GENETIC DRIFT AND COLLECTIVE DISPERSAL CAN RESULT IN CHAOTIC GENETIC PATCHINESS

Authors


Abstract

Chaotic genetic patchiness denotes unexpected patterns of genetic differentiation that are observed at a fine scale and are not stable in time. These patterns have been described in marine species with free-living larvae, but are unexpected because they occur at a scale below the dispersal range of pelagic larvae. At the scale where most larvae are immigrants, theory predicts spatially homogeneous, temporally stable genetic variation. Empirical studies have suggested that genetic drift interacts with complex dispersal patterns to create chaotic genetic patchiness. Here we use a coancestry model and individual-based simulations to test this idea. We found that chaotic genetic patterns (qualified by global FST and spatio-temporal variation in FST's between pairs of samples) arise from the combined effects of (1) genetic drift created by the small local effective population sizes of the sessile phase and variance in contribution among breeding groups and (2) collective dispersal of related individuals in the larval phase. Simulations show that patchiness levels qualitatively comparable to empirical results can be produced by a combination of strong variance in reproductive success and mild collective dispersal. These results call for empirical studies of the effective number of breeders producing larval cohorts, and population genetics at the larval stage.

No genetic structure builds up when dispersal randomly mixes individuals at each generation. This can be observed at a large spatial scale in species with great dispersal capabilities (e.g., in a terrestrial species such as the noctule bat, Petit and Mayer 1999; and in a marine species such as the bluehead wrasse, Purcell et al. 2006). In species without such extreme dispersal distances it is observed when one looks for fine scale genetic patterns, for instance below the minimum dispersal distance of individuals. However this study scale has revealed surprising results in a number of marine species with pelagic larval dispersal. The vast majority of marine invertebrates (> 70%, Mileikovsky 1971) and many reef fish disperse during a free-floating larval phase that can last from minutes to years (typically a few weeks, Shanks 2009). This larval phase is followed by metamorphosis into a sessile benthic adult (“bentho-pelagic” life cycle). Thus the dispersal range in these species is largely determined by the length of the larval phase, although realized dispersal distances can be affected by larval behavior (e.g., vertical migration or active swimming) and oceanic currents. In such species, classical population genetic theory predicts no spatial genetic structure between benthic samples of adults below the expected range of larval dispersal, because most if not all individuals are expected to be immigrants at this spatial scale.

Contrary to this prediction of spatial homogeneity, there has been an accumulation of empirical observations of low but significant levels of genetic structure between samples taken at distances far below the expected range of larval dispersal (e.g., Johnson and Black 1982, David et al. 1997b; for recent examples see our Table 1; for further references and discussion see Larson and Julian 1999; Hellberg et al. 2002; Bierne et al. 2003; Arnaud-Haond et al. 2008). This micro-geographic structure shows no isolation by distance: the differentiation among samples does not build up with increasing study scale. Furthermore it shows rapid fluctuations in time, as measured through changes in the level of spatial structure across generations, or changes in allelic frequencies at a given sampling point. These observations led Johnson and Black to coin the term “apparently chaotic genetic patchiness” in their seminal work looking at the genetic structure of pulmonate limpets (Siphonaria sp., Johnson and Black 1982). David et al. 1997b later referred to similar patterns as “fluctuating genetic mosaics.” These puzzling observations raise questions regarding the micro-evolutionary processes at work in marine ecosystems: what forces drive such small-scale population divergence? Is this divergence important to consider for the evolution of local adaptation? Clarifying the origin of this divergence is also important if genetic data are to be applied to questions in ecology and conservation (e.g., how can we use molecular data to infer demographic trends, to estimate dispersal and population connectivity, or to detect the signature of selection?).

Table 1. Examples of empirical genetic data showing apparently chaotic genetic patchiness. These data were obtained from multilocus microsatellite genotypes (additional examples from the many studies based on enzyme polymorphism can be found in Larson and Julian 1999). Planktonic larval duration (PLD) is the length of the larval phase, the spatial scale of the study area is the approximate spatial extent encompassing all sampling sites (in some cases isolated sites were excluded to present results for a homogeneously sampled study area). Hs is within-site gene diversity, global FST gives the differentiation across all sampling sites, pairwise FST measures the differentiation across pairs of sites, and temporal FSS refers to the differentiation of a deme with itself one generation earlier (see text). These statistics are calculated using samples from adults
SpeciesPLDSpatial scaleHsGlobal FSTPairwise FSTTemporal FSSReference
  1. 1Data from the Venice lagoon only.

  2. 2Excluding isolated sites 1, 2 and 16.

  3. 3Data from the ‘Eastern Baie de Seine’ only.

Stegastes partitus (coral reef fish)24–40 daysca. 200 Km0.78–0.84−0.001–0.02−0.001–0.020 Hogan et al. (2010)
Carcinus aestuarii (shore crab, crustacean)ca. 60 daysca. 40 Km10.78–0.82 0.000−0.004–0.015 Marino et al. (2010)
Centrostephanus rodgersii (urchin, echinoderm)ca. 4 monthsca. 1000 Km20.57–0.65 0.008 0.000–0.114 Banks et al. (2007)
Paralabrax clathratus (temperate reef fish)25–36 daysca. 360 Km0.73 0.003 0.002–0.003Selkoe et al. 2006
Pectinaria koreni (tubicolous polychaete)ca. 15 daysca. 30 Km30.87–0.97 0.006 0.001–0.012 Jolly et al. (2003)

Setting aside sampling artifacts, two classes of hypotheses have been proposed to explain the origin of chaotic genetic patchiness. First, heterogeneous environmental conditions could drive selection for locally beneficial alleles through differential survival of recruits (referred to as postsettlement selection, Johnson and Black 1982). This idea was supported by observations of enzyme allele frequency clines correlated with environmental gradients (such as temperature or salinity, e.g., Koehn et al. 1980). But the consistency of observed genetic patterns across multiple loci conflicts strongly with this hypothesis. The selection hypothesis also seems difficult to reconcile with rapid temporal fluctuations of genetic structure and with the absence of discernible micro-geographic environmental heterogeneity (David et al. 1997b). The same arguments also rule out a related hypothesis involving presettlement selection during the dispersal phase (but see Johnson and Black 1984a). While direct or indirect selection generates spatial genetic differentiation at specific loci (e.g., Calderon and Turon 2010; Toonen and Grosberg 2010), it can hardly be considered a driver of the multilocus patterns generally described as apparently chaotic genetic patchiness.

