Cytoplasmic incompatibility in social insects

Ed Long, CoMPLEx

Supervisors: Dr. Max Reuter & Dr. Greg Hurst

Word count: 3697

April 15, 2007

Wolbachia is a maternally-inherited, obligate endosymbiont which occurs in a large number of arthro-pod species. The effect of the bacterium on its host varies with different genotypes but a common traitis manipulation of host reproductive biology, including inducing cytoplasmic incompatibility.

In this essay, I briefly summarise reproductive phenotypes associated with Wolbachia in insects, presenta model of the dynamics of infection frequency for maternally-inherited microorganisms and deter-mine the effect of changing costs associated with incompatibility, infection-related fecundity loss andnumber of mates in ideal populations of social insects. The model predicts that infections of this typespread to fixation most easily in populations with a high cost due to incompatibility, little fecundityloss and where females have a large number of mates.

Contents

1 Introduction 11.1 Symbionts in nature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Wolbachia infection in insects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Cytoplasmic incompatibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Infection in Hymenoptera . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Models 42.1 Classical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Worker productivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 Imperfect maternal transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4 Multiple matings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Discussion 10

A Mathematica notebook 12

B Graphs of threshold values against mate number 12B.1 Directly proportional model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12B.2 Escalating model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13B.3 Diminishing model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1 Introduction

1.1 Symbionts in nature

Symbiosis is a broad term which describes a close association between two organisms from differentspecies. Commonly, at least one of the organisms derives a benefit from this association. If both organ-isms benefit, the relationship is described as mutualistic. If one organism benefits at the expense of theother, the relationship is termed parasitic. Other classes of symbiosis are also possible and organismsneed not be affected by the relationship.

A symbiotic relationship may be between two organisms who live nearby in the same environment.For example, many species of ant share habitats with aphids and protect them from predators in returnfor a supply of honeydew—an energy-rich substance which the aphids secrete (Figure 1)1. Another pos-sibility is that one of the organisms may live inside the body or cells of the other. This relationship iscalled endosymbiosis, with the internal organism referred to as an endosymbiont. Endosymbionts whichcannot survive without a host are termed obligate.

Figure 1: Ants collect honeydew from a group of aphids

Infection with bacteria is a form of endosymbiosis. We are all aware of this relationship in its parasitic(or pathogenic) form, responsible for food poisoning, pneumonia, sepsis and other diseases. Recent ad-vertising campaigns have also raised public awareness of bacteria which have a mutualistic relationshipwith humans, aiding digestion of milk proteins and complex carbohydrates and synthesising vitamins(the nominally “friendly bacteria”). The symbiotic bacterium Rhizobia aids plants in absorbing nitrogenfrom the soil and a number of symbiotic bacteria and protists are essential for enabling insects to digestthe cellulose in plant matter.

Symbiotic relationships can drive evolutionary development. Mutualistic relationships form a posi-tive feedback loop whereby the success of one species improves the fitness of the other, and hence itsown. This is believed to be the driving force behind the sudden radiation of both angiosperms (flower-ing plants) and pollinating insects in the mid-Cretaceous period. Parasitic symbionts can also influenceselection pressures on their hosts in favour of genotypes which minimise the negative effect of the par-asite.

1.2 Wolbachia infection in insects

Wolbachia is a genus of bacteria which infects a substantial number of arthropod and filarial nematodespecies and frequently influences the reproductive development and behaviour of its host. It lives within

1Photo ©Alex Wild

1

the cells of its host, in highest concentrations in the reproductive tissues and is typically vertically inher-ited rather than horizontally transmitted between generations. Inheritance is exclusively via the mother:although the testes of a male host contain the infection, it cannot by passed on in the sperm (see Table1). Wolbachia is an obligate endosymbiont and its relationship with the host ranges from parasitismto mutualism depending on host genotype, Wolbachia strain, location in the host and environmentalinfluences[11].

U♂ I♂

U♀ U U

I♀ I I

Table 1: Uninfected (U) or infected (I) offspring of U or I parents

Given that male hosts are “dead ends” for the bacterium, it has evolved mechanisms which skew the sexratios of infected populations in its favour. Although ratios would naturally be around 50:50 male andfemale, some infected arthropods have entirely or almost entirely female populations. The bacteriummanages this in a number of ways.

