Coevolutionary dynamics of polyandry and
sex-linked meiotic drive
Short title: Polyandry and selfish genetic elements
Luke Holman1, Thomas A.R. Price2, Nina Wedell3 and Hanna Kokko1,4
[email protected]
1
Centre of Excellence in Biological Interactions,
Division of Ecology, Evolution & Genetics,
Research School of Biology,
Australian National University,
Canberra, ACT 0200, Australia.
2
Institute of Integrative Biology,
University of Liverpool,
Liverpool, L69 7ZB, UK.
Biosciences,
University of Exeter,
Cornwall Campus,
Penryn TR10 9FE, UK.
3
4
Institute of Evolutionary Biology and Environmental Sciences,
University of Zurich,
Winterthurerstrasse 190,
CH-8057 Zurich, Switzerland
Word count: 6267 in main text
Keywords: Drosophila, extinction, meiotic drive, sex chromosome, t haplotype, sperm
competition
1
2
Abstract
3
fixation and cause extinction via a shortage of one sex, but in nature they are
4
often found at low, stable frequencies. One potential resolution to this long-
5
standing puzzle involves female multiple mating (polyandry). Because many
6
meiotic drivers severely reduce the sperm competitive ability of their male
7
carriers, females are predicted to evolve more frequent polyandry and thereby
8
promote sperm competition when a meiotic driver invades. Consequently, the
9
driving chromosome’s relative fitness should decline, halting or reversing its
10
spread. We used formal modeling to show that this initially appealing hypothesis
11
cannot resolve the puzzle alone: other selective pressures (e.g. low fitness of
12
drive homozygotes) are required to establish a stable meiotic drive
13
polymorphism. However, polyandry and meiotic drive can strongly affect one
14
another’s frequency, and polyandrous populations may be resistant to the
15
invasion of rare drive mutants.
Segregation distorters located on sex chromosomes are predicted to sweep to
16
17
Introduction
18
increase their representation in subsequent generations, often at the expense of
19
the fitness of the rest of the genome (Burt and Trivers 2006). They are
20
ubiquitous in living organisms, and the intragenomic conflicts they create have
21
major impacts on the evolution of sex, genetic systems, host ecology and
22
population dynamics (Hurst and Werren 2001; Burt and Trivers 2006; Charlat et
23
al. 2007a; Werren 2011; Wedell 2013).
24
Selfish genetic elements subvert normal patterns of DNA replication in ways that
Segregation distorters, such as meiotic drivers, are selfish genetic
25
elements
that
manipulate
gametogenesis
and
thereby
enhance
their
26
representation in the gametes relative to non-distorting elements (Burt and
27
Trivers 2006). This transmission advantage is expected to cause meiotic drivers
28
to rapidly go to fixation, assuming that drive-bearing and non-drive-bearing
29
individuals have equivalent survival and reproductive success (Ardlie 1998; Burt
30
and Trivers 2006). Several species possess meiotic drivers located on an X
31
chromosome, which is transmitted to up to 100% of the progeny of male carriers
32
(Beckenbach 1978; James and Jaenike 1990; Presgraves et al. 1997; Ardlie 1998;
33
Cazemajor et al. 2000; Hurst and Werren 2001; Charlat et al. 2007a; Werren
34
2011; Wedell 2013). These X-linked drivers can create strongly female biased
35
population-wide sex ratios (Jaenike 2001; Burt and Trivers 2006), potentially
36
causing extinction due to a shortage of males. Although putative population
37
crashes caused by meiotic drive have occasionally been reported (Pinzone and
38
Dyer 2013), meiotic drive has often been found in nature at stable, intermediate
39
frequencies over wide geographic areas and long periods of time (Dobzhansky
40
1958; Huang et al. 2001; Dyer 2012). What maintains polymorphisms for meiotic
41
drive (i.e. co-existence of driving and non-driving chromosomes or alleles) is
42
therefore a long-standing puzzle, as is the lower-than-expected frequency of
43
drive in some natural populations (e.g. Lewontin 1968; Charlesworth and Hartl
44
1978; Taylor and Jaenike 2002; Safronova and Chubykin 2013; Auclair et al.
45
2013).
46
Why don’t all meiotic drivers spread to fixation? Past attempts to explain
47
polymorphisms have mainly sought sources of negative frequency-dependent
48
selection on drive-carrying individuals. Under negative frequency-dependent
49
selection, a meiotic driver might spread when at low frequencies because of its
50
transmission advantage in segregation, and decline at high frequencies because
51
of the reduced fitness of its bearers. Low fitness of meiotic drive homozygotes is
52
one possible source of negative frequency-dependent selection, because drive
53
homozygotes increase in frequency as the driver becomes more common.
54
Accordingly, several meiotic drivers, including the t haplotype of mice (Ardlie
55
1998) and meiotic drive in some Drosophila (Dyer et al. 2007; Larracuente and
56
Presgraves 2012), cause sterility, death or milder deleterious effects in
57
homozygotes. These effects might arise because genomic regions near meiotic
58
drivers often undergo little or no recombination (Silver and Artzt 1981; Dyer et
59
al. 2007; Larracuente and Presgraves 2012), allowing the buildup of deleterious
60
genetic material at sites linked to the driver. However, meiotic drive
61
homozygotes appear to have comparable fitness in some systems (Powell 1997;
62
Price et al. 2012a). Costs to homozygotes are therefore likely to be only part of
63
the answer, especially since meiotic drive is often found at even lower
64
frequencies than predicted under homozygote lethality or sterility (the so-called
65
"t-paradox"; Lewontin 1968; Ardlie and Silver 1998; Manser et al. 2011).
66
Taylor and Jaenike (2002; 2003) modeled another solution, based on
67
observations that drive-carrying males experience an especially severe decline in
68
their ability to fertilize females and outcompete other males’ sperm after mating
69
a number of times (Wu 1983; Jaenike 1996; Price et al 2008a). This hypothesis is
70
appealing because there is abundant evidence that drive-carrying males are
71
disadvantaged in sperm competition relative to non-carrier males, often because
72
their Y-bearing sperm fail to develop (e.g. Wilkinson and Fry 2001; Atlan et al.