The second class of hypotheses proposes that genetically differentiated groups of individuals are produced through neutral demographic processes. Central to this idea is the recognition that reproduction in benthic populations happens at a spatial scale orders of magnitude smaller than larval dispersal. In other words, mating in benthic populations occurs within breeding groups (Johnson and Black 1984b) constrained in size and spatial extent by the mobility of adult individuals and the generally short lifetime of gametes (e.g., Levitan and Petersen 1995). Although many bentho-pelagic species are broadcast spawners, breeding group size is a fortiori even more limited in case of internal fertilization (e.g., Johnson and Black 1984b). This subdivision of the population for reproduction may allow offspring from distinct areas of a population (or from distinct reproduction events) to differ though random genetic drift (Li and Hedgecock 1998).

In some species, local genetic drift can be strong simply because the size of breeding groups is small. An extreme case in point is the slipper limpet (Crepidula fornicata): irrespective of population size breeding occurs essentially within stacks (typically 2 to 20 individuals, Dupont et al. 2006, 2007; Proestou et al. 2008; Le Cam et al. 2009). In species where the definition of breeding groups is more elusive, parentage analyses have provided useful information on the number of breeders contributing to local cohorts of larvae (e.g., 19 to 28 breeders per daily cohort in an experiment featuring 62 adult flat oysters Ostrea edulis, Lallias et al. 2010).

The variance in reproductive success among individuals can also greatly contribute to reducing the effective size of local breeding groups. This is supported by a growing body of results from relatedness or parentage analyses and from adult/offspring comparisons (e.g., Selkoe et al. 2006; Hedgecock et al. 2007; Liu and Ely 2009; Christie et al. 2010). Reproductive success can be extremely skewed in marine species with high fecundity and high initial mortality and where only a minute fraction of larval cohorts find oceanographic conditions favorable for larval survival and recruitment (sweepstake reproductive success hypothesis, Hedgecock 1994; reviewed in Hedgecock and Pudovkin 2011). In summary, spatially localized reproduction and variance in reproductive success can cause strong genetic drift whereby genetically differentiated pulses of larvae may be emitted from different locations and/or at different times within a population.

This being said, larval dispersal should still homogenize these differences. If local groups of adults are effectively random samples from the pool of predispersal individuals, then adult samples should show genetic differences no greater than expected from sampling error. Thus drift alone cannot account for statistically significant genetic divergence in adults if these few cohorts that are successful diffuse randomly in space before settlement. Assuming that sampling artifacts are avoided, genetic patchiness requires local drift to be followed by an incomplete mixing of dispersing larvae. This can be achieved, even with unlimited dispersal distance, if pools of larvae released from a local breeding group do not diffuse randomly but instead remain aggregated to some extent during dispersal and settlement (Shapiro 1983). The idea of larval aggregations is supported by some results from larval transport models (Siegel et al. 2008), observations of half-sibs in samples of dispersing larvae (F. Riquet, pers. comm.) or recruits (Selkoe et al. 2006; Hedgecock et al. 2007), although the scale of these aggregations is ill-defined. Larval aggregations are also consistent with observations of heterozygosity–fitness relationships that are due to inbreeding (because highly inbred individuals can only be produced by parents that share a common ancestry and thus dispersed together, David et al. 1995; see also Hedgecock et al. 2007). Hereafter we will refer to the correlated dispersal path of pools of larvae as “collective dispersal” (Yearsley et al. submitted).

The hypothesis that local drift and collective dispersal might produce chaotic genetic patchiness is not new (e.g., it is explicit in Campton et al. 1992), but it is difficult to test empirically (see Selkoe et al. 2006). Moreover there is no specific model that can be used to obtain quantitative expectations for chaotic genetic patchiness.

A population genetic model for the bentho-pelagic life cycle should allow for variation both in (1) the variance of reproductive success among adults and (2) the proportion of dispersers that travel collectively within packets of larvae. The former can be achieved by varying the local effective size of each deme and/or random local extinction prior to reproduction (i.e., some demes will not participate to the next generation). The collective dispersal of individuals was introduced in theoretical population genetics in the context of recolonization of empty demes (propagule pool model, Slatkin 1977; Whitlock and McCauley 1990; reviewed in Rousset 2003). A generalization of Slatkin's propagule pool model that incorporates collective dispersal as a regular mode of dispersal is presented in a related paper (Yearsley et al. submitted).

Here we aim to give a theoretical perspective on the chaotic genetic patchiness debate. Our objective is to assess the feasibility of neutral demographic processes producing such patterns in marine species. We use analytical coancestry dynamics and simulations to provide quantitative predictions on the effect of variance in reproductive success and collective larval dispersal upon fine scale patterns of genetic structure.

Coancestry Model

POPULATION STRUCTURE AND LIFE-CYCLE

A general population genetic model that incorporates collective dispersal is presented elsewhere (Yearsley et al. submitted). Here we derive specific predictions suited to explore the effects of the bentho-pelagic life cycle in the context of chaotic genetic patchiness.

We are interested in the coancestry dynamics of a metapopulation where reproduction takes place within a large number of randomly mating units (which we call demes). We therefore consider an infinite island model (Wright 1931) where each deme has a constant effective size, N (Fig. 1). We derive the coancestry dynamics for a simple life-cycle, where individuals mate randomly within demes (but selfing is not allowed), generations do no overlap, and offspring immigrate with probability m from any other deme without spatial constraint (see the following paragraph for more details about dispersal). The model assumes no demographic or genetic change from settlement to adults. New alleles occur at rate μ (infinite allele model, IAM, Kimura and Crow 1964).