Male killing (often abbreviated to MK) is one of these methods and has been observed in infected but-terflies (Acraea encedon) and ladybirds (Adalia bipunctata)[9] amongst other arthropod species. The sonsof infected females die during embryogenesis [3] and only daughters develop. The benefit to the bac-teria is that all offspring will be able to pass the infection on. There are also potential advantages forthe female hosts: a food source (they may eat their dead brothers); reduced sibling competition; andreduced chances of inbreeding. Wolbachia is not the only bacterium with this trait: [8] lists a number ofother clades of vertically-transmitted bacteria in which MK is used to promote infection frequency.

Wolbachia may also interfere with sex determination, inducing feminisation of male offspring. In somepopulations of Armadillidium vulgare, the pill woodlouse, Wolbachia is solely responsible for sex deter-mination. To counteract skewed sex ratios, there is an incentive on the woodlice to produce more maleoffspring. In some cases this happened to the extent that the entire population became genetically maleand the factor determining sex determination switched to whether or not the individual was infectedwith an active bacterium.

In haplodiploid species, females develop from fertilised (diploid) eggs, whereas unfertilised (haploid)eggs will become males. If there is a low frequency of males in a population, there is a lower chance thatfemales will be fertilised and so the next generation will produce more males. In order to guarantee itstransmission, Wolbachia is able to double the chromosome number of unfertilised eggs so a mother canproduce daughters asexually by parthogenesis.

1.3 Cytoplasmic incompatibility

A fourth mechanism used by Wolbachia to promote its transmission is cytoplasmic incompatibility (CI).Since the offspring of an infected male and an uninfected female will not carry the infection, Wolbachiaexacts a cost on pairings of this type. This is commonly a reduced egg hatch rate: Turelli and Hoffmannrecorded hatch rates from incompatible matings of Drosophila simulans at 30–70% as high as those fromcompatible matings[15]. In a separate study by Fry et al., however, the observed effect in D. melanogasterwas a reduced fecundity in the female offspring, with no change in hatch rate[4].

CI does not affect the sex ratios in the population, but gives a reproductive advantage to the infectedfemales by diminishing the reproductive success of uninfected females: effectively using the infectedmales, which cannot pass on the infection, as agents to sterilise their competition.

2

U♂ I♂

U♀ xxxx x

I♀ xxxx xxxx

A♂ B♂

A♀ xxxx x

B♀ x xxxx

Table 2: Illustration of unidirectional and bidirectional CI effects on number of offspring

When two or more strains of Wolbachia exist in a population the CI effect is bidirectional between a maleand female carrying different strains. Since there is selection against pairings between individuals carry-ing different strains the population may eventually partition into two separate species. Bidirectional CIcan also inhibit hybridisation between neighbouring species carrying different forms of the symbiont.

1.4 Infection in Hymenoptera

The insect order Hymenoptera includes wasps, bees, ants and sawflies and many species within theorder live in eusocial colonies: groups of individuals in which a proportion of the population are sterileand work for the benefit of the reproductive individuals in the colony. Eusociality is common in ants,bees and wasps and also in the more distantly related termites (order Isoptera).

Hymenoptera are haplodiploid: fertilised eggs develop into females and unfertilised into males. Af-ter mating, the queen controls which of the eggs are fertilised although the workers are responsible forrearing the larvae (see Fig 2)2 and determine which develop into workers and which into sexual femalesor gynes. Since workers are more closely related to their sisters they should theoretically favour a biastowards investment of the colony’s resources in females. This creates a conflict with the queen overcolony sex ratio. In [13], Reuter and Keller calculate evolutionarily stable levels for the proportion ofresources allocated to male or female production.

Figure 2: Worker ants tend larvae in the nursery

Infection with Wolbachia is widespread in Hymenoptera. In [16], Wenseleers et al. report that ants from25 out of 50 species of ant in Java and Sumatra screened positive for a particular strain of Wolbachia. In astudy of a single Swiss population of the ant Formica exsecta, Reuter and Keller found that all ants testedwere multiply infected with four or five different strains of Wolbachia[14].