73
2004; Wilkinson et al. 2006; Angelard et al. 2008; Price and Wedell 2008; Price
74
et al. 2008a; Manser et al. 2011). Male mating rate is expected to increase as the
75
driving X (and hence females) becomes more common, so the comparatively
76
poor ability of drive-carrying males to fertilize multiple females might impose
77
negative frequency-dependent selection on meiotic drive. However, Taylor and
78
Jaenike (2002; 2003) also noted that sperm competition will generally become
79
less frequent as the sex ratio shifts towards females, because fewer females will
80
encounter multiple potential mates. This should impose positive frequency-
81
dependent selection on drive, because the growing rarity of female multiple
82
mating should increase the relative fitness of the driving X as it invades. Because
83
of these conflicting effects of population-wide sex ratio on drive males’ fitness,
84
polyandry only prevented the fixation of drive in a limited set of conditions in
85
Taylor
86
spermatogenesis in drive males is a general resolution to the current problem.
and
Jaenike’s
model,
making
it
unclear
whether
impaired
87
Recent data suggest another potential source of negative frequency-
88
dependent selection arising from sperm competition, which relies on female
89
evolutionary responses to the presence of drive-carrying males (Wedell 2013).
90
When drive males are disadvantaged in sperm competition, females can reduce
91
the proportion of their eggs that are fertilized by meiotic drive males by mating
92
with multiple males (Haig and Bergstrom 1995; Wilkinson and Fry 2001; Price et
93
al. 2010; Manser et al. 2011). An experimental evolution study found that
94
females indeed evolved higher frequencies of multiple mating when a meiotic
95
driver that negatively affects sperm competitive ability in male carriers was
96
introduced into the population (Price et al. 2008b). Moreover, geographical
97
clines in the frequency of X-linked meiotic drivers in North American
98
populations of Drosophila pseudoobscura and D. neotestacea correlate with the
99
local frequency of female multiple mating (Pinzone and Dyer 2013; Price et al.
100
2014). Together with abundant evidence that polyandry can rapidly respond to
101
selection (e.g. Harano and Miyatake 2007; Price et al. 2008b), this correlation is
102
consistent with coevolution between polyandry and meiotic drive in the wild.
103
It is not straightforward to assess whether coevolution between
104
polyandry and meiotic drive provides a sufficiently strong source of negative
105
frequency-dependent selection to prevent the fixation of meiotic drive.
106
Intuitively, as the meiotic driver increases in frequency in a population due to its
107
transmission advantage, females might evolve to be more polyandrous, causing
108
the driver to go into decline due to its disadvantage in sperm competition. As the
109
meiotic driver falls in frequency, females might evolve lower polyandry, since
110
multiple mating often has costs (e.g. Wigby and Chapman 2005) and therefore
111
should provide weaker net fitness returns when drive-bearing males are scarce.
112
Whether these coevolutionary dynamics are able to protect genetic
113
polymorphism for driving X chromosomes is difficult to predict without formal
114
modeling. To address these questions, we used genetically explicit models to
115
examine the potential for (and evolutionary consequences of) coevolution
116
between polyandry and driving X chromosomes. Model 1 is analytical and hence
117
can be investigated more thoroughly, but it makes several simplifying
118
assumptions. These assumptions are relaxed in Model 2, which is a stochastic
119
individual-based simulation.
120
121
Model 1: A deterministic simulation
122
determination and non-overlapping generations. The population contains non-
123
driving X chromosomes, driving X chromosomes and Y chromosomes, which we
124
denote X, X* and Y respectively. XY males have fair meiosis, but X*Y males
125
transmit an X* chromosome with probability (1 + d)/2 and a Y with probability (1
126
− d)/2. Therefore, d = 0 denotes even segregation of sex chromosomes in X*Y
127
males, and d = 1 complete meiotic drive (0 ≤ d ≤ 1). When d = 1, all eggs fertilized
128
by X*Y males develop into females.
We consider a panmictic population composed of two sexes with XX/XY sex
129
To be able to derive analytical expressions for the evolving frequency of
130
polyandrous or monandrous females, we assume the simplest possible genetic
131
control of polyandry: a single, haploid autosomal locus with two alleles A and a.
132
We assume random mating, and that females mate either once or twice
133
depending on whether they carry allele A or a respectively. Males carry the A/a
134
locus but do not express it. Since there are two possible genotypes at the A/a
135
locus and five sex chromosomes genotypes (XX, X*X and X*X* females, plus XY
136
and X*Y males), the model must track ten genotypes. The assumption of
137
haploidy at the A/a locus is artificial but greatly simplifies the model, and is
138
unlikely to affect its qualitative conclusions; this assumption is relaxed in Model
139
2 below.
140
We further assume that polyandrous females (those carrying the A allele)
141
pay a fecundity cost such that their fecundity is multiplied by 1 – p, where 0 ≤ p ≤
142
1. The parameter p represents the costs to females of mating multiply (e.g. due to
143
extra mate search costs or additional harm from contact with males). We also
144
implemented a fecundity cost h (0 ≤ h ≤ 1) to females with the genotype X*X*, in
145
order to investigate the joint evolutionary consequences of polyandry and
146
homozygote disadvantage. Females with the genotype aX*X* thus have fecundity
147
1 – h and those with the genotype AX*X* have fecundity (1 – p)(1 – h); that is, we
148
assume multiplicative fitness costs of polyandry and X* homozygosity. All other
149
female genotypes had a fecundity of 1 or 1 – p, depending on whether they
150
carried the a or A allele respectively.
151
Tracking offspring genotypes
152
For singly mating females, the expected frequency of each type of offspring can
153
be readily predicted from the parental genotypes, assuming random mating,
154
Mendelian segregation in females and XY males, and biased segregation in X*Y
155
males (when d > 0). For example, fertilizations of the eggs of an aXX female by an
156
AX*Y male produce aX*X and AX*X daughters with frequency (1 + d)/4 each,
157
and aXY and AXY sons with frequency (1 − d)/4 each. Other frequencies are
158
derived analogously. Our assumption of random mating follows the finding that
159
drive-carrying males often appear to have equivalent mating success (e.g. Price
160
and Wedell 2008; Price et al. 2012b).
161
The expected genotypes of the offspring of twice-mated females also
162
depend on the outcome of sperm competition. The frequency of a given type of
163
“mating trio” equals 2fm1m2, where f, m1 and m2 refer to the genotype frequencies
164
of the female and her two mates. We assume that males of different genotypes
165
are equally likely to mate in the first male or second male roles, meaning that
166
sperm precedence effects (Parker et al. 1990) are inconsequential for our model.
167
There are 3 polyandrous female genotypes and 4 male genotypes, so we must
168
track 30 possible mating trios.
