mating ecology explains patterns of genome eliminationletter mating ecology explains patterns of...

LETTER Mating ecology explains patterns of genome elimination

Andy Gardner1* and Laura Ross2

1School of Biology University of

St Andrews Dyers Brae St Andrews,

KY16 9TH, UK2Institute of Evolutionary Biology

University of Edinburgh King’s

Buildings, Edinburgh, EH9 3JT, UK

*Correspondence:

E-mail: andy.gardner@st-andrews.

ac.uk

Abstract

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 animalspecies. 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 patternremain unclear. We develop a mathematical model to determine how degree of inbreeding, sexdetermination, genomic location, pattern of gene expression and parental origin of the eliminatedgenome interact to determine the fate of genome-elimination alleles. We find that: inbreeding pro-motes paternal genome elimination in the heterogametic sex; this may incur population extinctionunder female heterogamety, owing to eradication of males; and extinction is averted under maleheterogamety, owing to countervailing sex-ratio selection. Thus, we explain the observed patternof genome elimination. Our results highlight the interaction between mating system, sex-ratioselection and intragenomic conflict.

Keywords

Extinction, genomic imprinting, haplodiploidy, inbreeding, meiotic drive, paternal genome elimi-nation, paternal genome loss, sex determination, sex ratio, sib-mating.

Ecology Letters (2014) 17: 1602–1612

INTRODUCTION

Under standard mendelian inheritance, individuals receive oneset of chromosomes from each of their parents, and transmitone set of chromosomes to each of their offspring, withoutbias according to each chromosome’s parent of origin. How-ever, across thousands of animal species, some individuals(typically members of one sex) systematically transmit onlythose chromosomes that they inherited from a particular par-ent (Table 1 and Fig. 1; Burt & Trivers 2006). For example,in the citrus mealybug Planococcus citri, a male’s sperm carryonly the chromosomes he inherited from his mother, all pater-nal chromosomes having been eliminated, whereas a female’soocytes carry chromosomes from both of her parents. This‘genome elimination’ (GE) is a whole-genome form of meioticdrive and, accordingly, its evolutionary rationale makes sensefrom a selfish-gene perspective: a gene that ensures it is passedon to all – as opposed to only half – of an individual’s off-spring enjoys a two-fold selective advantage, and may increasein frequency unless the number of surviving offspring is morethan halved as a consequence (Bull 1979). Indeed, the realevolutionary puzzle is to explain why GE occurs only in somespecies and not in others (Normark 2004).There is a clear pattern in the incidence and type of GE that

occurs in the animal kingdom: it is commonly found in associa-tion with inbreeding, under male heterogamety, in males, andin the form of paternal genome elimination (Table 1 andFig. 1). This suggests that a species’ mating ecology is animportant factor in predisposing it to GE. A rich literaturespanning a century of work in ecology, population geneticsand cytology has yielded several hypotheses as to how inbreed-ing impacts upon the evolution of GE, both directly and in itsinteraction with a species’ sex determination system (Table 2).However, these ideas lead to different – indeed, sometimes dia-

metrically opposite – predictions, and the complexity of theproblem means that a full, quantitative analysis is lacking (Bull1983; Burt & Trivers 2006).Here, we develop a mathematical kin selection model to

determine how degree of inbreeding, mode of sex determina-tion, genomic location, pattern of gene expression and paren-tal origin of the eliminated genome interact to determine thefate of GE alleles. We identify those scenarios under whichGE may arise in the population by performing invasion analy-ses, and we identify those scenarios in which GE may bemaintained in the population by performing equilibriumanalyses and assessing the impact of GE upon population via-bility. Our aim is to assess the constraints that a species’ mat-ing ecology imposes upon its ability to evolve – and survive –GE, thereby explaining its pattern of incidence in the animalworld. Although specifically focusing upon GE, our analysisyields general insights into how mating system and sex-ratioselection shape conflicts within and between individuals, withapplication to sex-chromosome meiotic drive, endosymbioticparasitism and haplodiploidy.

MATERIALS AND METHODS

Mathematical model

We assume a gonochoristic, diploid population, subdividedinto a large number of mating groups, each containing a largenumber of individuals. We denote by a the probability that afemale and a male, randomly chosen from the same matinggroup, share the same mother. We assume that each femalemates with a large number of males, and at random withinher mating group, such that the probability of mating part-ners sharing the same father is effectively zero. Thus, a repre-sents the incidence of mating between maternal siblings, and

© 2014 The Authors. Ecology Letters published by John Wiley & Sons Ltd and CNRS.This is an open access article under the terms of the Creative Commons Attribution License, which permits use,

distribution and reproduction in any medium, provided the original work is properly cited.

Ecology Letters, (2014) 17: 1602–1612 doi: 10.1111/ele.12383

Table

1Overview

ofalltaxonomic

groupswithPGE

Class

Order

Clade

Number

ofspecies

TypeofPGE

Male

soma

Sex

chromosomes

Ancestral

SD

Sib-m

ating/

Inbreeding*

References

Acari

Mesostigmata

Phytoseiidae,

Otopheidomenidae,

Ascoidea

2000(10)

Embryonic

PGE

Haploid

(genomeloss)

Nosexchr.

XX/X

Y

orXX/X

0

Strongevidence

(WM,WF,

SB,SM,PS)

(Nelson-R

eeset

al.1980;

Nortonet

al.1993;

Nagelkerke&

Sabelis

1998;Burt

&Trivers2006)

Collem

bola

Symphypleona

Symphypleona

1180(6)

GermlinePGE

Diploid

withX

elim

ination

X1X2X1X2/

X1X200

XX/X

0Someevidence

(WM,WF,

PG,SB)

(Dallaiet

al.2001;Burt

&Trivers2006)

Insecta

Coleoptera

Cryphalini

190(1)

GermlinePGE

Diploid

with

paternal

genome

silencing

Nosexchr.

XX/X

YStrongevidence

(WM,SB,

SM,PG)

(Brunet

al.1995;Borsa

&Kjellberg1996;

Gauthier2010)

Insecta

Diptera

Cecidomyiidae

6168(22)

GermlinePGE

Diploid

withX

elim

ination

XX/X

0XX/X

YNoevidence

(Painter1966;Stuart

&Hatchett1991;Burt

&Trivers2006;Benatti

etal.2010)

Insecta

Diptera

Sciaridae

2300(10)

GermlinePGE

Diploid

withX

elim

ination

XX/X

0XX/X

YNoevidence

(Metz1938;Crouse

1960;

Haig

1993b;Goday

&Esteban2001;

S� an

chez

2010)

Insecta

Hem

iptera

Neococcoidea

4000(270)

GermlinePGE

Diploid

with

paternal

genome

silencing

Nosexchr.

XX/X

0Someevidence

(WF,PS,PG)

(Brown&

Nur1964;

Nur1990;Gavrilov2007;

Ross

etal.2010a;

Martinset

al.2012)

Insecta$

Hem

iptera

Diaspididae

2650(139)

Embryonic

PGE

Haploid

(genomeloss)

Nosexchr.

XX/X

0Someevidence

(WF,PS)

(Brown&

Nur1964;

Nur1990)

Insecta

Phthiraptera

Pediculushumanus

humanus

?(1)

GermlinePGE

Unknown

Nosexchr.

XX/X

Y?

Someevidence

(PS,PG,SB,

WM,WF)

(Perottiet

al.2004;

McM

enim

an&

Barker

2005;Ascunce

etal.2013)

RowsrepresentindependentoriginsofPGE,exceptforDiaspididae(indicated

by$),

which

represents

changein

typeofPGE

(germline?

embryonic).

Number

ofspeciescolumn:estimate

assumes

PGE

isconserved

across

whole

clade;

number

ofspeciesforwhichthereis

directevidence

ofPGE

given

inparentheses.Types

ofPGE:‘embryonic’when

complete

paternalgenomeis

elim

inatedfrom

both

somaandgermlinein

earlyem

bryogenesis;‘germline’

when

paternalgenomeonly

completely

elim

inatedin

thegermlineduringspermatogenesis.Male

somaandsex-chro-

mosomeinform

ationbasedonkaryotypeanalysisofspecieswithPGE.Ancestral-SD

system

inferred

from

karyotypedata

fordiploid

sister

groups.

Inbreedinginferred

from

FIS,matingsystem

andsex-ratiodata

(WM,wingless

males;

WF,wingless

females;

SB,female-biasedsexratios;

PG,population-genetic

evidence

oflow

genetic

diversity

andexcess

homozygosity;PS,life

history

leadingto

strongmeta-populationstructure;SM,frequentsib-m

ating).

© 2014 The Authors. Ecology Letters published by John Wiley & Sons Ltd and CNRS.

Letter Mating ecology explains genome elimination 1603

varying this key ecological parameter allows us to explore thewhole continuum from outbreeding (a = 0) to chronicinbreeding (a = 1). After mating, males die and mated femalesdisperse to new patches, as in Hamilton’s (1967) classic modelof local mate competition (LMC).We consider that GE alleles may be located on autosomes

or sex chromosomes, and may induce maternal genome elimi-nation (MGE) or paternal genome elimination (PGE). Weallow for the genes to be active in females or males, to induceeffects either in the carrier or the carrier’s daughters or sons,and – for genes inducing GE in the carrier – to have differenteffects if the gene is inherited from the mother or the father(i.e. genomic imprinting; Haig 2002). We assume ‘germline’GE, in which one parent’s genome is excluded from the focalindividual’s gametes, but is present and active in the individ-ual’s somatic tissue, as is this is likely the ancestral form ofGE (see Discussion). We assume that GE in females reducestheir number of surviving offspring by a fraction a and thatGE in males reduces their number of surviving offspring by a

fraction b, as a consequence of upsetting normal chromo-somal segregation. We assume that sex is determined by theindividual’s own genotype, and consider both male (XY andXO) and female (ZW and ZO) heterogamety. An importantfeature of the model is that GE in the heterogametic sexresults in offspring sex ratio bias, towards males for MGEand towards females for PGE (Fig. 2). No sex-ratio biasobtains if GE is absent or restricted to the homogametic sex(Fig. 2).

Invasion analysis

We analyse our model using the neighbour-modulated-fitnessmethodology of Taylor & Frank (1996; Supporting Infor-mation). This is a recipient-centred approach to kin selection,which considers the impact of social partners on a focalindividual’s fitness, and gives the same results as theactor-centred inclusive-fitness approach, which considersthe impact of a focal individual on the fitness of her social

Parents Gametes Soma Springtails

Fungus gnats

Armored scale insects

Mites

Ger

mlin

e P

GE

E

mbr

yoni

c P

GE

Mealybugs

Germline

XX

XX

XXXX XX XXXX

XX

XX X

Coffee borer beetle

XL

XXLL

Zygote

XXX LLL

XX LL

XX LL

XXXX

Human body louse y

?

Mites

Figure 1 Parent-of-origin-specific genome

elimination (GE), whereby an individual discards

the chromosomes inherited from one parent, and

transmits only those inherited from the other

parent. We include only those cases where GE is

sex-limited. Rows represent independent origins

of GE. Column 1: adult generation. Column 2:

gametes produced. Column 3: embryos shortly

after fertilisation. Column 4: offspring soma.

Column 5: offspring germline. Blue: male. Red:

female. X: presence of X-chromosome (colour

indicates parental origin). L in fungus-gnat

entry: a germline-linked chromosome. Blue circle

in mealybugs and coffee-borer-beetles entries:

complete heterochromatisation of paternal

genome. ‘?’ in body-louse entry: lack of

information about somatic effects. ‘Germline

PGE’: eliminated genome retained throughout

development but not transmitted to offspring.

‘Embryonic PGE’: eliminated genome lost early

in development, resulting in haploidy.


1604 A. Gardner and L. Ross Letter

partners (Hamilton 1964; Gardner et al. 2011). Here, therecipients are genes in potential zygotes, and they are sepa-rated into classes according to the sex of the individual, thesex of the parent of origin and the sex of the grandparent oforigin. The condition for natural selection to favour an allelefor GE is that instances of that allele are fitter (i.e. send moregene copies into future generations) than instances of alterna-tive alleles in the population. That is, dW=dgjg¼�g [ 0, whereW is expected relative fitness, g is genetic predisposition forGE, and �g is the population frequency of GE (Taylor &Frank 1996). A zygote’s expected fitness depends on its proba-bility of conception and its probability of surviving viabilityselection (both of which depend on the action of GE in thezygote’s parents) and expected mating success conditionalupon surviving to adulthood (which may depend on theaction of GE among those parents contributing offspring tothe zygote’s future mating group).Invasion of GE occurs when the corresponding allele

increases in frequency from rarity, that is dW=dgjg¼�g¼0 [ 0.We reformulate this invasion condition to determine thepotential for GE (Supporting Information). Specifically, forGE in females, the invasion condition may be expressed in theform a < c, where a is the actual cost of GE, in terms ofnumber of surviving offspring, and c is the maximum costthat is tolerated with GE still able to invade, and is a functionof model parameters. A positive potential for GE (c > 0) indi-cates that costly GE may invade, a negative potential for GE(c < 0) indicates that GE would need to provide a benefit, interms of number of surviving offspring, in order for it to beable to invade, and zero potential for GE (c = 0) indicatesthat GE cannot invade if it incurs any cost and will invade ifit provides any benefit. The potential for GE may also bedefined for males, by reformulating the invasion condition asb < c. In addition to allowing comparisons and contrastsacross different mating ecologies, the potential for GE

provides a way of gauging a particular genic actor’s interestswith respect to GE and, hence, allows quantification ofgenetic conflicts of interest.

Equilibrium analysis

Focusing attention upon scenarios in which GE invades, wenext investigate its subsequent evolutionary fate (SupportingInformation). The condition for increase in an allele’s fre-quency is computed in the same way as in the invasion analy-sis. However, we now assume that GE incurs no viability cost(a = b = 0), mainly for simplicity and also because we con-sider that natural selection may fine-tune the mechanism toreduce such deleterious by-product effects (in reality, these areunlikely to be completely absent). Having established that GEcan invade, we check for an intermediate equilibrium level bysolving dW=dgjg¼�g¼g� ¼ 0. And we determine when GE maygo to fixation according to the condition dW=dgjg¼�g¼1 [ 0.Straightforward fixation of GE in the heterogametic sex leadsto the eradication of all individuals of one sex from the popu-lation (males for PGE, females for MGE; Fig. 2). Accord-ingly, we consider that fixation leads to population extinction(Hamilton 1967). Finally, for scenarios in which GE is main-tained at an intermediate level, we consider its long-term fatefollowing the establishment of a novel mechanism of sexdetermination, which eliminates any impact of GE upon thesex ratio (Supporting Information).

RESULTS

Origin of genome elimination

We assess the impact of the degree of inbreeding, mode ofsex determination, genomic location, pattern of gene expres-sion and parental origin of the eliminated genome on the

Table 2 An overview of adaptive hypotheses for the evolution of paternal genome elimination (PGE)

Hypothesis Prediction References Notes

Meiotic drive I Inbreeding inhibits GE, because drive is only worthwhile in

heterozygotes

Brown (1964), Bull (1979) This effect is captured in the

present model.

Meiotic drive II Inbreeding promotes GE, as eliminated genome is more likely

to acquiesce to the driving genome’s interests

Burt & Trivers (2006) This effect is captured in the

present model.

Local mate competition I Inbreeding promotes GE, as it favours female bias, and GE

may enable maternal control of sex allocation

Hamilton (1967), Borgia

(1980), Normark (2004)

Although straightforward for

evolution of male haploidy

per se, the argument is obscure

for GE per se. This effect is

neglected in the present model.

Local mate competition II Inbreeding promotes GE, as it favours female bias, and GE

may lead to female bias in some scenarios

Bull (1979), Haig (1993a) Bull (apparently incorrectly)

attributed this to Hamilton

& Borgia, and dismissed it as

lacking generality. This effect

is captured in the present model.

Maternally transmitted

endosymbiont

Inbreeding promotes GE, because it favours female bias,

brought about by GE induced by endosymbiont in order

to enhance its own transmission

Normark (2004), Kuijper

& Pen (2010)

Formal analysis by Kuijper &

Pen considered the interests

of endosymbiont and maternal

genes only. Analysis only

applies to embryonic, not to

germline, GE. Endosymbionts

are neglected in the present

model.



evolutionary invasion of GE alleles. For each scenario, we cal-culate the potential for GE (Fig. 3). This is mediated by twokey effects: first, GE is a form of genetic drive, and as such itprovides a two-fold direct transmission advantage for drivinggenes and a corresponding direct transmission disadvantagefor those genes that are driven against; second, GE may leadto sex-ratio bias among an individual’s offspring, and thismay provide a selective advantage or disadvantage dependingupon the direction of bias and the population’s matingsystem.

DriveThe immediate consequences of drive are illustrated by con-sidering PGE in females under XY sex determination (Fig. 3panel a). The qualitative results for this permutation of the

model readily generalise to all forms of GE in the homoga-metic sex, which has no impact on offspring sex ratio (PGEand MGE in females under XY/XO and in males under ZW/ZO; Fig. 3 panels a, c, f and h). Here, there are four types ofgenic actor. First, there are genes that directly benefit, onaverage, from drive: the female’s maternal-origin autosomaland maternal-origin X-chromosomal genes, and her mother’sautosomal and X-chromosomal genes. These genes have posi-tive potential for PGE: in particular, c = 0.5 under outbreed-ing (a = 0), reflecting how the two-fold benefit of driving canoffset even a halving of offspring number. Second, there aregenes who suffer, on average, from drive: the female’s pater-nal-origin autosomal and paternal-origin X-chromosomalgenes, and her father’s autosomal and X-chromosomal genes.These genes have negative potential for PGE: in particular,

(a) (b)

(c) (d)

(e) (f)

(g) (h)

Figure 2 The consequences of genome

elimination (GE) for offspring sex ratio. GE in

the homogametic sex – that is, females under

XX or XO sex determination (panels a & c) and

males under ZW or ZO sex determination

(panels f & h) – has no impact on offspring sex

ratio. Paternal genome elimination (PGE) in the

heterogametic sex – that is, males under XX or

XO sex determination (panel b) and females

under ZW or ZO sex determination (panel e) –leads to a female-biased offspring sex ratio.

Maternal genome elimination (MGE) in the

heterogametic sex – that is, males under XX or

XO sex determination (panel d) and females

under ZW or ZO sex determination (panel g) –leads to a male-biased offspring sex ratio.



(a) (b)

(c) (d)

(e) (f)

(g) (h)

Figure 3 The origin of genome elimination (GE). Results of the invasion analyses, for: (a) Paternal genome elimination (PGE) in females under XY or XO

sex determination; (b) PGE in males, XY/XO; (c) Maternal genome elimination (MGE) in females, XY/XO; (d) MGE in males, XY/XO; (e) PGE in

females, ZW/ZO; (f) PGE in males, ZW/ZO; (g) MGE in females, ZW/ZO; (h) MGE in males, ZW/ZO. In each case, potential for GE is shown for each

class of genic actor, for whole-range of sib-mating (0 ≤ a ≤ 1). A: autosomal gene. X: X-linked gene. Y: Y-linked gene. Z: Z-linked gene. W: W-linked

gene. Unimprinted autosomal genes (black lines) have positive potential for GE only under sib-mating (a > 0), and for PGE in heterogametic males (panel

b) or PGE in heterogametic females (panel e), and so these are the scenarios in which GE robustly invades.



c = �∞ under outbreeding (a = 0), reflecting how no increasein offspring number can compensate for their eliminationfrom the individual’s gametes. Third, there are genes that areequally likely to directly benefit or suffer from drive: thefemale’s unimprinted autosomal and X-chromosomal genes.These genes have zero potential for PGE, c = 0, reflectinghow they are favoured to prevent PGE if this reduces off-spring number (a > 0) and to promote PGE if this increasesoffspring number (a < 0). Fourth, there are genes that are notdirectly affected by drive: the female’s father’s Y-chromo-somal genes. Their potential for PGE is undefined, meaningthat PGE is a neutral trait (West 2009).A gene’s inclusive fitness is not solely governed by its own

replicative success but also by that of its homologues, to theextent that they share identity by descent (Gardner & Welch2011). Accordingly, inbreeding (a > 0) changes the aboveresults quantitatively (but not qualitatively), such that a geneof the first type, who enjoys a direct benefit owing to drive,also suffers an indirect, kin-selected cost, owing to the disad-vantage incurred by the identical-by-descent genes it drivesagainst. Consequently, its potential for PGE falls fromc = 0.5 in the extreme of outbreeding (a = 0) to c = 0 in theextreme of chronic inbreeding (a = 1). To the extent that PGEis controlled by such ‘maternal’ genes, sib-mating inhibits theevolution of PGE (Brown 1964; Bull 1979). Conversely, agene of the second type, who suffers a direct cost owing todrive, also enjoys an indirect benefit, owing to the advantageaccrued by its identical-by-descent homologues. Accordingly,its potential for PGE increases from c = �∞ in the extreme ofoutbreeding to c = 0 in the extreme of chronic inbreeding. Tothe extent that PGE is controlled by such ‘paternal’ genes,inbreeding promotes its evolution (Burt & Trivers 2006).Inbreeding has no effect on the potential for PGE for genesof the third and fourth types, who are unaffected by drive onaverage or at all.To summarise, and to generalise to other instances of GE

in the homogametic sex, there is extensive scope for geneticconflicts of interest over GE, diminishing as individuals areincreasingly inbred, and no clear overall advantage or disad-vantage of GE (Fig. 3 panels a, c, f and h). Definite predic-tions of the outcomes of such conflicts are impossible withoutspecific information as to the additive genetic variance in GEcontributed by each type of gene. However, a ‘parliament-of-genes’ approach (Leigh 1971; Supporting Information) sug-gests the outcome will be close to that favoured by the indi-vidual’s unimprinted autosomal genes, owing to thenecessarily close involvement of the individual’s genome withthe process of GE, the numerical superiority of autosomalgenes (MacArthur 1961; Leigh 1971), and the cancelling-outof the interests of maternal-origin and paternal-originimprinted genes (Haig 2002). Applying this pragmaticapproach, GE appears relatively implausible in females undermale heterogamety (Fig. 3 panels a and c) and in males underfemale heterogamety (Fig. 3 panels f and h). Accordingly, wediscard these model permutations from further consideration.

