Mating ecology explains patterns of genome elimination

Genome elimination – whereby an individual discards chromosomes inherited from one parent, and transmits only those inherited from the other parent – is found across thousands of animal species. It is more common in association with inbreeding, under male heterogamety, in males, and in the form of paternal genome elimination. However, the reasons for this broad pattern remain unclear. We develop a mathematical model to determine how degree of inbreeding, sex determination, genomic location, pattern of gene expression and parental origin of the eliminated genome interact to determine the fate of genome-elimination alleles. We find that: inbreeding promotes paternal genome elimination in the heterogametic sex; this may incur population extinction under female heterogamety, owing to eradication of males; and extinction is averted under male heterogamety, owing to countervailing sex-ratio selection. Thus, we explain the observed pattern of genome elimination. Our results highlight the interaction between mating system, sex-ratio selection and intragenomic conflict.

Gʹ′ is the average genic value among those genes at the same locus whose expression controls PGE in the males contributing offspring to the focal individual's mating group. Here: dµ/dG = dµʹ′/dGʹ′ = 1; dG/dgsTUv = qsTUv is the consanguinity between the focal class--sTUv gene and the genes, residing at the same locus, who control PGE in the father of the focal gene's carrier; and dGʹ′/dgsTUv = qʹ′sTUv is the consanguinity between the focal class--sTUv gene and the genes, residing at the same locus, who control PGE in the males who contribute offspring to the focal individual's mating group. Note that, because dWfTUv/dµʹ′ = 0 for all classes of genes in females, we need only calculate consanguinities qʹ′mTUv for classes of genes in males. Again, the consanguinities will depend upon which class or classes of genes control PGE in males ----see the Consanguinity section below, and Table A1.1.2, for details. Consanguinity -The coefficient of inbreeding is the consanguinity of mating partners. This may be expressed as ρ = a((1/4) × (1+ρ)/2 + (1/2) × ρ + (1/4) × ρ). That is: with probability a the male and female share the same mother; so with probability ¼ we pick the maternal--origin genes from both individuals and hence their consanguinity is simply the consanguinity of their mother to herself, or (1+ρ)/2; and with probability ½ we pick the maternal--origin gene from one of the mating partners and the paternal--origin gene from the other, in which case their consanguinity is that of mating partners, or ρ; and with probability ¼ we pick the paternal--origin genes from both individuals, in which case their consanguinity is that of two males in the same mating group, or ρ. Solving this equation obtains ρ = a/ (8--7a), so that there is full outbreeding in the absence of sib mating (ρ = 0 when a = 0) and there is full inbreeding if all mating is between sibs (ρ = 1 when a = 1). From this coefficient can be defined other consanguinities between mating partners: ρM--= a((1/2) × (1+ρ)/2 + (1/2) × ρ), from the perspective of the female's maternal-origin gene; ρP--= a ρ, from the perspective of the female's paternal--origin gene; ρ--M = a((1/2) × (1+ρ)/2 + (1/2) × ρ), from the perspective of the male's maternal--origin gene; ρ--P = a ρ, from the perspective of the male's paternal--origin gene; ρMM = a (1+ρ)/2, from the perspective of both mating partners' maternal genes; ρMP = a ρ, from the perspective of the female's maternal gene and the male's paternal gene; ρPM = a ρ, from the perspective of the female's paternal gene and the male's maternal gene; and ρPP = a ρ, from the perspective of both mating partners' paternal genes. And these coefficients define all the consanguinities needed to solve the model (listed in Tables A1.1.1 & A1.1.2). Potential for PGE -The condition for increase in PGE is dW/dg > 0. The left--hand side of this inequality is the marginal fitness and, above, we have calculated these for female PGE and for male PGE, for different scenarios regarding which genes control PGE. Evaluating the marginal fitnesses at φ = φʹ′ = φ = µ = µʹ′ = µ = 0, the inequalities give the condition for invasion of female or male PGE in a non--PGE population. Setting the marginal fitness for female PGE equal to zero, and solving for α, obtains the threshold cost of female PGE γ such that invasion will occur if α < γ and invasion will not occur if α > γ. This defines the potential for female PGE. If γ > 0, then it is possible for costly PGE to invade (provided the cost is sufficiently small), Actor Class Recipient Class ρ--P ρMP ρPP a ρ a ρ Table A1.1.1. Consanguinities for the female PGE invasion analysis: autosomal genes. The five autosomal actor classes are the female's autosomal genes (A), her maternal--origin autosomal genes (AMat), her paternal--origin autosomal genes (APat), her mother's autosomal genes (AMot) and her father's autosomal genes (AFat). Shown here are their consanguinities psTUv to each of the recipient classes sTUv among the focal female's offspring. and if γ < 0, then it is possible that beneficial PGE (i.e. providing a viability benefit rather than a viability cost) may not invade (provided the benefit is sufficiently small). Thus, the potential γ provides a quantitative measure of the extent to which a particular class of genes desires PGE in females. Similarly, setting the marginal fitness for male PGE to zero, and solving for β, obtains the potential for male PGE γ. These potentials are illustrated in Figure 3 of the main text. 1.2. X--linked genes Classes -There are 4 classes of individual, classified according to their sex s ∈ {f,m} and the grandparent of origin of their maternal--origin gene T ∈ {M,P}. If the individual is female, then all of her paternal--origin X--linked genes are derived from her paternal grandmother. And if the individual is male, he has no paternal--origin X-linked genes. Thus, each class is notated in the form sTU, where U ∈ {M,--}, i.e. fMM, fPM, mM--and mP--. Survival -The probabilities of survival for each class are: SfMM(φ,µ,α,β) = (1/4)(1-φ)(1--µ) + (1/2)φ(1--µ)(1--α) + (1/2)(1--φ)µ(1--β) + φ µ (1--α)(1--β); SfPM (φ,µ,α,β