Figure 1.

Representation of the metapopulation model described in text. Dispersal is described by forward-time parameters (emigration rate math formula and probability that an emigrant disperses as part of a collective pool math formula, see main text). All demes (extinct or not) may receive immigrants that dispersed collectively or independently from each other. In our model, each deme can receive only one set of collectively dispersing larvae (see main text). Refer to Wakeley (2004, Fig. 1) for a comparable graphical representation of the classical metapopulation model where collective dispersal affects only the recolonization of extinct demes (propagule pool model).

We introduce two additional features that are typical of a bentho-pelagic life-cycle (Fig. 1). First, local deme extinction occurs before reproduction with probability e. This allows us to introduce variance in reproductive success between demes. Combined with the finite size of demes, this variance can increase drift effects in the metapopulation. Second, we consider collective dispersal, where two immigrant individuals sampled within a deme (whether or not it was previously extinct) have a probability ϕ of originating from the same source (i.e., co-dispersing individuals). To simplify we consider that each deme produces and receives at most a single pool of co-dispersing individuals.

It is important to note that such models should consider effectively small demes. Effective deme size is driven by the census size of breeding groups where all adults may mate randomly (in the sense that the distance among sedentary adults is not limiting mating opportunities) and also by any factor affecting the variance of reproductive success between adults in each group. Such factors stem from a species’ breeding system, but also from other aspects such as family correlated survival in the larval phase. Furthermore, most free-floating larvae that spend more than a few days in the plankton (as in the vast majority of all living invertebrates) will be unable to metamorphose and settle in exactly the same breeding group from which they originated. This means that population genetic models should explore the region of parameters where nearly all individuals are migrants (a situation radically different from classical works in population genetics theory). However, recent results show that self-recruitment can occur in some populations of marine invertebrates or reef fish (e.g., Pinsky et al. 2010), although self-recruitment in these studies may be defined at a spatial scale (e.g., reef) that is larger than the one we consider here. We will thus start with a general framework considering small breeding groups and large dispersal parameters to assess whether drift and gene flow may interact to create chaotic genetic patchiness. Then we will explore the effect of increased group size or reduced migration to enlarge the range of situations that can be compared to empirical observations from case studies.

RECURSION EQUATIONS FOR COANCESTRY

We derive recursion equations for coancestry coefficients math formula (probability that the two alleles at one locus within an individual are identical by descent) and math formula (probability that two alleles sampled from homologous genes in two different individuals within a deme are identical by descent). In the infinite island model, the expected level of differentiation among demes as measured by FST is given by math formula. These coancestry coefficients may be estimated at different points in the life cycle (Fig. 1). To illustrate clearly the respective effects of drift and gene flow, we will calculate coancestry coefficients for adults (i.e., before reproduction but after dispersal, math formula and math formula) and offspring (i.e., after reproduction but before dispersal, math formula and math formula). This will also allow us to highlight the expected differences in genetic structure when sampling from the benthic or the larval phase.

We divide the life cycle into two independent steps (namely, reproduction and dispersal). Offspring coancestry coefficients can be obtained from their parental generation following:

display math(1a)
math image(1b)

where math formula

Adult coancestry coefficients can be obtained from the offspring of the same generation by:

display math(2a)
display math(2b)

Equation (2b) assumes that two randomly sampled alleles can only be identical by descent if they originated from the same deme (the coancestry between demes is negligible because there is an infinite number of demes). This happens in nonextinct demes when two alleles are philopatric (probability math formula in the first term of equation (2b)) and when two alleles immigrated together from the same deme (probability math formula in the second term of equation (2b)). The third term in equation (2b) states that two alleles taken at random within an extinct deme may be identical following re-colonization only if immigrant alleles came collectively (probability math formula).

Equation (2b) can be rewritten as:

display math(3)

where math formula can be interpreted as a mixing parameter. Then, combining equations (1a), (1b), (2a), (2b), (3), we obtain the recursion equations:

display math(4a)
math image(4b)

Note that similar expressions for math formula and math formula can be obtained from equations (1a), (1b), (2a), (2b), (3). We find the following general solution at equilibrium:

math image(5a)
display math(5b)
display math(5c)

The mixing parameter a can take a value between 0 and 1. When a = 1 two individuals originating from a given deme will always reproduce in two different demes (complete reshuffling of the genetic lineages among demes at each generation). In this case, genetic divergence between demes is generated by drift from one round of reproduction (e.g., from eqs. (4b) or (5a): math formula) and is then immediately lost during dispersal (eq. (3): math formula). At the other extreme when a = 0 drift is unimportant as the divergence is entirely governed by mutation (eq. (5a): math formula). In biological terms, “mixing” results mainly from an interaction between total migration, collective dispersal, and extinction. Collective dispersal tends to reduce mixing. Migration and extinction increase mixing in absence of collective dispersal but have complex effects otherwise (Yearsley et al. submitted).

Focusing on genetic mosaics of marine benthic species, we start with cases where the dispersal rate m is close to 1 (i.e., none or very few pelagic larvae are expected to return to the exact breeding group in which they were born). Furthermore, mutation rate will often be small enough that its effect on the dynamics of the system is negligible, math formula, (math formula). In the limit of math formula and math formula we obtain the following approximation to equation (5a):

math image(6)

and math formula.

With this approximation the mixing term a is a simple function of ϕ, explicitly showing that a low proportion of collective migrants will produce the strongest homogenization of the genetic structure among demes. With math formula and math formula the system can be thought of as a classical island model where the identity of demes would be randomized at each generation (demes simply swap positions). When math formula is less than 1, collective dispersal can thus effectively be seen as the level of philopatry in the system. This result can be visualized in Figure 2, where we see that math formula and math formula increase with math formula. In this figure, we see also that the difference between offspring and adult coancestries disappears as math formula, because in this limit there is no mixing (math formula and thus math formula). Equation (6) also shows that in the limit of math formula extinction has no effect, because the number of demes is infinite so that extinction does not reduce the number of sources of collective migrants (this result follows from model assumptions, see simulations below and online material from Yearsley et al. submitted, for the effect of a finite number of demes). Thus in the absence of collective dispersal, divergence is entirely governed by drift.