Sex-ratio biasIn addition to the immediate effects of drive, GE may alsoaccrue fitness effects owing to its consequences for offspring

sex ratio. Sex-ratio bias arises from GE only when this occursin the heterogametic sex (males under XY/XO, females underZW/ZO; Fig. 3 panels b, d, e and g). In each case, PGE yieldsfemale bias, and MGE yields male bias (Fig. 2). As a conse-quence of LMC, sib-mating (a > 0) may favour a relativelyfemale-biased sex ratio (Hamilton 1967). This gives a selectiveadvantage to PGE, and a selective disadvantage to MGE, inthe heterogametic sex, relative to that obtained from consider-ation of the immediate effects of drive only (see above). Thissex-ratio-selection effect need not be uniform across all genicactors, as the coefficient of inbreeding depends upon mode ofinheritance, and parents may be in conflict with their off-spring and each other over desired sex allocation (Hamilton1967; Trivers 1974; Werren & Hatcher 2000). These detailsare captured by the model.The resulting potential for GE is illustrated by considering it

operating in heterogametic males. The potential for PGE isincreased for almost all genic actors as a consequence of theLMC effect (Fig. 3 panel b). In particular, the male’s unim-printed autosomal genes exhibit positive potential for PGE, ris-ing from c = 0 under the extreme of outbreeding (a = 0) toc = 0.5 under the extreme of chronic inbreeding (a = 1). Theexception is the Y-chromosomal genes that maintain a poten-tial for PGE of c = �∞ irrespective of the extent of sib-mating.This owes to our assumption of polyandry, which leads mater-nal-brothers to be unrelated through their fathers. Similarly,the potential for MGE is decreased for almost all geneticactors, because the resulting male bias is selectively disadvanta-geous under LMC (Fig. 3 panel d). Again, the exception is forthe Y-chromosomal genes, which gain the usual two-fold drivebenefit and hence exhibit a potential for MGE of c = 0.5.To summarise, and to generalise to other instances of GE in

the heterogametic sex, there is extensive scope for genetic con-flicts of interest, diminishing as populations are increasinglyinbred, but with a robust overall advantage accruing to PGE,and a robust overall disadvantage accruing to MGE, undersib-mating, owing to the impact of GE on offspring sex ratio.Thus, MGE is relatively implausible in males under male het-erogamety (Fig. 3 panel d) and in females under female hetero-gamety (Fig. 3 panel g), because it leads to a male-biasedoffspring sex ratio (Fig. 2d and g). Accordingly, we discardthese model permutations from further consideration.

Maintenance of genome elimination

We have narrowed the invasion of GE to two scenarios: PGEin males under male heterogamety (Fig. 3 panel b); and PGEin females under female heterogamety (Fig. 3 panel e). More-over, we have identified inbreeding as an important driver ofPGE in both instances. We now consider the scope for evolu-tionary maintenance of GE in each of these scenarios (Fig. 4).

PGE in heterogametic malesAlthough inbreeding facilitates the invasion of PGE in hetero-gametic males, this trait cannot generally increase to fixation.This is because, upon reaching a certain frequency, it willhave brought the population sex ratio to its ‘optimal’ level,and further increases in the frequency of PGE are preventedby countervailing selection for reduced female bias acting



directly on the PGE alleles (Burt & Trivers 2006). The equilib-rium point depends upon the degree of inbreeding and thegenomic location and mode of action of the genes underlyingPGE (Fig. 4 panel a). Applying the parliament-of-genesapproach, such that it is the interests of the unimprinted auto-somal genes that are expected to win out, the equilibrium levelof PGE increases approximately linearly, from g� ¼ 0 underoutbreeding (a = 0) to g� ¼ 1 under the chronic inbreeding(a = 1). That is, an intermediate equilibrium obtains for allintermediate degrees of inbreeding (0 < a < 1).In the longer term, it is useful to consider how this PGE

may subsequently evolve should a new mode of sex determi-nation, that eliminates the sex-ratio effects of GE, arise (Haig1993a,b). We find that the ‘maternal’ genic actors, who derivea direct benefit from the immediate effect of drive, arefavoured to increase PGE to fixation. In the event that theyachieve this outcome, population viability is maintainedbecause, under the new mode of sex determination, the fixa-tion of PGE does not eradicate males. The ‘paternal’ genicactors are, conversely, favoured to completely suppress PGE.From the perspective of the individual’s unimprinted autoso-mal genes, PGE is an entirely neutral trait. Accordingly, theparliament of genes is likely to be swayed by whicheverextremist faction happens to have most power, and the long-term evolutionary fate of PGE therefore depends on thedetails of whether control lies mostly with the ‘maternal’ genesor with the ‘paternal’ genes.

PGE in heterogametic femalesAlthough countervailing selection for reduced female bias pre-vents fixation of PGE in heterogametic males, this need nothappen for PGE in heterogametic females (Fig. 4 panel b).Indeed, if – following the parliament-of-genes logic that con-trol of PGE lies with the female’s unimprinted autosomalgenes – then our model predicts fixation of PGE in hetero-gametic females. This leads to the complete eradication of

males, and hence population extinction. The absence of coun-tervailing sex-ratio selection owes to PGE in heterogameticfemales leading to males having reduced reproductive value(Supporting Information). Indeed, in the limit of completePGE in females, males have zero reproductive value, and allreproductive value belongs to females’ maternal-origin genes.Hamilton (1967) provides more discussion of populationextinction caused by sex-ratio distorters.

DISCUSSION

The two-fold transmission advantage enjoyed by GE alleleshas raised the problem of why such biased inheritance isobserved only in some species, and in some forms, but notothers. In particular, it typically occurs in association withinbreeding, under male heterogamety, in males and in theform of PGE. Our analysis has clarified that, whilst somegenes do gain a transmission advantage from drive, otherssuffer a disadvantage from reduced transmission, and ecologi-cal factors are important in providing a robust advantage forGE. In particular, we find that: (1) inbreeding promotes theelimination of the paternal genome in the heterogametic sex(i.e. males in XY/XO systems and females in ZW/ZO sys-tems), (2) this may lead to population extinction under femaleheterogamety (i.e. ZW/ZO systems), owing to the eradicationof males and (c) extinction is averted under male heterogam-ety (i.e. XY/XO systems), owing to countervailing sex-ratioselection. That is, a species’ mating ecology imposes con-straints upon its predisposition to evolve and survive GE, andthis explains the widely observed pattern of PGE in heteroga-metic males under inbreeding.

Intragenomic conflict over genome elimination

Our analysis has considered genetic conflicts of interest withinthe nuclear family and within the individual’s own genome.

(a) (b)

Figure 4 The maintenance of genome elimination (GE). Results of the evolutionary equilibrium analyses, for: (a) Paternal genome elimination (PGE) in

heterogametic males (i.e. XY or XO sex determination); and (b) PGE in heterogametic females (i.e. ZW or ZO sex determination). In each case, the

equilibrium level of PGE is shown for each class of genic actor, over the whole-range of sib-mating (0 ≤ a ≤ 1). A indicates an autosomal gene, X indicates

an X-linked gene, Y indicates a Y-linked gene, Z indicates a Z-linked gene and W indicates a W-linked gene. Unimprinted autosomal genes (black lines)

favour an intermediate level of PGE under male heterogamety (panel a), owing to countervailing sex-ratio selection, and fixation of PGE – leading to

population extinction – under female heterogamety (panel b), owing to a lack of countervailing sex-ratio selection, so PGE in heterogametic males is the

only scenario in which the population robustly survives the evolution of GE.



Separate consideration of the individual’s maternal-origingenes vs. the genes carried by the individual’s mother revealsthat these distinct sets of genes have distinct evolutionaryinterests, as do the individual’s paternal-origin genes vs. thegenes carried by the individual’s father. For example, genesresiding on a maternal-origin autosome enjoy the full trans-mission benefit of PGE whereas genes residing on one of themother’s autosomes enjoy a smaller benefit as they are notguaranteed to have been passed on to the individual whoexhibits the PGE phenotype. This clarifies that intragenomicimprinting conflicts are conceptually distinct from (thoughmay arise in similar contexts to) conflicts of interest betweenparents, despite these two phenomena having often been con-flated (as discussed by Haig 2002).Within male-heterogametic systems, PGE appears to be

more common under XO than XY inheritance, as no PGEspecies with recognisable sex chromosomes has a Y chromo-some (Table 1). Burt & Trivers (2006) suggested that thisowes to the Y chromosome being strongly opposed to PGE inmales, and inhibiting its evolution. In support of this sugges-tion, we find PGE is never favoured by Y-chromosomalgenes, which suffer complete transmission failure under PGE,and this drive disadvantage cannot be offset by any offspring-survival or sex-ratio benefit, for any degree of inbreeding.Relaxing our assumption of polyandry could change modelpredictions in this respect, as this would lead to relatednessbetween mate competitors with respect to their Y-chromo-somal genes, enhancing selection for female bias (cf Hamilton1967), but the qualitative conclusion that the Y chromosomeis less inclined to PGE than are the autosomes appears to berobust. However, in many species the Y chromosome is rela-tively degenerate, containing few active genes, and hence mayhave insufficient power to inhibit PGE. Accordingly, loss ofthe Y chromosome more likely occurred subsequently to thetransition to PGE. Indeed, in most cases, the diploid sistergroups of PGE taxa have XY sex determination (Table 1).

Mating system and genome elimination

Our model of sib-mating is motivated by the empiricalobservation that GE often coincides with a life history thatleads to high levels of inbreeding (Table 1). Our modelshows that sib-mating and resulting selection for female-biased sex ratios can facilitate the evolution of GE, and wesuggest that this explains the apparent association. However,sib-mating need not be essential for the evolution of GE.Other forms of inbreeding, such as those arising from popu-lation viscosity, may also favour female-biased sex ratios(Gardner et al. 2009). Moreover, some dipteran species withPGE lack inbreeding altogether (Burt & Trivers 2006). It isalso possible that GE drives the evolution of inbreeding,rather than the reverse, yielding an alternative explanationfor their apparent association. For example, some specieswith PGE exhibit haploid gene expression in males (Table 1and Fig. 1), which could promote inbreeding by purgingrecessive deleterious alleles and hence ameliorating the costsof homozygosity. However, inbreeding appears frequentlyamong several PGE species that exhibit diploid-male geneexpression (e.g. lice and springtails).

Our model makes a number of other assumptions aboutancestral mating system. First, for analytical convenience, wehave assumed extreme female promiscuity, though monandrymay be more realistic for some species (Hamilton 1967). AsGE in males may lead to a reduction in the number of viablesperm, lower female promiscuity could promote GE in malesby reducing between-male sperm competition. Second, wehave assumed a classic LMC scenario in which females mateat their natal patch prior to dispersal (Hamilton 1967), whichis more realistic for some PGE species (lice, mites, coffee-borer beetles and springtails) than others (dipterans and scaleinsects). However, these details are not essential, and themajor purpose of the model is to illustrate that any matingecology that results in selection for sex-ratio bias may governthe evolution of GE.

Germline vs. embryonic genome elimination

The defining feature of PGE in males is the absence of thepaternal genome among the individual’s gametes, and ourmodel focuses upon this central feature. That is, it more clo-sely captures the ‘germline PGE’ of those species in whichelimination of the paternal genome occurs during spermato-genesis, rather than the ‘embryonic PGE’ of those species inwhich elimination occurs during early embryogenesis, whichpotentially involves additional fitness costs associated withsomatically haploid males (Table 1 and Fig. 1). Such costs ofhaploidy mean that germline PGE is likely the ancestral form,and embryonic PGE the more derived form. Indeed, embry-onic PGE has evolved from germline PGE at least in the scaleinsects (Nur 1990; Herrick & Seger 1999; Ross et al. 2010a).Among those species with germline PGE, there are various

somatic effects ranging from elimination of paternally derivedsex chromosomes to complete transcriptional silencing of thepaternal genome (Fig. 1 and Table 1). Our model providestwo non-mutually exclusive explanations for this. First, oncePGE has evolved, the paternally derived genes are understrong selection to evolve counteradaptations (Herrick &Seger 1999; Ross et al. 2010b). To avoid this, the maternal ormaternally derived genes are selected to either: disable thepaternal genome, leading to whole-genome heterochromatisa-tion, as seen in scale insects and beetles; or else completelyeliminate it, leading to embryonic PGE, as seen in mites andarmoured scale insects. Second, fixation of PGE in malesrequires the evolution of a new sex-determining system thatoverrides genetic sex determination. For example, in specieswhere sex is determined by X-dosage, the elimination of oneX chromosome would convert a female (XX) into a male(X0), as occurs in springtails (Dallai et al. 2001) and Sciaraflies (Haig 1993b). Indeed, sex-ratio selection might alsofavour the silencing or early elimination of the whole paternalgenome, converting females (XX) to hemizygous males (X).Our model explains the elimination of paternal – as

opposed to maternal – chromosomes owing to its impactupon offspring sex ratio. Another reason why the paternalgenome may be more vulnerable to elimination is anisogamy(Parker et al. 1972): an egg contains many more proteins andRNAs than does a sperm, so if elimination is under parentalcontrol, the mother’s interests may be expected to dominate.



This is likely to be important in the evolution of embryonicPGE, but less so in the evolution of germline PGE. Moreover,such imbalance in power fails to explain other patterns ofGE, such as its confinement to males.

Other forms of genome elimination

Our model has considered the most frequently occurring formof GE, whereby one haploid genome is eliminated accordingto its parent of origin. However, there are a number of repro-ductive systems that exhibit alternative forms of GE, typicallyinvolving hybridisation events where one species temporarily‘borrows’ a genome from a related species, to express in itssomatic tissues, but eliminates it from the germline (Beuke-boom & Vrijenhoek 1998; Burt & Trivers 2006). In many, butnot all, of these cases the hybridogenetic species is all-female.Our model shows that, in principle, this mating system couldhave evolved via PGE in heterogametic females, with extinc-tion avoided despite the eradication of males because femaleswere able to mate with males from a closely related species,but this is unlikely to be of empirical importance. First, if thiswere how hybridogenesis typically evolved, we would expectan overrepresentation of ancestral female heterogametyamong these taxa, and although ZW is present in the ances-tors of two hybridogenetic teleost fish, the ancestors ofhybridogenetic frogs and stick insects were clearly XY (TheTree of Sex Consortium 2014). Second, in all cases, hybrido-genesis apparently arose from an ancient hybridisation event,of which GE was a consequence rather than a cause (Burt &Trivers 2006).However, there is one other case where genome elimination

is dependent upon parent of origin. Androgenesis – found intwo species of ant, four species of clam and one species ofcypress tree – involves only sperm-contributing genes tozygotes, and maternal genomes being entirely eliminated(Beukeboom & Vrijenhoek 1998; Burt & Trivers 2006). Inants, it evolved from an already-asymmetric inheritance sys-tem (haplodiploidy) and resulted from selection pressuresparticular to eusociality. In clams and cypress, it evolvedfrom hermaphroditism and occurs in every offspring (unlikePGE, which is restricted to males; Burt & Trivers 2006), soit resembles a modified form of parthenogenesis (with allreproductive value belonging to males rather than females)and may be better captured by classic models for the evolu-tion of parthenogenesis.Finally, PGE shares several key features with haplodiploidy

(Hartl & Brown 1970; Bull 1979; Burt & Trivers 2006). Inparticular, all genes transmitted by males derive from theirmothers, and are passed on only by their daughters. PGE andhaplodiploidy often co-occur in closely related taxonomicgroups, including scale insects, mites and beetles, which sug-gest that similar selection pressures underlie the evolution ofboth genetic systems. In addition to the drive benefits that areenjoyed by maternal-origin genes, the offspring sex ratio biaseffect captured in our model may also apply to the evolutionof haplodiploidy, as mating with a haploid male results infemale bias that may be favoured under sib-mating (Hamilton1967; Borgia 1980; Bull 1983). Sib-mating may also promotethe evolution of haplodiploidy in other ways, unrelated to the

effects considered in the present model, for example by purg-ing recessive deleterious alleles and hence raising the viabilityof haploid males (Brown 1964). However, an alternativeexplanation for this empirical association is that haplodiploidymay evolve via PGE (Schrader & Hughes-Schrader 1931).This possibility remains to be formally explored, and presentsan exciting avenue for future theoretical and empirical study.

ACKNOWLEDGEMENTS

We thank Ben Normark and three anonymous reviewers forhelpful discussion. This research has been supported by aRoyal Society University Research Fellowship (AG), a RoyalSociety Newton International Fellowship (LR) and twoNERC Independent Research Fellowships (AG & LR).

AUTHORSHIP

AG and LR designed the study and wrote the manuscript,AG performed the mathematical analysis.

REFERENCES

Ascunce, M.S., Toups, M.A., Kassu, G., Fane, J., Scholl, K. & Reed,

D.L. (2013). Nuclear genetic diversity in human lice (Pediculus

humanus) reveals continental differences and high inbreeding among

worldwide populations. PLoS ONE, 8, e57619.

Benatti, T.R., Valicente, F.H., Aggarwal, R., Zhao, C., Walling, J.G.,

Chen, M.S. et al. (2010). A neo-sex chromosome that drives

postzygotic sex determination in the hessian fly (Mayetiola destructor).

Genetics, 184, 769–777.Beukeboom, L. & Vrijenhoek, R. (1998). Evolutionary genetics and

ecology of sperm-dependent parthenogenesis. J. Evol. Biol., 11, 755–782.

Borgia, G. (1980). Evolution of maplodiploidy - models for inbred and

outbred systems. Theor. Pop. Biol., 17, 103–128.Borsa, P. & Kjellberg, F. (1996). Experimental evidence for pseudo-

arrhenotoky in Hypothenemus hampei. Heredity, 76, 130–136.Brown, S.W. (1964). Automatic frequency response in evolution of

male haploidy and other coccid chromosome systems. Genetics, 49,

797–817.Brown, S.W. & Nur, U. (1964). Heterochromatic chromosomes in coccids

- process of heterochromatization and function of heterochromatin in

coccid insects are reviewed. Science, 145, 130–136.Brun, L.O., Stuart, J., Gaudichon, V., Aronstein, K. & Ffrench-Constant,

R.H. (1995). Functional haplodiploidy: a mechanism for the spread of

insecticide resistance in an important international insect pest. Proc.

Natl Acad. Sci. USA, 92, 9861–9865.Bull, J.J. (1979). An advantage for the evolution of male haploidy and

systems with similar genetic transmission. Heredity, 43, 361–381.Bull, J.J. (1983). The Evolution of Sex Determining Mechanisms. Benjamin

Cummings, Menlo Park, CA.

Burt, A. & Trivers, R. (2006). Genes in Conflict. Harvard University

Press, Cambridge.

Crouse, H.V. (1960). The controlling element in sex chromosome

behavior in Sciara. Genetics, 45, 1429–1443.Dallai, R., Fanciulli, P.P., Carapelli, A. & Frati, F. (2001). Aberrant

spermatogenesis and sex determination in Bourletiellidae (Hexapoda,

Collembola), and their evolutionary significance. Z Morph u Okol

Tiere, 120, 237–245.Gardner, A. & Welch, J.J. (2011). A formal theory of the selfish gene.

J. Evol. Biol., 24, 1801–1813.Gardner, A., Arce, A. & Alpedrinha, J. (2009). Budding dispersal and the

sex ratio. J. Evol. Biol., 22, 1036–1045.



Gardner, A., West, S.A. & Wild, G. (2011). The genetical theory of kin

selection. J. Evol. Biol., 24, 1020–1043.Gauthier, N. (2010). Multiple cryptic genetic units in Hypothenemus

hampei (Coleoptera: Scolytinae): evidence from microsatellite and

mitochondrial DNA sequence data. Biol. J. Linn. Soc., 101, 113–129.Gavrilov, I.A. (2007). A catalog of chromosome numbers and genetic

systems of scale insects (Homoptera: Coccinea) of the world. Israel J

Entomol, 37, 1–45.Goday, C. & Esteban, M. (2001). Chromosome elimination in sciarid

flies. BioEssays, 23, 242–250.Haig, D. (1993a). The evolution of unusual chromosomal systems in

coccoids: extraordinary sex ratios revisited. J. Evol. Biol., 6, 69–77.Haig, D. (1993b). The evolution of unusual chromosomal systems in

sciarid flies: intragenomic conflict and the sex ratio. J. Evol. Biol., 6,

249–261.Haig, D. (2002). Genomic Imprinting and Kinship. Rutgers University

Press, New Brunswick.

Hamilton, W.D. (1964). The genetical evolution of social behaviour. I &

II. J. Theor. Biol., 7, 1–52.Hamilton, W.D. (1967). Extraordinary sex ratios. Science, 156, 477–488.Hartl, D.L. & Brown, S.W. (1970). The origin of male haploid genetic

systems and their expected sex ratio. Theor Pop Biol, 1, 165–190.Herrick, G. & Seger, J. (1999). Imprinting and paternal genome

elimination in insects. In: Genomic Imprinting: An Interdisciplinary

Approach (ed. Ohlsson, R.). Springer-Verlag, New York, pp. 41–71.Kuijper, B. & Pen, I. (2010). The evolution of haplodiploidy by male-

killing endosymbionts: importance of population structure and

endosymbiont mutualisms. J. Evol. Biol., 23, 40–52.Leigh, E.G. (1971). Adaptation and Diversity. Freeman, Cooper and

Company, San Francisco.

MacArthur, R.H. (1961). Population effects of natural selection. Am.

Nat., 95, 195–199.Martins, R.F., Zina, V., Da Silva, E.B., Rebelo, M.T., Figueiredo, E.,

Mendel, Z. et al. (2012). Isolation and characterization of fifteen

polymorphic microsatellite loci for the citrus mealybug, Planococcus

citri (Hemiptera: Pseudococcidae), and cross-amplification in two other

mealybug species. J. Genet., 91, e75–e78.McMeniman, C.J. & Barker, S.C. (2005). Transmission ratio distortion in

the human body louse, Pediculus humanus (Insecta: Phthiraptera).

Heredity, 96, 63–68.Metz, C.W. (1938). Chromosome behavior, inheritance and sex

determination in Sciara. Am. Nat., 72, 485–520.Nagelkerke, C.J. & Sabelis, M. (1998). Precise control of sex allocation in

pseudo-arrhenotokous phytoseiid mites. J. Evol. Biol., 11, 649–684.Nelson-Rees, W.A., Hoy, M.A. & Roush, R.T. (1980).

Heterochromatinization, chromatin elimination and haploidization in

the parahaploid mite Metaseiulus occidentalis (Nesbitt) (Acarina:

Phytoseiidae). Chromosoma, 77, 263–276.Normark, B.B. (2004). Haplodiploidy as an outcome of coevolution

between male-killing cytoplasmic elements and their hosts. Evolution,

58, 790–798.Norton, R.A., Kethley, J.B., Johnston, D.E. & O’Connor, B.M. (1993).