Actor Class Recipient Class
a ρ*M--a 2 (1+ρ*)/2 a 2 ρ* mP--1ʹ′ a ρ*P--a 2 ρ* a 2 ψ* Table A1.2.2. Consanguinities for the male PGE invasion analysis: X--linked genes. The three X-linked actor classes are the male's X--linked genes (X), his mother's X--linked genes (XMot) and his father's X--linked genes (XFat). Shown here are their consanguinities qsTUv to each of the recipient classes sTUv among the male's offspring, and the consanguinities qʹ′mTUv to each of the recipient classes mTUvʹ′, among the males competing for mates with the focal male's sons. Reproductive value -Since there is only one class of Y--linked gene, all reproductive value belongs to this class: cm = 1. PGE in females -Following the same procedure as before, the condition for increase in female PGE is cm ((dWm/dφ) × (dφ/dG) × (dG/dgm)) > 0. Once again: dφ/dG is an arbitrary mapping between genic value and phenotypic value, and can be set to 1; and dG/dgm = pm is the consanguinity between the focal gene and the genes, residing at the same locus, who control PGE in the mother of the focal gene's carrier. PGE in males -Similarly, the condition for increase in male PGE is cm ((dWm/dµ) × (dµ/dG) × (dG/dgm) + (dWm/dµʹ′) × (dµʹ′/dGʹ′) × (dGʹ′/dgm)) > 0, where: dµ/dG = dµʹ′/dGʹ′ = 1; dG/dgm = qm is the consanguinity between the focal class--sTUv gene and the genes, residing at the same locus, who control PGE in the father of the focal gene's carrier; and dGʹ′/dgm = qʹ′m is the consanguinity between the focal class--sTUv gene and the genes, residing at the same locus, who control PGE in the males who contribute offspring to the focal individual's mating group. Consanguinity -Coefficients of inbreeding are undefined for Y--linked genes, because females do not carry any such genes. The consanguinity between two males in the same mating group is ψ** = aψ**; that is, with probability a they share the same mother, and hence their consanguinity is that of their fathers, i.e. two males in the same mating group. Solving yields ψ** = 0 for all 0 < a < 1. This coefficient defines all the consanguinities needed to solve the model (listed in Tables A1.3.1 & A1.3.2). Potential for PGE -The potentials for female and male PGE are calculated as described above. These are illustrated in Figure 3 of the main text.

