Stepping-stone colonisation model

We construct a stepping-stone colonisation model with the following basic assumptions mimicking how L. saxatilis colonised the post-glacial archipelago of western Sweden (and continues to colonise new emerging islands). First, colonisation was occasional and rapid, as rafted fertilised females release a few hundred offspring already in the first generation [12]. Second, small skerries are likely to be colonised within a few years after emergence, and hence all newly established populations are limited by the small size of the habitat, resulting in census sizes of [12]. Third, colonisation is likely to take place in a stepping-stone manner with smaller and more distantly related islands being colonised from their closest already colonised islands. For simplicity, we consider a system consisting of a mainland and of linearly arranged islands of equal carrying capacities (substantially smaller than that of the mainland).

In our model, islands are linearly arranged and numbered from to , with being furthest away from the mainland (see Fig. 1 C). We include high values of in our model (such as ) in order to be able to assess saturation effects. The mainland is labelled by . Generations are assumed to be non-overlapping. Before the process of colonisation starts, the mainland is the only populated habitat, and the population heterozygosity on the mainland is stationary (that is, the mainland population is assumed to be old). We assume that mutations accumulate according to the infinite-alleles model [16].

Within our model, an empty island becomes fully colonised in a single generation after the arrival of one or more founder females from the nearest neighbouring island. This is motivated by the very large capacity for population growth of L. saxatilis in a suitable habitat [12], [13]. In our model the founder females give rise to offspring in total, with equal sex ratio, where denotes the carrying capacity of the island. Upon establishing a given island population, its population size remains constant over time. In our model, mating takes place before migration, which allows us to trace only the movement of females (males also migrate, but since they will not mate after migration, they do not contribute to the progeny on the island they migrated to). Individuals are equally likely to migrate to each of their closest neighbours (but the mainland and the last island have only a single neighbour, Fig. 1 C). On average, females migrate per generation from one island to a neighbouring island, except for empty islands that only receive migrants.

In addition to the above, we assume that the population size on the mainland is much larger than the population size of a colonised island. In our computer simulations we set the mainland heterozygosity to unity. This simplifies the simulations, since the dynamics of the mainland does not need to be simulated explicitly.

An important aspect of our model is the mainland, providing a source of genetic variation through repeated founder events. We remark that our model differs essentially from the mainland-island model analysed in [17] (see also references cited in this work). In [17] it is assumed that all island populations (called sink populations in [17]) either receive migrants from a source simultaneously, or are equally likely to receive migrants. In this case the degree of genetic variation in the steady state turns out to be the same for all islands. This is also true for stepping-stone models without a mainland (see for example Refs. [18], [19]).

Mating model

In order to study the consequences of multiple paternity for genetic diversity, we introduce a mating model to describe different levels of multiple paternity in mating systems.

Based on known life-history traits [13], we assume that the duration of the reproductive cycle of females is the same for all females. Each female carries beneath her shell juveniles of varying degrees of maturity, and juveniles are released at an approximately steady rate. We also assume that after a successful mating, the mated female obtains a sperm package able to fertilise female eggs during its persistence time. Our observations show that sperm can be stored and used up to a year after mating. The number of eggs fertilised by a single sperm package is assumed to be the same for all sperm packages that a female receives during the reproductive cycle. The total number of active sperm packages received by each female during the reproductive cycle is denoted by , and is assumed not to depend on time (it has the same value in every generation).

The probability that two eggs are fertilised by the same sperm package is assuming that all sperm packages a female received during her reproductive cycle persist until the end of the reproductive cycle, that all eggs are fertilised after all sperm packages have been collected, and that sperm packages are chosen with replacement to fertilise eggs.

The scheme presented above models the process of mating at an individual level. We assume that individuals belong to a well mixed diploid population of males and females, and we take and . In our model, each female encounters different males during the reproductive cycle (thus, is assumed to be the same for all females, and it does not depend on time). Having reflects the limited movement of snails during the reproductive cycle. Unlike under random mating, where all males are treated as on average equally successful mates, it is possible that females exhibit cryptic choice, or that some of the available mates of each female are more dominant than others. We include this in our model by assigning different levels of dominance to the available mates: we assume that one of the males (randomly chosen) has on average a higher success in mating, whereas the remaining males have lower success. The dominant male mates the female with probability , whereas the remaining males mate the female with probability . We also assume that all females are on average equally successful mothers.