Figure 2.

Coancestry math formula expected at equilibrium in an infinite island model with extinction and collective dispersal when all individuals are immigrants (m = 1). Gray and black curves give the coancestry measured in offspring (predispersal) and adult (postdispersal), respectively. Local effective deme size (N) is 10. Even with such an extremely small effective deme size, equilibrium values of math formula are expected to remain low unless a very large proportion of offspring migrate collectively (e.g., >40%).

ROLE OF EFFECTIVE DEME SIZE

Our coancestry model considers a metapopulation composed of an infinite number of demes. As noted above, demes represent distinct randomly mating units that may or may not be discretely distributed in space. This is a useful representation of many bentho-pelagic species that occur in large (and potentially continuous) populations but where pulses of larvae are produced by small instantaneous breeding groups, and where the effective size of such groups can be further reduced by skewed reproductive success among adults. However, while the effective size of such local reproductive groups is an important driver of the system, it remains an elusive parameter with scarce empirical quantitative data (see discussion). In Figure 3 we show how the mixing term a (that captures the effects of m, e and ϕ) affects math formula and math formula equilibrium for local effective deme sizes N = 10, 100, or 1000. Except in the limit of math formula, increasing effective deme size reduces the expected differentiation between demes at equilibrium (Fig. 3). In the context of marine genetic patchiness, we aim to understand the effect of large e and low-to-moderate math formula values (e.g., giving a ≥ 0.8). In such cases math formula is strongly driven by drift (eq. (6)) and an increase in effective deme size translates into a nearly proportional decrease in offspring genetic structure (Fig. 3, in gray). At the adult stage, the effect of N is slightly less severe as a approaches 1 because in this parameter region math formula is primarily affected by a (Fig. 3, in black). The effect of deme size will be further explored using simulations.

Figure 3.

Effect of parameter a on equilibrium coancestry math formula in an infinite island model (taken before dispersal: offspring, or after dispersal: adults). The parameter a integrates the effect of extinction rate, migration rate, and the frequency of collective dispersal. It can be seen as a mixing factor that reduces the global genetic structure in the system. Effective deme size N is 10 (plain lines), 100 (dashed), or 1000 (dotted). Increasing the effective size of local demes from N = 10 to N = 1000 reduces the intensity of genetic drift and thus lowers the global level of differentiation math formula expected at equilibrium.

ROLE OF MIGRATION RATE

The effect of migration rate on equilibrium genetic structure depends on the rates of extinction and collective dispersal. Less migration generally results in lower mixing and thus stronger genetic structure (for an example without extinction see Fig. S1). Note however that decreasing migration can increase mixing when collective dispersal is strong (i.e., a combination of high ϕ and low e, Fig. S1. See also Yearsley et al. submitted). This is because migration rate enhances the clustering effect of collective dispersal (i.e., opposite to the effect of extinction shown in Fig. 2). In bentho-pelagic species we can rather expect math formula to take low or moderate values, in which case reducing m increases the genetic structure (particularly at the adult stage).

To summarize, we find that with math formula (i.e., when all larvae are thought to disperse away from their parental group, and thus all recruits are immigrants) genetic differentiation can still develop among groups if collective dispersal is frequent. However, we are interested in scenarios that are more realistic for bentho-pelagic species, where we expect low-to-moderate collective dispersal. In such cases, the model predicts (on average) very low levels of global genetic differentiation among reproductive groups. This prediction is in agreement with the large body of data from bentho-pelagic species (Shanks 2009; Weersing and Toonen 2009, our Table 1).

Individual-Based Simulations

We use individual-based simulations to quantify the variance around our coancestry predictions, assess whether such globally homogeneous populations may harbor chaotic genetic patchiness, and determine if sampling effects can produce patterns in the genetic data that mimic chaotic patchiness. In addition to low levels of global genetic structure, chaotic patchiness is characterized by a high heterogeneity in the genetic differentiation among samples taken at different locations and different times. Individual-based simulations are useful to identify the potential drivers of this variability, because (1) they produce multilocus genotype data that imitate empirical data, and (2) each replicate of a simulation scenario provides a random realization of the stochastic evolutionary processes.

SIMULATION FRAMEWORK

We used the simulation tool Nemo (version 2.1.3, Guillaume and Rougemont 2006) to produce individual multilocus genotypes for the population structure and life cycle from our coancestry model. However, there are important differences between the simulation and coancestry approaches. Most obviously, the number of demes is finite in the simulations (here we take nd = 1000). Another important point is the distinction between backwards and forward migration. Coancestry models consider backwards dispersal, characterized by the proportion of immigrants in a deme (m) and the probability that two immigrants came together math formula. In simulations dispersal is, as in the real world, a forward in time process (emigration rate math formula and the probability that an individual emigrates within a collective pool math formula). In the Appendix, we derive the relationships between forward and backward parameters for comparing simulation results to predictions from the coancestry model.

Other technical differences between our two modeling approaches are as follows. Although the number of gametes or offspring is implicitly infinite in the coancestry model, in the simulations we chose the fecundity to be large enough for the number of offspring in any given generation not to limit population size. The genetic information from the simulations is represented by individual multilocus genotypes from 20 loci where mutation creates new alleles (infinite allele mutation IAM) at rate μ = 10−4. Simulations were run for 15,000 generations, which was long enough for within-deme gene diversity Hs and FST to reach mutation–migration–drift equilibrium in all scenarios (details not shown). Each scenario was replicated 100 times, which allowed us to capture most of the variability due to the stochastic nature of the simulations (the standard deviation of all genetic statistics used below reached a plateau with less than 100 cumulated replicates, data not shown).