169
Paternity is shared equally when both males are XY or both are X*Y.
170
However when one male is XY and the other is X*Y, the XY male sires a
171
proportion 1/(1 + c) of the offspring and the X*Y males sires the remaining c/(1 +
172
c). The parameter c (0 ≤ c ≤ 1) thus measures the relative competitiveness of X*Y
173
males in sperm competition (averaged across the first and last male roles): c = 0
174
means X*Y males gain no offspring if the female has also mated with an XY male,
175
and c = 1 means X*Y and XY males share paternity equally when they mate with
176
the same female.
177
Model 1 also assumes that monandrous females always mate once, and
178
polyandrous females always mate twice, irrespective of the population sex ratio.
179
Although potentially unrealistic, this assumption is important if Model 1 is to
180
exclude the sources of frequency dependent selection modeled by Taylor and
181
Jaenike (2002); this assumption is relaxed in Model 2. Because we assume no
182
sperm limitation and an invariant number of mates per female for all frequencies
183
of X*, any instances of X chromosome polymorphism in Model 1 must result from
184
distinct evolutionary processes to those previously studied.
185
Although the model yields analytical expressions for the frequencies of
186
each genotype from one generation to the next, the solutions are very complex.
187
We therefore determined equilibrium gene frequencies numerically, by
188
iteratively determining the predicted offspring genotype frequencies from the
189
parental ones given the model’s four global parameters: d, c, p and h. Each
190
generation, we normalized offspring genotype frequencies to sum to unity by
191
dividing them by the sum of the frequencies. The model therefore assumes a
192
large population in which genetic drift is negligible (again, this assumption is
193
relaxed in Model 2).
194
Note that although Model 1 does not include mate or sperm limitation (i.e.
195
all females were assumed to be fully fertile even when males were rare; this
196
assumption is removed in Model 2), the output of the model can be interpreted
197
such that parameter spaces that drastically reduce the frequency of males would
198
likely result in extinction in the real world.
199
200
Model 2: A stochastic simulation
201
inheritance at the polyandry locus, infinite population size (negating genetic
202
drift), discrete time, and that mating and fertilization occur just as efficiently
203
when males are scare. To relax these assumptions and verify the generality of
204
Model 1’s conclusions, we constructed an individual-based stochastic simulation
205
in continuous time, termed Model 2. Model 2 uses the Gillespie algorithm, which
206
allows us to model discrete stochastic events (e.g. matings, deaths) that occur at
207
different rates in continuous time, by considering the exponentially distributed
208
time it takes for a next event to occur, as well as events that occur after a fixed
209
time has elapsed (e.g. maturation). For details of the algorithm see Kokko and
210
Heubel (2011). The life cycle proceeds as follows.
Model 1 has a number of limitations. For simplicity, it assumed haploid
211
Individuals carry two diploid loci: the A/a female mating behavior locus,
212
and sex chromosomes, which can be X, X* or Y. In half the simulations, we
213
assumed that the polyandry allele A is dominant; in the other half it was
214
recessive. Individuals begin life as pre-reproductive individuals (this includes
215
eggs and larvae), which become mature adults if they survive for 0.05 time units.
216
Adults of both sexes live for one time unit and then die. Pre-reproductive
217
individuals die in a density-dependent fashion: the rate of deaths in the pre-
218
reproductive population is (Ne)3/104 where Ne is the number of pre-reproductive
219
individuals. Density-dependent egg-to-adult survival thus limits population
220
growth in the model.
221
The mating rate of a female who is available to mate (i.e. is mature, but
222
has not yet reached her maximum number of matings) is mNm, where m is a
223
constant determining the mating rate and Nm is the number of males in the
224
population. Matings occur one at a time, and a female is immediately categorized
225
as no longer available to mate if she has reached her maximum number of
226
matings, which is one mating for aa females, two matings for AA females, and
227
either one or two for Aa females depending on the dominance of allele A. Note
228
that by “mating rate”, we always mean the rate at which matings occur, not the
229
total number of matings per female.
230
Males that have mated become sperm depleted, modeled by
231
instantaneously setting the focal male’s “sperm depletedness” (si) to one (s = 0
232
for virgin males). Sperm depletedness then declines exponentially towards zero
233
at rate r over time: thus if a non-virgin male mates, for example, 0.5613 time
234
units after his previous mating, his depletedness at the time of the new mating is
235
exp(–0.5613r). Note that a male is not less likely to mate if he has little sperm
236
available: we assume that males mate at every opportunity and that females
237
cannot discriminate among males with varying sperm stores. In order to exclude
238
the source of negative frequency-dependent selection proposed by Taylor and
239
Jaenike (2002; 2003) and limit the number of parameters in the model, we
240
assumed that the sperm replenishment rate was the same for XY and X*Y males.
241
Mated females lay all their eggs immediately before their death if they
242
acquired one or two matings during their lifetime, and unmated females die
243
without leaving offspring. Thus, when males are rare and/or m is low, many
244
potentially polyandrous females will mate only once before reproducing, and
245
females of any genotype may also die without mating at all.
246
The maximum possible number of eggs produced is f (f = 10 in all
247
simulations). In Model 2, three processes can reduce female fecundity from this
248
maximum value. First, f is multiplied by (1 – p) for females that mated twice (the
249
cost of polyandry), and second, by (1 – h) for X*X* females (the cost of
250
homozygous drive chromosomes; double-mated X*X* females pay both costs)
251
just as in Model 1. Third, female fecundity can fall lower still because of sperm
252
limitation. The fecundity of singly mated females depends on their mate’s sperm-
253
depletedness (si) at the time of mating. Fecundity is multiplied by (1 – si)k, where
254
k is a constant governing the shape of the relationship between fecundity and the
255
amount of sperm received. The fecundity of doubly-mated females is multiplied
256
by either (1 – s1)k + (1 – s2)k (where the subscripts refer to the first and second
257
mate), or 1, whichever is lower. That is, we assume that a female mating with one
258
virgin male reproduces at her own maximum possible fecundity (which depends
259
on her genotype, p and h), but that females who obtain even more sperm than
260
that of a virgin male by mating twice can equal but not exceed their maximum
261
fecundity. We assume k = 5 in all simulations, meaning that additional sperm is
262
especially valuable to females whose first mate was strongly sperm depleted.