In our model the probability that two offspring come from the same female, , is given by(1)Note that this implies that the brood sizes of females are multinomially distributed, similarly to the commonly used Wright-Fisher population [20], [21]. Furthermore, the probability that two offspring come from the same male, , is(2)In Eq. (2), the first term corresponds to the probability that two offspring share both a mother and a father, and the second term is the probability that two offspring come from different mothers but they share a father. The factor stands for the probability that two offspring having the same mother share a father, and it is given by(3)Here, the first term is the probability that two eggs are fertilised by the same sperm package, and the second term stands for the probability that two eggs are fertilised by two distinct sperm packages, both coming from the same male. The probability that two packages come from the same male has two contributions. The first contribution, , stands for the probability that both packages come from the dominant male, and the second contribution, , is the probability that two packages come from one of the remaining males. Note that, according to Eq. (3), decreases as the number of available mates, , of a female increases (keeping the values of and of fixed). Therefore, we take as a measure of the degree of multiple paternity.

Using Eqs. (1)–(3) we find that the effective size of a freely mixing population under the model described is given by:(4)The mathematical details of this derivation are given in Appendix S1. Our model reduces to the model in [22] in the case a female encounters all males present in the population, , upon assuming that all males are on average equally successful mates, , and upon identifying the number of matings in [22] with . Furthermore, if , , and the probability that two eggs are fertilised by the same sperm package is , our model reduces to random mating. We also note that in two particular cases, our model reduces to two mating models analysed in [23]. First, for , , and (or ), our model is analogous to the model of lottery polygyny in [23] in the case all females mate with certainty, and assuming non-overlapping generations (that is, upon setting , and in [23]). Second, for , , and , our model reduces to the model of dominance polygyny in [23] in the case of non-overlapping generations, and upon assuming that all males are dominant in this model (that is, for , and in [23]).

We show in Fig. 2 how the effective population size under our model depends upon and . As expected, by increasing the degree of multiple paternity, the effective population size becomes larger (as found in [24] for a different mating model). For the parameters set in Fig. 2, the maximum value of is equal to , which corresponds to the effective population size under random mating. The increasing trend of saturates at for a given value of and .

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Figure 2. Effective population size.

For different number of available mates, , and probability that two eggs are fertilised by the same sperm package, , we show the ratio of the effective population size () to the total population size (). Parameters: , .

https://doi.org/10.1371/journal.pone.0075587.g002

We used empirical data to test our mating model from two experiments. In one experiment, six virgin females of similar sizes were placed in separate aquaria, and each was accompanied by ten equally sized males. The snails were allowed to mate freely during eight weeks, after which the males were removed, and offspring were collected from each female over a one-year period following the mating period. The sire of each offspring was determined by genotyping the males, females and juveniles at eight microsatellite loci and performing parentage analysis following an exclusion method (Saltin, S. H. et al, unpublished data). Only the offspring that could uniquely be assigned to a given sire were kept. Hence, out of offspring were removed from the analysis.

In the second experiment, genotype data from five microsatellite loci were collected in four wild females and their offspring [10]. The brood sizes of the four wild females were , , , and [10]. Since the potential sires were unknown, COLONY [25] was used to estimate the sibships of each female's offspring. In order to test the goodness of our model, we fitted our mating model to the distribution of male family sizes in the empirical data, using a chi-square test to assess goodness of fit (details of binning and degrees of freedom are given in the results section). Because the circumstances in the controlled mating experiments differ from that of females mated in the wild, we fitted the two datasets separately.

The brood sizes of the six females mated under experimental conditions in the lab was , , , , , and (Saltin, S. H. et al, unpublished data). For the offspring of each female it was determined how many came from the same father, that is, male family sizes. For the snails mated in the lab, the maximum male family size was ( male). Remaining male families were of sizes ( male), ( male), ( male), ( male), ( males), ( male), ( males), ( males), ( males), ( males), and ( males). To avoid ambiguity in paternity, offspring were excluded from male family assignment, yielding broods of sizes , , (two broods), , and for further analysis. Similarly, for the snails mated in the wild, the male families were of sizes ( males), ( males), ( males), ( males), ( males), ( males), ( males), ( males), and ( males). For the details on parental assignment in this case, see [10].