Phylogenetic perspectives on genetic systems and reproductive modes of

mites. In: Evolution and diversity of sex ratio in insects and mites (eds.

Wrensch, D.L. & Ebbert, M.A.). Chapman & Hall, New York, pp. 8–99.

Nur, U. (1990). Heterochromatization and euchromatization of whole

genomes in scale insects (Coccoidea, Homoptera). Development

Supplement, 1990, 29–34.Painter, T.S. (1966). The role of the E-chromosomes in Cecidomyiidae.

Proc. Natl Acad. Sci. USA, 56, 853–855.Parker, G.A., Baker, R.R. & Smith, V. (1972). The origin and evolution

of gamete dimorphism and the male-female phenomenon. J. Theor.

Biol., 36, 529–553.Perotti, M.A., Catal�a, S.S., del V Orme~no, A., Zelazowska, M., Bilinski,

S.M. & Braig, H.R. (2004). The sex ratio distortion in the human head

louse is conserved over time. BMC Genet., 5, 10.

Ross, L., Pen, I. & Shuker, D.M. (2010a). Genomic conflict in scale

insects: the causes and consequences of bizarre genetic systems. Biol.

Rev. Camb. Philos. Soc., 85, 807–828.Ross, L., Shuker, D.M. & Pen, I. (2010b). The evolution and suppression

of male suicide under paternal genome elimination. Evolution, 65, 554–563.

S�anchez, L. (2010). Sciara as an experimental model for studies on the

evolutionary relationships between the zygotic, maternal and

environmental primary signals for sexual development. J. Genet., 89,

325–331.Schrader, F. & Hughes-Schrader, S. (1931). Haploidy in metazoa. Q Rev.

Biol., 6, 411–438.Stuart, J.J. & Hatchett, J.H. (1991). Genetics of sex determination in the

Hessian fly, Mayetiola destructor. J. Hered., 82, 43–52.Taylor, P.D. & Frank, S.A. (1996). How to make a kin selection model.

J. Theor. Biol., 180, 27–37.The Tree of Sex Consortium (2014). Tree of sex: a database of sexual

systems. Sci Data, 1, 140015.

Trivers, R.L. (1974). Parent-offspring conflict. Am. Zool., 14, 249–264.Werren, J.H. & Hatcher, M. (2000). Maternal-zygotic gene conflict over

sex determination: effects of inbreeding. Genetics, 155, 1469–1479.West, S.A. (2009). Sex Allocation. Princeton University Press

(Monographs in Population Biology Series), Princeton.

SUPPORTING INFORMATION

Additional Supporting Information may be downloaded viathe online version of this article at Wiley Online Library(www.ecologyletters.com).

Editor, Minus van BaalenManuscript received 11 June 2014First decision made 10 July 2014Second decision made 8 September 2014Manuscript accepted 18 September 2014



! 1!

Gardner,(A.(&(Ross,(L.(“Mating(ecology(explains(patterns(of(genome(elimination”(Supplementary(Material(1(–(Extended(Methods.((1.(PGE(in(females(and/or(males(under(XY/XO(inheritance:(invasion(analysis((1.1.!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:!SfMM(φ,µ,α,β)!=!(1/8)(1>φ)(1>µ)!+!(1/4)φ(1>µ)(1>α)!+!(1/2)(1>φ)µ(1>β)!+!φ!µ!(1>α)(1>β);!SfMP(φ,µ,α,β)!=!(1/8)(1>φ)(1>µ)!+!(1/4)φ(1>µ)(1>α);!SfPM(φ,µ,α,β)!=!(1/8)(1>φ)(1>µ)!+!(1/2)(1>φ)µ(1>β);!SfPP(φ,µ,α,β)!=!(1/8)(1>φ)(1>µ);!SmMM(φ,µ,α,β)!=!(1/8)(1>φ)(1>µ)!+!(1/4)φ(1>µ)(1>α);!SmMP(φ,µ,α,β)!=!(1/8)(1>φ)(1>µ)!+!(1/4)φ(1>µ)(1>α);!SmPM(φ,µ,α,β)!=!(1/8)(1>φ)(1>µ);!and!SmPP(φ,µ,α,β)!=!(1/8)(1>φ)(1>µ).!!!

Reproductive1success!–!Upon!survival!to!maturity,!every!female!achieves!a!unit!of!expected!reproductive!success,!and!every!male!achieves!(1>z(φ%,µ%,α,β))/z(φ%,µ%,α,β)!units!of!expected!reproductive!success,!where:!z(φ%,µ%,α,β)!=!(∑T∈{M,P},U∈{M,P}!

SmTU(φ%,µ%,α,β))!/!(∑s∈{f,m},T∈{M,P},U∈{M,P}!SsTU(φ%,µ%,α,β))!is!the!sex!ratio!(proportion!male)!in!his!mating!group;!φ%!is!the!level!of!PGE!among!the!females!contributing!offspring!to!the!mating!group;!and!µ%!is!the!level!of!PGE!among!the!fathers!contributing!offspring!to!the!mating!group.!Thus,!contingent!upon!survival,!expected!

reproductive!success!is!RfTU(φ%,µ%,α,β)!=!1!for!females!and!RmTU(φ%,µ%,α,β)!=!(1>z(φ%,µ%,α,β))/z(φ%,µ%,α,β)!for!males.!!Fitness1–!Expected!fitness!is!given!by!the!product!of!survival!to!maturity!and!expected!reproductive!success!contingent!upon!surviving!to!maturity,!i.e.!

wsTU(φ,φ%,µ,µ%,α,β)!=!SsTU(φ,µ,α,β)×RsTU(φ%,µ%,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,φ%,µ,µ%,α,β)!=!wsTU(φ,φ%,µ,µ%,α,β)/�wsTU(α,β).!!

! 2!

!

Gene1fitness!–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,φ%,µ,µ%,α,β)!=!WsTU2(φ,φ%,µ,µ%,α,β)!=!WsTU(φ,φ%,µ,µ%,α,β).!!

Reproductive1value!–!The!flow!of!genes!between!classes!determines!each!class’s!reproductive!value.!Each!gene’s!class!in!any!generation!may!provide!some!

information!and!some!uncertainty!about!that!gene’s!origin!in!the!previous!

generation.!For!example,!genes!of!class!fMM1!in!the!present!generation!are!derived!from!genes!of!class!fTU1!in!the!previous!generation,!where!T!=!M!or!P!with!equal!

probability!and!U!=!M!or!P!with!equal!probability.!Consequently,!the!fMM1!gene!

derived!from!a!fMM1!gene!with!probability!℘fMM1←fMM1!=!¼,!from!a!fMP1!gene!with!

probability!℘fMM1←fMP1!=!¼,!from!a!fPM1!gene!with!probability℘fMM1←fPM1!=!¼!and!

from!a!fPP1!gene!with!probability!℘fMM1←fPP1!=!¼.!The!reproductive!value!of!a!given!

class!is!shared!among!all!the!classes!who!contribute!genes!to!that!class,!in!

proportion!to!the!contributions!that!they!make:!csTUv!=!∑s*∈{f,m},T*∈{M,P},U*∈{M,P},v*∈{1,2}!

℘s*T*U*v*←sTUv!cs*T*U*v*.!This!defines!a!system!of!16!linear!equations,!which!can!be!written!as!c!=(c•G,!where!c!=!{cfMM1,cfMM2,cfMP1,…,cmPP2}!is!the!vector!of!class!reproductive!values!and!G!is!the!gene>flow!matrix.!Solving!(i.e.!finding!the!left!eigenvector!of!G!corresponding!to!the!eigenvalue!1)!obtains!csTUv!=!1/16!for!all!gene!classes.!!

PGE1in1females!–!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!(dWsTUv/dφ)!×!(dφ/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!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!PGE!in!the!mother!of!the!focal!gene’s!carrier.!The!consanguinities!

will!depend!upon!which!class!or!classes!of!genes!control!PGE!in!females!>>!see!the!Consanguinity!section!below,!and!Table!A1.1.1,!for!details.!

!

PGE1in1males!–!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)!+!(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,!and!

! 3!

G%!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!=!dµ%/dG%!=!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!dG%/dgsTUv!=!q!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/dµ%!=!0!for!all!classes!of!genes!in!females,!we!need!only!calculate!consanguinities!q!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).!!

Potential1for1PGE1–!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!=!φ%!=!�φ!=!µ!=!µ%!=!�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!γ1such!that!invasion!will!occur!if!α!<!γ1and!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),!!

! 4!

! Actor(Class( ( ! ! !

Recipient(Class(

A! AMat! APat! AMot! AFat!

fMM1! (1+ρ)/2! 1! ρ! (1+ρ)/2! ρ!fMM2! ρ>M! ρMM! ρPM! a(1+ρ)/2! a!ρ!fMP1! (1+ρ)/2! 1! ρ! (1+ρ)/2! ρ!fMP2! ρ>P! ρMP! ρPP! a!ρ! a!ρ!fPM1! (1+ρ)/2! ρ! 1! ρ! (1+ρ)/2!fPM2! ρ>M! ρMM! ρPM! a(1+ρ)/2! a!ρ!fPP1! (1+ρ)/2! ρ! 1! ρ! (1+ρ)/2!fPP2! ρ>P! ρMP! ρPP! a!ρ! a!ρ!mMM1! (1+ρ)/2! 1! ρ! (1+ρ)/2! ρ!mMM2! ρ>M! ρMM! ρPM! a(1+ρ)/2! a!ρ!mMP1! (1+ρ)/2! 1! ρ! (1+ρ)/2! ρ!mMP2! ρ>P! ρMP! ρPP! a!ρ! a!ρ!mPM1! (1+ρ)/2! ρ! 1! ρ! (1+ρ)/2!mPM2! ρ>M! ρMM! ρPM! a(1+ρ)/2! a!ρ!mPP1! (1+ρ)/2! ρ! 1! ρ! (1+ρ)/2!mPP2! 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)!=!!!

! 5!


Recipient(Class(


fMM1! ρM>! ρMM! ρMP! a(1+ρ)/2! a!ρ!fMM2! (1+ρ)/2! 1! ρ! (1+ρ)/2! ρ!fMP1! ρM>! ρMM! ρMP! a(1+ρ)/2! a!ρ!fMP2! (1+ρ)/2! ρ! 1! ρ! (1+ρ)/2!fPM1! ρP>! ρPM! ρPP! a!ρ! a!ρ!fPM2! (1+ρ)/2! 1! ρ! !(1+ρ)/2! ρ!fPP1! ρP>! ρPM! ρPP! a!ρ! a!ρ!fPP2! (1+ρ)/2! ρ! 1! ρ! (1+ρ)/2!mMM1! ρM>! ρMM! ρMP! a(1+ρ)/2! a!ρ!mMM2! (1+ρ)/2! 1! ρ! (1+ρ)/2! ρ!mMP1! ρM>! ρMM! ρMP! a(1+ρ)/2! a!ρ!mMP2! (1+ρ)/2! ρ! 1! ρ! (1+ρ)/2!mPM1! ρP>! ρPM! ρPP! a!ρ! a!ρ!mPM2! (1+ρ)/2! 1! ρ! !(1+ρ)/2! ρ!mPP1! ρP>! ρPM! ρPP! a!ρ! a!ρ!mPP2! (1+ρ)/2! ρ! 1! ρ! (1+ρ)/2!mMM1%! a!ρM>! a!ρMM! a!ρMP! a2(1+ρ)/2! a2!ρ!mMM2%! a!ρ>M! a!ρMM! a!ρMP! a2(1+ρ)/2! a2!ρ!mMP1%! a!ρM>! a!ρMM! a!ρMP! a2(1+ρ)/2! a2!ρ!mMP2%! a!ρ>P! a!ρMP! a!ρPP! a2!ρ! a2!ρ!mPM1%! a!ρP>! a!ρPM! a!ρPP! a2!ρ! a2!ρ!mPM2%! a!ρ>M! a!ρMM! a!ρMP! a2(1+ρ)/2! a2!ρ!mPP1%! a!ρP>! a!ρPM! a!ρPP! a2!ρ! a2!ρ!mPP2%! a!ρP>! a!ρMP! a!ρPP! a2!ρ! a2!ρ!!

Table(A1.1.2.!Consanguinities(for(the(male(PGE(invasion(analysis:(autosomal(genes.!The!five!autosomal!actor!classes!are!the!male’s!autosomal!genes!(A),!his!maternal>origin!autosomal!genes!

(AMat),!his!paternal>origin!autosomal!genes!(APat),!his!mother’s!autosomal!genes!(AMot)!and!his!

father’s!autosomal!genes!(AFat).!Shown!here!are!their!consanguinities!qsTUv!to!each!of!the!recipient!classes!sTUv!among!the!male’s!offspring,!and!the!consanguinities!q%mTUv!to!each!of!the!recipient!classes!mTUv%,!among!the!males!competing!for!mates!with!the!focal!male’s!sons.!!

(1/4)(1>φ)(1>µ)!+!(1/2)(1>φ)µ(1>β);!SmM>(φ,µ,α,β)!=!(1/4)(1>φ)(1>µ)!+!(1/2)φ(1>µ)(1>α);!and!SmP>(φ,µ,α,β)!=!(1/4)(1>φ)(1>µ).!!1Reproductive1success!–!Contingent!upon!survival,!expected!reproductive!success!is!RfTU(φ%,µ%,α,β)!=!1!for!females!and!RmTU(φ%,µ%,α,β)!=!(1>z(φ%,µ%,α,β))/z(φ%,µ%,α,β)!for!males,!where!z(φ%,µ%,α,β)!=!(SmM>(φ%,µ%,α,β)!+1SmP>(φ%,µ%,α,β))/(SfMM(φ%,µ%,α,β)!+1SfPM(φ%,µ%,α,β)!+1SmM>(φ%,µ%,α,β)!+1SmP>(φ%,µ%,α,β)).!!Fitness1–!Expected!fitness!is!given!by!the!product!of!survival!to!maturity!and!expected!reproductive!success!contingent!upon!surviving!to!maturity,!i.e.!

! 6!

wsTU(φ,φ%,µ,µ%,α,β)!=!SsTU(φ,µ,α,β)×RsTU(φ%,µ%,α,β).!As!before,!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,φ%,µ,µ%,α,β)!=!wsTU(φ,φ%,µ,µ%,α,β)/�wsTU(α,β).!!!

Gene1fitness!–There!are!6!classes!of!gene,!because!individuals!of!each!of!the!two!female!classes!carry!separate!maternal>origin!and!paternal>origin!genes,!whereas!

individuals!of!each!of!the!two!male!classes!carry!only!maternal>origin!genes.!As!

before,!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!relative!fitness!

of!a!gene!in!a!female!is!given!by!WfTP1(φ,φ%,µ,µ%,α,β)!=!WfTP2(φ,φ%,µ,µ%,α,β)!=!WfTP(φ,φ%,µ,µ%,α,β)!and!the!relative!fitness!of!a!gene!in!a!male!is!given!by!WmT>

1(φ,φ%,µ,µ%,α,β)!=!WmT>(φ,φ%,µ,µ%,α,β).!!Reproductive1value!–!As!before,!each!gene’s!class!in!any!generation!may!provide!some!information!and!some!uncertainty!about!that!gene’s!origin!in!the!previous!

generation.!For!example,!genes!of!class!fMM1!in!the!present!generation!are!derived!from!genes!of!class!fTM1!in!the!previous!generation,!where!T!=!M!or!P!with!equal!

probability.!Consequently,!the!fMM1!gene!derived!from!a!fMM1!gene!with!

probability!℘fMM1←fMM1!=!½!and!from!a!fPM1!gene!with!probability℘fMM1←fPM1!=!½.!

The!corresponding!system!of!6!linear!equations!may!be!written!in!linear!algebraic!

form!as!c!=(c•G,!where!c!=!{cfMM1,cfMM2,cfPM1,cfPM2,cmM>1,cmP>1}!is!the!vector!of!class!reproductive!values!and!G!is!the!gene>flow!matrix.!Solving!obtains!csTUv!=!1/6!for!all!gene!classes.!

!

PGE1in1females!–!Following!the!same!procedure!as!before,!the!condition!for!increase!in!female!PGE!is!cfMM1!((dWfMM1/dφ)!×!(dφ/dG)!×!(dG/dgfMM1))!+!cfMM2!((dWfMM2/dφ)!×!(dφ/dG)!×!(dG/dgfMM2))!+!cfPM1!((dWfPM1/dφ)!×!(dφ/dG)!×!(dG/dgfPM1))!+!cfPM2!((dWfPM2/dφ)!×!(dφ/dG)!×!(dG/dgfPM2))!+!cmM>1!((dWmM>1/dφ)!×!(dφ/dG)!×!(dG/dgmM>1))!+!cmP>1!((dWmP>1/dφ)!×!(dφ/dG)!×!(dG/dgmP>1))!>!0.!Once!again:!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!PGE!in!the!mother!of!the!focal!gene’s!

carrier.!!

PGE1in1males!–!Similarly,!the!condition!for!increase!in!male!PGE!is!cfMM1!((dWfMM1/dµ)!×!(dµ/dG)!×!(dG/dgfMM1))!+!cfMM2!((dWfMM2/dµ)!×!(dµ/dG)!×!(dG/dgfMM2))!+!cfPM1!((dWfPM1/dµ)!×!(dµ/dG)!×!(dG/dgfPM1))!+!cfPM2!((dWfPM2/dµ)!×!(dµ/dG)!×!(dG/dgfPM2))!+!cmM>1!((dWmM>1/dµ)!×!(dµ/dG)!×!(dG/dgmM>1)!+!(dWmM>

1/dµ%)!×!(dµ%/dG%)!×!(dG%/dgmM>1))!+!cmP>1!((dWmP>1/dµ)!×!(dµ/dG)!×!(dG/dgmP>1)!+!(dWmP>1/dµ%)!×!(dµ%/dG%)!×!(dG%/dgmP>1))!>!0,!where:!dµ/dG!=!dµ%/dG%!=!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!dG%/dgmTUv!=!q!mTUv!is!the!consanguinity!between!the!focal!class>sTUv!gene!and!

! 7!

the!genes,!residing!at!the!same!locus,!who!control!PGE!in!the!males!who!contribute!

offspring!to!the!focal!individual’s!mating!group.!!!

Consanguinity!–!The!coefficient!of!inbreeding!for!X>linked!genes!is!ρ*!=!a((1/2)!×!(1+ρ*)/2!+!(1/2)!×!ρ*).!That!is:!with!probability!a!the!male!and!female!share!the!same!mother;!so!with!probability!½!we!pick!the!maternal>origin!genes!from!the!female!and!hence!their!consanguinity!is!simply!the!consanguinity!of!their!mother!to!

herself,!or!(1+ρ*)/2;!and!with!probability!½!we!pick!the!paternal>origin!gene!from!the!female,!in!which!case!their!consanguinity!is!that!of!mating!partners,!or!ρ*.!Solving!this!equation!obtains!ρ*!=!a/(4>3a).!From!this!coefficient!can!be!defined!other!consanguinities!between!mating!partners:!ρ*M>!=!a(1+ρ*)/2,!from!the!perspective!of!the!female’s!maternal>origin!gene;!and!ρ*P>!=!a!ρ*,!from!the!perspective!of!the!female’s!paternal>origin!gene.!Moreover,!the!consanguinity!

between!two!males!in!the!same!mating!group!is!ψ*!=!a(1+ρ*)/2.!These!coefficients!define!all!the!consanguinities!needed!to!solve!the!model!(listed!in!Tables!A1.2.1!&!A1.2.2).!

!

Potential1for1PGE!–!The!potentials!for!female!and!male!PGE!are!calculated!as!described!above.!These!are!illustrated!in!Figure!3!of!the!main!text.!

!1.3.!Y>linked!genes!

1Classes!–!There!are!2!classes!of!individual,!classified!according!to!their!sex!s!∈!{f,m}.!If!the!individual!is!female,!then!she!has!no!Y>linked!genes.!And!if!the!individual!is!

male,!then!all!of!his!Y>linked!genes!are!paternal!in!origin!(and!came!from!his!

paternal!grandfather).!Thus,!class!is!notated!in!the!form!s!∈!{f,m}.!!

Survival!–!The!probabilities!of!survival!for!each!class!are!Sf(φ,µ,α,β)!=!(1/2)(1>φ)(1>µ)!+!(1/2)φ(1>µ)(1>α)!+!(1>φ)µ(1>β)!+!φ!µ!(1>α)(1>β)!and!Sm(φ,µ,α,β)!=!(1/2)(1>φ)(1>µ)!+!(1/2)φ(1>µ)(1>α).!!

Reproductive1success!–!Contingent!upon!survival,!expected!reproductive!success!is!Rf(φ%,µ%,α,β)!=!1!for!females!and!Rm(φ%,µ%,α,β)!=!(1>z(φ%,µ%,α,β))/z(φ%,µ%,α,β)!for!males,!where!z(φ%,µ%,α,β)!=!Sm(φ%,µ%,α,β)/(Sf(φ%,µ%,α,β)1+1Sm(φ%,µ%,α,β)).!!

Fitness1–!Expected!fitness!is!given!by!the!product!of!survival!to!maturity!and!expected!reproductive!success!contingent!upon!surviving!to!maturity,!i.e.!

ws(φ,φ%,µ,µ%,α,β)!=!Ss(φ,µ,α,β)×Rs(φ%,µ%,α,β).!As!before,!because!PGE!is!vanishingly!rare!in!the!population,!the!average!fitness!among!all!individuals!of!a!particular!class!

is!given!by�ws(α,β)!=!ws(0,0,0,0,α,β),!and!each!individual’s!fitness!can!be!expressed!relative!to!the!average!of!their!class,!by!Ws(φ,φ%,µ,µ%,α,β)!=!ws(φ,φ%,µ,µ%,α,β)/�ws(α,β).!!!

Gene1fitness!–!There!is!only!one!class!of!Y>linked!gene,!because!females!carry!no!Y>linked!genes!and!males!carry!one!class!of!Y>linked!genes.!Accordingly,!the!relative!

fitness!of!a!Y>linked!gene!in!a!male!is!simply!Wm(φ,φ%,µ,µ%,α,β).!

! 8!

!

