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 arthropod species. The effect of the bacterium on its host varies with different genotypes but a common trait
is manipulation of host reproductive biology, including inducing cytoplasmic incompatibility.
In this essay, I briefly summarise reproductive phenotypes associated with Wolbachia in insects, present
a model of the dynamics of infection frequency for maternally-inherited microorganisms and determine the effect of changing costs associated with incompatibility, infection-related fecundity loss and
number of mates in ideal populations of social insects. The model predicts that infections of this type
spread to fixation most easily in populations with a high cost due to incompatibility, little fecundity
loss and where females have a large number of mates.
Contents
1
2
3
Introduction
1.1 Symbionts in nature . . . . .
1.2 Wolbachia infection in insects
1.3 Cytoplasmic incompatibility
1.4 Infection in Hymenoptera .
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1
1
1
2
3
Models
2.1 Classical model . . . . . . . . . .
2.2 Worker productivity . . . . . . .
2.3 Imperfect maternal transmission
2.4 Multiple matings . . . . . . . . .
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4
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4
7
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Discussion
10
A Mathematica notebook
12
B Graphs of threshold values against mate number
B.1 Directly proportional model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.2 Escalating model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.3 Diminishing model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
12
13
13
1
Introduction
1.1
Symbionts in nature
Symbiosis is a broad term which describes a close association between two organisms from different
species. Commonly, at least one of the organisms derives a benefit from this association. If both organisms benefit, the relationship is described as mutualistic. If one organism benefits at the expense of the
other, the relationship is termed parasitic. Other classes of symbiosis are also possible and organisms
need 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 return
for a supply of honeydew—an energy-rich substance which the aphids secrete (Figure 1)1 . Another possibility is that one of the organisms may live inside the body or cells of the other. This relationship is
called endosymbiosis, with the internal organism referred to as an endosymbiont. Endosymbionts which
cannot 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 advertising campaigns have also raised public awareness of bacteria which have a mutualistic relationship
with humans, aiding digestion of milk proteins and complex carbohydrates and synthesising vitamins
(the nominally “friendly bacteria”). The symbiotic bacterium Rhizobia aids plants in absorbing nitrogen
from the soil and a number of symbiotic bacteria and protists are essential for enabling insects to digest
the cellulose in plant matter.
Symbiotic relationships can drive evolutionary development. Mutualistic relationships form a positive feedback loop whereby the success of one species improves the fitness of the other, and hence its
own. This is believed to be the driving force behind the sudden radiation of both angiosperms (flowering plants) and pollinating insects in the mid-Cretaceous period. Parasitic symbionts can also influence
selection pressures on their hosts in favour of genotypes which minimise the negative effect of the parasite.
1.2
Wolbachia infection in insects
Wolbachia is a genus of bacteria which infects a substantial number of arthropod and filarial nematode
species and frequently influences the reproductive development and behaviour of its host. It lives within
1 Photo
©Alex Wild
1
the cells of its host, in highest concentrations in the reproductive tissues and is typically vertically inherited 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 Table
1). Wolbachia is an obligate endosymbiont and its relationship with the host ranges from parasitism
to mutualism depending on host genotype, Wolbachia strain, location in the host and environmental
influences[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 sex
ratios of infected populations in its favour. Although ratios would naturally be around 50:50 male and
female, some infected arthropods have entirely or almost entirely female populations. The bacterium
manages this in a number of ways.