SIMULATION SCENARIOS

We followed the same logic as the coancestry model, first setting N = 10 and math formula = 1 in all simulation scenarios. We ran three main scenarios to explore how the dynamics of the genetic variance was affected by extinction, collective dispersal of larval pools, or both (parameter values in Table 2). Each parameter set resulted in a unique value for the mixing parameter a (ranging 0.19–1) except when math formula = 0 (a = 1). Additional sets of simulations were then run to assess the effects of deme size and subsampling of individuals (10 individuals vs. all individuals sampled per deme) in three situations characterized by [e = 0, math formula = 0], [e = 0, math formula = 0.5], or [e = 0.5, math formula = 0.5]. Finally, we explored the effect of reduced migration in an intermediate situation where e = 0.5 and math formula = 0.5.

Table 2. Parameter values used in individual-based simulations
Simulation parametersSymbolDefaultOther simulated values
Number of demesnd1000 
Number of individuals per demeN1050, 100
Mutation rateμ10−4 
Extinction ratee00.1, 0.5, 0.9
Migration rate (emigration)math formula10.99, 0.9, 0.5
Probability that an emigrant disperses within a collective poolmath formula00.1, 0.25, 0.5, 0.75, 0.9

SIMULATION ANALYSES AND RESULTS

To assess whether extinction and collective dispersal can generate chaotic genetic patchiness we looked first at the global structure of the population at equilibrium, we then examined the spatial variability in pairwise FST's, and finally we looked at temporal variation in the genetic composition of local demes. The within-deme gene diversity HS and observed number of alleles k are presented in Figures S2 and S3.

To estimate the global equilibrium FST, we recorded the genotypes of offspring and adults from the final generation, and calculated Weir & Cockerham's estimator of θ (Weir and Cockerham 1984, calculated in Nemo). Mean FST values calculated over all replicates were in good agreement with the predictions from the coancestry model (Fig. 4). We found that independent stochastic realizations deviate from these expectations only with increasing extinction rate (Fig. 4A vs. B). As e increases, stochasticity increases most quickly in the offspring global FST, because fewer demes contribute to the production of offspring at any given generation. In adults, some of the variation observed in global FST is due to FST being measured after extinction (and thus what we see is the effect of sampling randomly (1 – e) demes where the expected FST is 0). Taken together, extinction and collective dispersal result in low but variable global genetic structure at the adult stage (Fig. 4C).

Figure 4.

Global FST at mutation—migration–drift equilibrium in individual-based simulations (100 replicates) of a structured population (nd = 1000, N = 10) with extinction and/or collective dispersal. White and gray box-plots are for adults and offspring, respectively (i.e., post- and predispersal sampling). White dots and triangles give expected equilibrium values from our deterministic coancestry model for adults and offspring, respectively (see Fig. 1A). In panel A, larvae disperse independently (math formula = 0, math formula = 1). In panel B, there is no extinction. Both extinction and collective dispersal do occur in panel C. Note the differences in Y-axis scaling. At each parameter value on the x-axis the results for adults are to the left (white boxes, empty circles), and offspring are to the right (gray boxes, empty triangles). The variance among replicates of each scenario indicates that independent stochastic realizations of the simulated life cycle produce very limited departures from theoretical expectations.

To quantify the spatial variability in pairwise differentiation among demes (Fig. 5), we calculated pairwise FST values for a subset of 100 distinct pairs of demes taken from one simulation at the final generation (hence 100 pairs from each simulation scenario; sampling 10 pairs from 100 independent replicates of each scenario produced almost exactly the same results, data not shown). These data were calculated in R (version 2.12.1, R Development Core Team 2010) using the package hierfstat (Goudet 2005). The baseline distribution for pairwise FST obtained in the classical island model is shown in Figure 5A (first box-plot, e = 0). The variance in this distribution stems from the stochastic nature of demographic and evolutionary processes, the small size of demes (after dispersal each deme is effectively a random sample of 10 individuals from the global migrant pool), and from genetic sampling effects (FST estimates were based upon simulated genotypes of 20 microsatellite loci). Further simulation scenarios show that increases in extinction rate (Fig. 5A) and collective dispersal (Fig. 5B) both increase the spatial variability of pairwise FST. For instance, the range of pairwise FST roughly triples when e = 0.9 and math formula = 0.5 (Fig. 5C) compared to math formula (Fig. 5A). In agreement with the predictions from the reproductive variance and collective dispersal hypotheses, decreasing the proportion of demes that participate to the next generation and increasing the aggregation of larvae during dispersal both produce additional variability in the spatial genetic structure.

Figure 5.

Distribution of pairwise FST estimates for a subset of 100 population pairs sampled at mutation–migration–drift equilibrium from the simulation of a structured population (nd = 1000, N = 10) with extinction and/or collective dispersal. In panel A, larvae disperse independently (math formula = 0, math formula = 1). In panel B, there is no extinction. Both extinction and collective dispersal do occur in panel C. Note the differences among panels in Y-axis scaling.

To visualize the dynamics of the differentiation between pairs of demes, we plotted pairwise FST for four randomly chosen pairs of demes across 10 consecutive generations (Fig. 6). This figure illustrates that pairwise FST's are equally variable in space and time. Because all individuals are migrants, pairwise FST can vary greatly from one generation to the next. Strong variations in space and time are produced with any configuration for e and math formula shown in Figure 6. With these simulations the range of pairwise FST's is approximately one order of magnitude higher than that of global FST's (e.g., Figs. 6 and 4C).

Figure 6.

Sample of the spatio-temporal variation in pairwise FST estimates from individual-based simulations of a structured population (nd = 1000, N = 10) with extinction and collective dispersal. Dots represent FST between a pair of demes (four pairs represented) measured at each generation during 10 generations (sampled after mutation–migration–drift equilibrium was reached). The spatial variation in pairwise genetic structure can therefore be seen from the differences among the four dots at any generation, while the temporal variation is shown by the trajectory of each line.