! Actor(Class( ( ! ! !Recipient(Class(

X! XMat! XPat! XMot! XFat!

fMM1! (1+ρ*)/2! 1! ρ*! (1+ρ*)/2! ρ*!fMM2! ρ*! ρ*M>! ρ*P>! a(1+ρ*)/2! a!ρ*!fPM1! (1+ρ*)/2! ρ*! 1! ρ*! 1!

fPM2! ρ*! ρ*M>! ρ*P>! a(1+ρ*)/2! a!ρ*!mM>1! (1+ρ*)/2! 1! ρ*! (1+ρ*)/2! ρ*!mP>1! (1+ρ*)/2! ρ*! 1! ρ*! 1!!

Table(A1.2.1.!Consanguinities(for(the(female(PGE(invasion(analysis:(XNlinked(genes.!The!five!X>linked!actor!classes!are!the!female’s!X>linked!genes!(X),!her!maternal>origin!X>linked!genes!(XMat),!her!

paternal>origin!X>linked!genes!(XPat),!her!mother’s!X>linked!genes!(X!Mot)!and!her!father’s!X>linked!genes!(XFat).!Shown!here!are!their!consanguinities!psTUv!to!each!of!the!recipient!classes!sTUv!among!the!focal!female’s!offspring.!

!

! Actor(Class( ! !Recipient(Class(

X! XMot! XFat!

fMM1! ρ*M>! a(1+ρ*)/2! a!ρ*!fMM2! 1! (1+ρ*)/2! ρ*!fPM1! ρ*P>! a!ρ*! a!ψ*!fPM2! 1! (1+ρ*)/2! ρ*!mM>1! ρ*M>! a(1+ρ*)/2! a!ρ*!mP>1! ρ*P>! a!ρ*! a!ψ*!mM>1%! a1ρ*M>! a2(1+ρ*)/2! a2!ρ*!mP>1%! a1ρ*P>! a2!ρ*! a2!ψ*!!

Table(A1.2.2.!Consanguinities(for(the(male(PGE(invasion(analysis:(XNlinked(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!q%mTUv!to!each!of!the!recipient!classes!mTUv%,!among!the!males!competing!for!mates!with!the!focal!male’s!sons.!!Reproductive1value!–!Since!there!is!only!one!class!of!Y>linked!gene,!all!reproductive!value!belongs!to!this!class:!cm!=!1.11PGE1in1females!–!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.!

!

PGE1in1males!–!Similarly,!the!condition!for!increase!in!male!PGE!is!cm!((dWm/dµ)!×!(dµ/dG)!×!(dG/dgm)!+!(dWm/dµ%)!×!(dµ%/dG%)!×!(dG%/dgm))!>!0,!where:!dµ/dG!=!

! 9!

dµ%/dG%!=!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!dG%/dgm!=!q!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).!!

Potential1for1PGE!–!The!potentials!for!female!and!male!PGE!are!calculated!as!described!above.!These!are!illustrated!in!Figure!3!of!the!main!text.!(2.(MGE(in(females(and/or(males(under(XY/XO(inheritance:(invasion(analysis((2.1.!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!–!Denoting!the!probability!that!the!zygote’s!mother!exhibited!MGE!by!φ,!the!probability!that!the!zygote’s!father!exhibited!MGE!by!µ,!the!viability!cost!associated!with!female!MGE!by!α!and!the!viability!cost!associated!with!male!MGE!by!β,!the!probability!that!each!potential!zygote!survives!to!maturity!is:!SfMM(φ,µ,α,β)!=!(1/8)(1>φ)(1>µ);!SfMP(φ,µ,α,β)!=!(1/8)(1>φ)(1>µ);!SfPM(φ,µ,α,β)!=!(1/8)(1>φ)(1>µ)!+!(1/4)φ(1>µ)(1>α);!SfPP(φ,µ,α,β)!=!(1/8)(1>φ)(1>µ)!+!(1/4)φ(1>µ)(1>α);!SmMM(φ,µ,α,β)!=!(1/8)(1>φ)(1>µ);!SmMP(φ,µ,α,β)!=!(1/8)(1>φ)(1>µ)!+!(1/2)(1>φ)µ(1>β);!SmPM(φ,µ,α,β)!=!(1/8)(1>φ)(1>µ)!+!(1/4)φ(1>µ)(1>α);!and!SmPP(φ,µ,α,β)!=!(1/8)(1>φ)(1>µ)!+!(1/4)φ(1>µ)(1>α)!+!(1/2)(1>φ)µ(1>β)!+!φµ(1>α)(1>β).!!!

Reproductive1success!–!Upon!survival!to!maturity,!every!female!achieves!a!unit!of!expected!reproductive!success,!and!every!male!achieves!(1>z(φ%,µ%,α,β))/z(φ%,µ%,α,β)!units!of!expected!reproductive!success,!where:!z(φ%,µ%,α,β)!=!(∑T∈{M,P},U∈{M,P}!

SmTU(φ%,µ%,α,β))!/!(∑s∈{f,m},T∈{M,P},U∈{M,P}!SsTU(φ%,µ%,α,β))!is!the!sex!ratio!(proportion!male)!in!his!mating!group;!φ%!is!the!level!of!MGE!among!the!females!contributing!offspring!to!the!mating!group;!and!µ%!is!the!level!of!MGE!among!the!fathers!contributing!offspring!to!the!mating!group.!Thus,!contingent!upon!survival,!expected!

reproductive!success!is!RfTU(φ%,µ%,α,β)!=!1!for!females!and!RmTU(φ%,µ%,α,β)!=!(1>z(φ%,µ%,α,β))/z(φ%,µ%,α,β)!for!males.!

! 10!

!

! Actor(Class(Recipient(Class(

YFat!

m! a!ψ**!!

Table(A1.3.1.!Consanguinities(for(the(female(PGE(invasion(analysis:(YNlinked(genes.!The!only!Y>linked!actor!class!is!the!female’s!father’s!Y>linked!genes!(YFat).!Shown!here!are!their!consanguinity!pm!to!the!single!recipient!class!m!among!the!focal!female’s!sons.!

!

! Actor(Class( !

Recipient(Class(

Y! YFat!

m! 1! 1!

m%! a!ψ**! a!ψ**!!

Table(A1.3.2.!Consanguinities(for(the(male(PGE(invasion(analysis:(YNlinked(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!q%m!to!the!recipient!class!m%!in!the!males!who!compete!for!mates!with!the!focal!male’s!sons.!

!Fitness1–!Expected!fitness!is!given!by!the!product!of!survival!to!maturity!and!expected!reproductive!success!contingent!upon!surviving!to!maturity,!i.e.!

wsTU(φ,φ%,µ,µ%,α,β)!=!SsTU(φ,µ,α,β)×RsTU(φ%,µ%,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,φ%,µ,µ%,α,β)!=!wsTU(φ,φ%,µ,µ%,α,β)/�wsTU(α,β).!!!Gene1fitness!–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!


WsTU1(φ,φ%,µ,µ%,α,β)!=!WsTU2(φ,φ%,µ,µ%,α,β)!=!WsTU(φ,φ%,µ,µ%,α,β).!!Reproductive1value!–!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.!!

MGE1in1females!–!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!

! 11!

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.!

!

MGE1in1males!–!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/dµ%)!×!(dµ%/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!father,!

and!G%!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!=!dµ%/dG%!=!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!dG%/dgsTUv!=!q!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/dµ%!=!0!for!all!classes!of!genes!in!females,!we!need!only!calculate!consanguinities!q!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.!

!Potential1for1MGE1–!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.!

(2.2.!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>


!

Survival!–!The!probabilities!of!survival!for!each!class!are:!SfMM(φ,µ,α,β)!=!(1/4)(1>φ)(1>µ);!SfPM(φ,µ,α,β)!=!(1/4)(1>φ)(1>µ)!+!(1/2)φ(1>µ)(1>α);!SmM>(φ,µ,α,β)!=!(1/4)(1>φ)(1>µ)!+!(1/2)(1>φ)µ(1>β);!and!SmP>(φ,µ,α,β)!=!(1/4)(1>φ)(1>µ)!+!(1/2)φ(1>µ)(1>α)!+!(1/2)(1>φ)µ(1>β)!+!(1/2)φµ(1>α)(1>β).!!!

! 12!

Reproductive1success!–!Contingent!upon!survival,!expected!reproductive!success!is!RfTU(φ%,µ%,α,β)!=!1!for!females!and!RmTU(φ%,µ%,α,β)!=!(1>z(φ%,µ%,α,β))/z(φ%,µ%,α,β)!for!males,!where!z(φ%,µ%,α,β)!=!(SmM>(φ%,µ%,α,β)!+1SmP>(φ%,µ%,α,β))/(SfMM(φ%,µ%,α,β)!+1SfPM(φ%,µ%,α,β)!+1SmM>(φ%,µ%,α,β)!+1SmP>(φ%,µ%,α,β)).!!Fitness1–!Expected!fitness!is!given!by!the!product!of!survival!to!maturity!and!expected!reproductive!success!contingent!upon!surviving!to!maturity,!i.e.!

wsTU(φ,φ%,µ,µ%,α,β)!=!SsTU(φ,µ,α,β)×RsTU(φ%,µ%,α,β).!As!before,!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,φ%,µ,µ%,α,β)!=!wsTU(φ,φ%,µ,µ%,α,β)/�wsTU(α,β).!!!Gene1fitness!–There!are!6!classes!of!gene,!because!individuals!of!each!of!the!two!female!classes!carries!separate!maternal>origin!and!paternal>origin!genes,!whereas!



of!a!gene!in!a!female!is!given!by!WfTP1(φ,φ%,µ,µ%,α,β)!=!WfTP2(φ,φ%,µ,µ%,α,β)!=!WfTP(φ,φ%,µ,µ%,α,β)!and!the!relative!fitness!of!a!gene!in!a!male!is!given!by!WmT>

1(φ,φ%,µ,µ%,α,β)!=!WmT>(φ,φ%,µ,µ%,α,β).!!

Reproductive1value!–!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!1.2,!i.e.!csTUv!=!1/6!for!all!gene!classes.!!MGE1in1females!–!Following!the!same!procedure!as!before,!the!condition!for!increase!in!female!MGE!is!cfMM1!((dWfMM1/dφ)!×!(dφ/dG)!×!(dG/dgfMM1))!+!cfMM2!((dWfMM2/dφ)!×!(dφ/dG)!×!(dG/dgfMM2))!+!cfPM1!((dWfPM1/dφ)!×!(dφ/dG)!×!(dG/dgfPM1))!+!cfPM2!((dWfPM2/dφ)!×!(dφ/dG)!×!(dG/dgfPM2))!+!cmM>1!((dWmM>1/dφ)!×!(dφ/dG)!×!(dG/dgmM>1))!+!cmP>1!((dWmP>1/dφ)!×!(dφ/dG)!×!(dG/dgmP>1))!>!0.!Once!again:!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.!

!

MGE1in1males!–!Similarly,!the!condition!for!increase!in!male!MGE!is!cfMM1!((dWfMM1/dµ)!×!(dµ/dG)!×!(dG/dgfMM1))!+!cfMM2!((dWfMM2/dµ)!×!(dµ/dG)!×!(dG/dgfMM2))!+!cfPM1!((dWfPM1/dµ)!×!(dµ/dG)!×!(dG/dgfPM1))!+!cfPM2!((dWfPM2/dµ)!×!(dµ/dG)!×!(dG/dgfPM2))!+!cmM>1!((dWmM>1/dµ)!×!(dµ/dG)!×!(dG/dgmM>1)!+!(dWmM>

1/dµ%)!×!(dµ%/dG%)!×!(dG%/dgmM>1))!+!cmP>1!((dWmP>1/dµ)!×!(dµ/dG)!×!(dG/dgmP>1)!+!(dWmP>1/dµ%)!×!(dµ%/dG%)!×!(dG%/dgmP>1))!>!0,!where:!dµ/dG!=!dµ%/dG%!=!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!dG%/dgmTUv!=!q!mTUv!is!the!consanguinity!between!the!focal!class>sTUv!gene!and!

! 13!

the!genes,!residing!at!the!same!locus,!who!control!MGE!in!the!males!who!contribute!



those!listed!in!Tables!A1.2.1!&!A1.2.2.!

!Potential1for1MGE1–!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.!!

2.3.!Y>linked!genes!(Classes!–!There!are!2!classes!of!individual,!classified!according!to!their!sex!s!∈!{f,m}.!If!the!individual!is!female,!then!she!has!no!Y>linked!genes.!And!if!the!individual!is!



Survival!–!The!probabilities!of!survival!for!each!class!are!Sf(φ,µ,α,β)!=!(1/2)(1>φ)(1>µ)!+!(1/2)φ(1>µ)(1>α)!and!Sm(φ,µ,α,β)!=!(1/2)(1>φ)(1>µ)!+!(1/2)φ(1>µ)(1>α)!+!(1>φ)µ(1>β)!+!φ!µ!(1>α)(1>β).!!Reproductive1success!–!Contingent!upon!survival,!expected!reproductive!success!is!Rf(φ%,µ%,α,β)!=!1!for!females!and!Rm(φ%,µ%,α,β)!=!(1>z(φ%,µ%,α,β))/z(φ%,µ%,α,β)!for!males,!where!z(φ%,µ%,α,β)!=!Sm(φ%,µ%,α,β)/(Sf(φ%,µ%,α,β)1+1Sm(φ%,µ%,α,β)).!!Fitness1–!Expected!fitness!is!given!by!the!product!of!survival!to!maturity!and!expected!reproductive!success!contingent!upon!surviving!to!maturity,!i.e.!

ws(φ,φ%,µ,µ%,α,β)!=!Ss(φ,µ,α,β)×Rs(φ%,µ%,α,β).!As!before,!because!MGE!is!vanishingly!rare!in!the!population,!the!average!fitness!among!all!individuals!of!a!particular!class!

is!given!by�ws(α,β)!=!ws(0,0,0,0,α,β),!and!each!individual’s!fitness!can!be!expressed!relative!to!the!average!of!their!class,!by!Ws(φ,φ%,µ,µ%,α,β)!=!ws(φ,φ%,µ,µ%,α,β)/�ws(α,β).!!!


fitness!of!a!Y>linked!gene!in!a!male!is!simply!Wm(φ,φ%,µ,µ%,α,β).!!Reproductive1value!–!Since!there!is!only!one!class!of!Y>linked!gene,!all!reproductive!value!belongs!to!this!class:!cm!=!1.!!!

MGE1in1females!–!Following!the!same!procedure!as!before,!the!condition!for!increase!in!female!MGE!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!MGE!in!the!mother!of!the!focal!gene’s!carrier.!

! 14!

!

MGE1in1males!–!Similarly,!the!condition!for!increase!in!male!MGE!is!cm!((dWm/dµ)!×!(dµ/dG)!×!(dG/dgm)!+!(dWm/dµ%)!×!(dµ%/dG%)!×!(dG%/dgm))!>!0,!where:!dµ/dG!=!dµ%/dG%!=!1;!dG/dgm!=!qm!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!dG%/dgm!=!q!m!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.!!

!


those!listed!in!Tables!A1.3.1!&!A1.3.2.!!

Potential1for1MGE1–!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.!

(3.(PGE(in(females(and/or(males(under(ZW(inheritance:(invasion(analysis((3.1.!Autosomal!genes!!


!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:!SfMM(φ,µ,α,β)!=!(1/8)(1>φ)(1>µ)!+!(1/2)φ(1>µ)(1>α)!+!(1/4)(1>φ)µ(1>β)!+!φ!µ!(1>α)(1>β);!SfMP(φ,µ,α,β)!=!(1/8)(1>φ)(1>µ)!+!(1/2)φ(1>µ)(1>α);!SfPM(φ,µ,α,β)!=!(1/8)(1>φ)(1>µ)!+!(1/4)(1>φ)µ(1>β);!SfPP(φ,µ,α,β)!=!(1/8)(1>φ)(1>µ);!SmMM(φ,µ,α,β)!=!(1/8)(1>φ)(1>µ)!+!(1/4)(1>φ)µ(1>β);!SmMP(φ,µ,α,β)!=!(1/8)(1>φ)(1>µ);!SmPM(φ,µ,α,β)!=!(1/8)(1>φ)(1>µ)+!(1/4)(1>φ)µ(1>β);!and!SmPP(φ,µ,α,β)!=!(1/8)(1>φ)(1>µ).!!!Reproductive1success!–!Upon!survival!to!maturity,!every!female!achieves!a!unit!of!expected!reproductive!success,!and!every!male!achieves!(1>z(φ%,µ%,α,β))/z(φ%,µ%,α,β)!units!of!expected!reproductive!success,!where:!z(φ%,µ%,α,β)!=!(∑T∈{M,P},U∈{M,P}!

SmTU(φ%,µ%,α,β))!/!(∑s∈{f,m},T∈{M,P},U∈{M,P}!SsTU(φ%,µ%,α,β))!is!the!sex!ratio!(proportion!male)!

! 15!

in!his!mating!group;!φ%!is!the!level!of!PGE!among!the!females!contributing!offspring!to!the!mating!group;!and!µ%!is!the!level!of!PGE!among!the!fathers!contributing!offspring!to!the!mating!group.!Thus,!contingent!upon!survival,!expected!

reproductive!success!is!RfTU(φ%,µ%,α,β)!=!1!for!females!and!RmTU(φ%,µ%,α,β)!=!(1>z(φ%,µ%,α,β))/z(φ%,µ%,α,β)!for!males.!!


wsTU(φ,φ%,µ,µ%,α,β)!=!SsTU(φ,µ,α,β)×RsTU(φ%,µ%,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,φ%,µ,µ%,α,β)!=!wsTU(φ,φ%,µ,µ%,α,β)/�wsTU(α,β).!!!



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,φ%,µ,µ%,α,β)!=!WsTU2(φ,φ%,µ,µ%,α,β)!=!WsTU(φ,φ%,µ,µ%,α,β).!!

Reproductive1value!–!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.!!PGE1in1females!–!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!((dWsTUv/dφ)!×!(dφ/dG)!×!(dG/dgsTUv)!+!(dWsTUv/dφ%)!×!(dφ%/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!mother,!and!G%!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!=!dφ%/dG%!=!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!dG%/dgsTUv!=!p!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/dφ%!=!0!for!all!classes!of!genes!in!females,!we!need!only!calculate!consanguinities!p!mTUv!for!classes!of!genes!in!males.!Again,!the!consanguinities!will!

! 16!

depend!upon!which!class!or!classes!of!genes!control!PGE!in!females!>>!see!the!

Consanguinity!section!below.!!

PGE1in1males!–!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>!=!a((1/2)!×!(1+ρ)/2!+!(1/2)!×!ρ),!ρP>!=!a!ρ,!ρ>M!=!a((1/2)!×!(1+ρ)/2!+!(1/2)!×!ρ),!ρ>P!=!a!ρ,!ρMM!=!a!(1+ρ)/2,!ρMP!=!a!ρ,!ρPM!=!a!ρ!and!ρPP!=!a!ρ.!These!coefficients!define!all!the!consanguinities!needed!to!solve!the!model!(listed!in!Tables!A3.1.1!&!A3.1.2).!

!Potential1for1PGE1–!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.!!

3.2.!Z>linked!genes!!

Classes!–!There!are!4!classes!of!individual,!classified!according!to!their!sex!s!∈!{f,m}!and!the!grandparent!of!origin!of!their!paternal>origin!gene!U!∈!{M,P}.!If!the!individual!is!female,!she!has!no!maternal>origin!Z>linked!genes.!And!if!the!individual!is!male,!then!all!of!his!maternal>origin!Z>linked!genes!are!derived!from!his!maternal!

grandfather.!Thus,!each!class!is!notated!in!the!form!sTU,!where!T!∈!{M,>},!i.e.!f>M,!f>P,!mPM!and!mPP.!

!

Survival!–!The!probabilities!of!survival!for!each!class!are:!Sf>M(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);!Sf>P(φ,µ,α,β)!=!(1/4)(1>φ)(1>µ)!+!(1/2)φ(1>µ)(1>α);!SmPM(φ,µ,α,β)!=!(1/4)(1>φ)(1>µ)!+!(1/2)(1>φ)µ(1>β);!and!SmPP(φ,µ,α,β)!=!(1/4)(1>φ)(1>µ).!!!

Reproductive1success!–!Contingent!upon!survival,!expected!reproductive!success!is!RfTU(φ%,µ%,α,β)!=!1!for!females!and!RmTU(φ%,µ%,α,β)!=!(1>z(φ%,µ%,α,β))/z(φ%,µ%,α,β)!for!males,!where!z(φ%,µ%,α,β)!=!(SmPM(φ%,µ%,α,β)!+1SmPP(φ%,µ%,α,β))/(Sf>M(φ%,µ%,α,β)!+1Sf>P!(φ%,µ%,α,β)!+1SmPM(φ%,µ%,α,β)!+1SmPP(φ%,µ%,α,β)).!!


wsTU(φ,φ%,µ,µ%,α,β)!=!SsTU(φ,µ,α,β)×RsTU(φ%,µ%,α,β).!As!before,!because!PGE!is!!

! 17!

!

! Actor(Class( ( ! ! !Recipient(Class(


fMM1! (1+ρ)/2! 1! ρ! (1+ρ)/2! ρ!fMM2! ρ>M! ρMM! ρPM! a(1+ρ)/2! a!ρ!fMP1! (1+ρ)/2! 1! ρ! (1+ρ)/2! ρ!fMP2! ρ>P! ρMP! ρPP! a!ρ! a!ρ!fPM1! (1+ρ)/2! ρ! 1! ρ! (1+ρ)/2!fPM2! ρ>M! ρMM! ρPM! a(1+ρ)/2! a!ρ!fPP1! (1+ρ)/2! ρ! 1! ρ! (1+ρ)/2!fPP2! ρ>P! ρMP! ρPP! a!ρ! a!ρ!mMM1! (1+ρ)/2! 1! ρ! (1+ρ)/2! ρ!mMM2! ρ>M! ρMM! ρPM! a(1+ρ)/2! a!ρ!mMP1! (1+ρ)/2! 1! ρ! (1+ρ)/2! ρ!mMP2! ρ>P! ρMP! ρPP! a!ρ! a!ρ!mPM1! (1+ρ)/2! ρ! 1! ρ! (1+ρ)/2!mPM2! ρ>M! ρMM! ρPM! a(1+ρ)/2! a!ρ!mPP1! (1+ρ)/2! ρ! 1! ρ! (1+ρ)/2!mPP2! ρ>P! ρMP! ρPP! a!ρ! a!ρ!mMM1%! a(1+ρ)/2! a1 a!ρ! a(1+ρ)/21 a!ρ1mMM2%! aρ>M! a!ρMM! a!ρPM! a2(1+ρ)/21 a2!ρ1mMP1%! a(1+ρ)/2! a!! a!ρ! a(1+ρ)/21 a!ρ1mMP2%! aρ>P! a!ρMP! a!ρPP! a2!ρ1 a2!ρ1mPM1%! a(1+ρ)/2! a!ρ! a!! a!ρ1 a(1+ρ)/21mPM2%! aρ>M! a!ρMM! a!ρPM! a2(1+ρ)/21 a2!ρ1mPP1%! a(1+ρ)/2! a!ρ! a!! a!ρ1 a(1+ρ)/21mPP2%! aρ>P! a!ρMP! a!ρPP! a2!ρ1 a2!ρ1!