263
The paternity of eggs laid by doubly mated females depends on sperm
264
competition. When both males are XY or both are X*Y, the focal male has a
265
paternity probability of si/(si + sj); that is, we assume that sperm-depleted males
266
are worse in sperm competition. This formula is modified to csi/(csi + sj) if the
267
focal male is X*Y and the other XY, or si/(si + csj) if the focal male is XY and the
268
other X*Y. The genotypes of the eggs produced are decided by Mendelian
269
segregation, and eggs become adults 0.05 time units after being laid if they
270
survive pre-reproductive density-dependent mortality. Note that although Model
271
2 could incorporate non-overlapping generations, in practice generations were
272
discrete because we initiated the population with adults only, all of which laid
273
eggs and died at the same time. Therefore, adults did not mate with their own
274
offspring.
275
Simulations were initialized with the polyandry allele at 50% frequency
276
and the X* chromosome at either 10% or 90% frequency (allowing us to
277
estimate its likely evolutionary endpoint in parameter spaces where evolution
278
was very slow; see Results and Figure S8). We ran the simulation until the
279
population went extinct, or until a total of 100,000 eggs had been produced (10
280
replicates per parameter space for a total of 17,280 runs).
281
Results of Model 1
282
X* has trouble spreading in a fully polyandrous population
283
First, we investigated whether polyandry can prevent X* from invading when all
284
females in the population are polyandrous and when there are no costs to X*
285
homozygotes (h = 0). Note also that if we had assumed that polyandry confers a
286
net direct benefit (p < 0) in Model 1 rather than a cost, the polyandry allele
287
would always increase and fix, resulting in a uniformly polyandrous population.
288
In a population where all females mate twice, an analytical approximation exists
289
that permits simpler solutions than the general case, provided that we assume
290
X* is rare (see Online Supplementary Material). In short, a rare X* mutant can
291
invade and go to fixation in a population in which all females mate twice
292
provided that c > 1/(1+2d) (Figure 1), i.e. when drive males are sufficiently
293
successful in sperm competition relative to the strength of meiotic drive.
294
However, this analytical approximation does not accurately predict the
295
fixation criteria for drive mutants that have managed to become fairly common,
296
for example via genetic drift or a large influx of migrants carrying X*. Because
297
the complete analytical model is complex, we numerically investigated whether
298
X* was fixed or lost when the population was initiated with a range of X* initial
299
frequencies between 0.01 and 0.99, for a range of values of c and d (Figure 1). We
300
thereby found the minimum initial frequency of X* required for invasion, which
301
predicts the degree to which polyandrous populations are protected against X*
302
(note that X* can invade a monandrous population from any non-zero initial
303
frequency assuming h = 0 and d > 0 in Model 1).
304
Figure 1 shows that polyandry causes X* to decline from all starting
305
frequencies if X*Y males are sufficiently disadvantaged in sperm competition
306
relative to the strength of meiotic drive. Therefore, polyandry creates selection
307
against meiotic drivers that reduce the bearer’s sperm competitive ability.
308
However, when c was sufficiently high relative to d, X* went to fixation from all
309
starting frequencies. The boundary between the zones of inevitable fixation or
310
inevitable loss of X* was separated by a zone in which the initial frequency of X*
311
determined its fate. This zone was especially large when c was low, i.e. when
312
drive males do not fare well in sperm competition. This result illustrates that
313
fixed levels of polyandry produce positive frequency-dependent selection on X*
314
when carriers are disadvantaged in sperm competition. As X* becomes more
315
common, sperm competition involving only X*Y males becomes increasingly
316
frequent, so the X* chromosome suffers its sperm competition disadvantage less
317
often.
318
Polyandry alone cannot maintain genetic variation for meiotic drive
319
We next examined whether freely evolving polyandry can maintain
320
polymorphism for meiotic drive, assuming that polyandry has a direct cost p to
321
females (assuming p = 0 generally causes polyandry to fix, simplifying the model
322
to the case in Figure 1). Figure 2 illustrates three contrasting simulation
323
outcomes. In Figure 2A (which assumes no costs to X*X* homozygotes), the
324
meiotic driver went to fixation despite an increase in the frequency of polyandry
325
as X* increased in frequency, leading to the extinction of males. In Figure 2B, we
326
assumed strong costs to X*X* homozygotes, which prevented X* from going to
327
fixation, and also selected for polyandry. The presence of X* resulted in a
328
moderately female-biased sex ratio. Lastly in Figure 2C, X* and the polyandry
329
allele oscillate in frequency, because the frequency of each determines the fitness
330
of the other.
331
We then set p = 0.01 and varied h, d and c, to determine the effects of the
332
latter parameters on the evolutionary outcome. Each run of the model began
333
with a population in which the sex ratio was even, alleles A and a had
334
frequencies of 0.5, X* had frequency 0.001, and A and X* were in linkage
335
equilibrium. Simulations were terminated after 105 generations, and we present
336
the average allele frequencies in the last 10,000 generations (thereby finding the
337
value around which allele frequencies oscillated, in cases where allele
338
frequencies were not stable).
339
Figure 3 shows that when h = 0 (i.e. X*X* homozygotes had normal
340
fitness), X* typically went to fixation. However for other values of h,
341
polymorphism for drive was possible, with higher h resulting in lower
342
frequencies of X*. As expected, strong meiotic drive (d) and high competitiveness
343
of drive male sperm (c) generally had a positive effect on the frequency of X*.
344
However, co-evolution between meiotic drive and the polyandry allele (Figure 4)
345
resulted in a complex picture. For example, one might expect low c to favor the
346
evolution of polyandry, since this condition increases the difference in the
347
proportion of sons produced by polyandrous and monandrous females, all else
348
equal. However, low c also causes X* to be less common, which favors monandry
349
(since polyandry is costly, and females rarely benefit from screening out X*
350
sperm when X* is rare). Consequently, polyandry was especially strongly
351
selected for when drive males were bad, but not too bad, in sperm competition (c
352
= 0.3-0.7; i.e. close to many real-world estimates of drive males’ sperm
353
competitive ability; Price and Wedell 2008; Price et al. 2008a). These higher
354
rates of polyandry tended to depress the frequency of X*.
355
An important result of Model 1 is that we found no evidence that co-
356
evolving polyandry alone is sufficient to maintain genetic polymorphism for
357
drive, because X* always either went to fixation or went extinct assuming h = 0.
358
To confirm that this result was not specific to the limited parameter space
359
chosen for Figures 2 and 3, we searched for parameter values that allowed X*
360
polymorphism to persist by randomly generating 40,000 parameter spaces in
361
which h, d and c varied independently between 0 and 1, and p varied between 0
362
and 0.2 (initial tests showed that higher p always drove allele A to extinction).