Male killing (often abbreviated to MK) is one of these methods and has been observed in infected butterflies (Acraea encedon) and ladybirds (Adalia bipunctata)[9] amongst other arthropod species. The sons
of infected females die during embryogenesis [3] and only daughters develop. The benefit to the bacteria is that all offspring will be able to pass the infection on. There are also potential advantages for
the female hosts: a food source (they may eat their dead brothers); reduced sibling competition; and
reduced chances of inbreeding. Wolbachia is not the only bacterium with this trait: [8] lists a number of
other 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 some
populations of Armadillidium vulgare, the pill woodlouse, Wolbachia is solely responsible for sex determination. To counteract skewed sex ratios, there is an incentive on the woodlice to produce more male
offspring. In some cases this happened to the extent that the entire population became genetically male
and the factor determining sex determination switched to whether or not the individual was infected
with 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 that
females will be fertilised and so the next generation will produce more males. In order to guarantee its
transmission, Wolbachia is able to double the chromosome number of unfertilised eggs so a mother can
produce 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, Wolbachia
exacts a cost on pairings of this type. This is commonly a reduced egg hatch rate: Turelli and Hoffmann
recorded hatch rates from incompatible matings of Drosophila simulans at 30–70% as high as those from
compatible matings[15]. In a separate study by Fry et al., however, the observed effect in D. melanogaster
was 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 infected
females by diminishing the reproductive success of uninfected females: effectively using the infected
males, 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 male
and female carrying different strains. Since there is selection against pairings between individuals carrying different strains the population may eventually partition into two separate species. Bidirectional CI
can 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 the
order live in eusocial colonies: groups of individuals in which a proportion of the population are sterile
and 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. After mating, the queen controls which of the eggs are fertilised although the workers are responsible for
rearing the larvae (see Fig 2)2 and determine which develop into workers and which into sexual females
or gynes. Since workers are more closely related to their sisters they should theoretically favour a bias
towards investment of the colony’s resources in females. This creates a conflict with the queen over
colony sex ratio. In [13], Reuter and Keller calculate evolutionarily stable levels for the proportion of
resources 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 from
25 out of 50 species of ant in Java and Sumatra screened positive for a particular strain of Wolbachia. In a
study of a single Swiss population of the ant Formica exsecta, Reuter and Keller found that all ants tested
were 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 association between prevalence of infection and colony sex ratio in the expected direction[10] and a later
2 Photo
©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 honeybee
Apis mellifera capensis[7], fig wasp (family Agaonidae)[5] and termite species Zootermopsis angusticollis
and Z. nevadensis[1].
In this essay I will give a theoretical analysis of the spread of a vertically transmitted endosymbiont
such as Wolbachia and attempt to determine what effect the social structure and mating behaviour of
eusocial insects such as ants might have on the ability of the infection to spread through the population
over successive generations.
2
2.1
Models
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 frequency
of infection in the population is written as pt for generation t. We assume that CI causes a reduction in
the number of viable offspring when an infected male mates with an uninfected female. In this case, we
denote the number of viable offspring (as a proportion of the maximum number with no incompatibility) by h (0 < h < 1). Infected females may also have a reduced fecundity, also resulting in reduced
brood 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 individual
and I denotes infection.
U♂
I♂
U♀
(1 − p t )2
p t (1 − p t ) h
I♀
p t (1 − p t ) f
p2t f
Table 3: Outcomes of random pairing (perfect maternal transmission)
The infection frequency in the next generation is then given by:
Infected individuals
= F ( pt )
All individuals
=
=
pt (1 − pt ) f + p2t f
pt (1 − pt ) f + p2t f + (1 − pt )2 + pt (1 − pt )h
pt f
2
(1 − p t ) + p t f + p t h (1 − p t )
The infection frequency reaches an equilibrium at p = 0, p = 1 and p =
f = 0.7, h = 0.5).
(1)
(2)
1− f
(see Figure 3 for the case
1−h
dF
< 1. As noted in [6], the equilibria at 0 and 1 are
dpt
stable, while the intermediate point is unstable and the final infection frequency reached in this model
depends on whether the initial frequency is above or below this point.
An equilibrium point is locally stable if −1 <
2.2
Worker productivity
In eusocial colonies, a queen mates and produces an initial brood of workers who then raise the next
generation of sexual individuals. Because of this two-stage process, the number of sexual individuals
4
1
F(p)
0.75
0.5
0.25
p
0
0.25
0.5
0.75
1
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 available 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 maximum
and let b(w) be a productivity function describing the proportion of sexual individuals which are raised
successfully based on the number of workers. Productivity is at a maximum when w = 1 and zero when
w = 0. Taking this into account gives a new mating table as shown in Table 4.
U♂
I♂
U♀
b (1) · (1 − p t )2
b ( h ) · p t (1 − p t ) h
I♀
b ( f ) · p t (1 − p t ) 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:
p t +1 = F ( p t )
=
=
b( f ) · pt (1 − pt ) f + b( f ) · p2t f
b( f ) · pt (1 − pt ) f + b( f ) · p2t f + b(1) · (1 − pt )2 + b(h) · pt (1 − pt )h
b( f ) · pt f
b(1) · (1 − pt )2 + b( f ) · pt f + b(h) · hpt (1 − pt )
(3)
(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 case
we have:
f 2 pt
p t +1 = 2
(5)
f p t + (1 − p t )2 + h2 p t (1 − p t )
Equilibrium points are the roots of the cubic p3 (1 − h2 ) + p2 ( f 2 + h2 − 2) + p(1 − f 2 ): 0,1, and
1 − f2
.