Table(A3.1.1.!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!the!consanguinities!p%mTUv!to!each!of!the!recipient!classes!mTUv%,!among!the!males!competing!for!mates!with!the!focal!female’s!sons.!

fitness!can!be!expressed!relative!to!the!average!of!their!class,!by!WsTU(φ,φ%,µ,µ%,α,β)!=!wsTU(φ,φ%,µ,µ%,α,β)/�wsTU(α,β).!!!

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!!!Gene1fitness!–There!are!6!classes!of!gene,!because!individuals!of!each!of!the!two!female!classes!carry!only!paternal>origin!genes,!whereas!individuals!of!each!of!the!

two!male!classes!carry!separate!maternal>origin!and!paternal>origin!genes.!As!

before,!gene!classes!are!denoted!in!the!form!sTUv,!where!v!∈!{1,2}!according!to!!

! 18!


Recipient(Class(


fMM1! ρM>! ρMM! ρMP! a(1+ρ)/2! a!ρ!fMM2! (1+ρ)/2! 1! ρ! (1+ρ)/2! ρ!fMP1! ρM>! ρMM! ρMP! a(1+ρ)/2! a!ρ!fMP2! (1+ρ)/2! ρ! 1! ρ! (1+ρ)/2!fPM1! ρP>! ρPM! ρPP! a!ρ! a!ρ!fPM2! (1+ρ)/2! 1! ρ! !(1+ρ)/2! ρ!fPP1! ρP>! ρPM! ρPP! a!ρ! a!ρ!fPP2! (1+ρ)/2! ρ! 1! ρ! (1+ρ)/2!mMM1! ρM>! ρMM! ρMP! a(1+ρ)/2! a!ρ!mMM2! (1+ρ)/2! 1! ρ! (1+ρ)/2! ρ!mMP1! ρM>! ρMM! ρMP! a(1+ρ)/2! a!ρ!mMP2! (1+ρ)/2! ρ! 1! ρ! (1+ρ)/2!mPM1! ρP>! ρPM! ρPP! a!ρ! a!ρ!mPM2! (1+ρ)/2! 1! ρ! !(1+ρ)/2! ρ!mPP1! ρP>! ρPM! ρPP! a!ρ! a!ρ!mPP2! (1+ρ)/2! ρ! 1! ρ! (1+ρ)/2!!

Table(A3.1.2.!Consanguinities(for(the(male(PGE(analysis:(autosomal(genes.!The!five!autosomal!actor!classes!are!the!male’s!autosomal!genes!(A),!his!maternal>origin!autosomal!genes!(AMat),!his!

paternal>origin!autosomal!genes!(APat),!his!mother’s!autosomal!genes!(AMot)!and!his!father’s!

autosomal!genes!(AFat).!Shown!here!are!their!consanguinities!qsTUv!to!each!of!the!recipient!classes!sTUv!among!the!male’s!offspring.!

!

whether!the!gene!is!of!maternal!(v!=!1)!or!paternal!(v!=!2)!origin.!The!relative!fitness!

of!a!gene!in!a!female!is!given!by!Wf>U2(φ,φ%,µ,µ%,α,β)!=!Wf>U(φ,φ%,µ,µ%,α,β)!and!the!relative!fitness!of!a!gene!in!a!male!is!given!by!WmPU1(φ,φ%,µ,µ%,α,β)!=!WmPU2(φ,φ%,µ,µ%,α,β)!=!WmPU(φ,φ%,µ,µ%,α,β).!!Reproductive1value!–!As!before,!each!gene’s!class!in!any!generation!may!provide!some!information!and!some!uncertainty!about!that!gene’s!origin!in!the!previous!generation.!For!example,!genes!of!class!f>P2!in!the!present!generation!are!derived!

from!genes!of!class!mPU2!in!the!previous!generation,!where!U!=!M!or!P!with!equal!

probability.!Consequently,!the!f>P2!gene!derived!from!a!mPM2!gene!with!probability!

℘f>P2←mPM2!=!½!and!from!a!mPP2!gene!with!probability℘f>P2←mPP2!=!½.!The!

corresponding!system!of!6!linear!equations!may!be!written!in!linear!algebraic!form!

as!c!=(c•G,!where!c!=!{cf>M2,cf>P2,cmPM1,cmPM2,cmPP1,cmPP2}!is!the!vector!of!class!reproductive!values!and!G!is!the!gene>flow!matrix.!Solving!obtains!csTUv!=!1/6!for!all!gene!classes.!

!

PGE1in1females!–!Following!the!same!procedure!as!before,!the!condition!for!increase!in!female!PGE!is!cf>M2!((dWf>M2/dφ)!×!(dφ/dG)!×!(dG/dgf>M2))!+!cf>P2!((dWf>P2/dφ)!×!(dφ/dG)!×!(dG/dgf>P2))!+!cmPM1!((dWmPM1/dφ)!×!(dφ/dG)!×!(dG/dgmPM1)!+!

! 19!

(dWmPM1/dφ%)!×!(dφ%/dG%)!×!(dG%/dgmPM1))!+!cmPM2!((dWmPM2/dφ)!×!(dφ/dG)!×!(dG/dgmPM2)!+!(dWmPM2/dφ%)!×!(dφ%/dG%)!×!(dG%/dgmPM2))!+!cmPP1!((dWmPP1/dφ)!×!(dφ/dG)!×!(dG/dgmPP1)!+!(dWmPP1/dφ%)!×!(dφ%/dG%)!×!(dG%/dgmPP1))!+!cmPP2!((dWmPP2/dφ)!×!(dφ/dG)!×!(dG/dgmPP2)!+!(dWmPP2/dφ%)!×!(dφ%/dG%)!×!(dG%/dgmPP2))!>!0,!where:!dφ/dG!=!dφ%/dG%!=!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!dG%/dgmTUv!=!p!mTUv!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.!!

!

PGE1in1males!–!Similarly,!the!condition!for!increase!in!male!PGE!is!cf>M2!((dWf>M2/dµ)!×!(dµ/dG)!×!(dG/dgf>M2))!+!cf>P2!((dWf>P2/dµ)!×!(dµ/dG)!×!(dG/dgf>P2))!+!cmPM1!((dWmPM1/dµ)!×!(dφ/dG)!×!(dG/dgmPM1))!+!cmPM2!((dWmPM2/dµ)!×!(dµ/dG)!×!(dG/dgmPM2))!+!cmPP1!((dWmPP1/dµ)!×!(dµ/dG)!×!(dG/dgmPP1))!+!cmPP2!((dWmPP2/dµ)!×!(dµ/dG)!×!(dG/dgmPP2))!>!0,!where!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.!!

Consanguinity!–!The!coefficient!of!inbreeding!for!Z>linked!genes!is!ρ***!=!a((1/2)!×!ρ***!+!(1/2)!×!ψ***).!That!is:!with!probability!a!the!male!and!female!share!the!same!mother;!so!with!probability!½!we!pick!the!paternal>origin!gene!from!the!female!and!the!maternal>origin!gene!from!the!male!and!hence!their!consanguinity!is!simply!the!

consanguinity!of!mating!partners,!ρ***;!and!with!probability!½!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!ψ***.!Note!that!ψ***!=!a((1/4)!×!1!+!(1/2)!×!ρ***!+!(1/4)!×!ψ***).!That!is,!with!probability!a!the!two!males!share!the!same!mother;!so!with!probability!¼!we!pick!the!maternal>origin!genes!from!both!males,!so!their!consanguinity!is!simply!the!mother’s!consanguinity!to!herself,!i.e.!1;!

with!probability!½!we!pick!the!maternal>origin!gene!from!one!male!and!the!

paternal>origin!gene!from!the!other,!in!which!case!their!consanguinity!is!that!

between!mating!partners,!ρ***;!and!with!probability!¼!we!pick!the!paternal>origin!genes!from!both!males,!so!their!consanguinity!is!that!of!two!males!in!the!same!

mating!group,!ψ***.!Simultaneously!solving!both!equations!obtains!ρ***!=!a2/(8>a(6+a))!and!ψ***!=!a(2>a)/(8>a(6+a)).!From!these!coefficients!can!be!defined!other!consanguinities!between!mating!partners:!ρ***>M!=!a1ρ***,!from!the!perspective!of!the!male’s!maternal>origin!gene;!and!ρ***>P!=!a!ψ***,!from!the!perspective!of!the!male’s!paternal>origin!gene.!These!coefficients!define!all!the!consanguinities!needed!to!solve!the!model!(listed!in!Tables!A3.2.1!&!A3.2.2).!

!


!

!!

! 20!

! Actor(Class( ! !

Recipient(Class(

Z! ZMot! ZFat!

f>M2! ρ***>M! a1 a!ρ***!f>P2! ρ***>P! a!ρ***! a!ψ***!mPM1! 1! ρ***! (1+ρ***)/2!mPM2! ρ***>M! a! a!ρ***!mPP1! 1! ρ***! (1+ρ***)/2!mPP2! ρ***>P! a!ρ***! a!ψ***!mPM1%! a1 a!ρ***1 a(1+ρ***)/2!mPM2%! a!ρ***>M! a2! a2!ρ***!mPP1%! a1 a!ρ***1 a(1+ρ***)/2!mPP2%! a!ρ***>P! a2!ρ***1 a2!ψ***!!

Table(A3.2.1.!Consanguinities(for(the(female(PGE(invasion(analysis:(ZNlinked(genes.!The!three!Z>linked!actor!classes!are!the!female’s!Z>linked!genes!(Z),!her!mother’s!Z>linked!genes!(Z!Mot)!and!her!father’s!Z>linked!genes!(ZFat).!Shown!here!are!their!consanguinities!psTUv!to!each!of!the!recipient!classes!sTUv!among!the!focal!female’s!offspring,!and!the!consanguinities!p%mTUv!to!each!of!the!recipient!classes!mTUv%,!among!the!males!competing!for!mates!with!the!focal!female’s!sons.!!

! Actor(Class( ! ! ! !

Recipient(Class(

Z! ZMat! ZPat! ZMot! ZFat!

f>M2! (1+ρ***)/2! 1! ρ***! 1! ρ***!f>P2! (1+ρ***)/2! ρ***! 1! ρ***! (1+ρ***)/2!mPM1! ρ***! ρ***>M! ρ***>P! a1ρ***! a!ψ***!mPM2! (1+ρ***)/2! 1! ρ***! 1! ρ***!mPP1! ρ***! ρ***>M! ρ***>P! a1ρ***! a!ψ***!mPP2! (1+ρ***)/2! ρ***! 1! ρ***! (1+ρ***)/2!!Table(A3.2.2.!Consanguinities(for(the(male(PGE(invasion(analysis:(ZNlinked(genes.!The!five!Z>linked!actor!classes!are!the!male’s!Z>linked!genes!(Z),!his!maternal>origin!Z>linked!genes!(ZMat),!his!

paternal>origin!Z>linked!genes!(ZPat),!his!mother’s!Z>linked!genes!(ZMot)!and!his!father’s!Z>linked!

genes!(ZFat).!Shown!here!are!their!consanguinities!qsTUv!to!each!of!the!recipient!classes!sTUv!among!the!male’s!offspring.!

!3.3.!W>linked!genes!

1Classes!–!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}.!1Survival!–!The!probabilities!of!survival!for!each!class!are!Sf(φ,µ,α,β)!=!(1/2)(1>φ)(1>µ)!+!φ(1>µ)(1>α)!+!(1/2)(1>φ)µ(1>β)!+!φ!µ!(1>α)(1>β)!and!Sm(φ,µ,α,β)!=!(1/2)(1>φ)(1>µ)!+!(1/2)(1>φ)µ(1>β).!

! 21!

!



ws(φ,φ%,µ,µ%,α,β)!=!Ss(φ,µ,α,β)×Rs(φ%,µ%,α,β).!As!before,!because!PGE!is!vanishingly!rare!in!the!population,!the!average!fitness!among!all!individuals!of!a!particular!class!

is!given!by�ws(α,β)!=!ws(0,0,0,0,α,β),!and!each!individual’s!fitness!can!be!expressed!relative!to!the!average!of!their!class,!by!Ws(φ,φ%,µ,µ%,α,β)!=!ws(φ,φ%,µ,µ%,α,β)/�ws(α,β).!!!Gene1fitness!–!There!is!only!one!class!of!W>linked!gene,!because!females!carry!only!one!class!of!W>linked!gene!and!males!carry!no!W>linked!genes.!Accordingly,!the!

relative!fitness!of!a!W>linked!gene!in!a!female!is!simply!Wf(φ,φ%,µ,µ%,α,β).!!Reproductive1value!–!Since!there!is!only!one!class!of!W>linked!gene,!all!reproductive!value!belongs!to!this!class:!cf!=!1.!!!PGE1in1females!–!Following!the!same!procedure!as!before,!the!condition!for!increase!in!female!PGE!is!cf!((dWf/dφ)!×!(dφ/dG)!×!(dG/dgf))!>!0.!Once!again:!dφ/dG!is!an!arbitrary!mapping!between!genic!value!and!phenotypic!value,!and!can!be!set!to!1;!

and!dG/dgf!=!pf!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.!!!

PGE1in1males!–!Similarly,!the!condition!for!increase!in!male!PGE!is!cf!((dWf/dµ)!×!(dµ/dG)!×!(dG/dgf))!>!0,!where!dµ/dG!=!1!and!dG/dgf!=!qf!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.!

!Consanguinity!–!Coefficients!of!inbreeding!are!undefined!for!W>linked!genes,!because!males!do!not!carry!any!such!genes.!Only!three!consanguinities!feature!in!

the!W>linkage!analysis:!between!a!female’s!W>genes!and!her!mother’s!W>genes;!between!a!female’s!W>genes!and!her!maternal!grandmother’s!W>genes;!and!

between!a!female’s!W>genes!and!her!paternal!grandmother’s!W>genes!(listed!in!Tables!A3.3.1!&!A3.3.2).!

!


!

!!

!!

!

! 22!

! Actor(Class( (Recipient(Class(

W! WMot!

f! 1! 1!!

Table(A3.3.1.!Consanguinities(for(the(female(PGE(invasion(analysis:(WNlinked(genes.!The!two!W>linked!actor!classes!are!the!female’s!own!W>linked!genes!(W)!and!the!female’s!mother’s!W>linked!

genes!(WMot).!Shown!here!are!their!consanguinity!pf!to!the!single!recipient!class!f!among!the!focal!female’s!daughters.!

!

! Actor(Class(Recipient(Class(

WMot!

f! a1!

Table(A3.3.2.!Consanguinities(for(the(male(PGE(invasion(analysis:(WNlinked(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.!!4.(MGE(in(females(and/or(males(under(ZW(inheritance:(invasion(analysis((4.1.!Autosomal!genes!!


!

Survival!–!Denoting!the!probability!that!a!zygote’s!mother!exhibited!MGE!by!φ,!the!probability!that!the!zygote’s!father!exhibited!MGE!by!µ,!the!viability!cost!associated!with!female!MGE!by!α!and!the!viability!cost!associated!with!male!MGE!by!β,!the!probability!that!each!potential!zygote!survives!to!maturity!is:!SfMM(φ,µ,α,β)!=!(1/8)(1>φ)(1>µ);!SfMP(φ,µ,α,β)!=!(1/8)(1>φ)(1>µ)!+!(1/4)(1>φ)µ(1>β);!SfPM(φ,µ,α,β)!=!(1/8)(1>φ)(1>µ);!SfPP(φ,µ,α,β)!=!(1/8)(1>φ)(1>µ)!+!(1/4)(1>φ)µ(1>β);!SmMM(φ,µ,α,β)!=!(1/8)(1>φ)(1>µ);!SmMP(φ,µ,α,β)!=!(1/8)(1>φ)(1>µ)!+!(1/4)(1>φ)µ(1>β);!SmPM(φ,µ,α,β)!=!(1/8)(1>φ)(1>µ)!+!(1/2)φ(1>µ)(1>α);!and!SmPP(φ,µ,α,β)!=!(1/8)(1>φ)(1>µ)!+!(1/2)φ(1>µ)(1>α)!+!(1/4)(1>φ)µ(1>β)!+!φ!µ!(1>α)(1>β).!!!Reproductive1success!–!Upon!survival!to!maturity,!every!female!achieves!a!unit!of!expected!reproductive!success,!and!every!male!achieves!(1>z(φ%,µ%,α,β))/z(φ%,µ%,α,β)!units!of!expected!reproductive!success,!where:!z(φ%,µ%,α,β)!=!(∑T∈{M,P},U∈{M,P}!

SmTU(φ%,µ%,α,β))!/!(∑s∈{f,m},T∈{M,P},U∈{M,P}!SsTU(φ%,µ%,α,β))!is!the!sex!ratio!(proportion!male)!in!his!mating!group;!φ%!is!the!level!of!MGE!among!the!females!contributing!offspring!to!the!mating!group;!and!µ%!is!the!level!of!MGE!among!the!fathers!contributing!offspring!to!the!mating!group.!Thus,!contingent!upon!survival,!expected!

! 23!

reproductive!success!is!RfTU(φ%,µ%,α,β)!=!1!for!females!and!RmTU(φ%,µ%,α,β)!=!(1>z(φ%,µ%,α,β))/z(φ%,µ%,α,β)!for!males.!!Fitness1–!Expected!fitness!is!given!by!the!product!of!survival!to!maturity!and!expected!reproductive!success!contingent!upon!surviving!to!maturity,!i.e.!

wsTU(φ,φ%,µ,µ%,α,β)!=!SsTU(φ,µ,α,β)×RsTU(φ%,µ%,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,φ%,µ,µ%,α,β)!=!wsTU(φ,φ%,µ,µ%,α,β)/�wsTU(α,β).!!!Gene1fitness!–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!




WsTU1(φ,φ%,µ,µ%,α,β)!=!WsTU2(φ,φ%,µ,µ%,α,β)!=!WsTU(φ,φ%,µ,µ%,α,β).!!Reproductive1value!–!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!2.1.1,!i.e.!csTUv!=!1/16!for!all!gene!classes.!!

MGE1in1females!–!The!condition!for!increase!in!female!MGE!is!∑!s∈{f,m},T∈{M,P},U∈{M,P},v∈{1,2}!

csTUv!((dWsTUv/dφ)!×!(dφ/dG)!×!(dG/dgsTUv)!+!(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,!and!G%!is!the!average!genic!value!among!those!genes!at!the!same!locus!whose!expression!controls!

MGE!in!the!females!contributing!offspring!to!the!focal!individual’s!mating!group.!

Here:!dφ/dG!=!dφ%/dG%!=!1;!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;!and!dG%/dgsTUv!=!p!sTUv!is!the!consanguinity!between!the!focal!class>sTUv!gene!and!the!genes,!residing!at!the!same!locus,!who!

control!MGE!in!the!females!who!contribute!offspring!to!the!focal!individual’s!mating!

group.!Note!that,!because!dWfTUv/dφ%!=!0!for!all!classes!of!genes!in!females,!we!need!only!calculate!consanguinities!p!mTUv!for!classes!of!genes!in!males.!Again,!the!consanguinities!will!depend!upon!which!class!or!classes!of!genes!control!MGE!in!

females!>>!see!the!Consanguinity!section!below.!!

MGE1in1males!–!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))!>!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.!Here,!dµ/dG!=!1,!and!dG/dgsTUv!=!qsTUv!is!the!consanguinity!between!the!focal!class>sTUv!gene!and!the!genes,!residing!at!the!same!

! 24!

locus,!who!control!MGE!in!the!father!of!the!focal!gene’s!carrier.!Again,!the!

consanguinities!will!depend!upon!which!class!or!classes!of!genes!control!MGE!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!2.1:!the!consanguinity!of!mating!

partners!is!ρ!=!a/(8>7a).!Moreover,!as!before,!we!have!ρM>!=!a((1/2)!×!(1+ρ)/2!+!(1/2)!×!ρ),!ρP>!=!a!ρ,!ρ>M!=!a((1/2)!×!(1+ρ)/2!+!(1/2)!×!ρ),!ρ>P!=!a!ρ,!ρMM!=!a!(1+ρ)/2,!ρMP!=!a!ρ,!ρPM!=!a!ρ!and!ρPP!=!a!ρ.!These!coefficients!define!all!the!consanguinities!needed!to!solve!the!model!(listed!in!Tables!A3.1.1!&!A3.1.2).!!

Potential1for1MGE1–!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.!!

4.2.!Z>linked!genes!!

Classes!–!There!are!4!classes!of!individual,!classified!according!to!their!sex!s!∈!{f,m}!and!the!grandparent!of!origin!of!their!paternal>origin!gene!U!∈!{M,P}.!If!the!individual!is!female,!she!has!no!maternal>origin!Z>linked!genes.!And!if!the!individual!is!male,!then!all!of!his!maternal>origin!Z>linked!genes!are!derived!from!his!maternal!


!

Survival!–!The!probabilities!of!survival!for!each!class!are:!Sf>M(φ,µ,α,β)!=!(1/4)(1>φ)(1>µ);!Sf>P(φ,µ,α,β)!=!(1/4)(1>φ)(1>µ)!+!(1/2)(1>φ)µ(1>β);!SmPM(φ,µ,α,β)!=!(1/4)(1>φ)(1>µ)!+!(1/2)φ(1>µ)(1>α);!and!SmPP(φ,µ,α,β)!=!(1/4)(1>φ)(1>µ)!+!(1/2)φ(1>µ)(1>α)!+!(1/2)(1>φ)µ(1>β)!+!φµ(1>α)(1>β).!!!