Actor Class Recipient Class
Y YFat m 1 1 mʹ′ a ψ** a ψ** Table A1.3.2. Consanguinities for the male PGE invasion analysis: Y--linked genes. The two Y-linked actor classes are the male's own Y--linked genes (Y) and the Y--linked genes of his father (YFat). Shown here are their consanguinity qm to the recipient class m among the focal male's sons, and the consanguinity qʹ′m to the recipient class mʹ′ in the males who compete for mates with the focal male's sons. Fitness -Expected fitness is given by the product of survival to maturity and expected reproductive success contingent upon surviving to maturity, i.e. wsTU (φ,φʹ′,µ,µʹ′,α,β) = SsTU(φ,µ,α,β)×RsTU(φʹ′,µʹ′,α,β). Because MGE is vanishingly rare in the population, the average fitness among all individuals of a particular class is given by wsTU(α,β) = wsTU (0,0,0,0,α,β), and each individual's fitness can be expressed relative to the average of their class, by WsTU (φ,φʹ′,µ,µʹ′,α,β) = wsTU(φ,φʹ′,µ,µʹ′,α,β)/ wsTU(α,β). Gene fitness -There are 16 classes of gene, because every class of individual carries one maternal--origin gene and one paternal--origin gene. Thus, gene classes are denoted in the form sTUv, where v ∈ {1,2} according to whether the gene is of maternal (v = 1) or paternal (v = 2) origin. The fitness of a gene, defined in terms of its probability of being transmitted to a potential (rather than a realised) zygote, is proportional to the fitness of the individual who carries it. Consequently, the relative fitness of a gene is equal to the relative fitness of its carrier: WsTU1 (φ,φʹ′,µ,µʹ′,α,β) = WsTU2(φ,φʹ′,µ,µʹ′,α,β) = WsTU(φ,φʹ′,µ,µʹ′,α,β). Reproductive value -Because class reproductive values are calculated in a population in which genome elimination is vanishingly rare, these are exactly the same as calculated in section 1.1, i.e. csTUv = 1/16 for all gene classes. MGE in females -The condition for natural selection to favour an increase in MGE is ∑ s∈{f,m},T∈{M,P},U∈{M,P},v∈{1,2} csTUv (dWsTUv/dφ) × (dφ/dG) × (dG/dgsTUv) > 0, where G is the average genic value among those genes at the same locus whose expression controls MGE in the focal individual's mother. Consequently: dφ/dG is an arbitrary mapping between genic value and phenotypic value, and can be set to 1; and dG/dgsTUv = psTUv is the consanguinity between the focal class--sTUv gene and the genes, residing at the same locus, who control MGE in the mother of the focal gene's carrier. The consanguinities will depend upon which class or classes of genes control MGE in females -and these are exactly the same as those listed in Table A1.1.1. MGE in males -Similarly, the condition for increase in male MGE is ∑ s∈{f,m},T∈{M,P},U∈{M,P},v∈{1,2} csTUv ((dWsTUv/dµ) × (dµ/dG) × (dG/dgsTUv) + (dWsTUv/dµʹ′) × (dµʹ′/dGʹ′) × (dGʹ′/dgsTUv)) > 0, where G is the average genic value among those genes at the same locus whose expression controls MGE in the focal individual's father, and Gʹ′ is the average genic value among those genes at the same locus whose expression controls MGE in the males contributing offspring to the focal individual's mating group. Here: dµ/dG = dµʹ′/dGʹ′ = 1; dG/dgsTUv = qsTUv is the consanguinity between the focal class--sTUv gene and the genes, residing at the same locus, who control MGE in the father of the focal gene's carrier; and dGʹ′/dgsTUv = qʹ′sTUv is the consanguinity between the focal class--sTUv gene and the genes, residing at the same locus, who control MGE in the males who contribute offspring to the focal individual's mating group. Note that, because dWfTUv/dµʹ′ = 0 for all classes of genes in females, we need only calculate consanguinities qʹ′mTUv for classes of genes in males. Again, the consanguinities will depend upon which class or classes of genes control PGE in males ----and these are exactly the same as those listed in Table A1.1.2. Consanguinity -As noted above, because consanguinities are calculated in a population from which genomic elimination is absent, these are exactly the same as those listed in Tables A1.1.1 & A1.1.2. Potential for MGE -The potential for female MGE and the potential for male MGE are calculated in the same way as before, and these are illustrated in Figure 3 of the main text.

