Department of Genetics, University of Georgia, Life Science Building, Athens, Georgia 30602, USA

Department of Earth Sciences, University of New Hampshire, Durham, New Hampshire 03824, USA

Abstract

Background

Genetic estimates of effective population size often generate surprising results, including dramatically low ratios of effective population size to census size. This is particularly true for many marine species, and this effect has been associated with hypotheses of "sweepstakes" reproduction and selective hitchhiking.

Results

Here we show that in advective environments such as oceans and rivers, the mean asymmetric transport of passively dispersed reproductive propagules will act to limit the effective population size in species with a drifting developmental stage. As advection increases, effective population size becomes decoupled from census size as the persistence of novel genetic lineages is restricted to those that arise in a small upstream portion of the species domain.

Conclusion

This result leads to predictions about the maintenance of diversity in advective systems, and complements the "sweepstakes" hypothesis and other hypotheses proposed to explain cases of low allelic diversity in species with high fecundity. We describe the spatial extent of the species domain in which novel allelic diversity will be retained, thus determining how large an appropriately placed marine reserve must be to allow the persistence of endemic allelic diversity.

Background

The relationship between genetic diversity and population size offers a number of tantalizing insights into demographic influences on evolution _{e}) of a species much lower than the actual census size _{e }_{e }as estimated from genetic markers is several orders of magnitude lower than would be predicted based on census size (N) and a species' reproductive traits

Here, using the results of Pringle and colleagues _{e }for populations whose dispersal is subject to persistent directional flow and find a complementary mechanism for the reduction of N_{e}. We do this in a linear domain, such as a benthic population in a stream or a coastline, though the results can be easily generalized to different geometries. We find that _{e }in a given species domain. This mechanism for the reduction of N_{e }will be shown to be distinct from sweepstakes models. In the sweepstakes models, N_{e }is reduced by variability in reproductive success between individuals in the same region. In contrast, physical drift will be shown to reduce N_{e }by creating differential reproductive success between individuals in different regions.

Previous work [e.g. _{e }of the entire group of populations would tend to that of only the upstream population, given sufficiently large and asymmetric migration. While their work treated this effect in discrete demes, we examine a continuously distributed species along a coast or river in which there is no _{e }for the species over its entire domain. By identifying allelic retention as a spatially defined component of coastal diversity, this work has implications for the design of marine reserves. While genetic diversity has typically not been considered in the placement or size of marine reserves

An estimate of N_{e }in an advective environment

In order to determine N_{e}, it is necessary to divide the populations into source and sink regions. To do so, we define a "retentive population" as a demographically stable group of individuals that can persist without immigration from outside its domain. This definition is equivalent to that of Booke

_{allele}) > L^{2}_{adv}/(2L^{2}_{diff})

where N_{allele }is the mean number of a given allele (or class of alleles, _{allele }is equivalent to mean lifetime reproductive success per adult _{adv }is the average distance a successfully recruiting larva is moved downstream from its parent before recruiting, and L_{diff }is the standard deviation of that distance for all successful recruiting larvae an adult releases. These criteria assume that kurtosis of the dispersal kernel is close to that of a Gaussian; for other kernels, a correction has been developed (Pringle et al., in review).

Allelic diversity persists within this retentive population, as the population is not supported by migration from elsewhere. We define the concept of allelic "persistence" relative to the expectation for the rate at which neutral alleles are lost or go to fixation in a finite randomly mating population

With advection an entire species domain cannot be a retentive population, for the criteria in equation (1) above cannot be met throughout the species range if L_{adv }is not zero. At demographic equilibrium, each adult will (on average) generate one surviving offspring and thus one copy each for each copy of an allele it carries – thus the average of N_{allele }over the species range is 1 for neutral alleles, which does not satisfy equation (1). Retention of some allelic diversity occurs because the reproductive success per adult is not evenly distributed spatially, and so in some places is great enough to satisfy eq. 1 _{allele }for a neutral allele, is greater than that needed to satisfy eq. (1), the population will increase at that point. This suggests that the population at the upstream edge will increase until, due to density dependent effects, the average of N_{allele }over the upstream retention zone is reduced until it just satisfies eq. (1). At the upstream edge of the domain there will be a region where eq. (1) is satisfied, and novel allelic diversity can be retained. Since most larvae are transported downstream a mean distance L_{adv}, this upstream region also supplies migrants to downstream regions. Thus, the upstream edge is a retentive population where alleles will only change in frequency due to stochastic drift in allele frequency and the accompanying probability of fixation.

The size and census population of the region of enhanced reproductive success, and thus N_{e}, will depend on the nature of the spatial variation in habitat quality, L_{adv }and L_{diff}. Here we examine the case in which the habitat is spatially uniform downstream of the upstream edge of the habitat. The mean transport will move an average propagule nL_{adv }downstream of its parents after n generations, while the stochastic component of transport will move the propagule a standard deviation of n^{0.5}L_{diff }around that point ^{2}_{diff}/L^{2}_{adv }generations. Substituting this expression into either of the distances defined above gives the distance L_{reten }= L^{2}_{diff}/L_{adv}, suggesting L_{reten }is the fundamental length scale of this system, and is the distance over which the effects of mean and stochastic propagule transport are balanced. This suggestion is confirmed with dimensional analysis _{adv }and n^{0.5}L_{diff }suggest underlying parameters with units of velocity (time/distance) and diffusivity (distance^{2}/time). From these, only a single dimensionally consistent length scale can be formed, and it is L_{reten}. Multiplying this distance by the carrying capacity per unit length of the environment (H_{dens}) provides a scaling for N_{e }comparable to that of