Reproductive1success!–!Contingent!upon!survival,!expected!reproductive!success!is!RfTU(φ%,µ%,α,β)!=!1!for!females!and!RmTU(φ%,µ%,α,β)!=!(1>z(φ%,µ%,α,β))/z(φ%,µ%,α,β)!for!males,!where!z(φ%,µ%,α,β)!=!(SmPM(φ%,µ%,α,β)!+1SmPP(φ%,µ%,α,β))/(Sf>M(φ%,µ%,α,β)!+1Sf>P!(φ%,µ%,α,β)!+1SmPM(φ%,µ%,α,β)!+1SmPP(φ%,µ%,α,β)).!!


wsTU(φ,φ%,µ,µ%,α,β)!=!SsTU(φ,µ,α,β)×RsTU(φ%,µ%,α,β).!As!before,!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,φ%,µ,µ%,α,β)!=!wsTU(φ,φ%,µ,µ%,α,β)/�wsTU(α,β).!!!

Gene1fitness!–There!are!6!classes!of!gene,!because!individuals!of!each!of!the!two!female!classes!carry!only!paternal>origin!genes,!whereas!individuals!of!each!of!the!two!male!classes!carry!separate!maternal>origin!and!paternal>origin!genes.!As!

! 25!


of!a!gene!in!a!female!is!given!by!Wf>U2(φ,φ%,µ,µ%,α,β)!=!Wf>U(φ,φ%,µ,µ%,α,β)!and!the!relative!fitness!of!a!gene!in!a!male!is!given!by!WmPU1(φ,φ%,µ,µ%,α,β)!=!WmPU2(φ,φ%,µ,µ%,α,β)!=!WmPU(φ,φ%,µ,µ%,α,β).!!

Reproductive1value!–!The!flow!of!Z>linked!genes!between!classes!is!exactly!the!same!as!in!section!3.2.!Accordingly,!csTUv!=!1/6!for!all!gene!classes.!!

MGE1in1females!–!Following!the!same!procedure!as!before,!the!condition!for!increase!in!female!MGE!is!cf>M2!((dWf>M2/dφ)!×!(dφ/dG)!×!(dG/dgf>M2))!+!cf>P2!((dWf>P2/dφ)!×!(dφ/dG)!×!(dG/dgf>P2))!+!cmPM1!((dWmPM1/dφ)!×!(dφ/dG)!×!(dG/dgmPM1)!+!(dWmPM1/dφ%)!×!(dφ%/dG%)!×!(dG%/dgmPM1))!+!cmPM2!((dWmPM2/dφ)!×!(dφ/dG)!×!(dG/dgmPM2)!+!(dWmPM2/dφ%)!×!(dφ%/dG%)!×!(dG%/dgmPM2))!+!cmPP1!((dWmPP1/dφ)!×!(dφ/dG)!×!(dG/dgmPP1)!+!(dWmPP1/dφ%)!×!(dφ%/dG%)!×!(dG%/dgmPP1))!+!cmPP2!((dWmPP2/dφ)!×!(dφ/dG)!×!(dG/dgmPP2)!+!(dWmPP2/dφ%)!×!(dφ%/dG%)!×!(dG%/dgmPP2))!>!0,!where:!dφ/dG!=!dφ%/dG%!=!1;!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;!and!dG%/dgmTUv!=!p!mTUv!is!the!consanguinity!between!the!focal!class>sTUv!gene!and!the!genes,!residing!at!the!same!locus,!who!control!MGE!in!the!females!who!contribute!offspring!to!the!focal!individual’s!mating!

group.!!

!MGE1in1males!–!Similarly,!the!condition!for!increase!in!male!MGE!is!cf>M2!((dWf>

M2/dµ)!×!(dµ/dG)!×!(dG/dgf>M2))!+!cf>P2!((dWf>P2/dµ)!×!(dµ/dG)!×!(dG/dgf>P2))!+!cmPM1!((dWmPM1/dµ)!×!(dφ/dG)!×!(dG/dgmPM1))!+!cmPM2!((dWmPM2/dµ)!×!(dµ/dG)!×!(dG/dgmPM2))!+!cmPP1!((dWmPP1/dµ)!×!(dµ/dG)!×!(dG/dgmPP1))!+!cmPP2!((dWmPP2/dµ)!×!(dµ/dG)!×!(dG/dgmPP2))!>!0,!where!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!MGE!in!the!father!of!the!focal!gene’s!carrier.!

!Consanguinity!–!The!coefficients!of!consanguinity!are!exactly!the!same!as!described!in!section!3.2!(listed!in!Tables!A3.2.1!&!A3.2.2).!

!Potential1for1MGE!–!The!potentials!for!female!and!male!MGE!are!calculated!as!described!above.!These!are!illustrated!in!Figure!3!of!the!main!text.!!

4.3.!W>linked!genes!

1Classes!–!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}.!!

! 26!

Survival!–!The!probabilities!of!survival!for!each!class!are!Sf(φ,µ,α,β)!=!(1/2)(1>φ)(1>µ)!+!(1/2)(1>φ)µ(1>β)!and!Sm(φ,µ,α,β)!=!(1/2)(1>φ)(1>µ)!+!φ(1>µ)(1>α)!+!(1/2)(1>φ)µ(1>β)!+!φµ(1>α)(1>β).!!



ws(φ,φ%,µ,µ%,α,β)!=!Ss(φ,µ,α,β)×Rs(φ%,µ%,α,β).!As!before,!because!MGE!is!vanishingly!rare!in!the!population,!the!average!fitness!among!all!individuals!of!a!particular!class!

is!given!by�ws(α,β)!=!ws(0,0,0,0,α,β),!and!each!individual’s!fitness!can!be!expressed!relative!to!the!average!of!their!class,!by!Ws(φ,φ%,µ,µ%,α,β)!=!ws(φ,φ%,µ,µ%,α,β)/�ws(α,β).!!!Gene1fitness!–!There!is!only!one!class!of!W>linked!gene,!because!females!carry!only!one!class!of!W>linked!gene!and!males!carry!no!W>linked!genes.!Accordingly,!the!

relative!fitness!of!a!W>linked!gene!in!a!female!is!simply!Wf(φ,φ%,µ,µ%,α,β).!!

Reproductive1value!–!Since!there!is!only!one!class!of!W>linked!gene,!all!reproductive!value!belongs!to!this!class:!cf!=!1.!!!MGE1in1females!–!Following!the!same!procedure!as!before,!the!condition!for!increase!in!female!MGE!is!cf!((dWf/dφ)!×!(dφ/dG)!×!(dG/dgf))!>!0.!Once!again:!dφ/dG!is!an!arbitrary!mapping!between!genic!value!and!phenotypic!value,!and!can!be!set!to!1;!

and!dG/dgf!=!pf!is!the!consanguinity!between!the!focal!gene!and!the!genes,!residing!at!the!same!locus,!who!control!MGE!in!the!mother!of!the!focal!gene’s!carrier.!!!

MGE1in1males!–!Similarly,!the!condition!for!increase!in!male!MGE!is!cf!((dWf/dµ)!×!(dµ/dG)!×!(dG/dgf))!>!0,!where!dµ/dG!=!1!and!dG/dgf!=!qf!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.!

!

Consanguinity!–!The!consanguinities!are!exactly!the!same!as!those!derived!in!section!3.3!(listed!in!Tables!A3.3.1!&!A3.3.2).!

!Potential1for1MGE!–!The!potentials!for!female!and!male!MGE!are!calculated!as!described!above.!These!are!listed!in!illustrated!in!Figure!3!of!the!main!text.!

(5.(PGE(in(males(under(XY/XO(inheritance:(equilibrium(analysis((5.1.!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!

! 27!

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!–!The!zygote!survival!functions!are!obtained!by!substituting!φ!=!β!=!0!into!those!of!section!1.1.!This!yields!SfMM(µ)!=!(1/8)(1>µ)!+!(1/2)µ;!SfMP(µ)!=!(1/8)(1>µ);!SfPM(µ)!=!(1/8)(1>µ)!+!(1/2)µ;!SfPP(µ)!=!(1/8)(1>µ);!SmMM(µ)!=!(1/8)(1>µ);!SmMP(µ)!=!(1/8)(1>µ);!SmPM(µ)!=!(1/8)(1>µ);!and!SmPP(µ)!=!(1/8)(1>µ).!!!

Reproductive1success!–!Upon!survival!to!maturity,!every!female!achieves!a!unit!of!expected!reproductive!success,!and!every!male!achieves!(1>z(µ%))/z(µ%)!units!of!expected!reproductive!success,!where:!z(µ%)!=!(∑T∈{M,P},U∈{M,P}!SmTU(µ%))!/!(∑s∈{f,m},T∈{M,P},U∈{M,P}!SsTU(µ%))!is!the!sex!ratio!(proportion!male)!in!his!mating!group!and!µ%!is!the!level!of!PGE!among!the!fathers!contributing!offspring!to!the!mating!group.!Thus,!contingent!upon!survival,!expected!reproductive!success!is!RfTU(µ%)!=!1!for!females!and!RmTU(µ%)!=!(1>z(µ%))/z(µ%)!for!males.!!Fitness1–!Expected!fitness!is!given!by!the!product!of!survival!to!maturity!and!expected!reproductive!success!contingent!upon!surviving!to!maturity,!i.e.!wsTU(µ,µ%)!=!SsTU(µ)×RsTU(µ%).!The!average!fitness!among!all!individuals!of!a!particular!class!is!given!by�wsTU(�µ)!=!wsTU(�µ,�µ),!and!each!individual’s!fitness!can!be!expressed!relative!to!the!average!of!their!class,!by!WsTU(µ,µ%,�µ)!=!wsTU(µ,µ%)/�wsTU(�µ).!!!


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!


relative!fitness!of!a!gene!is!equal!to!the!relative!fitness!of!its!carrier:!WsTU1(µ,µ%,�µ)!=!WsTU2(µ,µ%,�µ)!=!WsTU(µ,µ%,�µ).!!


information!and!some!uncertainty!about!that!gene’s!origin!in!the!previous!

generation.!For!example,!genes!of!class!fMM1!in!the!present!generation!are!derived!from!genes!of!class!fTU1!in!the!previous!generation,!where!T!=!M!or!P!and!U!=!M!or!

P.!The!effects!of!PGE!in!a!male!are!accounted!for!in!the!survival!of!his!potential!

offspring:!for!example,!his!potential!male!offspring!are!eliminated!in!the!event!that!he!undergoes!PGE.!So!the!class!fTU1!donating!genes!to!the!class!fMM1!involves!T!=!

M!or!P!with!equal!probability,!because!there!is!no!bias!regarding!the!source!of!their!maternal!genes,!but!U!=!M!or!P!with!unequal!probability,!because!there!is!possible!

bias!regarding!the!source!of!their!paternal!genes.!Specifically,!U!=!M!with!probability!

�wfTM1/(1�wfTM1!+1�wfTP1)!=!(1+3�µ)/(2+2�µ),!and!U!=!P!with!probability!�wfTP1/(1�wfTM1!+1�wfTP1)!=!(1>�µ)/(2+2�µ).!Accordingly,!℘fMM1←fMM1!=!

(1+3�µ)/(4+4�µ),!℘fMM1←fPM1!=!(1+3�µ)/(4+4�µ),!℘fMM1←fMP1!=!(1>�µ)/(4+4�µ)!!

! 28!

!and!℘fMM1←fPP1!=!(1>�µ)/(4+4�µ).!Note!that,!making!the!substitution!�µ!=!0,!all!of!these!gene>flow!coefficients!are!equal!to!¼,!as!in!section!1.1.!Also!note!that,!if!the!

donor!class!involves!genes!in!males,!then!there!is!no!bias,!because!no!male!is!the!

product!of!a!PGE!father!(all!PGE!events!lead!to!the!production!of!daughters).!This!

defines!a!system!of!16!linear!equations,!which!can!be!written!as!c!=(c•G,!where!c!=!{cfMM1,cfMM2,cfMP1,…,cmPP2}!is!the!vector!of!class!reproductive!values!and!G!is!the!gene>flow!matrix.!Solving!(i.e.!finding!the!left!eigenvector!of!G!corresponding!to!the!eigenvalue!1)!obtains!cfTMv!=!(1+3�µ)/(16+8�µ),!cfTPv!=!(1>�µ)/(16+8�µ),!cmTU1!=!(1+�µ)/(16+8�µ)!and!cmTM2!=!(1>�µ)/(16+8�µ).!In!the!absence!of!male!PGE!(�µ!=!0),!all!gene!classes!have!reproductive!value!csTUv!=!1/16;!that!is,!a!gene!chosen!at!random!from!the!distant!future!has!an!equal!chance!of!tracing!back!to!any!of!the!16!

gene!classes!in!the!present!generation.!In!the!extreme!of!full!male!PGE!(�µ!=!1),1cfTMv!=!1/6,!cfTPv!=!0,!cmTU1!=!1/12!and!cmTM2!=!0;!that!is,!two!third!of!the!reproductive!value!of!the!population!belongs!to!genes!in!females!and!only!one!third!belongs!to!genes!in!males,!just!as!under!haplodiploidy.!

!

PGE1in1males!–!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)!+!(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,!and!G%!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!=!dµ%/dG%!=!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!dG%/dgsTUv!=!q!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/dµ%!=!0!for!all!classes!of!genes!in!females,!we!need!only!calculate!consanguinities!q!mTUv!for!classes!of!genes!in!males.!The!consanguinities!will!depend!upon!which!class!or!classes!of!genes!control!PGE!in!males!>>!see!the!Consanguinity!section!below,!and!Table!A5.1.1,!for!details.!

!

Consanguinity!–!The!coefficient!of!inbreeding!is!ρ!=!a((1/4)×((1/2)!+!(1/2)×(θ!ρ>M!+!(1>θ)ρ))!+!(1/4)×(θ!ρ>M!+!(1>θ)ρ)!+!(1/4)!×!ρ!+!(1/4)!×!(θ!ψM>!+!(1>θ)ψ)),!where!θ!=!2�µ/(1+�µ)!is!the!probability!that!a!focal!adult!female!derived!from!an!egg!that!was!fertilized!by!a!PGE!sperm.!Note!that!whilst!the!consanguinity!of!mating!partners!is!ρ,!the!consanguinity!of!the!genes!that!they!transmit!to!their!offspring!is!θ!ρ>M!+!(1>θ)ρ,!because!with!probability!θ!the!male!exhibits!PGE!and!so!only!transmits!his!maternal>origin!genes,!and!with!probability!1>θ!the!male!does!not!exhibit!PGE!and!hence!transmits!maternal>origin!and!paternal>origin!genes!with!equal!probability.!

The!conditional!consanguinities!of!mating!partners!are:!ρ>M!=!a((1/2)((1/2)+(1/2)(θ!ρ>M!+!(1>θ)ρ))!+(1/2)(θ!ρ>M!+!(1>θ)ρ)),!from!the!perspective!of!the!male’s!maternal>origin!gene;!ρM>!=!a((1/2)((1/2)+(1/2)(θ!ρ>M!+!(1>θ)ρ))!+(1/2)ρ),!from!the!perspective!of!the!female’s!maternal>origin!gene;!ρ>P!=!a((1/2)ρ!+(1/2)(θ!ψM>!+!(1>

! 29!

θ)ψ)),!from!the!perspective!of!the!male’s!paternal>origin!gene;!ρP>!=!a((1/2)(θ!ρ>M!+!(1>θ)ρ)!+!(1/2)(θ!ψM>!+!(1>θ)ψ)),!from!the!perspective!of!the!female’s!paternal>origin!gene;!ρMM!=!a((1/2)!+!(1/2)(θ!ρ>M!+!(1>θ)ρ)),!from!the!perspective!of!both!mating!partners’!maternal>origin!genes;!ρMP!=!aρ,!from!the!perspective!of!the!female’s!maternal>origin!gene!and!the!male’s!paternal>origin!gene;!ρPM!=!a(θ!ρ>M!+!(1>θ)ρ),!from!the!perspective!of!the!female’s!paternal>origin!gene!and!the!male’s!maternal>origin!gene;!and!ρPP!=!a(θ!ψM>!+!(1>θ)ψ),!from!the!perspective!of!both!mating!partners’!paternal>origin!genes.!The!consanguinity!of!two!males!in!the!same!

mating!group!is!ψ!=!a((1/4)((1/2)!+!(1/2)×(θ!ρ>M!+!(1>θ)ρ))!+!(1/2)ρ!+!(1/4)ψ),!and!the!conditional!consanguinities!between!males!are:!ψM>!=!a((1/2)((1/2)!+!(1/2)×(θ!ρ>M!+!(1>θ)ρ))!+!(1/2)ρ),!from!the!perspective!of!one!male’s!maternal>origin!gene;!ψP>!=!a((1/2)ρ!+!(1/2)ψ),!from!the!perspective!of!one!male’s!paternal>origin!gene;!ψMM!=!a((1/2)!+!(1/2)(θ!ρ>M!+!(1>θ)ρ)),!from!the!perspective!of!both!males’!maternal>origin!genes;!ψMP!=!aρ,!from!the!perspective!of!one!male’s!maternal>origin!gene!and!the!other’s!paternal>origin!gene;!and!ψPP!=!aψ,!from!the!perspective!of!both!males’!paternal>origin!genes.!The!consanguinity!of!two!females!in!the!same!

mating!group!is!ζ!=!a((1/4)((1/2)!+!(1/2)(θ!ρ>M!+!(1>θ)ρ))!+!(1/2)(θ!ρ>M!+!(1>θ)ρ)!!+!(1/4)(θ2!ψMM!+!2θ(1>θ)ψM>!+!(1>θ)2ψ)),!and!the!conditional!consanguinities!between!females!are:!ζM>!=!a((1/2)((1/2)!+!(1/2)(θ!ρ>M!+!(1>θ)ρ))!+!(1/2)(θ!ρ>M!+!(1>θ)ρ)),!from!the!perspective!of!one!female’s!maternal>origin!gene;!ζP>!=!a((1/2)(θ!ρ>M!+!(1>θ)ρ)!+!(1/2)!(θ2!ψMM!+!2θ(1>θ)ψM>!+!(1>θ)2ψ)),!from!the!perspective!of!one!female’s!paternal>origin!gene;!ζMM!=!a((1/2)!+!(1/2)(θ!ρ>M!+!(1>θ)ρ)),!from!the!perspective!of!both!females’!maternal>origin!genes;!ζMP!=!a(θ!ρ>M!+!(1>θ)ρ),!from!the!perspective!of!one!female’s!maternal>origin!gene!and!the!other’s!paternal>origin!gene;!and!ζPP!=!a(θ2!ψMM!+!2θ(1>θ)ψM>!+!(1>θ)2ψ),!from!the!perspective!of!both!females’!paternal>origin!genes.!Simultaneously!solving!these!equations!yields!ρ!=!a(2+(2+a(2>a))�µ)/(16(1+�µ)!>!a(14+(18>a(2+3a))�µ)),!and!each!of!the!other!consanguinities.!And!these!in!turn!define!all!the!consanguinities!needed!to!solve!the!model!(listed!in!Table!A5.1.1).!

!!

Equilibrium1level1of1PGE!–!If!the!condition!for!increase!in!PGE!is!not!satisfied!at!�µ!=!0,!then!PGE!cannot!invade!the!population!from!rarity,!and!its!equilibrium!level!is!�µ!=!g*!=!0.!If!the!condition!for!increase!in!PGE!is!satisfied!at!�µ!=!1,!then!non>PGE!cannot!invade!from!rarity,!and!hence!the!equilibrium!level!of!PGE!is!g*!=!1.!Setting!the!LHS!of!the!condition!equal!to!zero,!and!solving!for!�µ!=!g*,!obtains!the!intermediate!equilibrium!level!of!PGE.!The!equilibrium!level!of!PGE!is!illustrated!in!Figure!4!of!the!main!text.!

!5.2.!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!!

!

! 30!


Recipient(Class(


fMM1! ρM>! ρMM! ρMP! a1γ! a!ρ!fMM2! (1+ρ)/2! 1! ρ! (1+γ)/2! ρ!fMP1! ρM>! ρMM! ρMP! a(1+γ)/2! a!ρ!fMP2! (1+ρ)/2! ρ! 1! ρ! (1+ρ)/2!fPM1! ρP>! ρPM! ρPP! a!γ! a!η!fPM2! (1+ρ)/2! 1! ρ! !(1+γ)/2! ρ!fPP1! ρP>! ρPM! ρPP! a!γ! a!η!fPP2! (1+ρ)/2! ρ! 1! ρ! (1+ρ)/2!mMM1! ρM>! ρMM! ρMP! a(1+γ)/2! a!ρ!mMM2! (1+ρ)/2! 1! ρ! (1+γ)/2! ρ!mMP1! ρM>! ρMM! ρMP! a(1+γ)/2! a!ρ!mMP2! (1+ρ)/2! ρ! 1! ρ! (1+ρ)/2!mPM1! ρP>! ρPM! ρPP! a!γ! a!η!mPM2! (1+ρ)/2! 1! ρ! !(1+γ)/2! ρ!mPP1! ρP>! ρPM! ρPP! a!γ! a!η!mPP2! (1+ρ)/2! ρ! 1! ρ! (1+ρ)/2!mMM1%! a!ρM>! a!ρMM! a!ρMP! a2(1+γ)/2! a2!ρ!mMM2%! a!ψM>! a!ψMM! a!ψMP! a!ψMM! a!ψMP!mMP1%! a!ρM>! a!ρMM! a!ρMP! a2(1+γ)/2! a2!ρ!mMP2%! a!ψP>! a!ψPM! a!ψPP! a!ψMP! a!ψPP!mPM1%! a!ρP>! a!ρPM! a!ρPP! a!ρPM! a2!η!mPM2%! a!ψM>! a!ψMM! a!ψMP! a!ψMM! a!ψMP!mPP1%! a!ρP>! a!ρPM! a!ρPP! a!ρPM! a2!η!mPP2%! a!ψP>! a!ψPM! a!ψPP! a!ψMP! a!ψPP!!

Table(A5.1.1.!Consanguinities(for(the(male(PGE(equilibrium(analysis:(autosomal(genes.!The!five!autosomal!actor!classes!are!the!male’s!autosomal!genes!(A),!his!maternal>origin!autosomal!genes!


father’s!autosomal!genes!(AFat).!Shown!here!are!their!consanguinities!qsTUv!to!each!of!the!recipient!classes!sTUv!among!the!male’s!offspring,!and!the!consanguinities!q%mTUv!to!each!of!the!recipient!classes!mTUv%,!among!the!males!competing!for!mates!with!the!focal!male’s!sons.!Here,!γ!=!(θ!ρ>M!+!(1>θ)ρ)!and!η!=!(θ!ψ>M!+!(1>θ)ψ).!!!her!paternal!grandmother.!And!if!the!individual!is!male,!he!has!no!paternal>origin!X>!


!