Second, we examined the temporal variability in the genetic make-up of focal demes. For that we calculated temporal F-statistics (math formula) defined as the differentiation between a focal deme and itself one generation later. These temporal statistics were calculated for a subset of 100 demes sampled at mutation–migration–drift equilibrium and considering a single replicate per simulation scenario (Fig. S4). Predictions for temporal F-statistics across t generations can be obtained from coancestry dynamics as math formula, where math formula is the probability that a gene randomly sampled in a deme in a focal generation is identical-by-descent with a homologous gene randomly sampled from that deme t generations earlier, and math formula is the probability of identity between pairs of genes sampled in the same generation (following Lehmann 2007; Vogel et al. 2009). For t > 0 we have

math image(7)

This equation shows that math formula in the limit of math formula. This prediction is in line with the simulation results (Fig. S4) where the level of temporal differentiation appears to be similar to the level of spatial differentiation in all scenarios (Figs. 5 and S4). Note that the definition of temporal differentiation used here (Vogel et al. 2009) differs from the temporal F introduced by Nei and Tajima 1981 for estimating effective population size using temporally spaced samples.

ROLE OF EFFECTIVE DEME SIZE AND SAMPLING ERROR

We examined the effect of deme effective size greater than 10 (N = 50 and 100) by repeating three scenarios (namely [e = 0, math formula = 0]; [e = 0, math formula = 0.5]; [e = 0.5, math formula = 0.5]) that had shown no to moderate mosaic patterns for N = 10 (Fig. 7). As expected, increasing local effective size decreased genetic drift and buffered the variations of genetic structure in space and time. For instance, increasing N from 10 to 50 resulted in a 10-fold reduction in the range of pairwise FST's when e = 0.5 and math formula = 0.5 (Fig. 7C). Identical results were obtained for FSS (data not shown).

Figure 7.

Effect of deme effective size and sampling design on genetic differentiation between pairs of demes sampled at mutation–migration–drift equilibrium from the simulation of a structured population (nd = 1000, N = 10, math formula = 1) with extinction and/or collective dispersal. Each boxplot shows the distribution of pairwise FST estimates for a subset of 100 pairs of demes as a function of sample size and deme size (e.g., 10/50 means 10 individuals sampled of 50 in each deme). In panel A, there is no extinction and larvae disperse independently (e = 0, math formula = 0). In panel B, there is no extinction but collective dispersal occurs (e = 0, math formula = 0.5). Both extinction and collective dispersal occur in panel c (e = 0.5, math formula = 0.5). Panels A–C show pairwise FST estimates when all individuals are sampled within each deme. Panels D–F show the same data when only 10 individuals are sampled per deme.

The features of chaotic genetic patchiness (i.e., fine scale differentiation with no spatial or temporal patterns) are also expected features of the random noise associated with sampling effects. To investigate whether sampling could produce confounding patterns, we recalculated pairwise FST's using subsamples of 10 individuals per deme from all the above simulations (i.e., N = 10, 50, and 100, Fig. 7D–F). We also tested for the significance of spatial pairwise divergence for a subset of population pairs in each simulation described above. We subsampled 15 demes (i.e., 105 pairs) from each simulated data set and tested for genetic differentiation between pairs with a commonly used method (pairwise tests of genetic differentiation to the 5% level using G-statistics and Bonferroni correction in the software Fstat, Goudet 2005). The results show that using 10 genotypes from each deme systematically increases the variance in spatial pairwise FST's (compare panels d-f to a-c in Figure 7), resulting in spatial variations as strong as or stronger than created by drift and collective dispersal. However, no significant divergence was detected between any pair of demes in absence of collective dispersal, whatever the sampling design is (Fig. 7A and D). When all individuals were sampled in each deme, a number of pairwise comparisons were significant in all situations involving collective dispersal (from 3% of significant pairwise comparisons when N = 10, e = 0.5 and math formula = 0.5 to 94% when N = 100, e = 0 and math formula = 0.5, details not shown). The effect of subsampling (10 individuals per deme) almost completely eliminated the probability of detecting significant pairwise differences in all situations.

ROLE OF MIGRATION RATE

To suit our purpose of modeling the bentho-pelagic life cycle, we have set math formula = 1 in all simulation scenarios presented so far. With m < 1 spatial and temporal statistics are decoupled. Reducing migration resulted on average in more spatial divergence (Fig. S5A) but less temporal divergence (Fig. S5B), though with more stochastic variations around these expectations in both cases.

Discussion

The core result of this study is the confirmation that chaotic patchiness can be produced by neutral demographic processes alone. As hypothesized by a number of authors (Campton et al. 1992; Hedgecock 1994; David et al. 1997b), genetic drift combined with collective larval dispersal can produce chaotic genetic patchiness resembling the patterns observed in marine species with a bentho-pelagic life cycle. Below we describe these processes and highlight unanswered questions.

PROCESSES BEHIND THE CHAOS

The processes potentially creating patchiness are as follows. Genetic structure builds up if breeding groups produce pools of larvae that differ through genetic drift (ignoring mutations, the mean divergence is equal to 1/2N when m = 1 and ϕ = 0, see offspring in Figs 2 and 4A). Small effective deme size is thus the first component required to create chaotic genetic patchiness. Second, genetic drift at the scale of the metapopulation can then be enhanced by among-deme variance in reproductive success. Among-deme variance was modeled here using extinction prior to reproduction, but it could also result from an unbalanced contribution of nonextinct demes (i.e., varying effective deme size). The variance among demes is not required for creating chaotic genetic patchiness, but it increases strongly the stochasticity of the signal used to qualify such patterns. Moreover it decreases the total effective size of the metapopulation by reducing the number of larval sources. Finally, at a spatial scale where all or most larvae are immigrants, collective dispersal is required to yield significant genetic divergence after dispersal (i.e., in recruits or adults). Results from the individual-based simulations show that patchiness levels comparable to empirical results can be produced by a combination of strong variance in reproductive success and mild collective dispersal (e.g., case c of 4-6). As shown by the coancestry model, such a combination still produces low equilibrium levels of global differentiation, in agreement with empirical observations (Table 1).