Distribution of male family sizes

In Fig. 3 A we show the histogram of male family sizes obtained experimentally and in Fig. 3 B we show similar data collected from wild females in natural habitats. For both sets of data we use computer simulations in order to find the parameters in our model resulting in male family sizes that are closest to those empirically observed. In the computer simulations we vary the number of available mates , the probability , and the level of dominance of a dominant male, . For the females mated in the lab, we vary the parameter from to , and for the wild females, we vary from to . In both cases, we vary the parameter from to in steps of , and the inverse of parameter from to in steps of , and we test as well. For each set of test values of , , and , we simulate mating of six females (four for wild snails) and we generate broods of sizes corresponding to that in the empirical data. By simulating the process of mating times for each set of test parameters, we compute the average probability of obtaining a given male family size. Using these data, we apply a chi-square test to quantify the difference between the empirical and simulated data for each set of test parameters. The best fits obtained are shown in Fig. 3 A–B by circles, and they correspond to , , (Fig. 3 A), and to , , (Fig. 3 B). We discuss these fits in Section Discussion and conclusions. The results shown in Fig. 3 suggest that our mating model describes empirical data well, but the agreement is not perfect. Possible improvements of our model are discussed in Section Discussion and conclusions.

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Figure 3. Distribution of number of progeny attributed to each sire.

Bars in panel A show the empirical data obtained under experimental conditions; the data correspond to six broods, two of which have size , and the remaining four have sizes , , , and . Bars in panel B show the results from females mated in the field, data taken from [10]; the data correspond to four broods of sizes , , , and . The width of the bins are chosen so that the expected number of counts in each bin is not smaller than . The probability assigned to each bar is proportional to the bar area. Symbols and error bars show the result of the best fit to the experimental data, together with their 95% confidence intervals: in (A), and in (B). We simulated independent realisations of the mating process.

https://doi.org/10.1371/journal.pone.0075587.g003

To address the question of how multiple paternity affects genetic variation and structure in our subdivided population, we analyse genetic diversity under the colonisation model described above. We analyse separately two phases of population dynamics on each island: initial colonisation, and the steady state that develops once the colonisation phase is over. For a given island, we compute the expected heterozygosity in the generation when the island is colonised (colonisation phase), as well as the expected heterozygosity in the steady state. The corresponding analytical computations are described in detail in Appendix S2 and Appendix S3. We also study temporal changes of heterozygosity under our model by computer simulations. In the following two subsections we present separately the results for the colonisation phase and for the steady state.

Colonisation phase

We compute the heterozygosity in the colonisation phase analytically using a coalescent approach [26]. Recall that at a given island, the colonisation-phase heterozygosity corresponds to the island heterozygosity in the generation when the island is colonised. Note that the population-size history of the population on island at the time when this island is colonised can be represented as a sequence of bottlenecks. In our model, each of the bottlenecks lasted one generation, and the most recent bottleneck occurred when the first colonisers (founder females) came to island . Note that there are bottlenecks, because is the number of colonisation events that the population ancestral to that on island went through before the island was colonised. We derive an analytical result upon assuming that the migration rate is small, . Under this assumption, colonisation of empty islands occurs rarely, but when it does, an island typically receives a single founder female. The resulting heterozygosity in island in the colonisation phase, , is (see Appendix S2):(5)Here, is the mainland heterozygosity, and is the probability that the most recent common ancestor of two alleles sampled from the newly established population in island stems from an allele that was born on the mainland (see Appendix S2):(6)It follows from Eqs. (5) and (6) that the dependence of the heterozygosity in the colonisation phase on the degree of multiple paternity can not be expressed only in terms of the effective population size. This suggests that the degree of multiple paternity has possibly a larger effect on the colonisation-phase heterozygosity in subpopulations, than on the heterozygosity of freely mixing populations.

In Fig. 4 A we show how the colonisation-phase heterozygosity depends on distance from the mainland. We observe two expected features. First, the colonisation-phase heterozygosity decays as distance from the mainland increases. Second, at any distance from the mainland, multiple paternity results in higher heterozygosity than single paternity. We find that mating two males () increases the values of single-paternity heterozygosity by for the parameters used in Fig. 4 A, and mating ten males () increases the values of single-paternity heterozygosity by , depending on distance from the mainland (Fig. 4 B). The largest increase is observed at the island furthest from the mainland. It is interesting to note that mating more than about males only marginally increases the heterozygosity (results not shown), as found in the case of freely mixing populations (see our description of the mating model).