363
Simulations were terminated after 105 generations, or when X* reached a
364
frequency of >0.99, and we again recorded average allele frequencies in the last
365
10,000 generations.
366
Figures S1 and S2 show the effect of the model’s four parameters on the
367
frequency of X* and polyandry respectively. Figure S1 confirms that
368
polymorphism for X* never occurred when h was close to 0 (we additionally ran
369
10,000 other random parameter spaces all with h = 0 which confirmed this
370
result; not shown), and Figure S2 illustrates that costly polyandry is not selected
371
for if meiotic drive is too weak (low d) or if the costs of polyandry are too high
372
(high p). Figure S3 shows the effects of effects of the four parameters on the
373
range of the frequency of X* in the last 10,000 generations, showing which
374
parameters values can generate cycling allele frequencies as shown in Figure 2C.
375
Figure S3 suggests that cycles occur when h, c and p are low and d is high. That is,
376
cycling is more likely when X* can quickly spread from low frequencies (high d),
377
and polyandry can rapidly evolve and effectively cause its spread to reverse (low
378
c and p).
379
Finally, the model found some evidence that when polymorphism for
380
meiotic drive persists, the frequency of drive tends to be negatively correlated
381
with the frequency of polyandry. Figure S4 shows the final frequencies of X* and
382
the polyandry allele for the 40,000 randomly generated parameter spaces shown
383
in Figures S1 and S2. The regression line shows the linear relationship for those
384
parameter spaces for which polyandry and X* had a final mean frequency
385
between 0.01 and 0.09, and the frequency of males was greater than 1%.
386
Although variable values of the model’s four parameters created abundant
387
scatter, there was a net negative relationship, likely because polyandry reduces
388
the selective of advantage of X* whenever X* is disadvantaged in sperm
389
competition.
390
391
Results of Model 2
392
Figures S5-S8. Most of the conclusions are qualitatively identical to Model 1.
393
Strong costs of polyandry (p) disfavored the polyandry allele, allowing X* to
394
spread more readily, and increasing the probability of extinction. The X*
395
chromosome spread more easily if X*Y males were not disadvantaged in sperm
396
competition (c) because more eggs were then fertilized by X*Y males, and
397
probably also because polyandry was not as common when c was high.
398
Accordingly, high c also increased extinction risk. Strong meiotic drive (d) greatly
399
increased extinction probability, and had positive effects on X* and polyandry
400
frequency.
The results of Model 2 are summarized in Table 1 and shown graphically in
401
Costs to X*X* homozygotes (h) again hindered the spread and fixation of
402
X*, reducing the risk of extinction. Importantly, just as in Model 1 we found no
403
evidence that polyandry alone can selectively maintain X* polymorphism:
404
evidence of stabilizing selection on X* was only found when X*X* homozygotes
405
paid a fitness cost (h > 0). Specifically, there were no parameter spaces with h = 0
406
in which X* commonly increased from 10% and decreased from 90% initial
407
frequency (Figure S8).
408
The new parameter r, which controlled the rate at which mated males
409
replenished their sperm, had a strong negative effect on extinction probability.
410
This result is expected because when males can rapidly replenish their sperm,
411
female fecundity is less affected by a shortage of males. When r was low,
412
polyandry reached higher frequencies, because females derived more benefit
413
from the extra sperm gained by mating twice when their first mate was more
414
often sperm depleted. Likely as a consequence of its negative effect on polyandry
415
frequency, r positively affected the frequency of X*.
416
The other new parameter m, which scales the probability that an available
417
female mates at any given male density, had a positive effect on the evolution of
418
polyandry. This result is intuitive because with low m, comparatively few females
419
carrying one or two A alleles actually mated twice before reproducing,
420
diminishing the fitness difference between alleles A and a and thus weakening
421
selection on polyandry. Accordingly, the evolution of allele A was relatively
422
variable when m was low, suggesting drift was strong relative to selection
423
(bottom left, Figure S6). Because multiple mating causes sperm competition, high
424
m was associated with somewhat lower frequencies of X*. Interestingly, higher
425
mating rates were associated with more extinction despite resulting in lower
426
frequencies of X*. This counterintuitive relationship becomes explicable once
427
one notes that all else being equal, the average male is more sperm-depleted
428
when m is high, and females can therefore remain sperm limited even if they hit
429
their mating quota (1 or 2 matings, depending on genotype). The problem of
430
sperm limitation is exacerbated because it selects for polyandry, making sperm
431
even scarcer (for an analogous argument in a different system see Charlat et al.
432
2007b).
433
X* tended to be less common when the polyandry allele A was dominant
434
than when it was recessive, presumably because dominance increases the
435
number of females who mate twice for any given A frequency. As a likely
436
consequence, the polyandry allele reached lower frequencies when it was
437
dominant. The effect of allele A’s dominance on extinction probability was small
438
and inconsistent.
439
As in Model 1, the frequency of the polyandry allele negatively predicted
440
the frequency of X* across all non-extinct simulation runs (slope±95%
441
confidence limits: -0.18±0.02; intercept = 0.52±0.02, n = 8409). However, the
442
explanatory effect of polyandry was very weak (R2 = 0.023), reflecting the
443
plethora of interacting predictors affecting both frequencies.
444
445
Discussion
446
provided sufficiently strong negative frequency-dependent selection to maintain
447
polymorphism for an X-linked meiotic drive gene under the present
448
assumptions. However, when the X* chromosome was prevented from reaching
449
fixation by costs to X*X* homozygotes, polyandry had substantial effects on the
450
frequency and evolutionary dynamics of X*. Polyandry was also sometimes able
451
to purge X* if X*Y males were disadvantaged in sperm competition.
452
Why didn’t polyandry allow stable polymorphism of the driving X?
453
This result likely stems from variation in the strength and direction of selection
454
on polyandry at different stages of the invasion of the X* chromosome. For any
455
given sex ratio, selection for polyandry is strongest when X and X* are present in
456
exactly equal frequencies in males, because this condition maximizes the chance
457
that polyandrous females will mate with both XY and X*Y males. Therefore, the
458
reduction in the average number of X*-bearing offspring produced by
Contrary to our predictions, we found no evidence that co-evolving polyandry
459
polyandrous females relative to monandrous ones is also maximized. By
460
contrast, when the frequency of X* is close to 0 or 1, polyandry provides weaker
461
benefits (again, for any given sex ratio) because females will usually mate with
462
multiple males of the same type. Selection for polyandry will also tend to be
463
strong at high frequencies of X* relative to lower X* frequencies that are
464
similarly far from 0.5 (e.g. at 0.75 relative to 0.25). This is because higher X*
465
frequencies tend to be associated with a more female-biased population-wide
466
sex ratio; the extra sons produced by polyandrous females therefore tend to have
467
higher reproductive value (Fisher 1930) when X* is common.