1 − h2
dF
h2
= 2 so this point is stable as long as h < f . Heuristically, a population may maintain a
dpt
f
zero infection level whatever the effect of infection but can only maintain a population-wide infection if
the loss of offspring due to CI is more severe than the loss due to reduced fecundity.
At p = 1,
For the third equilibrium, since p is on [0, 1] we must have 0 ≤
5
1 − f2
≤ 1 and hence h ≤ f . But
1 − h2
for the equilibrium at this point to be stable we have:
dF
<1
dpt
⇒
f 4 − 2 f 2 + h2
<1
f 2 ( h2 − 1)
(6)
⇒
f 4 − 2 f 2 + h2 > f 2 h2 − f 2
(7)
2
⇒
⇒
2
2
2
f ( f − 1) > h ( f − 1)
f <h
(8)
(9)
So any equilibrium between 0 and 1 will be an unstable one.
In general, the equilibrium points are the solutions of:
n
o
n
o
n
o
p3 1 − b ( h ) · h + p2 b ( f ) · f + b ( h ) · h − 2 + p 1 − b ( f ) · f
which are 0, 1 and
(10)
1 − b( f ) · f
.
1 − b(h) · h
Two possibilities for alternative productivity functions are an escalating model: b(w) = w2 ; and a diminishing model: b(w) = 1 − (1 − w)2 (see Figure 4).
1
b(w)
0.75
0.5
0.25
w
0
0.25
0.5
0.75
1
Figure 4: Three possible functions for productivity b(w)
These have potential non-boundary equilibrium points at p =
1 − f3
1 − 2f2 + f3
and
p
=
respectively.
1 − h3
1 − 2h2 + h3
For b(w) = w2 , a similar working to (6)–(9) shows that the equilibrium is unstable again.
In the third case, we have:
f 2 (2 − f )(1 − 2 f 2 + f 3 ) + f 2 (2 − f ) − h2 (2 − h)
dF
=
dpt
f 2 (2 − f )(1 − 2h2 + h3 )
(11)
And so:
dF
dpt
< 1
⇔ f 2 (2 − f ) − h2 (2 − h ) <
⇔ f 2 (2 − f ) − h2 (2 − h ) <
(12)
n
f 2 (2 − f ) (1 − 2h2 + h3 ) − (1 − 2 f 2 + h3 )
n
o
f 2 (2 − f ) f 2 (2 − f ) − h2 (2 − h )
⇔ f 2 (2 − f ) < h2 (2 − h )
o
(13)
(14)
(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
1
F(p)
0.75
0.5
0.25
p
0
0.25
0.5
0.75
1
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 the
infection will spread through the whole population. In general, the threshold is lowest (the infection
reaches fixation most readily) if h is much lower than f : that is, the cost of CI is much greater than that
of fecundity loss.
The particular productivity model also has an effect on the invasion threshold. Figure 5 shows the
thresholds 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 escalating 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 − 2f2 + f3
1 − 2h2 + h3
1 − 2f2 + f3
⇔
1 − f2
≤
≤
1 − f2
1 − h2
1 − 2h2 + h3
1 − h2
(16)
(17)
1 − 2t2 + t3
is monotonic decreasing on [0,1], the condition h ≤ f gives the inequality
1 − t2
in (17). In the higher two cases, we have:
Since the function
1 − f2
1 − h2
1 − f2
⇔
1 − f3
As
2.3
≤
≤
1 − f3
1 − h3
1 − h2
1 − h3
(18)
(19)
1 − t2
is also monotonic decreasing on [0,1], h ≤ f again implies (19).