Wenseleers et al. also suggested that given the ability of Wolbachia to alter sex ratios in some of its hosts,it might have an effect on the equilibrium sex ratio reached in the queen-worker conflict in ant colonies.Studies of Swiss and Finnish populations of F. exsecta by Keller et al., however, found no significant as-sociation between prevalence of infection and colony sex ratio in the expected direction[10] and a later

2Photo ©Alex Wild

3

study by Wenseleers et al. of populations of F. truncorum also found no connection[17].

Infection with Wolbachia has also been found in other social insect species including the Cape honeybeeApis mellifera capensis[7], fig wasp (family Agaonidae)[5] and termite species Zootermopsis angusticollisand Z. nevadensis[1].

In this essay I will give a theoretical analysis of the spread of a vertically transmitted endosymbiontsuch as Wolbachia and attempt to determine what effect the social structure and mating behaviour ofeusocial insects such as ants might have on the ability of the infection to spread through the populationover successive generations.

2 Models

2.1 Classical model

A deterministic model for the spread of a CI-inducing infection in nonsocial insects is presented in [6]based on the 1959 model of Caspari and Watson[2]. We shall use a similar nomenclature: the frequencyof infection in the population is written as pt for generation t. We assume that CI causes a reduction inthe number of viable offspring when an infected male mates with an uninfected female. In this case, wedenote the number of viable offspring (as a proportion of the maximum number with no incompatibil-ity) by h (0 < h < 1). Infected females may also have a reduced fecundity, also resulting in reducedbrood size, which is represented by the parameter f (0 < f < 1); again a proportion of the maximum.

Assuming random pairing and 100% transmission of the infection from the mother to her offspring,the proportion of viable offspring of each type is shown in Table 3. U denotes an uninfected individualand I denotes infection.

U♂ I♂

U♀ (1− pt)2 pt(1− pt)h

I♀ pt(1− pt) f p2t f

Table 3: Outcomes of random pairing (perfect maternal transmission)

The infection frequency in the next generation is then given by:

Infected individualsAll individuals

= F(pt) =pt(1− pt) f + p2

t fpt(1− pt) f + p2

t f + (1− pt)2 + pt(1− pt)h(1)

=pt f

(1− pt)2 + pt f + pth(1− pt)(2)

The infection frequency reaches an equilibrium at p = 0, p = 1 and p =1− f1− h

(see Figure 3 for the case

f = 0.7, h = 0.5).

An equilibrium point is locally stable if −1 <dFdpt

< 1. As noted in [6], the equilibria at 0 and 1 are

stable, while the intermediate point is unstable and the final infection frequency reached in this modeldepends on whether the initial frequency is above or below this point.

2.2 Worker productivity

In eusocial colonies, a queen mates and produces an initial brood of workers who then raise the nextgeneration of sexual individuals. Because of this two-stage process, the number of sexual individuals

4

0 0.25 0.5 0.75 1

0.25

0.5

0.75

1

p

F(p)

Figure 3: Crossing points of p and F(p) showing equilibrium values

produced depends both on the ability of the queen to produce them and of the number of workers avail-able to help raise them. The effects of infection influence both of these stages.

Let w be a variable describing the proportion of workers produced by a queen out of the maximumand let b(w) be a productivity function describing the proportion of sexual individuals which are raisedsuccessfully based on the number of workers. Productivity is at a maximum when w = 1 and zero whenw = 0. Taking this into account gives a new mating table as shown in Table 4.

U♂ I♂

U♀ b(1) · (1− pt)2 b(h) · pt(1− pt)h

I♀ b( f ) · pt(1− pt) f b( f ) · p2t f

Table 4: Outcomes of random pairing (perfect maternal transmission)

Implementing this gives the expression for infection frequency in the next generation as:

pt+1 = F(pt) =b( f ) · pt(1− pt) f + b( f ) · p2

t fb( f ) · pt(1− pt) f + b( f ) · p2

t f + b(1) · (1− pt)2 + b(h) · pt(1− pt)h(3)

=b( f ) · pt f

b(1) · (1− pt)2 + b( f ) · pt f + b(h) · hpt(1− pt)(4)

The function b(w) should be a maximum when the number of workers is at a maximum (i.e. b(1) = 1)and also zero when the number of workers is zero. The simplest model is to set b(w) = w. In this casewe have:

pt+1 =f 2 pt

f 2 pt + (1− pt)2 + h2 pt(1− pt)(5)

Equilibrium points are the roots of the cubic p3(1− h2) + p2( f 2 + h2 − 2) + p(1− f 2): 0,1, and1− f 2

1− h2 .

