondriarelated genes and genes whose expression is affected by mitochondrial haplotype are affected as well by Y-chromosome variation (Guo, 2015;Rogell et al., 2014). Experimental studies in D. melanogaster have revealed Y-mt epistatic interactions for male reproduction, including effects on male fertility and mating success (Ågren et al., 2019b;Yee et al., 2015). Although many of the data cited here are from experiments involving between-population crosses, we note that many genetic phenomena were discovered and are studied in inter-population crosses, including cytoplasmic inheritance, the ubiquitous arthropod microbe Wolbachia, meiotic drive and its suppressors, p-elements and their suppressors, and Dobzhansky-Muller incompatibilities. Notably, the latter three