1 − t3
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 Drosophila
simulans. To model this we may consider a third parameter, µ in [6], giving the proportion of uninfected
eggs. 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♂
(1 − p t )2
p t (1 − p t ) h
I♀ (U eggs)
p t (1 − p t ) µ f
p2t µ f h
I♀ (I eggs)
pt (1 − pt )(1 − µ) f
p2t (1 − µ) f
U♀
Table 5: Outcomes of random pairing (imperfect maternal transmission)
In this case the proportion for the next generation of individuals is given by:
p t +1 = F ( p t )
=
=
p t (1 − µ ) f
pt (1 − µ) f + pt (1 − pt )µ f + p2t µ f h + (1 − pt )2 + pt (1 − pt )h
p t (1 − µ ) f
2
pt (1 − µ f )(1 − h) + pt ( f + h − 2) + 1
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 )
p=
2(1 − µ f )(1 − h)
(20)
(21)
(22)
Now, considering the case of our Hymenoptera model with the productivity function b(w), the mating
table is as shown in Table 6.
U♂
I♂
b (1) · (1 − p t )2
b ( h ) · p t (1 − p t ) 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 ) · p t (1 − p t ) µ f
b((1 − µ) f h + µ f ) · p2t µ f
U♀
Table 6: Outcomes of random pairing (imperfect maternal transmission)
Combining all these expressions gives the formula for the proportion of infected individuals in generation t + 1 as:
p t +1 =
b( f ) · pt (1 − pt )µ f + b((1 − µ) f h + µ f ) · p2t µ f
b(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 this
essay any further.
2.4
Multiple matings
A female may mate with multiple males. In this case, we must consider how many of her partners
are infected. In an uninfected non-social insect which mates a number of times, the proportion of her
offspring which are viable is given by:
No. of uninfected mates + h (No. of infected mates)
Total mates
In the case of a twice-mated female, we have a mating table as shown in Table 7.
8
(24)
U♂U♂
U♂I♂
I♂I♂
U♀
(1 − p t )3
p t (1 − p t )2 ( h + 1)
p2t (1 − pt )h
I♀
p t (1 − p t )2 f
p2t (1 − pt ) f
p3t 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
hk + (n − k )
infection, the proportion of her offspring which are viable is
.
n
The proportion of such matings taking place randomly in an infinite population is:
n
(1 − pt ) · pkt (1 − pt )n−k
k
(25)
and so the proportion of all viable offspring produced is given by:
n
n hk + (n − k ) k
p t (1 − p t ) n − k +1
∑ k
n
k =0
(26)
for all uninfected females and by:
n
n
∑ k f pkt +1 (1 − pt )n−k
k =0
(27)
for all infected females. Combining these results gives an expression for the infection frequency in the
subsequent generation:
n
p t +1 = F ( p t ) =
∑ ( k ) f pkt +1 (1 − pt )n−k
n
n hk + ( n − k ) k
p t (1 − p t ) n − k +1
∑ ( k ) f pkt +1 (1 − pt )n−k + ∑ ( k )
n
(28)
In social insects, we must again include the productivity function b(w) in our calculations. Here we
have the proportion of viable sexuals given by:
hk + (n − k)
hk + (n − k)
(29)
·b
n
n
for each uninfected female and so the proportion of viable offspring produced by all uninfected females
will be:
n hk + (n − k )
n hk + (n − k)
·b
pkt (1 − pt )n−k+1
(30)
∑ k
n
n
k =0
and the proportion produced by infected females will be:
n
n
∑ k f · b( f ) pkt +1 (1 − pt )n−k
k =0
for a female with the infection. If we define Vh (k; n, h) =
for viable offspring in generation t + 1 as:
p t +1 = F ( p t ) =
hk + (n − k)
then we may express the formula
n
n
∑ ( k ) f · b( f ) pkt +1 (1 − pt )n−k
n
n
∑ ( k ) f · b( f ) pkt +1 (1 − pt )n−k + ∑ ( k )Vh (k) · b(Vh (k)) pkt (1 − pt )n−k+1
9
(31)
(32)
The equilibria for this system are then at the roots of the equation:
n
o
n n
∑ k pk+1 (1 − p)n−k+1 Vh (k) · b(Vh (k)) − f · b( f ) = 0
k =0
(33)
Clearly, equilibria again exist at 0 and 1. As the number of mates n increases, the number of potential
equilibria also increase: up to a maximum of n + 2. The algebra is much more complex than in the single
mate case but numerical investigation suggests that for f and h on [0, 1] there is only one equilibrium in
between 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); select those lying within the open unit interval; and plot the value against the number of mates. Each
graph 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 for
values of f at 0.3, 0.5, 0.7 and 0.9. The output for the directly proportional, escalating and diminishing
productivity models is shown in Figures 7, 8 and 9 in Appendices B.1, B.2 and B.3 respectively. Figure 6
shows the results at a fixed value of f for all three models.