At p = 1,dFdpt

=h2

f 2 so this point is stable as long as h < f . Heuristically, a population may maintain a

zero infection level whatever the effect of infection but can only maintain a population-wide infection ifthe loss of offspring due to CI is more severe than the loss due to reduced fecundity.

For the third equilibrium, since p is on [0, 1] we must have 0 ≤ 1− f 2

1− h2 ≤ 1 and hence h ≤ f . But

5

for the equilibrium at this point to be stable we have:

dFdpt

< 1 ⇒ f 4 − 2 f 2 + h2

f 2(h2 − 1)< 1 (6)

⇒ f 4 − 2 f 2 + h2 > f 2h2 − f 2 (7)

⇒ f 2( f 2 − 1) > h2( f 2 − 1) (8)⇒ f < h (9)

So any equilibrium between 0 and 1 will be an unstable one.

In general, the equilibrium points are the solutions of:

p3{

1− b(h) · h}

+ p2{

b( f ) · f + b(h) · h− 2}

+ p{

1− b( f ) · f}

(10)

which are 0, 1 and1− b( f ) · f1− b(h) · h

.

Two possibilities for alternative productivity functions are an escalating model: b(w) = w2; and a di-minishing model: b(w) = 1− (1− w)2 (see Figure 4).

0 0.25 0.5 0.75 1

0.25

0.5

0.75

1

b(w)

w

Figure 4: Three possible functions for productivity b(w)

These have potential non-boundary equilibrium points at p =1− f 3

1− h3 and p =1− 2 f 2 + f 3

1− 2h2 + h3 respectively.

For b(w) = w2, a similar working to (6)–(9) shows that the equilibrium is unstable again.

In the third case, we have:

dFdpt

=f 2(2− f )(1− 2 f 2 + f 3) + f 2(2− f )− h2(2− h)

f 2(2− f )(1− 2h2 + h3)(11)

And so:

dFdpt

< 1 (12)

⇔ f 2(2− f )− h2(2− h) < f 2(2− f ){(1− 2h2 + h3)− (1− 2 f 2 + h3)

}(13)

⇔ f 2(2− f )− h2(2− h) < f 2(2− f ){

f 2(2− f )− h2(2− h)}

(14)

⇔ f 2(2− f ) < h2(2− h) (15)

As the form t2(2− t) is monotonic increasing on [0, 1], the condition f ≥ h implies that f 2(2− h) ≥h2(2− h) and hence the point is unstable for all valid values of the parameters f and h.

6

0 0.25 0.5 0.75 1

0.25

0.5

0.75

1

F(p)

p

Figure 5: Threshold values for b(w) = 1− (1− w)2 (green), w (orange) and w2 (blue)

In each case, the unstable equilibrium between 0 and 1 is an invasion threshold, above which level theinfection will spread through the whole population. In general, the threshold is lowest (the infectionreaches fixation most readily) if h is much lower than f : that is, the cost of CI is much greater than thatof fecundity loss.

The particular productivity model also has an effect on the invasion threshold. Figure 5 shows thethresholds for the three models with f = 0.7 and h = 0.6.

The threshold for the diminishing model is lowest, the directly proportional model next and the escalat-ing model highest. In fact this is always the case for fixed values of the parameters f and h.

Compare the thresholds in the lower two cases. We have:

1− 2 f 2 + f 3

1− 2h2 + h3 ≤ 1− f 2

1− h2 (16)

⇔ 1− 2 f 2 + f 3

1− f 2 ≤ 1− 2h2 + h3

1− h2 (17)

Since the function1− 2t2 + t3

1− t2 is monotonic decreasing on [0,1], the condition h ≤ f gives the inequality

in (17). In the higher two cases, we have:

1− f 2

1− h2 ≤ 1− f 3

1− h3 (18)

⇔ 1− f 2

1− f 3 ≤ 1− h2

1− h3 (19)

As1− t2

1− t3 is also monotonic decreasing on [0,1], h ≤ f again implies (19).