Autosomal genes
Classes -There are 8 classes of individual, classified according to their sex s ∈ {f,m}, the grandparent of origin of their maternal--origin gene T ∈ {M,P} and the grandparent of origin of their paternal--origin gene U ∈ {M,P}. Each class is notated in the form sTU, i.e. fMM, fMP, fPM, fPP, mMM, mMP, mPM and mPP. Survival -Picking a zygote at random, it has some probability of belonging to each of the 8 classes, depending on whether its parents exhibited genome elimination. We model this as if there were 8 potential zygotes, of which one is chosen to come into existence. Moreover, the realised zygote may not survive to maturity, owing to viability costs associated with the genome elimination behaviour of its parents. Denoting the probability that the zygote's mother exhibited PGE by φ, the probability that the zygote's father exhibited PGE by µ, the viability cost associated with female PGE by α and the viability cost associated with male PGE by β, the probability that each potential zygote survives to maturity is: Reproductive success -Upon survival to maturity, every female achieves a unit of expected reproductive success, and every male achieves (1--z(φʹ′,µʹ′,α,β))/z(φʹ′,µʹ′,α,β) units of expected reproductive success, where: z(φʹ′,µʹ′,α,β) = (∑T∈{M,P},U∈{M,P} SmTU(φʹ′,µʹ′,α,β)) / (∑s∈{f,m},T∈{M,P},U∈{M,P} SsTU (φʹ′,µʹ′,α,β)) is the sex ratio (proportion male) in his mating group; φʹ′ is the level of PGE among the females contributing offspring to the mating group; and µʹ′ is the level of PGE among the fathers contributing offspring to the mating group. Thus, contingent upon survival, expected reproductive success is RfTU(φʹ′,µʹ′,α,β) = 1 for females and RmTU(φʹ′,µʹ′,α,β) = (1-z(φʹ′,µʹ′,α,β))/z(φʹ′,µʹ′,α,β) for males. Fitness -Expected fitness is given by the product of survival to maturity and expected reproductive success contingent upon surviving to maturity, i.e. wsTU (φ,φʹ′,µ,µʹ′,α,β) = SsTU(φ,µ,α,β)×RsTU(φʹ′,µʹ′,α,β). Because PGE is vanishingly rare in the population, the average fitness among all individuals of a particular class is given by wsTU(α,β) = wsTU (0,0,0,0,α,β), and each individual's fitness can be expressed relative to the average of their class, by WsTU (φ,φʹ′,µ,µʹ′,α,β) = wsTU(φ,φʹ′,µ,µʹ′,α,β)/ wsTU(α,β). Gene fitness -There are 16 classes of gene, because every class of individual carries one maternal--origin gene and one paternal--origin gene. Thus, gene classes are denoted in the form sTUv, where v ∈ {1,2} according to whether the gene is of maternal (v = 1) or paternal (v = 2) origin. The fitness of a gene, defined in terms of its probability of being transmitted to a potential (rather than a realised) zygote, is proportional to the fitness of the individual who carries it. Consequently, the relative fitness of a gene is equal to the relative fitness of its carrier: (φ,φʹ′,µ,µʹ′,α,β). Reproductive value -The flow of autosomal genes between classes is exactly the same under XY/XO and ZW inheritance, so the class reproductive values are identical to those derived in section 1.1.1, i.e. csTUv = 1/16 for all gene classes. PGE in females -Genes can be assigned genic values g according to their heritable tendency for any trait of interest. The condition for natural selection to favour an increase in this trait is dW/dg > 0, where W = ∑ s∈{f,m},T∈{M,P},U∈{M,P},v∈{1,2} csTUv WsTUv. Here, g is the genic value associated with an autosomal gene in a potential zygote. Thus, the condition for increase in female PGE is ∑ s∈{f,m},T∈{M,P},U∈{M,P},v∈{1,2} csTUv where G is the average genic value among those genes at the same locus whose expression controls PGE in the focal individual's mother, and Gʹ′ is the average genic value among those genes at the same locus whose expression controls PGE in the females contributing offspring to the focal individual's mating group. Here: dφ/dG = dφʹ′/dGʹ′ = 1; dG/dgsTUv = psTUv is the consanguinity between the focal class--sTUv gene and the genes, residing at the same locus, who control PGE in the mother of the focal gene's carrier; and dGʹ′/dgsTUv = pʹ′sTUv is the consanguinity between the focal class--sTUv gene and the genes, residing at the same locus, who control PGE in the females who contribute offspring to the focal individual's mating group. Note that, because dWfTUv/dφʹ′ = 0 for all classes of genes in females, we need only calculate consanguinities pʹ′mTUv for classes of genes in males. Again, the consanguinities will depend upon which class or classes of genes control PGE in females ----see the Consanguinity section below. PGE in males -Similarly, the condition for increase in male PGE is ∑ s∈{f,m},T∈{M,P},U∈{M,P},v∈{1,2} csTUv ((dWsTUv/dµ) × (dµ/dG) × (dG/dgsTUv)) > 0, where G is the average genic value among those genes at the same locus whose expression controls PGE in the focal individual's father. Here, dµ/dG = 1, and dG/dgsTUv = qsTUv is the consanguinity between the focal class--sTUv gene and the genes, residing at the same locus, who control PGE in the father of the focal gene's carrier. Again, the consanguinities will depend upon which class or classes of genes control PGE in males ----see the Consanguinity section below. Consanguinity -The inheritance of autosomal genes is exactly the same under XY/XO and ZW inheritance so, just as in section 1.1: the consanguinity of mating partners is ρ = a/ (8--7a). Moreover, as before, we have ρM-- Potential for PGE -The potential for female PGE and the potential for male PGE are calculated in the same way as before, and these are illustrated in Figure 3 of the main text.

Actor Class Recipient Class
WMot f a Table A3.

Consanguinities for the male PGE invasion analysis: W--linked genes.
The only W-linked actor class is the male's mother's W--linked genes (WMot). Shown here are their consanguinity qf to the recipient class f among the focal male's daughters.

W--linked genes
Classes -There are 2 classes of individual, classified according to their sex s ∈ {f,m}. If the individual is female, then all of her W--linked genes are maternal in origin (and came from her maternal grandmother). And if the individual is male, he has no W-linked genes. Thus, class is notated in the form s ∈ {f,m}.