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
2
4
6
8
2
10
4
6
8
10
(b) b(w) = w2
(a) b(w) = w
1
0.8
0.6
0.4
0.2
2
4
6
8
10
(c) b(w) = 1 − (1 − w)2
Figure 6: Invasion threshold against number of mates with f = 0.5 and h ranging from
(blue)
f
10
(orange) to
9f
10
The visible trends in the directly proportional model are that the threshold value is lower when the female has a larger number of mates, and when f is high and h is low. Even as low as around 0.1 as seen
in 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 quite
different: when h is close to f , the number of mates makes little difference to the threshold value and for
very 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 infection
will spread to complete profusion most readily if the reproductive costs are greater for incompatible
matings 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 form
1 − (1 − w )2 .
In a population where the queen mates multiple times, an increased number of mates nearly always
implies a lower frequency threshold for the infection to reach complete profusion. The only exceptions
in the cases considered were extremal parameter values of the diminishing productivity model where
there is little fecundity loss or CI cost.
The numerical investigation also shows more clearly the difference in sensitivity to the parameter h
in the singly-mated case: the threshold varies very little as h decreases in the escalating model, but is
more sensitive to h in the directly proportional model and still more in the diminishing model. Thus
in the latter model, not only is the threshold value lower than other models for a fixed CI cost, a more
severe 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. With
f = 0.7 and h = 0.07 there is a dramatic drop in the threshold frequency from around 0.7 (one mate) to
below 0.4 (ten mates). The effect of increased mate number is less pronounced in the directly proportional 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, it
was assumed that the infection was passed on to all of the offspring. A model for imperfect maternal transmission was presented in §2.3 and, although analytical investigation of the model would be
tricky given the number of parameters, numerical calculations of the type already discussed could have
yielded useful results given more time. In particular, models with perfect maternal transmission always
tend towards zero or total infection frequency whereas imperfect transmission allows the persistence of
a partially-infected or polymorphic population.
Infection may also be lost over time because of resistance of the host or exposure to conditions which kill
the infection. This occurs frequently in ant populations: Wenseleers et al. measured infection frequency
in adult and pupal F. truncorum workers and found that they dropped from 0.87 at the pupal stage to
0.45 in the adult, suggesting that the infection is lost over time. Reuter et al. also measure infection
frequency in colonies of an invasive species of ant, Linepithema humile and suggested that colonisation
of new habitats was a mechanism driving infection loss, since established colonies had higher infection
rates than introduced ones[12]. Turelli and Hoffmann found a decrease in CI effects in D. simulans with
increased 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 transmission in infections such as Wolbachia. Although the bacterium is primarily maternally inherited, there
must also be a degree of horizontal transmission or the infection would not be able to enter a population. Based on the model in this essay, the threshold infection frequency must initially be reached via
horizontal transmission—this could be from other infected species living in the same habitat or infected
individuals 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 individuals
were more likely to find mates nearby. This would allow the infection to spread more quickly in a local
area since threshold values would be reached more easily in a small neighbourhood from which the
infection could radiate.
In conclusion, although the model formulated in this essay is too simple to describe realistic bacterial
population dynamics, it gives clear and informative predictions as to how easily infection can spread
in an idealised population according to variations in fitness parameters, social structure and mating
behaviour and could hopefully be used as a theoretical basis for more detailed investigation.
11
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 = 0.9
graphval = {};
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
B.1
Graphs of threshold values against mate number
Directly proportional model: b(w) = w
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
2
4
6
8
2
10
(a) f=0.3
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
2
4
4
6
8
10
(b) f=0.5
6
8
10
2
(c) f=0.7
4
(d) f=0.9
Figure 7
12
6
8
10
B.2
Escalating model: b(w) = w2
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
2
4
6
8
10
2
(a) f=0.3
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
2
4
4
6
8
10
6
8
10
6
8
10
6
8
10
(b) f=0.5
6
8
10
2
(c) f=0.7
4
(d) f=0.9
Figure 8
B.3
Diminishing model: b(w) = 1 − (1 − w)2
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
2
4
6
8
2
10
(a) f=0.3
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
2
4
4
(b) f=0.5
6
8
10
2
(c) f=0.7
4
(d) f=0.9
Figure 9
13
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14