EFFECTIVE DEME SIZE: HOW SMALL?

Our baseline scenario used a very small effective size for breeding groups (N = 10). Such small effective sizes can be obtained in principle when the census number of adults exchanging gametes is locally limited and the reproductive success among these adults varies widely (in line with the sweepstake reproductive success hypothesis, Hedgecock and Pudovkin 2011). A situation with small N is useful to illustrate the neutral processes at work in the bentho-pelagic life cycle, and to test the general idea that these processes can result in chaotic genetic patchiness. Yet real-world effective sizes will frequently exceed our baseline scenario. We found that increasing effective deme size decreases strongly the spatio-temporal variations in genetic structure (Figs. 3 and 7). Such variations will then be detected only provided a large number of individuals can be sampled within demes.

Empirical estimates of effective size of breeding groups are scarce. Hedgecock et al. 2007 found that only 10–20 breeders contributed to a cohort of juvenile flat oysters O. edulis (defined as a group of juveniles that recruited at one site within a 12-day period). In this study, the effective number of breeders was computed by comparing adults and offspring taken postdispersal, which is thus not strictly comparable to the size of breeding groups in our model. Parentage analysis will be useful for estimating the effective number of breeders in other natural populations (e.g., maximum number of 10–15 parents in C. fornicata, Dupont et al. 2006; Proestou et al. 2008; Le Cam et al. 2009).

COLLECTIVE LARVAL DISPERSAL … SERIOUSLY?

In many species the metamorphosis of competent larvae is triggered by chemical signals. Such mechanisms may favor the collective recruitment of batches of larvae (Pineda et al. 2010). Yet it is not trivial to accept that pools of larvae released by a local breeding group migrate collectively and settle together in an area tight enough so that they have a chance of reproducing together once they become adults. The strongest evidence for this comes from the observation of half-sibs in recruits (e.g., Selkoe et al. 2006; Hedgecock et al. 2007) but more empirical data at different stages of larval dispersal are required to explore this hypothesis. Along the same lines, when this is possible one could potentially examine the distribution of relatedness between benthic individuals of the same age and test whether observed data are compatible with random isotropic dispersal (Avise and Shapiro 1986; Planes and Lenfant 2002; Veliz et al. 2006). Additionally, indirect information is given by bio-physical models of larval transport (Siegel et al. 2008), which could be used to specifically test this question in species where collective larval dispersal is suspected.

LIMITS OF THE MODEL AND PERSPECTIVES FOR EMPIRICAL RESEARCH

Here we used a theoretical framework to show that chaotic genetic patchiness can arise as a consequence of neutral processes in the bentho-pelagic life cycle. However, precise quantitative comparisons between empirical case studies and modeling results are limited. First, different theoretical scenarios can lead to indiscernible differences in genetic patterns. Second, our model makes a number of necessary simplifications (population structured into identifiable breeding groups, distinct breeding and dispersal steps, nonoverlapping generations). Overlapping generations for instance has the potential to buffer the stochasticity produced by the bentho-pelagic life cycle (Dupont et al. 2007). Finally, here we provide genetic results applying to clearly identified breeding groups and cohorts of larvae. In nature samples will more often be composed by a mixture of adults or larvae from different demes. Nevertheless, the modeling results presented here bring additional support for two aspects of empirical research: (1) the benthic phase and the pelagic dispersal phase have equally important roles in generating chaotic genetic patchiness. Empirical estimates of the effective number of breeders producing larval cohorts are thus important to understand a species’ genetic structure (Hedgecock et al. 2007; Liu and Ely 2009); (2) the genetic structure of larvae before dispersal is rarely studied but it would, in combination with adult data, allow the effects of drift and gene flow to be largely distinguished. In this paper, we have highlighted the differences in genetic structure expected when one samples from offspring or adults. In particular, it is important to consider whether sampled individuals have experienced dispersal. In predispersal offspring, the genetic structure observed will reflect the genetic divergence of the parental population accentuated by one round of genetic drift. This observation was termed the “Allendorf-Phelps effect” by Waples 1998, by reference to a situation described by Allendorf and Phelps 1981 where dispersal results in perfect mixing of individuals but offspring issued from a limited number of parents are sampled before dispersal. In this situation Waples 1998 found that the genetic structure of offspring is weighted by a factor 1/2N, in line with the model described here.

Interestingly, Domingues et al. 2011 found that samples of shore crab megalopae larvae (collected within 1 to 4 tidal excursions on a unique sampling site) were homogeneous in time and no different from adult samples, which is in disagreement with the hypothesized drivers of chaotic genetic patchiness. Additional empirical studies on the larval phase before settlement will help determine if other species show similar patterns. A number of studies compare the genetic structure of recruited juveniles (i.e., postdispersal) with adults (Flowers et al. 2002; Liu and Ely 2009; Muths et al. 2009; Christie et al. 2010; Burford et al. 2011). It is important to note that under the conditions of our model, recruited juveniles and adults are predicted to have identical genetic structure (e.g., global FST, pairwise FST and FSS, local gene diversity HS). This is because (1) we consider a population where all genetic metrics have reached mutation–migration–drift equilibrium, (2) reproduction and dispersal are modeled as distinct, successive steps in the life cycle, and (3) we assume no drift, selection, mutation, or migration once juveniles have been recruited. This situation is a good fit to short-lived species with one major period of reproduction. On the other hand, many species have a long reproductive period where the emission of larvae is continuous. With these species there can be successive episodes of recruitment yielding adult breeding groups composed by a mixture of individuals from different origins. In these conditions, a sample of recruits at a given time potentially captures the contribution of a particular set of successfully reproducing adults. Point sweepstake effects can thus be detected (the effective number of breeders of a particular cohort), but it would not be informative about global effective population size. Yet even rapid successions of recruitment do not necessarily superimpose spatially, because fluctuating settlement limitations (e.g., local changes in habitat suitability) or the passive transport of larvae can still lead to spatio-temporal mosaics (e.g., Fig. 3 from David et al. 1997a). Such mosaics should be reasonably well described by our model.