Survival!–!The!probabilities!of!survival!for!each!class!are:!SfMM(φ,µ,α,β)!=!(1/4)(1>µ)!+!(1/2)µ;!SfPM(φ,µ,α,β)!=!(1/4)(1>µ)!+!(1/2)µ;!SmM>(φ,µ,α,β)!=!(1/4)(1>µ);!and!SmP>(φ,µ,α,β)!=!(1/4)(1>µ).!

! 31!

Reproductive1success!–!Contingent!upon!survival,!expected!reproductive!success!is!RfTU(µ%)!=!1!for!females!and!RmTU(µ%)!=!(1>z(µ%))/z(µ%)!for!males,!where!z(µ%)!=!(SmM>(µ%)!+1SmP>(µ%))/(SfMM(µ%)!+1SfPM(µ%)!+1SmM>(µ%)!+1SmP>(µ%)).!!

Fitness1–!Expected!fitness!is!given!by!the!product!of!survival!to!maturity!and!expected!reproductive!success!contingent!upon!surviving!to!maturity,!i.e.!wsTU(µ,µ%)!=!SsTU(µ)×RsTU(µ%).!Average!fitness!among!all!individuals!of!a!particular!class!is!given!by�wsTU(�µ)!=!wsTU(�µ,�µ),!and!each!individual’s!fitness!can!be!expressed!relative!to!the!average!of!their!class,!by!WsTU(µ,µ%,�µ)!=!wsTU(µ,µ%)/�wsTU(�µ).!!!

Gene1fitness!–There!are!6!classes!of!gene,!because!individuals!of!each!of!the!two!female!classes!carry!separate!maternal>origin!and!paternal>origin!genes,!whereas!



of!a!gene!in!a!female!is!given!by!WfTP1(µ,µ%,�µ)!=!WfTP2(µ,µ%,�µ)!=!WfTP(µ,µ%,�µ)!and!the!relative!fitness!of!a!gene!in!a!male!is!given!by!WmT>1(µ,µ%,�µ)!=!WmT>(µ,µ%,�µ).!!

Reproductive1value!–!PGE!in!males!does!not!impact!upon!the!gene>flow!coefficients!for!X>linked!genes,!because!males!can!only!transmit!maternal>origin!X>linked!genes!anway.!Accordingly,!all!of!the!non>zero!gene>flow!coefficients!are!equal!to!½,!and!

reproductive!value!is!csTUv!=!1/6!for!all!gene!classes,!as!in!section!A1.2.!!!

PGE1in1males!–!As!before,!the!condition!for!increase!in!male!PGE!is!cfMM1!((dWfMM1/dµ)!×!(dµ/dG)!×!(dG/dgfMM1))!+!cfMM2!((dWfMM2/dµ)!×!(dµ/dG)!×!(dG/dgfMM2))!+!cfPM1!((dWfPM1/dµ)!×!(dµ/dG)!×!(dG/dgfPM1))!+!cfPM2!((dWfPM2/dµ)!×!(dµ/dG)!×!(dG/dgfPM2))!+!cmM>1!((dWmM>1/dµ)!×!(dµ/dG)!×!(dG/dgmM>1)!+!(dWmM>

1/dµ%)!×!(dµ%/dG%)!×!(dG%/dgmM>1))!+!cmP>1!((dWmP>1/dµ)!×!(dµ/dG)!×!(dG/dgmP>1)!+!(dWmP>1/dµ%)!×!(dµ%/dG%)!×!(dG%/dgmP>1))!>!0,!where:!dµ/dG!=!dµ%/dG%!=!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!dG%/dgmTUv!=!q!mTUv!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!


Consanguinity!–!Again,!PGE!in!males!does!not!impact!upon!the!transmission!of!X>linked!genes,!because!males!can!only!transmit!maternal>origin!X>linked!genes!anway.!Accordingly,!the!coefficients!of!consanguinity!are!exactly!the!same!as!those!

given!in!Table!A1.2.2.!!

Equilibrium1level1of1PGE!–!If!the!condition!for!increase!in!PGE!is!not!satisfied!at!�µ!=!0,!then!PGE!cannot!invade!the!population!from!rarity,!and!its!equilibrium!level!is!g*!=!0.!If!the!condition!for!increase!in!PGE!is!satisfied!at!�µ!=!1,!then!non>PGE!cannot!invade!from!rarity,!and!hence!the!equilibrium!level!of!PGE!is!g*!=!1.!Setting!the!LHS!of!the!condition!equal!to!zero,!and!solving!for!�µ!=!g*,!obtains!the!intermediate!

! 32!

equilibrium!level!of!PGE.!The!equilibrium!level!of!PGE!is!illustrated!in!Figure!4!of!the!

main!text.!!

5.3.!Y>linked!genes!1Classes!–!There!are!2!classes!of!individual,!classified!according!to!their!sex!s!∈!{f,m}.!If!the!individual!is!female,!then!she!has!no!Y>linked!genes.!And!if!the!individual!is!



Survival!–!The!probabilities!of!survival!for!each!class!are!Sf(µ)!=!(1/2)(1>µ)!+!µ!and!Sm(µ)!=!(1/2)(1>µ).!!

Reproductive1success!–!Contingent!upon!survival,!expected!reproductive!success!is!Rf(µ%)!=!1!for!females!and!Rm(µ%)!=!(1>z(µ%))/z(µ%)!for!males,!where!z(µ%)!=!Sm(µ%)/(Sf(µ%)1+1Sm(µ%)).!!

Fitness1–!Expected!fitness!is!given!by!the!product!of!survival!to!maturity!and!expected!reproductive!success!contingent!upon!surviving!to!maturity,!i.e.!ws(µ,µ%)!=!Ss(µ)×Rs(µ%).!The!average!fitness!among!all!individuals!of!a!particular!class!is!given!by�ws(�µ)!=!ws(�µ,�µ),!and!each!individual’s!fitness!can!be!expressed!relative!to!the!average!of!their!class,!by!Ws(µ,µ%,�µ)!=!ws(µ,µ%,�µ)/�ws(�µ,�µ).!!!


fitness!of!a!Y>linked!gene!in!a!male!is!simply!Wm(µ,µ%,�µ).!!

Reproductive1value!–!Since!there!is!only!one!class!of!Y>linked!gene,!all!reproductive!value!belongs!to!this!class:!cm!=!1.!!!

PGE1in1males!–!As!before,!the!condition!for!increase!in!male!PGE!is!cm!((dWm/dµ)!×!(dµ/dG)!×!(dG/dgm)!+!(dWm/dµ%)!×!(dµ%/dG%)!×!(dG%/dgm))!>!0,!where:!dµ/dG!=!dµ%/dG%!=!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!dG%/dgm!=!q!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!–!PGE!in!males!does!not!impact!upon!consanguinity!of!Y>linked!genes,!which!are!exactly!the!same!as!those!listed!in!Table!A1.3.2.!

!

Equilibrium1level1of1PGE!–!If!the!condition!for!increase!in!PGE!is!not!satisfied!at!�µ!=!0,!then!PGE!cannot!invade!the!population!from!rarity,!and!its!equilibrium!level!is!g*!=!0.!If!the!condition!for!increase!in!PGE!is!satisfied!at!�µ!=!1,!then!non>PGE!cannot!invade!from!rarity,!and!hence!the!equilibrium!level!of!PGE!is!g*!=!1.!Setting!the!LHS!

! 33!

of!the!condition!equal!to!zero,!and!solving!for!�µ!=!g*,!obtains!the!intermediate!equilibrium!level!of!PGE.!The!equilibrium!level!of!PGE!is!illustrated!in!Figure!4!of!the!

main!text.!

!6.(PGE(in(females(under(ZW(inheritance:(equilibrium(analysis((6.1.!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!–!The!zygote!survival!functions!are!obtained!by!substituting!µ!=!α!=!0!into!those!of!section!3.1.!This!yields:!SfMM(φ)!=!(1/8)(1>φ)!+!(1/2)φ;!SfMP(φ)!=!(1/8)(1>φ)!+!(1/2)φ;!SfPM(φ)!=!(1/8)(1>φ);!SfPP(φ)!=!(1/8)(1>φ);!SmMM(φ)!=!(1/8)(1>φ);!SmMP(φ)!=!(1/8)(1>φ);!SmPM(φ)!=!(1/8)(1>φ);!and!SmPP(φ)!=!(1/8)(1>φ).!!!Reproductive1success!–!Upon!survival!to!maturity,!every!female!achieves!a!unit!of!expected!reproductive!success,!and!every!male!achieves!(1>z(φ%))/z(φ%)!units!of!expected!reproductive!success,!where:!z(φ%)!=!(∑T∈{M,P},U∈{M,P}!SmTU(φ%))!/!(∑s∈{f,m},T∈{M,P},U∈{M,P}!SsTU(φ%))!is!the!sex!ratio!(proportion!male)!in!his!mating!group.!Thus,!contingent!upon!survival,!expected!reproductive!success!is!RfTU(φ%)!=!1!for!females!and!RmTU(φ%)!=!(1>z(φ%))/z(φ%)!for!males.!!Fitness1–!Expected!fitness!is!given!by!the!product!of!survival!to!maturity!and!expected!reproductive!success!contingent!upon!surviving!to!maturity,!i.e.!wsTU(φ,φ%)!=!SsTU(φ)×RsTU(φ%).!The!average!fitness!among!all!individuals!of!a!particular!class!is!given!by�wsTU(�φ)!=!wsTU(�φ,�φ),!where!�φ!is!the!average!level!of!female!PGE!in!the!population.!Each!individual’s!fitness!can!be!expressed!relative!to!the!average!of!their!

class,!by!WsTU(φ,φ%,�φ)!=!wsTU(φ,φ%)/�wsTU(�φ).!!!



its!probability!of!being!transmitted!to!a!potential!(rather!than!a!realised)!zygote,!is!


relative!fitness!of!a!gene!is!equal!to!the!relative!fitness!of!its!carrier:!WsTU1(φ,φ%,�φ)!=!WsTU2(φ,φ%,�φ)!=!WsTU(φ,φ%,�φ).!!


information!and!some!uncertainty!about!that!gene’s!origin!in!the!previous!generation.!For!example,!genes!of!class!fMM1!in!the!present!generation!are!derived!

! 34!

from!genes!of!class!fTU1!in!the!previous!generation,!where!T!=!M!or!P!and!U!=!M!or!

P.!The!effects!of!PGE!in!a!female!are!accounted!for!in!the!survival!of!her!potential!offspring:!for!example,!her!potential!male!offspring!are!eliminated!in!the!event!that!

she!undergoes!PGE.!So!the!class!fTU1!donating!genes!to!the!class!fMM1!involves!T!=!M!or!P!with!unequal!probability,!because!there!is!possible!bias!regarding!the!source!

of!the!latter!class’s!maternal>origin!genes,!and!U!=!M!or!P!with!equal!probability,!

because!there!is!no!bias!regarding!the!source!of!their!paternal!genes.!Specifically,!T!

=!M!with!probability!�wfMU1/(1�wfMU1!+1�wfPU1)!=!(1+3�φ)/(2+2�φ),!and!T!=!P!with!probability!�wfPU1/(1�wfMU1!+1�wfPU1)!=!(1>�φ)/(2+2�φ).!Accordingly,!℘fMM1←fMM1!=!

(1+3�φ)/(4+4�φ),!℘fMM1←fMP1!=!(1+3�φ)/(4+4�φ),!℘fMM1←fPM1!=!(1>�φ)/(4+4�φ)!!!and!℘fMM1←fPP1!=!(1>�φ)/(4+4�φ).!Note!that,!making!the!substitution!�φ!=!0,!all!of!these!gene>flow!coefficients!are!equal!to!¼,!as!in!section!3.1.!Also!note!that,!if!the!donor!class!involves!genes!in!males,!then!there!is!no!bias,!because!no!male!is!the!

product!of!a!PGE!mother!(all!PGE!events!lead!to!the!production!of!daughters).!This!

defines!a!system!of!16!linear!equations,!which!can!be!written!as!c!=(c•G,!where!c!=!{cfMM1,cfMM2,cfMP1,…,cmPP2}!is!the!vector!of!class!reproductive!values!and!G!is!the!gene>flow!matrix.!Solving!(i.e.!finding!the!left!eigenvector!of!G!corresponding!to!the!eigenvalue!1)!obtains!cfMU1!=!(1+3�φ)/(16>8�φ),!cfMU2!=!((1>�φ)(1+3�φ))/(8(2>�φ)(1+�φ)),!cfPU1!=!(1>�φ)/(8(2>�φ)),!cfPU2!=!(1>�φ)2/(8(2>�φ)(1+�φ))!and!cmTUv!=!(1>�φ)/(8(2>�φ)).!In!the!absence!of!female!PGE!(�φ!=!0),!all!gene!classes!have!reproductive!value!csTUv!=!1/16.!In!the!extreme!of!full!female!PGE!(�φ!=!1),1cfMU1!=!½,!cfMU2!=!0,!cfPU1!=!0,!cfPU2!=!0!and!cmTUv!=!0,!i.e.!all!of!the!population’s!reproductive!value!belongs!to!maternal!grandmaternal!genes!in!females!(classes!fMM1!and!fMP1),!

and!no!gene!in!any!male!has!any!reproductive!value.!

!

PGE1in1females!–!The!condition!for!increase!in!female!PGE!is!∑!s∈{f,m},T∈{M,P},U∈{M,P},v∈{1,2}!

csTUv!((dWsTUv/dφ)!×!(dφ/dG)!×!(dG/dgsTUv)!+!(dWsTUv/dφ%)!×!(dφ%/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!mother,!and!G%!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!=!dφ%/dG%!=!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!dG%/dgsTUv!=!p!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.!!!

Consanguinity!–!The!coefficient!of!inbreeding!is!ρ!=!a((1/4)×((1/2)!+!(1/2)×(λ!ρM>!+!(1>λ)ρ))!+!(1/4)×(λ!ρM>!+!(1>λ)ρ)!+!(1/4)!×!ρ!+!(1/4)!×!ψ),!where!λ!=!2�φ/(1+�φ)!is!the!probability!that!a!focal!adult!female!derived!from!a!PGE!egg.!The!conditional!

consanguinities!of!mating!partners!are:!ρM>!=!a((1/2)((1/2)+(1/2)(λ!ρM>!+!(1>λ)ρ))!+!(1/2)(λ!ρM>!+!(1>λ)ρ)),!from!the!perspective!of!the!female’s!maternal>origin!gene;!ρ>M!=!a((1/2)((1/2)+(1/2)(λ!ρM>!+!(1>λ)ρ))!+(1/2)ρ),!from!the!perspective!of!the!male’s!maternal>origin!gene;!ρP>!=!a((1/2)ρ!+(1/2)ψ),!from!the!perspective!of!the!

! 35!

female’s!paternal>origin!gene;!ρ>P!=!a((1/2)(λ!ρM>!+!(1>λ)ρ)!+!(1/2)ψ),!from!the!perspective!of!the!male’s!paternal>origin!gene;!ρMM!=!a((1/2)!+!(1/2)(λ!ρM>!+!(1>λ)ρ)),!from!the!perspective!of!both!mating!partners’!maternal>origin!genes;!ρMP!=!a(λρM>!+!(1>λ)ρ),!from!the!perspective!of!the!female’s!maternal>origin!gene!and!the!male’s!paternal>origin!gene;!ρPM!=!aρ,!from!the!perspective!of!the!female’s!paternal>origin!gene!and!the!male’s!maternal>origin!gene;!and!ρPP!=!a!ψ,!from!the!perspective!of!both!mating!partners’!paternal>origin!genes.!The!consanguinity!of!two!males!in!

the!same!mating!group!is!ψ!=!a((1/4)((1/2)!+!(1/2)×(λ!ρM>!+!(1>λ)ρ))!+!(1/2)ρ!+!(1/4)ψ),!and!the!conditional!consanguinities!between!males!are:!ψM>!=!a((1/2)((1/2)!+!(1/2)×(λ!ρM>!+!(1>λ)ρ))!+!(1/2)ρ),!from!the!perspective!of!one!male’s!maternal>origin!gene;!ψP>!=!a((1/2)ρ!+!(1/2)ψ),!from!the!perspective!of!one!male’s!paternal>origin!gene;!ψMM!=!a((1/2)!+!(1/2)(λ!ρM>!+!(1>λ)ρ)),!from!the!perspective!of!both!males’!maternal>origin!genes;!ψMP!=!aρ,!from!the!perspective!of!one!male’s!maternal>origin!gene!and!the!other’s!paternal>origin!gene;!and!ψPP!=!aψ,!from!the!perspective!of!both!males’!paternal>origin!genes.!The!consanguinity!of!two!

females!in!the!same!mating!group!is!ζ!=!a((1/4)(λ2!+!(1>λ2)((1/2)!+!(1/2)(λ!ρM>!+!(1>λ)ρ))!+!(1/2)(λ!ρM>!+!(1>λ)ρ)!!+!(1/4)ψ),!and!the!conditional!consanguinities!between!females!are:!ζM>!=!a((1/2)(λ2!+!(1>λ2)((1/2)!+!(1/2)(λ!ρM>!+!(1>λ)ρ))!+!(1/2)(λ!ρM>!+!(1>λ)ρ)),!from!the!perspective!of!one!female’s!maternal>origin!gene;!ζP>!=!a((1/2)(λ!ρM>!+!(1>λ)ρ)!!+!(1/2)ψ),!from!the!perspective!of!one!female’s!paternal>origin!gene;!ζMM!=!a((λ2!+!(1>λ2)((1/2)!+!(1/2)(λ!ρM>!+!(1>λ)ρ))),!from!the!perspective!of!both!females’!maternal>origin!genes;!ζMP!=!a(λ!ρM>!+!(1>λ)ρ),!from!the!perspective!of!one!female’s!maternal>origin!gene!and!the!other’s!paternal>origin!

gene;!and!ζPP!=!a!ψ,!from!the!perspective!of!both!females’!paternal>origin!genes.!Simultaneously!solving!these!equations!yields!ρ!=!a(4+(4>a2)�φ)/(32(1+�φ)!>!a(28+(2>a(26+3a))�φ)),!and!each!of!the!other!consanguinities.!And!these!in!turn!define!all!the!consanguinities!needed!to!solve!the!model!(listed!in!Table!A6.1.1).!!

Equilibrium1level1of1PGE!–!If!the!condition!for!increase!in!PGE!is!not!satisfied!at!�φ!=!0,!then!PGE!cannot!invade!the!population!from!rarity,!and!its!equilibrium!level!is!g*!=!0.!If!the!condition!for!increase!in!PGE!is!satisfied!at!�φ!=!1,!then!non>PGE!cannot!invade!from!rarity,!and!hence!the!equilibrium!level!of!PGE!is!g*!=!1.!Setting!the!LHS!of!the!condition!equal!to!zero,!and!solving!for!�φ!=!g*,!obtains!the!intermediate!equilibrium!level!of!PGE.!The!equilibrium!level!of!PGE!is!illustrated!in!Figure!4!of!the!

main!text.!!

6.2.!Z>linked!genes!

!

Classes!–!There!are!4!classes!of!individual,!classified!according!to!their!sex!s!∈!{f,m}!and!the!grandparent!of!origin!of!their!paternal>origin!gene!U!∈!{M,P}.!If!the!individual!is!female,!she!has!no!maternal>origin!Z>linked!genes.!And!if!the!individual!

is!male,!then!all!of!his!maternal>origin!Z>linked!genes!are!derived!from!his!maternal!


! 36!


Recipient(Class(


fMM1! (1+ν)/2! 1! ν! (1+ν)/2! ν!fMM2! ρ>M! ρMM! ρPM! a(1+ν)/2! a!ρ!fMP1! (1+ν)/2! 1! ν! (1+ν)/2! ν!fMP2! ρ>P! ρMP! ρPP! a!ρ! a!ψ!fPM1! (1+ν)/2! ν! 1! ρ! (1+ρ)/2!fPM2! ρ>M! ρMM! ρPM! a(1+ν)/2! a!ρ!fPP1! (1+ν)/2! ν! 1! ρ! (1+ρ)/2!fPP2! ρ>P! ρMP! ρPP! a!ρ! a!ψ!mMM1! (1+ν)/2! 1! ν! (1+ν)/2! ν!mMM2! ρ>M! ρMM! ρPM! a(1+ν)/2! a!ρ!mMP1! (1+ν)/2! 1! ν! (1+ν)/2! ν!mMP2! ρ>P! ρMP! ρPP! a!ρ! a!ψ!mPM1! (1+ν)/2! ν! 1! ρ! (1+ρ)/2!mPM2! ρ>M! ρMM! ρPM! a(1+ν)/2! a!ρ!mPP1! (1+ν)/2! ν! 1! ρ! (1+ρ)/2!mPP2! ρ>P! ρMP! ρPP! a!ρ! a!ψ!mMM1%! a(1+ν)/2! a1 a!ν! a(1+ν)/21 a!ν1mMM2%! aρ>M! a!ρMM! a!ρPM! a2(1+ν)/21 a2!ρ1mMP1%! a(1+ν)/2! a!! a!ν! a(1+ν)/21 a!ν1mMP2%! aρ>P! a!ρMP! a!ρPP! a2ρ1 a2!ψ1mPM1%! a(1+ν)/2! a!ν! a!! a!ρ1 a(1+ρ)/21mPM2%! aρ>M! a!ρMM! a!ρPM! a2(1+ν)/21 a2!ρ1mPP1%! a(1+ν)/2! a!ν! a!! a!ρ1 a(1+ρ)/21mPP2%! aρ>P! a!ρMP! a!ρPP! a2!ρ1 a2!ψ1!

Table(A6.1.1.!Consanguinities(for(the(female(PGE(equilibrium(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!the!consanguinities!p%mTUv!to!each!of!the!recipient!classes!mTUv%,!among!the!males!competing!for!mates!with!the!focal!female’s!sons.!Here,!ν!=!(λ!ρM>!+!(1>λ)ρ).!!!

Survival!–!The!probabilities!of!survival!for!each!class!are:!Sf>M(φ)!=!(1/4)(1>φ)!+!(1/2)φ;!Sf>P(φ)!=!(1/4)(1>φ)!+!(1/2)φ;!SmPM(φ)!=!(1/4)(1>φ);!and!SmPP(φ)!=!(1/4)(1>φ).!!!

Reproductive1success!–!Contingent!upon!survival,!expected!reproductive!success!is!RfTU(φ%)!=!1!for!females!and!RmTU(φ%)!=!(1>z(φ%))/z(φ%)!for!males,!where!z(φ%)!=!(SmPM(φ%)!+1SmPP(φ%))/(Sf>M(φ%)!+1Sf>P!(φ%)!+1SmPM(φ%)!+1SmPP(φ%)).!!