468
Together, these effects mean that the “extra-son” benefits to polyandrous
469
females will remain small until partway through the invasion of the X*
470
chromosome. Although polyandrous females get increasingly more benefits as X*
471
begins to invade, these benefits might start to decline sometime after X* has
472
passed 50% frequency, since polyandrous females then more often fail to also
473
mate with an XY male (this is especially true if the declining abundance of males
474
causes females to mate with fewer males, as in Model 2 and Taylor and Jaenike
475
2002; 2003). Given that meiotic drivers can spread very quickly, polyandry will
476
often fail to evolve rapidly enough to reach a sufficiently high level to stop or
477
reverse the spread of X*, assuming that negative frequency-dependent selection
478
on X* from other factors (e.g. a cost to X*X* homozygotes) does not stop or slow
479
its spread. This was true even in Model 2, which introduced an additional benefit
480
to polyandry (extra sperm) that grew more valuable as males became rare.
481
In short, polyandry needs to be present at high frequencies to stop the
482
spread of X*. A growing invasion of X* does not generate enough selection on
483
polyandry to boost polyandry to high enough frequencies to stop the invasion,
484
and thus unexplained incidences of meiotic drive polymorphism in the wild (see
485
Introduction) are unlikely to be explained solely by co-evolution with polyandry.
486
Polyandry can purge meiotic drive, and influences drive frequency when
487
other factors maintain polymorphism
488
Although polyandry alone could not protect the X chromosome polymorphism,
489
we found that populations with a high frequency of polyandry are resistant to
490
invasion by X-linked meiotic drivers (and sometimes even purge them from
491
some high initial frequency, such as might occur following an immigration
492
event), provided that drive-carrying males are disadvantaged in sperm
493
competition. This result is similar to that of a previous model of cytoplasmic
494
incompatibility-inducing (CI) Wolbachia. Champion de Crespigny et al. (2008)
495
showed that if CI-inducing Wolbachia harm the sperm competitive ability of male
496
carriers, Wolbachia can be prevented from invading from low initial frequencies.
497
As in our model and those of Taylor and Jaenike (2002; 2003), polyandry can
498
generate positive frequency-dependent selection on Wolbachia: once the
499
bacterium becomes sufficiently common, Wolbachia-carrying males tend to
500
compete against each other, reducing the importance of selection from their
501
inferior sperm competitive ability against uninfected males (Champion de
502
Crespigny et al. 2008). This suggests that our other results generalize to other
503
kinds of selfish genetic element: for example, it is likely that coevolution between
504
polyandry and Wolbachia prevalence cannot explain puzzling polymorphisms for
505
Wolbachia infection under complete maternal transmission or complete CI.
506
We also found that when costs to X*X* females prevented X* from fixing,
507
costly polyandry was often selectively maintained because it provided benefits in
508
terms of extra sons (which is beneficial in a population containing X*). In these
509
situations, the equilibrium mean frequency of polyandry predicted that of X*. In
510
some cases, polyandry remained at a stable frequency, while in others, polyandry
511
and X* tracked each other in stable cycles. Similar cycles have been found in
512
other models of meiotic drive and drive suppressors (Charlesworth and Hartl
513
1978; Hall 2004), and of Wolbachia causing cytoplasmic incompatibility (Hurst
514
and McVean 1996). Additionally, correlations between polyandry and drive
515
frequency were found in previous models of CI-inducing Wolbachia (Champion
516
de Crespigny et al. 2008) and the mouse t haplotype (Manser et al. 2011), both of
517
which assumed that males carrying segregation distorters were disadvantaged
518
in sperm competition as in our model. Available theoretical work therefore
519
suggests that observed correlations between polyandry and meiotic drive in
520
natural populations (Pinzone and Dyer 2013; Price et al. 2014) may result from a
521
combination of adaptation in female mating behavior to the local frequency of
522
drive males and negative effects of polyandry on drive allele fitness.
523
What other factors might stabilize meiotic drive?
524
Given that polyandry and sperm competition appear unlikely to be the full story,
525
what other factors might be acting to stabilize meiotic drive frequencies in
526
natural populations?
527
One possibility is that meiotic drivers are strongly affected by the
528
environment, and that seasonal reductions in drive efficacy and/or the fitness of
529
carriers reduce drive frequency to such an extent that drivers cannot spread to
530
fixation over the course of a year. Laboratory estimates of segregation distortion
531
and fitness of carriers may overestimate the selective advantage of meiotic drive,
532
if these parameters are impaired in less benign environments (Feder et al. 1999).
533
Consistent with an effect of climate on the strength of segregation distortion or
534
the relative fitness of drive-bearing individuals, several meiotic drivers appear to
535
be distributed in latitudinal clines, being rarer at high latitudes (Krimbas 1993;
536
Powell 1997; Dyer 2012; Price et al. 2014). This correlation implies that meiotic
537
drive carriers might have reduced survival, fecundity, fertility, mating success
538
and/or drive efficacy in colder climates. However, direct evidence for decreased
539
drive efficacy or fitness of drive-carrying individuals in cooler environments
540
appears to be absent. D. pseudoobscura males carrying a driving X chromosome
541
have impaired fertility at high temperatures (Price et al. 2012a), but this finding
542
cannot explain the observed latitudinal cline because the driving X is commoner
543
at warmer latitudes (Price et al. 2014). Wallace (1948) found that D.
544
pseudoobscura females homozygous for drive had reduced larval survival,
545
especially at high temperatures, but again this result predicts the opposite
546
latitudinal cline to that which is observed in nature.
547
Lastly, meiotic drivers might persist in metapopulations, where the drive
548
chromosomes can spread within patches and cause local extinctions.
549
Polymorphism might be maintained at the metapopulation level because patches
550
containing the driver produce fewer migrants, e.g. because females in these
551
patches are more often sperm limited (Taylor and Jaenike 2002; cf. Kokko et al.
552
2008). However, evidence for local extinctions caused by meiotic drive in natural
553
populations is equivocal (Pinzone and Dyer 2013).