2.3 Imperfect maternal transmission

It is possible that the infection in a mother will not be passed on to the entirety of her offspring. In[15], Turelli and Hoffmann record an average of 3–4% uninfected ova produced by infected Drosophilasimulans. To model this we may consider a third parameter, µ in [6], giving the proportion of uninfectedeggs. This gives a different proportion of infected individuals in the second generation (see Table 5).Note that we assume that CI occurs between the sperm of an infected male and an uninfected egg,rather than an uninfected female.

7

U♂ I♂

U♀ (1− pt)2 pt(1− pt)h

I♀ (U eggs) pt(1− pt)µ f p2t µ f h

I♀ (I eggs) pt(1− pt)(1− µ) f p2t (1− µ) f

Table 5: Outcomes of random pairing (imperfect maternal transmission)

In this case the proportion for the next generation of individuals is given by:

pt+1 = F(pt) =pt(1− µ) f

pt(1− µ) f + pt(1− pt)µ f + p2t µ f h + (1− pt)2 + pt(1− pt)h

(20)

=pt(1− µ) f

p2t (1− µ f )(1− h) + pt( f + h− 2) + 1

(21)

This system has an equilibrium point at p = 0 and also at the pair of values:

p =2− ( f + h)±

√(2− f − h)2 − 4(1− µ f )(1− h)(1− f − µ f )

2(1− µ f )(1− h)(22)

Now, considering the case of our Hymenoptera model with the productivity function b(w), the matingtable is as shown in Table 6.

U♂ I♂

U♀ b(1) · (1− pt)2 b(h) · pt(1− pt)h

I♀ (U eggs) b( f ) · pt(1− pt)(1− µ) f b((1− µ) f h + µ f ) · p2t (1− µ) f h

I♀ (I eggs) b( f ) · pt(1− pt)µ f b((1− µ) f h + µ f ) · p2t µ f

Table 6: Outcomes of random pairing (imperfect maternal transmission)

Combining all these expressions gives the formula for the proportion of infected individuals in genera-tion t + 1 as:

pt+1 =b( f ) · pt(1− pt)µ f + b((1− µ) f h + µ f ) · p2

t µ fb(1)(1− pt)2 + b((1− µ) f h + µ f ) f p2(h + µ− hµ) + b( f ) f p(1− p) + b(h)hp(1− p)

(23)

Because of time constraints, I will not consider models with imperfect maternal transmission in thisessay any further.

2.4 Multiple matings

A female may mate with multiple males. In this case, we must consider how many of her partnersare infected. In an uninfected non-social insect which mates a number of times, the proportion of heroffspring which are viable is given by:

No. of uninfected mates + h (No. of infected mates)Total mates

(24)

In the case of a twice-mated female, we have a mating table as shown in Table 7.

8

U♂U♂ U♂I♂ I♂I♂

U♀ (1− pt)3 pt(1− pt)2(h + 1) p2t (1− pt)h

I♀ pt(1− pt)2 f p2t (1− pt) f p3

t f

Table 7: Offspring of twice-mated non-social females

More generally, given an uninfected female which mates n times and where k of her mates carry the

infection, the proportion of her offspring which are viable ishk + (n− k)

n.

The proportion of such matings taking place randomly in an infinite population is:(nk

)(1− pt) · pk

t (1− pt)n−k (25)

and so the proportion of all viable offspring produced is given by:

n

∑k=0

(nk

)hk + (n− k)

npk

t (1− pt)n−k+1 (26)

for all uninfected females and by:n

∑k=0

(nk

)f pk+1

t (1− pt)n−k (27)

for all infected females. Combining these results gives an expression for the infection frequency in thesubsequent generation:

pt+1 = F(pt) =∑ (n

k) f pk+1t (1− pt)n−k

∑ (nk) f pk+1

t (1− pt)n−k + ∑ (nk)

hk + (n− k)n

pkt (1− pt)n−k+1

(28)