ALTERNATIVE EXPLANATIONS TO CHAOTIC PATCHINESS

Apparently chaotic genetic patchiness could arise simply through sampling effects. First, with m = 1, groups of adults are effectively random samples from the pool of predispersal individuals, meaning their allelic frequencies will reflect sampling error in drawing from the global migrant pool. This will be especially visible in pairwise FST and temporal FSS (e.g., Fig. S4A with e = 0). Second, the statistics used to estimate genetic divergence suffer sampling variance. Simulations show that this variation can be as high as or higher than that produced by collective dispersal (e.g., Fig. 7C vs. d) and thus may lead to apparent patterns of chaotic genetic patchiness with no need to refer to specific demographic processes in the bentho-pelagic life cycle. Unsurprisingly, our results show that both sampling effects will be correctly identified by a commonly used statistical testing procedure. Paradoxical observations of statistically significant genetic structure at a spatial scale where migration is not limited should therefore not be due to sampling artifacts.

A different situation occurs if dispersal has been overestimated. We have shown that philopatry (m < 1) can enhance the stochasticity in the dynamics of the genetic variance (Fig. S5). The fact that pelagic larvae may settle in the breeding group where they were produced may appear unlikely, but recent empirical results suggest that self-recruitment at small spatial scales is possible (Pinsky et al. 2010). In such cases collective larval dispersal is not required to produce chaotic genetic patchiness: some of the divergence created by genetic drift may persist after dispersal because offspring are not completely mixed in the pelagic phase. In this paper, we have adopted a framework where space has no effect on the genetic structure simply because it has no effect on larval dispersal. Under the hypothesis that dispersal is somewhat spatially limited, the chaotic genetic patchiness paradox shifts from understanding fine scale genetic structure to understanding why this structure does not build up with distance. Eldon and Wakeley (2006, 2009) have shown using coalescence theory that severe skews in the distributions of offspring numbers can enhance a population's genetic structure even in the face of strong gene flow. Eldon and Wakeley's model allows some individuals to have an extremely large reproductive success. This situation fits Hedgecock's sweepstakes reproductive success hypothesis where some adults may contribute to a very large fraction of all descendants. Such severe skews yield chaotic genetic patterns provided offspring do not disperse across the whole population within one generation (illustrated in Fig. 5 from Hedgecock and Pudovkin 2011). Such effects may explain why chaotic genetic patchiness can be observed not only locally but at much larger spatial scales (i.e., beyond typical larval dispersal distance but where gene flow is still high).

CHAOTIC GENETIC PATCHINESS AND HETEROZYGOSITY–FITNESS CORRELATIONS

What are the consequences of chaotic genetic patchiness? The dynamics of selected alleles is one aspect that needs further consideration (see Der et al. 2012 for a relevant study based upon Eldon and Wakeley's model cited above). Regarding neutral processes, the collective movement of individuals will affect gene flow estimates that are based upon indirect methods (Yearsley et al. submitted). With respect to our understanding of genetic patterns in marine species, one interesting consequence of small deme size and collective dispersal is to increase the variance in within-individual coancestry. In particular, some highly inbred offspring can be produced by parents that share a common ancestry and dispersed together. Such offspring will have a reduced genome-wide heterozygosity, with two independent consequences: (1) reduced heterozygosity at marker loci, (2) reduced fitness if deleterious alleles at fitness loci are exposed as homozygotes (inbreeding depression). Hence a statistical correlation between heterozygosity at marker loci and fitness measurements can emerge (associative overdominance sensus Szulkin et al. 2010). The combination of drift and collective dispersal described here could thus lie behind some observations of heterozygosity–fitness correlations in marine species (as advocated by David et al. 1995; Bierne et al. 2000).

ACKNOWLEDGMENTS

We thank P. David, D. Roze, R. Waples, an anonymous referee, and Editor O. Ronce for their comments on previous versions of this manuscript. We are grateful to F. Guillaume for his help with the simulation software Nemo. JY was supported by a Ulysses grant from IRCSET. TB was supported by a Ulysses grant from the French ministers ‘Affaires étrangères et européennes’ and ‘Enseignement supérieur de la recherche’. TB and FV were supported by the “Marine Aliens and Climate Change” programme funded by AXA Research Funds.

Appendix

Dispersal is commonly thought of as a process moving forwards in time (e.g., the emigration rate, math formula, and the probability that an emigrant disperses collectively, ψ). By contrast, models of coancestry dynamics use parameters that look at dispersal backwards in time (e.g., the immigration rate, m, and the probability that two immigrants came from the same deme, ϕ). The relationships between forward-time and backward-time processes can help us to think about dispersal process in the real-world, and are necessary to compare coancestry model predictions to forward-time simulations.

Here we present the relationship between the forward-time parameters (math formula, ψ) and the backward-time parameters (m, ϕ) used in this paper. To do this, we consider the three classes of individuals that can occupy a focal deme: individuals that were born locally (Nr), individuals that immigrated collectively from a single source (Np), and individuals that immigrated from any other deme (No). In a model with extinction and collective dispersal, the contribution of these three classes depends upon whether the focal patch and its source of collective migrants experienced extinction.

By definition math formula. For any nonextinct focal patch (first term of eq. (2b)) there are two cases: case 1 with probability e the source of collective migration is extinct (math formula); case 2 with probability 1 – e the source of collective migration is not extinct (math formula). In both cases we have math formula and math formula. For an extinct focal patch m is implicitly equal to 1 (second term of eq. (2b)). Combining these cases gives

display math(A1)

where math formula and math formula.

Similarly, we have math formula, which we find to be equal to:

math image(A2)

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