554
Conclusions
555
Our models tested whether coevolution with polyandry could provide a
556
sufficiently strong source of negative frequency-dependent selection to prevent
557
the fixation of an X-linked meiotic drive gene that also reduces the sperm
558
competitive ability of male carriers. Although polyandry often became more
559
common after the meiotic driver began to spread, reducing the driver’s relative
560
fitness, co-evolving polyandry was never sufficient to prevent the driver from
561
continuing to spread to fixation. However, when we additionally assumed that
562
meiotic drive homozygotes had low fitness, preventing drive fixation, polyandry
563
often had large effects on the frequency and evolutionary dynamics of meiotic
564
drive. We also confirmed that polyandrous populations are more resistant than
565
monandrous populations to invasion by X-linked meiotic drivers that reduce
566
sperm competitive ability.
567
568
Acknowledgements
569
fellowship to LH (DE140101481), Natural Environmental Research Council
570
(NERC) grants to NW (NE/I027711/1) and TARP (NE/H015604/1), and ARC and
571
Finnish Academy (project 252411) funds to HK.
572
573
Data archiving
574
575
576
577
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703
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Tables
705
706
Table 1: Summary of the qualitative results of Model 2. See Figures S5-S8 for
further details.
Parameter
Effect on X*
frequency
Drive strength (d)
Polyandry cost (p)
Competitiveness
of X*Y sperm (c)
Cost to X*X*
females (h)
Mating rate (m)
Sperm
replenishment
rate (r)
Dominance of
polyandry allele
707
Positive
Positive
Positive
Effect on
polyandry
frequency
Positive
Negative
Negative
Effect on
extinction
probability
Positive
Positive
Positive
Negative
Inconsistent
Negative
Negative, up to a
point
Positive
Positive, up to a
point
Negative
Inconsistent
Negative
Negative
Negligible
Negative
708
Figure legends
709
710
711
Figure 1: In a 100% polyandrous population, the invasion of meiotic drive
is difficult or impossible if X*Y males are sufficiently disadvantaged in
sperm competition, especially when meiotic drive is weak.
712
The color shows the minimum initial frequency of X* required for X* to invade
713
for each combination of c and d. Darker colors indicate that X* can invade even
714
from a low initial frequency, while white areas indicate that X* would decline
715
from any initial frequency less than one (i.e. a population fixed for X* is invasible
716
by X). Intermediate color shows regions in which X* can invade, but only if it
717
exceeds a certain initial frequency. The line shows the approximate threshold at
718
which X* can invade from any non-zero initial frequency (see Online
719
Supplementary Material). These results assume that all females mate with two
720
males, and h = 0.
721
722
723
Figure 2: Three contrasting simulation runs, respectively showing fixation
of X*, stable polymorphism, and cyclic polymorphism.
724
In Figure 2A, X* invades from low frequency and goes to fixation, causing
725
extinction of males; the polyandry allele also began to (temporarily) increase in
726
frequency as X* became common, but X* nevertheless continued to fixation. In
727
2B, strong costs to X*X* homozygotes prevented X* from fixing, and the
728
subsequent increase in the frequency of polyandry affected the equilibrium
729
frequency of X*. Lastly in 2C, stable co-evolutionary cycles of polyandry and X*
730
occurred. We assumed d = 0.9 and starting frequencies of 0.5 for polyandry and
731
0.001 for X* in all simulation runs.
732
733
Figure 3: The effect of the parameters c, d, and h on the frequency of X*.
734
When X*X* homozygotes paid no fitness cost (h = 0), X* fixed for all values of c
735
(assuming d > 0), indicating that co-evolving rates of polyandry alone cannot
736
stop a meiotic driver from invading and fixing. By contrast when h = 0.2, X* only
737
fixed when it did not seriously impact its bearer’s sperm competitive ability and
738
meiotic drive was strong; otherwise, X* remained polymorphic. Higher values of
739
h yielded a similar picture, but X* reached lower frequencies due to the reduced
740
fitness of X*X* homozygotes. Grey areas indicate parameter spaces in which the
741
frequency of males dropped below 1%, suggesting that these parameter
742
combinations would lead to extinction in reality. We assumed p = 0.01 and
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starting frequencies of 0.5 for polyandry and 0.001 for X* in all simulation runs.
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Figure 4: The effect of the parameters c, d, and h on the frequency of
polyandry.
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Polyandry was always lost when h = 0, since X* always went to fixation, meaning
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there were no XY males to favor using sperm competition. For other h values,
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polyandry became most common when c was intermediate and d was high.
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Intermediate h favored polyandry more than high h, since X* was present at
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lower frequencies if h was high. Grey areas indicate parameter spaces in which
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the frequency of males dropped below 1%, suggesting that these parameter
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combinations would lead to extinction in reality. We assumed p = 0.01 and
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starting frequencies of 0.5 for polyandry and 0.001 for X* in all simulation runs.
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Online supplementary material
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Analytical approximation when h = 0 and X* is rare: polyandry can prevent
the invasion of X*
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Here we consider a simplified case of Model 1, in which all females mate with
two males (i.e. allele A is fixed), X*X* females have normal fecundity (i.e. h = 0),
and X* is rare. As in the main model, XY and X*Y males fertilize a fraction of the
eggs 1/(1 + c) and c/(1 + c), respectively, when they mate with the same female.
Again, X*Y males transmit an X* chromosome to a fraction (1 + d)/2 of their
offspring, and a Y chromosome to the remaining (1 – d)/2.
The X* chromosome has an initial frequency of q, where q ≈ 0. The
frequencies of matings involving more than one individual carrying an X*
chromosome, and the frequency of X*X* females, are therefore negligible and
can be ignored when calculating the invasion condition for X*. We therefore have
to consider just three mating types: 1) an XX female mating with two XY males,
2) an XX female mating with an XY male and an X*Y male, and 3) an X*X female
mating with two XY males. The frequencies of these mating types are (1 – q)3,
2q(1 – q)2, and q, respectively, which sum to 1–q2+q3 ≈ 1, since q ≈ 0. We can then
calculate the frequencies of each of the four possible types of offspring in the
next generation given the values of c and d by summing the columns of Table S1.
The frequency of X* in the next generation, q’, is approximately equal to
the sum of the frequencies of X*Y and X*X individuals in the offspring cohort,
which can be found by multiplying the above mating type frequencies by the
frequencies with which they produce X*Y and X*X offspring (Table S1). After
simplifying and ignoring higher order terms of q, which are negligible when q is
small, we find that q’ > q approximately when c > 1/(1 + 2d). This threshold is
plotted on Figure 1.