In social insects, we must again include the productivity function b(w) in our calculations. Here wehave the proportion of viable sexuals given by:

hk + (n− k)n

· b(

hk + (n− k)n

)(29)

for each uninfected female and so the proportion of viable offspring produced by all uninfected femaleswill be:

n

∑k=0

(nk

)hk + (n− k)

n· b

(hk + (n− k)

n

)pk

t (1− pt)n−k+1 (30)

and the proportion produced by infected females will be:

n

∑k=0

(nk

)f · b( f )pk+1

t (1− pt)n−k (31)

for a female with the infection. If we define Vh(k; n, h) =hk + (n− k)

nthen we may express the formula

for viable offspring in generation t + 1 as:

pt+1 = F(pt) =∑ (n

k) f · b( f )pk+1t (1− pt)n−k

∑ (nk) f · b( f )pk+1

t (1− pt)n−k + ∑ (nk)Vh(k) · b(Vh(k))pk

t (1− pt)n−k+1(32)

9

The equilibria for this system are then at the roots of the equation:

n

∑k=0

(nk

)pk+1(1− p)n−k+1

{Vh(k) · b(Vh(k))− f · b( f )

}= 0 (33)

Clearly, equilibria again exist at 0 and 1. As the number of mates n increases, the number of potentialequilibria also increase: up to a maximum of n + 2. The algebra is much more complex than in the singlemate case but numerical investigation suggests that for f and h on [0, 1] there is only one equilibrium inbetween 0 and 1 and all others are either complex or outside the unit interval.

I used the Mathematica script in Appendix A to calculate numerically all roots of equation (33); se-lect those lying within the open unit interval; and plot the value against the number of mates. Eachgraph has a fixed value of f , and h is a fraction of f ranging from 1/10 f to 9/10 f . I ran the script forvalues of f at 0.3, 0.5, 0.7 and 0.9. The output for the directly proportional, escalating and diminishingproductivity models is shown in Figures 7, 8 and 9 in Appendices B.1, B.2 and B.3 respectively. Figure 6shows the results at a fixed value of f for all three models.

2 4 6 8 10

0.2

0.4

0.6

0.8

1

(a) b(w) = w2 4 6 8 10

0.2

0.4

0.6

0.8

1

(b) b(w) = w2

2 4 6 8 10

0.2

0.4

0.6

0.8

1

(c) b(w) = 1− (1− w)2

Figure 6: Invasion threshold against number of mates with f = 0.5 and h ranging from f10 (orange) to 9 f

10(blue)

The visible trends in the directly proportional model are that the threshold value is lower when the fe-male has a larger number of mates, and when f is high and h is low. Even as low as around 0.1 as seenin Figure 7d, where the female has ten partners and the values of f and h are 0.9 and 0.09 respectively.

In the escalating model the trends are similar but the characteristics in the diminishing case are quitedifferent: when h is close to f , the number of mates makes little difference to the threshold value and forvery high values of f the threshold even increases with increased number of mates.

3 Discussion

In populations with singly-mated queens, the results clearly show that a maternally-inherited infectionwill spread to complete profusion most readily if the reproductive costs are greater for incompatiblematings than they are because of infection-related fecundity loss. Of the three models, infection also

10

spreads most easily if the number of offspring raised by workers is given by a function of the form1− (1− w)2.

In a population where the queen mates multiple times, an increased number of mates nearly alwaysimplies a lower frequency threshold for the infection to reach complete profusion. The only exceptionsin the cases considered were extremal parameter values of the diminishing productivity model wherethere is little fecundity loss or CI cost.

The numerical investigation also shows more clearly the difference in sensitivity to the parameter hin the singly-mated case: the threshold varies very little as h decreases in the escalating model, but ismore sensitive to h in the directly proportional model and still more in the diminishing model. Thusin the latter model, not only is the threshold value lower than other models for a fixed CI cost, a moresevere cost also produces a greater reduction in the threshold value in this model than in the others.

The effect of increasing mate number, however, is most pronounced in the escalating model. Withf = 0.7 and h = 0.07 there is a dramatic drop in the threshold frequency from around 0.7 (one mate) tobelow 0.4 (ten mates). The effect of increased mate number is less pronounced in the directly propor-tional model and there is little to no effect in the diminishing model.