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Figure 4. Analytically computed heterozygosity during colonisation and in the steady state.

(A) Heterozygosity during colonisation as a function of distance from the mainland. The lines shown from top to bottom correspond to: , , , , and . (B) Heterozygosity during colonisation relative to its corresponding value for as a function of distance from the mainland. The order of lines corresponds to that in A. (C) and (D) Same as in (A) and (B), respectively, but for the steady state. Remaining parameters: all available males are on average equally successful, , the mainland heterozygosity is set to unity, the scaled female migration rate is , the number of females in each populated island is , the probability that two eggs are fertilised by the same sperm package is , and the number of islands is .

https://doi.org/10.1371/journal.pone.0075587.g004

We note that the results of our computer simulations (see Fig. S1 A–C) agree well with the analytical results for low migration rates. For large migration rates (, i. e. females on average per generation) by contrast, the simulations assume somewhat higher values than the theory. The reason for this deviation is that for it is probable that more than one founder female come to an empty island to establish the population, and, consequently, will contribute with more genetic variation than just one founder female. For natural populations of L. saxatilis it has been estimated that of empty islands receive a migrant each year [12]. This estimate is close to the lower value of used in our simulations ().

Steady state

We show in Appendix S3 how to compute the heterozygosity in the steady state at distance from the mainland using recursion relations (this is a standard procedure, see for example [19], [21]). Note that this derivation does not require to be small. In Fig. 4 C, we show how the resulting steady-state heterozygosity depends on distance from the mainland. As in the colonisation phase, the steady-state heterozygosity decreases as distance from the mainland increases. Also, by increasing the degree of multiple paternity, the heterozygosity at a given island increases (this effect saturates at , results not shown). In contrast to the strong effect of multiple paternity during colonisation, the effect is substantially smaller in the steady state. We find that the single-paternity heterozygosity in the steady state increases by for , and by for (Fig. 4 D). As in the colonisation phase, the largest increase is observed at the island furthest from the mainland.

In addition, we examined the variation in heterozygosity over consecutive generations in a particular realisation of our model. We find that the heterozygosity shows strong temporal fluctuations (Fig. 5 A). Notably, the fluctuations are strongest furthest from the mainland, with periods of high diversity separated by long periods of near or complete fixation. Hence the distribution of heterozygosity at large distances from the mainland is bimodal. The heterozygosity is expected to have a bimodal distribution if genetic variation is introduced at a very small rate (see, for example, [27] and references cited therein).

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Figure 5. Temporal fluctuations of heterozygosity.

(A) Heterozygosity as a function of distance from the mainland and of time (single realisation of the model described). Mainland is not shown. The data correspond to generations after the initial generations. The number of available mates is . (B) Analytically computed durations of low-and high-heterozygosity phases (blue, and red) relative to their corresponding values for . (C) Analytically computed steady-state heterozygosity (black) relative to its corresponding value for . Remaining parameters are as in Fig. 4 C.

https://doi.org/10.1371/journal.pone.0075587.g005

In what follows, we analyse how the durations of the phases of low and high heterozygosity are affected by multiple paternity. We consider values of the heterozygosity smaller than to be low, and values of the heterozygosity larger than to be high. The minimum value for the high phase () is chosen since the typical maximum value that the heterozygosity has at the island furthest from the mainland is for the parameters chosen in Fig. 5 A. Note that the maximum value of the heterozygosity at a locus with only allelic types is equal to .

Using the results of our computer simulations, we compute the average durations of low- and high-heterozygosity phases at the island furthest from the mainland. We also derive corresponding analytical results under the assumption that the scaled migration rate is small (see Appendix S4). For island we show in Fig. 5 B (see also Fig. S2 B, D, F) the durations of low- and high-heterozygosity phases relative to their corresponding values for a single mate (). Fig. 5 B shows that multiple paternity prolongs the duration of the high-heterozygosity phase, and decreases the duration of the low-heterozygosity phase. For example, the high-heterozygosity phase for the highest level of multiple paternity shown () is prolonged by around compared to its value under single paternity (). The low-heterozygosity phase is shortened by only around for (Fig. 5 B). For comparison, Fig. 5 C shows the steady-state heterozygosity relative to its corresponding value for a single mate (). In conclusion, multiple paternity promotes heterozygosity by prolonging the duration of peaks of variation that reach the most distant islands.