Fitness1–!Expected!fitness!is!given!by!the!product!of!survival!to!maturity!and!expected!reproductive!success!contingent!upon!surviving!to!maturity,!i.e.!wsTU(φ,φ%)!

! 37!

=!SsTU(φ)×RsTU(φ%).!The!average!fitness!among!all!individuals!of!a!particular!class!is!!given!by�wsTU(�φ)!=!wsTU(�φ,�φ).!Each!individual’s!fitness!can!be!expressed!relative!to!the!average!of!their!class,!by!WsTU(φ,φ%,�φ)!=!wsTU(φ,φ%)/�wsTU(�φ).!!

Gene1fitness!–There!are!6!classes!of!gene,!because!individuals!of!each!of!the!two!female!classes!carry!only!paternal>origin!genes,!whereas!individuals!of!each!of!the!

two!male!classes!carry!separate!maternal>origin!and!paternal>origin!genes.!As!


of!a!gene!in!a!female!is!given!by!Wf>U2(φ,φ%,�φ)!=!Wf>U(φ,φ%,�φ)!and!the!relative!fitness!of!a!gene!in!a!male!is!given!by!WmPU1(φ,φ%,�φ)!=!WmPU2(φ,φ%,�φ)!=!WmPU(φ,φ%,�φ).!!

Reproductive1value!–!PGE!in!females!does!not!impact!upon!the!gene>flow!coefficients!for!Z>linked!genes,!because!females!can!only!transmit!paternal>origin!Z>linked!genes!anway.!Accordingly,!all!of!the!non>zero!gene>flow!coefficients!are!equal!to!½,!and!

reproductive!value!is!csTUv!=!1/6!for!all!gene!classes,!as!in!section!A3.2.!!

PGE1in1females!–!Following!the!same!procedure!as!before,!the!condition!for!increase!in!female!PGE!is!cf>M2!((dWf>M2/dφ)!×!(dφ/dG)!×!(dG/dgf>M2))!+!cf>P2!((dWf>P2/dφ)!×!(dφ/dG)!×!(dG/dgf>P2))!+!cmPM1!((dWmPM1/dφ)!×!(dφ/dG)!×!(dG/dgmPM1)!+!(dWmPM1/dφ%)!×!(dφ%/dG%)!×!(dG%/dgmPM1))!+!cmPM2!((dWmPM2/dφ)!×!(dφ/dG)!×!(dG/dgmPM2)!+!(dWmPM2/dφ%)!×!(dφ%/dG%)!×!(dG%/dgmPM2))!+!cmPP1!((dWmPP1/dφ)!×!(dφ/dG)!×!(dG/dgmPP1)!+!(dWmPP1/dφ%)!×!(dφ%/dG%)!×!(dG%/dgmPP1))!+!cmPP2!((dWmPP2/dφ)!×!(dφ/dG)!×!(dG/dgmPP2)!+!(dWmPP2/dφ%)!×!(dφ%/dG%)!×!(dG%/dgmPP2))!>!0,!where:!dφ/dG!=!dφ%/dG%!=!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!dG%/dgmTUv!=!p!mTUv!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.!!!

Consanguinity!–!Again,!PGE!in!females!does!not!impact!upon!the!transmission!of!Z>linked!genes,!because!females!can!only!transmit!maternal>origin!Z>linked!genes!

anway.!Accordingly,!the!coefficients!of!consanguinity!are!exactly!the!same!as!those!

given!in!Table!A3.2.1.!!


main!text.!!

! 38!

!

6.3.!W>linked!genes!1Classes!–!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}.!!

Survival!–!The!probabilities!of!survival!for!each!class!are!Sf(φ)!=!(1/2)(1>φ)!+!φ!and!Sm(φ)!=!(1/2)(1>φ).!!

Reproductive1success!–!Contingent!upon!survival,!expected!reproductive!success!is!Rf(φ%)!=!1!for!females!and!Rm(φ%)!=!(1>z(φ%))/z(φ%)!for!males,!where!z(φ%)!=!Sm(φ%)/(Sf(φ%)1+1Sm(φ%)).!!

Fitness1–!Expected!fitness!is!given!by!the!product!of!survival!to!maturity!and!expected!reproductive!success!contingent!upon!surviving!to!maturity,!i.e.!wsTU(φ,φ%)!=!SsTU(φ)×RsTU(φ%).!The!average!fitness!among!all!individuals!of!a!particular!class!is!given!by�wsTU(�φ)!=!wsTU(�φ,�φ).!Each!individual’s!fitness!can!be!expressed!relative!to!the!average!of!their!class,!by!WsTU(φ,φ%,�φ)!=!wsTU(φ,φ%)/�wsTU(�φ).!!

Gene1fitness!–!There!is!only!one!class!of!W>linked!gene,!because!females!carry!only!one!class!of!W>linked!gene!and!males!carry!no!W>linked!genes.!Accordingly,!the!

relative!fitness!of!a!W>linked!gene!in!a!female!is!simply!Wf(φ,φ%,�φ).!!

Reproductive1value!–!Since!there!is!only!one!class!of!W>linked!gene,!all!reproductive!value!belongs!to!this!class:!cf!=!1.!!!

PGE1in1females!–!Following!the!same!procedure!as!before,!the!condition!for!increase!in!female!PGE!is!cf!((dWf/dφ)!×!(dφ/dG)!×!(dG/dgf))!>!0.!Once!again:!dφ/dG!is!an!arbitrary!mapping!between!genic!value!and!phenotypic!value,!and!can!be!set!to!1;!and!dG/dgf!=!pf!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.!!

1Consanguinity!–!PGE!in!females!does!not!impact!upon!consanguinity!of!W>linked!genes,!which!are!exactly!the!same!as!those!listed!in!Table!A3.3.1.!!


main!text.!

!

! 39!

7.(PGE(in(males(under(XY/XO(inheritance:(equilibrium(analysis(with(a(new(mode(of(sex(determination((7.1.!Autosomal!genes!!


!Survival!–!If!sex!allocation!z!is!determined!independently!of!genotype,!then!the!zygote!survival!functions!are!SfMM(µ,z)!=!(1>z)((1/4)(1>µ)!+!(1/2)µ);!SfMP(µ,z)!=!(1>z)(1/4)(1>µ);!SfPM(µ,z)!=!(1>z)((1/4)(1>µ)!+!(1/2)µ);!SfPP(µ,z)!=!(1>z)(1/4)(1>µ);!SmMM(µ,z)!=!z((1/4)(1>µ)+(1/2)µ);!SmMP(µ,z)!=!z(1/4)(1>µ);!SmPM(µ,z)!=!z((1/4)(1>µ)+(1/2)µ);!and!SmPP(µ,z)!=!z(1/4)(1>µ).!!!Reproductive1success!–!Upon!survival!to!maturity,!every!female!achieves!a!unit!of!expected!reproductive!success,!and!every!male!achieves!(1>z%)/1z%!units!of!expected!reproductive!success,!where!z%!is!the!sex!ratio!of!the!group!in!which!they!compete!for!mates.!Thus,!contingent!upon!survival,!expected!reproductive!success!is!RfTU(z%)!=!1!for!females!and!RmTU(z%)!=!(1>1z%)/z%!for!males.!!


wsTU(µ,z,z%)!=!SsTU(µ,z)×RsTU(z%).!The!average!fitness!among!all!individuals!of!a!particular!class!is!given!by�wsTU(�µ,�z)!=!wsTU(�µ,�z,�z),!and!each!individual’s!fitness!can!be!expressed!relative!to!the!average!of!their!class,!by!WsTU(µ,z,z%,�z)!=!wsTU(µ,z,z%)/�wsTU(�µ,�z).!!!Gene1fitness!–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!



relative!fitness!of!a!gene!is!equal!to!the!relative!fitness!of!its!carrier:!WsTU1(µ,z,z%,�z)!=!WsTU2(µ,z,z%,�z)!=!WsTU(µ,z,z%,�z).!!Reproductive1value!–!The!flow!of!genes!between!classes!determines!each!class’s!reproductive!value.!Each!gene’s!class!in!any!generation!may!provide!some!information!and!some!uncertainty!about!that!gene’s!origin!in!the!previous!

generation.!For!example,!genes!of!class!fMM1!in!the!present!generation!are!derived!

from!genes!of!class!fTU1!in!the!previous!generation,!where!T!=!M!or!P!and!U!=!M!or!P.!The!effects!of!PGE!in!a!male!are!accounted!for!in!the!survival!of!his!potential!

offspring:!for!example,!his!potential!male!offspring!are!eliminated!in!the!event!that!

! 40!

he!undergoes!PGE.!So!the!class!fTU1!donating!genes!to!the!class!fMM1!involves!T!=!

M!or!P!with!equal!probability,!because!there!is!no!bias!regarding!the!source!of!their!maternal!genes,!but!U!=!M!or!P!with!unequal!probability,!because!there!is!possible!

bias!regarding!the!source!of!their!paternal!genes.!Specifically,!U!=!M!with!probability!

�wfTM1/(1�wfTM1!+1�wfTP1)!=!(1+�µ)/2,!and!U!=!P!with!probability!�wfTP1/(1�wfTM1!+1B�wfTP1)!=!(1>�µ)/2.!Accordingly,!℘fMM1←fMM1!=!(1+�µ)/4,!℘fMM1←fPM1!=!(1+�µ)/4,!℘fMM1←fMP1!=!(1>�µ)/4!and!℘fMM1←fPP1!=!(1>�µ)/4.!Note!that,!making!the!substitution!�µ!=!0,!all!of!these!gene>flow!coefficients!are!equal!to!¼,!as!in!section!1.1.!This!defines!a!system!of!16!linear!equations,!which!can!be!written!as!c!=(c•G,!where!c!=!{cfMM1,cfMM2,cfMP1,…,cmPP2}!is!the!vector!of!class!reproductive!values!and!G!is!the!gene>flow!matrix.!Solving!(i.e.!finding!the!left!eigenvector!of!G!corresponding!to!the!eigenvalue!1)!obtains!cfTMv!=!(1+�µ)2/(16+8�µ),!cfTPv!=!(1>�µ2)/(16+8�µ),!cmTM1!=!(1+�µ)2/(16+8�µ),!cmTM2!=!(1>�µ2)/(16+8�µ),!cmTP1!=!(1>�µ2)/(16+8�µ),!and!cmTP2!=!(1>�µ)2/(16+8�µ).!!!

PGE1in1males!–!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.!The!consanguinities!will!depend!upon!which!class!or!classes!of!genes!control!PGE!in!males!>>!see!the!Consanguinity!

section!below,!and!Table!A7.1.1,!for!details.!

!Consanguinity!–!Note!that!there!are!no!sex!differences!in!genes!carried!(but!there!are!sex!differences!in!genes!transmitted),!so!ψ!=!ζ!=!ρ.!The!coefficient!of!inbreeding!is!ρ!=!a((1/4)×((1/2)!+!(1/2)×(�µ!ρ>M!+!(1>�µ)ρ))!+!(1/2)×(!�µ!ρ>M!+!(1>�µ)ρ)!+!(1/4)!×!(�µ2!ρMM!+!2�µ(1>�µ)ρM>!+!(1>�µ)2!ρ)).!Note!that,!whilst!the!consanguinity!of!mating!partners!is!ρ,!the!consanguinity!of!the!genes!that!they!transmit!to!their!offspring!is!�µ!ρ>M!+!(1>�µ)ρ,!because!with!probability!�µ!the!male!exhibits!PGE!and!so!only!transmits!his!maternal>origin!genes,!and!with!probability!1>�µ!the!male!does!not!exhibit!PGE!and!hence!transmits!maternal>origin!and!paternal>origin!genes!with!

equal!probability.!The!conditional!consanguinities!of!mating!partners!are:!ρ>M!=!ρM>!=!a((1/2)×((1/2)!+!(1/2)×(�µ!ρ>M!+!(1>�µ)ρ))!+!(1/2)×(!�µ!ρ>M!+!(1>�µ)ρ)),!from!the!perspective!of!one!partner’s!maternal>origin!gene;!ρ>P!=!ρP>!=!a((1/2)×(!�µ!ρ>M!+!(1>�µ)ρ)!+!(1/2)!×!(�µ2!ρMM!+!2�µ(1>�µ)ρM>!+!(1>�µ)2!ρ)),!from!the!perspective!of!one!partner’s!paternal>origin!gene;!ρMM!=!a((1/2)!+!(1/2)×(�µ!ρ>M!+!(1>�µ)ρ)),!from!the!perspective!of!both!mating!partners’!maternal>origin!genes;!ρMP!=!ρPM!=!a(!�µ!ρ>M!+!(1>�µ)ρ),!from!the!perspective!of!one!partner’s!maternal>origin!gene!and!the!other!partner’s!paternal>origin!gene;!and!ρPP!=!a(�µ2!ρMM!+!2�µ(1>�µ)ρM>!+!(1>�µ)2!ρ),!from!the!perspective!of!both!mating!partners’!paternal>origin!genes.!Simultaneously!

solving!these!equations!yields!ρ!=!a(2+a(3>a�µ2)�µ)/(16>!a(14>(6>a(3>�µ)(1>�µ)>4�µ)�µ)),!and!each!of!the!other!consanguinities.!And!these!in!turn!define!all!the!consanguinities!needed!to!solve!the!model!(listed!in!Table!A7.1.1).!

! 41!


Recipient(Class(


fMM1! ρM>! ρMM! ρMP! a1(1+σ)/2! a!ρ!fMM2! (1+σ)/2! 1! σ! (1+σ)/2! ρ!fMP1! ρM>! ρMM! ρMP! a(1+σ)/2! a!ρ!fMP2! (1+σ)/2! σ! 1! σ! (1+σ)/2!fPM1! ρP>! ρPM! ρPP! a!σ! a1σ!fPM2! (1+σ)/2! 1! σ! !(1+σ)/2! ρ!fPP1! ρP>! ρPM! ρPP! a!σ! a!σ!fPP2! (1+σ)/2! σ! 1! σ! (1+σ)/2!mMM1! ρM>! ρMM! ρMP! a(1+σ)/2! a!ρ!mMM2! (1+σ)/2! 1! σ! (1+σ)/2! ρ!mMP1! ρM>! ρMM! ρMP! a(1+σ)/2! a!ρ!mMP2! (1+σ)/2! σ! 1! σ! (1+σ)/2!mPM1! ρP>! ρPM! ρPP! a!σ! a!σ!mPM2! (1+σ)/2! 1! σ! !(1+σ)/2! ρ!mPP1! ρP>! ρPM! ρPP! a!σ! a!σ!mPP2! (1+σ)/2! σ! 1! σ! (1+σ)/2!!

Table(A7.1.1.!Consanguinities(for(the(male(PGE(equilibrium(analysis:(autosomal(genes.!The!five!autosomal!actor!classes!are!the!male’s!autosomal!genes!(A),!his!maternal>origin!autosomal!genes!


father’s!autosomal!genes!(AFat).!Shown!here!are!their!consanguinities!qsTUv!to!each!of!the!recipient!classes!sTUv!among!the!male’s!offspring,!and!the!consanguinities!q%mTUv!to!each!of!the!recipient!classes!mTUv%,!among!the!males!competing!for!mates!with!the!focal!male’s!sons.!Here,!σ!=!(�µ!ρ>M!+!(1>�µ)ρ).!!!

Equilibrium1level1of1PGE!–!If!the!condition!for!increase!in!PGE!is!not!satisfied!at!�µ!=!0,!then!PGE!cannot!invade!the!population!from!rarity,!and!its!equilibrium!level!is!g*!=!0.!If!the!condition!for!increase!in!PGE!is!satisfied!at!�µ!=!1,!then!non>PGE!cannot!invade!from!rarity,!and!hence!the!equilibrium!level!of!PGE!is!g*!=!1.!Setting!the!LHS!of!the!condition!equal!to!zero,!and!solving!for!�µ!=!g*,!obtains!the!intermediate!equilibrium!level!of!PGE.!!

8.(Combined(action(of(multiple(genic(actors(with(conflicting(interests(!8.1.!Selection!under!conflict!

!

Total1selective1change!–!As!outlined!in!the!above!sections,!a!gene!that!promotes!GE!is!favoured!by!natural!selection!if!dW/dg1>!0!and!disfavoured!by!natural!selection!if!dW/dg1<!0.!The!corresponding!selective!change!in!the!frequency!of!GE!is!(dW/dg)!×!var(g),!where!var(g)!is!the!additive!genetic!variance!in!GE!contributed!by!that!gene!position.!If!several!genes!of!the!same!type!(e.g.!unimprinted!autosomal!genes)!impact!upon!the!GE!phenotype,!then!the!corresponding!selective!change!in!

the!frequency!of!GE!is!(dW/dg)!×!var(g),!where!var(g)!is!the!total!additive!genetic!

! 42!

variance!in!GE!contributed!by!those!gene!positions.!And!if!several!genes!of!different!

types!(e.g.!unimprinted!autosomal!genes!and!imprinted!maternal>origin!genes!and!paternal>origin!genes)!impact!upon!the!GE!phenotype,!then!the!corresponding!

selective!change!in!the!frequency!of!GE!is!∑t∈T(dW/dg)t!πt!var(g),!where!T!is!the!set!of!all!types!of!gene!position!impacting!upon!the!GE!phenotype,!var(g)!is!the!total!additive!genetic!variance!in!GE!contributed!by!those!gene!positions,!(dW/dg)t!!is!the!marginal!fitness!associated!with!gene!positions!of!type!t,!and!πt!is!the!proportion!of!the!total!additive!genetic!variance!that!is!contributed!by!gene!positions!of!type!t.!!!

Condition1for1increase!–!The!condition!for!natural!selection!to!favour!an!increase!in!the!frequency!of!GE!is!∑t∈T(dW/dg)t!πt!>!0,!where!the!LHS!of!the!condition!represents!a!weighted!average!of!the!interests!of!the!different!genic!types.!!

1Equilibrium1point!–!Note!that!(dW/dg)t°!≤!∑t∈T(dW/dg)t!πt!≤!(dW/dg)t*,!where!t°!∈!T!is!the!type!of!gene!position!least>strongly!favouring!(or!most>strongly!disfavouring)!

GE!–!i.e.!(dW/dg)t°!≤!(dW/dg)t!for!all!t!∈!T!–!and!t*!∈!T!is!the!type!of!gene!position!most>strongly!favouring!(or!least>strongly!disfavouring)!GE!–!i.e.!(dW/dg)t*!≥!(dW/dg)t!for!all!t!∈!T.!Hence,!∑t∈T(dW/dg)t!πt!≥!0!when!(dW/dg)t°!=!0,!and!∑t∈T(dW/dg)t!πt!≤!0!when!(dW/dg)t*!=!0,!such!that!the!equilibrium!level!of!GE!under!conflict!is!bounded!by!the!equilibria!favoured!by!the!extremist!parties.!!

8.2.!The!parliament!of!genes!

!Interests1of1maternalBorigin1and1paternalBorigin1genes1cancel1–1The!marginal!fitness!for!a!maternal>origin!gene!at!a!diploid!locus!may!be!expressed!as!(dW/dg)M1=!>c!+!∑i∈I1bi1piM,!i.e.!its!inclusive>fitness!effect,!where!–c!is!the!impact!an!increase!in!gene!expression!has!on!its!own!replicative!success,!bi!is!the!impact!on!the!replicative!success!of!its!social!partner!i!∈!I,!and!piM!is!its!consanguinity!with!this!social!partner.!The!marginal!fitness!for!the!corresponding!paternal>origin!gene!may!be!expressed!

as!(dW/dg)P1=!>c!+!∑i∈I1bi1piP,!where!piP!need!not!be!equal!to!piM!on!account!of!social!partners!being!more!related!through!one!parent!than!through!the!other.!This!

asymmetry!in!consanguinity!coefficients,!and!hence!inclusive>fitness!effects,!is!the!reason!for!intragenomic!conflict!between!maternal>origin!and!paternal>origin!genes.!

If!maternal>origin!and!paternal>origin!genes!contribute!equally!to!additive!genetic!

variance!(πM!=!πP!=!π),!then!∑t∈{M,P}(dW/dg)t!πt!>!0!is!equivalent!to!>c!+!∑i∈I1bi1pi!>!0!where!pi!=!(piM+piP)/2!is!the!consanguinity!that!has!not!been!conditioned!upon!parent!of!origin,!and!this!inequality!is!equivalent!to!(dW/dg)U!>!0,!where!U!denotes!the!corresponding!genes!that!lack!information!as!to!their!parent!of!origin.!For!

example,!considering!the!evolution!of!PGE!in!males!under!XY/XO!sex!determination!(section!1!above,!and!SM2!section!1),!the!marginal!fitness!for!maternal>origin!

autosomal!genes!is!(dW/dg)M!=!(4>8β>a(4>5β>a(4>2a(1>β)>3β)))/(2(8>7a))!and!the!marginal!fitness!for!paternal>origin!autosomal!genes!is!(dW/dg)P!=!>(4>a(4>β+a(2a(1>β)>β)))/(2(8>7a))!and!the!average!of!these!two!exactly!recovers!the!marginal!fitness!for!unimprinted!autosomal!genes,!(dW/dg)U!=!(a2(1>β)>2β+aβ)/(8>7a).!Accordingly,!the!combined!action!of!multiple!maternal>origin!and!paternal>

! 43!

origin!genes!is!expected!to!give!an!equilibrium!level!of!GE!that!approximately!

coincides!with!that!favoured!by!their!unimprinted!counterparts.!!

Majority1party1dominates1the1conflict!–!The!weighted>average!marginal!fitness!∑t∈T(dW/dg)t!πt!→!(dW/dg)t%!!as!πt%!→!1.!That!is,!the!interests!of!the!majority!party!

dominate!the!conflict.!If!the!individual’s!own!autosomal!genes!(including!unimprinted!genes!and!imprinted!genes!of!maternal>origin!and!paternal>origin;!see!

above)!contribute!most!of!the!additive!genetic!variance!in!GE,!for!example!on!

account!of!their!numerical!superiority!over!the!sex!chromosomes!and!also!because!the!individual’s!own!genome!is!necessarily!closely!involved!with!the!process!of!GE!

whereas!its!parents’!genomes!can!act!only!at!a!distance!and!so!have!less!control,!

then!the!majority!interest!will!be!equivalent!to!that!of!the!individual’s!autosomal!genes!lacking!information!as!to!their!parent!of!origin.1!

mating ecology explains patterns of genome eliminationletter mating ecology explains patterns of...

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