Informative as these results are, there are a large number of factors not taken into account. Firstly, itwas assumed that the infection was passed on to all of the offspring. A model for imperfect mater-nal transmission was presented in §2.3 and, although analytical investigation of the model would betricky given the number of parameters, numerical calculations of the type already discussed could haveyielded useful results given more time. In particular, models with perfect maternal transmission alwaystend towards zero or total infection frequency whereas imperfect transmission allows the persistence ofa partially-infected or polymorphic population.

Infection may also be lost over time because of resistance of the host or exposure to conditions which killthe infection. This occurs frequently in ant populations: Wenseleers et al. measured infection frequencyin adult and pupal F. truncorum workers and found that they dropped from 0.87 at the pupal stage to0.45 in the adult, suggesting that the infection is lost over time. Reuter et al. also measure infectionfrequency in colonies of an invasive species of ant, Linepithema humile and suggested that colonisationof new habitats was a mechanism driving infection loss, since established colonies had higher infectionrates than introduced ones[12]. Turelli and Hoffmann found a decrease in CI effects in D. simulans withincreased male age, which may also be a result of loss of the infection.

An important issue which I have not addressed is the interplay of horizontal and vertical transmis-sion in infections such as Wolbachia. Although the bacterium is primarily maternally inherited, theremust also be a degree of horizontal transmission or the infection would not be able to enter a popula-tion. Based on the model in this essay, the threshold infection frequency must initially be reached viahorizontal transmission—this could be from other infected species living in the same habitat or infectedindividuals which are eaten—before selection based on CI increases infection to the whole population.

Other possible considerations are the presence of multiple strains of the infection in the population,and how the dynamics would behave in a finite population or if locality were considered so individualswere more likely to find mates nearby. This would allow the infection to spread more quickly in a localarea since threshold values would be reached more easily in a small neighbourhood from which theinfection could radiate.

In conclusion, although the model formulated in this essay is too simple to describe realistic bacterialpopulation dynamics, it gives clear and informative predictions as to how easily infection can spreadin an idealised population according to variations in fitness parameters, social structure and matingbehaviour and could hopefully be used as a theoretical basis for more detailed investigation.

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A Mathematica notebook

<< Graphics‘MultipleListPlot‘G[n_] := Sum[Binomial[n, k]*p^(k+1)*(1-p)^(n-k+1)*((1+k*((h-1)/n))^2-f^2),{k,0,n}]f = 0.9graphval = {};For[t = 1, t < 10, t++,coords = {};For[i = 1, i < 11, i++,

h = t*f/10;solution = Solve[G[i] == 0, p];For[j = 1, j < Length[solution] + 1, j++,tvalue = Evaluate[p /. solution[[j]]];If[Im[tvalue] == 0 && 0 < tvalue < 0.999,AppendTo[coords, {i, tvalue}]]

]]

AppendTo[graphval, coords];]

colourvals = Table[{Thickness[0.004],RGBColor[1-i,0.5,i]},{i,0.1,0.9,0.1}]MultipleListPlot[graphval,

AxesOrigin -> {0, 0}, PlotJoined -> True, PlotRange -> {0,1},SymbolShape -> None, PlotStyle -> colourvals]

B Graphs of threshold values against mate number

B.1 Directly proportional model: b(w) = w

2 4 6 8 10

0.2

0.4

0.6

0.8

1

(a) f=0.32 4 6 8 10

0.2

0.4

0.6

0.8

1

(b) f=0.5

2 4 6 8 10

0.2

0.4

0.6

0.8

1

(c) f=0.72 4 6 8 10

0.2

0.4

0.6

0.8

1

(d) f=0.9

Figure 7

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B.2 Escalating model: b(w) = w2