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Supplementary table
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Table S1: Frequencies of each type of offspring produced when X* has frequency
q, c scales the competitiveness of X*Y males in sperm competition, and d is the
strength of meiotic drive.
Frequency of each offspring type
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Mating type
XX
XY
X*X
X*Y
XX × XY × XY
(1–q)3/2
(1–p)3/2
0
0
𝑐
1+d
}
2 + 2𝑐 2
0
XX × X*Y × XY
2𝑞(1 − 𝑞)2 (
X*X × XY × XY
q/4
1
)
2 + 2𝑐
2q(1– q)2 {
1
c
1−d
+
}
2 + 2c 2 + 2c 2
q/4
2𝑞(1 − 𝑞)2 {
q/4
q/4
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Supplementary figure legends
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Figure S1: The cost of X*X* homozygosity (h) define whether X*
polymorphism can persist.
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The plot shows the frequency of X* for individual model runs (averaged over the
final 10,000 generations, except when simulations were terminated because X*
reached 0.99 frequency, in which case 0.99 is shown; total: 40,000), paneled by
the model’s four parameters. Areas empty of points imply that X* never reached
a particular final frequency when the parameter takes the value indicated on the
x axis. For example, h = 0 did not allow polymorphism for meiotic drive, and X*
never reached high frequencies when h was high. We assumed starting
frequencies of 0.5 for polyandry and 0.001 for X* in all simulation runs.
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Figure S2: The effects of the model’s parameters on the frequency of
polyandry.
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The plot shows the frequency of allele A for individual model runs (averaged
over the final 10,000 generations, except when simulations were terminated
because X* reached 0.99 frequency, in which case the frequency of allele A at
termination was used; total: 40,000), paneled by the model’s four parameters.
Areas empty of points imply that allele A cannot evolve to a particular frequency
when the parameter takes that value. For example, polyandry never increased
from its initial frequency of 0.5 when meiotic drive was weak or the costs of
polyandry were large. We assumed starting frequencies of 0.5 for polyandry and
0.001 for X* in all simulation runs.
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Figure S3: The effects of the model’s parameters on the range of X*
frequencies in the last 10,000 generations.
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The plot shows the difference between the maximum and minimum frequencies
of X* observed in the last 10% of generations, for those simulations that ran for
the full 105 generations without being terminated due to X* fixation. Areas with a
non-zero range in frequencies displayed co-evolutionary cycles of the kind
shown in Figure 2C. Cycles were only observed for intermediate h, low c, high d
and low p. We assumed starting frequencies of 0.5 for polyandry and 0.001 for
X* in all simulation runs.
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Figure S4: There was a negative relationship between the frequency of
polyandry and X*.
The red line and its grey 95% confidence intervals are from a linear regression of
polyandry on X* frequency across the 3472 simulation runs in which both
polyandry and X* had a final frequency in the range 0.01-0.99, and there were at
1% males. These runs are shown by black dots; grey dots mark the remaining
36528 simulation runs (not used to calculate the regression line). The points
were randomly jittered to reduce over-plotting.
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Figure S5: Effect of Model 2’s parameters on the final frequency of X*.
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The points show means and their 95% confidence limits; red points show data
from runs in which the A allele was dominant, while blue points denote
recessivity of A. The semi-transparent points show individual simulation
outcomes. Runs that ended with extinction are not shown, and were not used to
calculate the means. The simulation runs covered every permutation of the
following parameter lists, and each parameter combination was replicated 10
times: Initial frequency of X* = 10% and 90%; Allele A dominant or recessive; h =
0, 0.2 or 0.99; p = 0, 0.05, 0.2 and 0.6; d = 0.5 and 1; r = 0.1, 1 and 10; m = 0.1, 1
and 10; c = 0.1 and 0.9. There were thus 10 replicate runs of each of 1728
parameter spaces (8409 simulations did not go extinct, and are thus plotted
here).
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Figure S6: Effect of Model 2’s parameters on the final frequency of the
polyandry allele.
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The points show means and their 95% confidence limits; red points show data
from runs in which the A allele was dominant, while blue points denote
recessivity of A. The semi-transparent points show individual simulation
outcomes. Runs that ended with extinction are not shown, and were not used to
calculate the means. The data are from the same simulations as those shown in
Figure S1.
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Figure S7: Effect of Model 2’s parameters on extinction probability.
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Extinction probability was estimated as the proportion of the 10 replicate
simulation runs per parameter space that went extinct. The points show mean
extinction probabilities and their 95% confidence limits; red points show data
from runs in which the A allele was dominant, while blue points denote
recessivity of A. The semi-transparent points show extinction probabilities for
individual parameter spaces. The data are from the parameter space described in
the legend of Figure S1.
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Figure S8: Effect of Model 2’s parameters on the frequency of four possible
evolutionary outcomes for X*.
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This plot considers four possible evolutionary trajectories for X*. First, X* could
increase from both 10% and 90% initial frequencies for a given parameter space,
implying that those conditions allow X* to invade when rare and proceed to
fixation (red). Second, X* could increase from 90% initial frequency but decline
from 10%, suggesting that it can fix but only if present at a sufficiently high
initial frequency (green). Third, X* could increase from 10% and decrease from
90%, implying that selection moves X* to some intermediate frequency (blue).
Finally, X* could decline from both 10% and 90% frequencies (purple). The
frequency of each of these four outcomes was determined for each of the 864
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parameter spaces shown in Figures S5-S7 by multiplying the proportion of
replicates in which the frequency of X* changed in the predicted direction in the
10% runs by the frequency of X* changed in the predicted direction in the 90%
runs. For example, if X* increased from an initial frequency of 10% and declined
from 90% in 2/10 and 6/10 replicate simulations respectively, the blue outcome
was assigned a probability of 0.2×0.6=0.12 for that parameter space (the points
show the average probability and its 95% confidence limits). A key result to note
is that there was no evidence for stabilizing selection on X* when homozygotes
pay no costs, which is shown by the third, blue outcome having a probability of
zero when h = 0. Thus, Model 2 reaffirms a major conclusion of Model 1, namely
that co-evolving rates of polyandry cannot selectively maintain variation for
meiotic drive under the present assumptions. Nevertheless, polyandry is
important to the evolution of X* whenever c < 1.