2 4 6 8 10

0.2

0.4

0.6

0.8

1

(a) f=0.32 4 6 8 10

0.2

0.4

0.6

0.8

1

(b) f=0.5

2 4 6 8 10

0.2

0.4

0.6

0.8

1

(c) f=0.72 4 6 8 10

0.2

0.4

0.6

0.8

1

(d) f=0.9

Figure 8

B.3 Diminishing model: b(w) = 1− (1− w)2

2 4 6 8 10

0.2

0.4

0.6

0.8

1

(a) f=0.32 4 6 8 10

0.2

0.4

0.6

0.8

1

(b) f=0.5

2 4 6 8 10

0.2

0.4

0.6

0.8

1

(c) f=0.72 4 6 8 10

0.2

0.4

0.6

0.8

1

(d) f=0.9

Figure 9

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References

[1] Seth Bordenstein and Rebeca B. Rosengaus. Discovery of a novel Wolbachia supergroup in Isoptera.Current Microbiology, 51:393–398, 2005.

[2] Ernst Caspari and G.S. Watson. On the evolutionary importance of cytoplasmic sterility inmosquitoes. Evolution, 13(4):568–570, 1959.

[3] Sylvain Charlat, Gregory D.D. Hurst, and Herve Mercot. Evolutionary consequences of Wolbachiainfections. TRENDS in Genetics, 19(4):217–223, 2003.

[4] A.J. Fry, M.R. Palmer, and D.M. Rand. Variable fitness effects of Wolbachia infection in Drosophilamelanogaster. Heredity, 93:379–389, 2004.

[5] Eleanor R. Haine and James M. Cook. Convergent incidences of Wolbachia infection in fig waspcommunities from two continents. Proc. R. Soc. B, 272:421–429, 2005.

[6] A.A. Hoffmann and M. Turelli. Influential Passengers. Oxford University Press, New York, 1997.

[7] Marjorie A. Hoy, Ayyamperumal Jeyaprakash, Juan M. Alvarez, and Michael H. Allsopp. Wolbachiais present in Apis mellifera capensis, A. m. scutellata and their hybrid in southern africa. Apidologie,34:53–60, 2003.

[8] Gregory D.D. Hurst and Francis M. Jiggins. Male-killing bacteria in insects: Mechanisms, incidence,and implications. Emerging Infectious Diseases, 6(4), 2000.

[9] Gregory D.D. Hurst, Francis M. Jiggins, J. Hinrich Graf von der Schulenberg, Dominique Bertrand,Stuart A. West, Irina I. Goriacheva, Ilia A. Zakharov, John H. Werren, Richard Stouthamer, andMichael E.N. Majerus. Male-killing Wolbachia in two species of insect. Proc. R. Soc. Lond. B., 266:735–740, 1999.

[10] Laurent Keller, Cathy Liautard, Max Reuter, William D. Brown, Lotta Sundstrom, and Michel Cha-puisat. Sex ratio and Wolbachia infection in the ant Formica exsecta. Heredity, 87:227–233, 2001.

[11] Elizabeth A. McGraw and Scott L. O’Neill. Wolbachia pipientis: intracellular infection and patho-genesis in Drosophila. Current Opinion in Microbiology, 7:67–70, 2004.

[12] M. Reuter, J.S. Pedersen, and L. Keller. Loss of Wolbachia infection during colonisation in the inva-sive Argentine ant Linepithema humile. Heredity, 94:364–369, 2005.

[13] Max Reuter and Laurent Keller. Sex ratio conflict and worker production in eusocial hymenoptera.Am. Nat., 158:166–177, 2001.

[14] Max Reuter and Laurent Keller. High levels of multiple Wolbachia infection and recombination inthe ant Formica exsecta. Mol. Biol. Evol., 20(5):748–753, 2003.

[15] Michael Turelli and Ary A. Hoffmann. Cytoplasmic incompatibility in Drosophila simulans: Dynam-ics and parameter estimates from natural populations. Genetics, 140:1319–1338, 1995.

[16] T. Wenseleers, F. Ito, S. Van Borm, R. Huybrechts, F. Volckaert, and J. Billen. Widespread occurrenceof the micro-organism Wolbachia in ants. Proc. R. Soc. Lond. B, 265:1447–1452, 1998.

[17] T. Wenseleers, L. Sundstrom, and J.Billen. Deleterious Wolbachia in the ant Formica truncorum. Proc.R. Soc. Lond. B, 269:623–629, 2002.

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