_{e }= H_{dens}L^{2}_{diff}/L_{adv},

and the numerical modeling described below confirms the appropriateness of this scale. (We assume that eq. (1) can be satisfied even when the population is close to its carrying capacity. When this is not true, the population is marginal at this location _{e }will be further reduced). We expect this estimate of N_{e }to be reduced relative to standard drift expectations by increasing mean propagule transport (L_{adv}), and that this effect is diminished by increased stochastic transport (L_{diff}). Since L_{adv }and L_{diff }are significantly smaller than species ranges for most coastal species _{e }should be much less than the census population size of the entire domain or metapopulation [as in

Downstream of the retentive population that defines N_{e}, N_{allele }will not satisfy (1) – and alleles are not retained – if L_{adv }is non-zero. Allelic diversity in downstream regions will be set by the allelic composition of migrants from upstream. So, in an advective environment the evolution of allelic diversity in the entire population will be governed by the allelic diversity in the retentive population, and the N_{e }for the entire population should approach the census population of the retentive population given by eq. (2). However, heterogeneity in abiotic (

Methods & results

To test these ideas, we use a simple numerical model, similar to those used by Pringle and colleagues _{adv }downstream, with standard deviation of L_{diff }with a Gaussian dispersal kernel

To illustrate how advection reduces genetic diversity in a population, two domains are initialized with five different alleles each in different parts of the domain in the numerical model (figure _{adv }is zero; in the other it is 4 km/generation to the right. In both, the stochastic component of larval transport L_{diff }is 10 km/generation. In the case with no mean larval transport, all genetic diversity is retained. However, when there is mean larval transport, only the upstream allele persists and the other alleles are lost downstream, for only the upstream allele begins in the retentive region that lies within L_{reten }= 25 km of the upstream edge of the domain.

In both (A) and (B), the domain is initialized with haploid adults containing 5 different alleles, each geographically isolated to 1/5 of the domain, and each adult colored according to its allelic composition

**In both (A) and (B), the domain is initialized with haploid adults containing 5 different alleles, each geographically isolated to 1/5 of the domain, and each adult colored according to its allelic composition.** The model is run for 400 generations. In (A), L_{adv }= 0 km and L_{diff }= 10 km. The allelic composition diffuses isotropically away from initial positions, and no allele is favored over others. In (B), L_{adv }= 4, so larvae preferentially disperse towards positive x (to the right) and the upstream allele quickly dominates the entire domain. L_{reten }in (B) is 25 km.

A second numerical experiment illustrates how the presence of directional larval dispersal changes the spatial structure of the population system. In these model runs, L_{diff }is fixed to 100 km and L_{adv }is varied from 0 to 116 km. There is a mutation rate μ = 10^{-3 }such that larvae randomly carry a new allele with this frequency (a smaller, more realistic μ does not change the results, but dramatically increases computation time). In these model runs, N_{allele }is uniform in the interior and small near the edges when L_{adv }is zero, but as L_{adv }increases, N_{allele }becomes largest near the upstream edge of the species domain, and is one in the interior of the model domain (figure _{allele }divided by the value that just satisfies eq. (1) (figure _{allele }just satisfies eq. (1) in the retention zone that lies within L_{reten }from the upstream edge of the domain, and does not elsewhere. In the model, time to fixation or extinction of all novel alleles is tracked as a function of their origin. As discussed above, enhanced reproductive success within a distance L_{reten }from the upstream edge allows novel alleles to persist longer in the upstream retentive population, for a time appropriate to N_{e }as given by Eq. (1), while those in downstream regions are lost much more quickly (figure _{adv }increases (figure _{reten}.

(A) N_{allele }for a 1-dimensional ocean with a mean current from left to right as a function of the alongshore distance and the mean larval transport distance, L_{adv}

**(A) N _{allele }for a 1-dimensional ocean with a mean current from left to right as a function of the alongshore distance and the mean larval transport distance, L_{adv}.** The heavy black line represents the width of the retention zone L

To determine N_{e }as a function of L_{adv }and L_{diff}, we calculate the inbreeding effective population size N_{e }_{e }_{e }from the average fixation time of 100 model runs. In Figure _{domain }= 10^{3}, 4 × 10^{3}, and 1.6 × 10^{4 }km. In each domain, we fix L_{diff }to 200 km, and vary L_{adv }from 0 to 110km, and compare the estimated N_{e }from (2) to the estimation from fixation time in an upstream region of the model L_{reten }in size. Once there is fixation in this upstream region, the allele fixes rapidly in the rest of the species domain in approximately L_{domain}/L_{adv }generations. When the size of the domain is less than L_{reten }in extent, N_{e }is limited to the population census size (figure _{adv }is small, N_{e }is nearly equal to the census population of the entire population, though somewhat smaller due to loss of larvae from the edges of the domain caused by stochastic larval transport. When the domain size is greater than L_{reten}, eq. (2) captures the variability of N_{e }with L_{adv }very well, capturing the several order of magnitude decline in N_{e }with increasing L_{adv}. As mentioned above, the estimate of N_{e }from (2) is, for most values of L_{adv}, very much smaller than – and not dependent upon – the census population size. When the model is re-run with Laplace's dispersal kernel, the results shown in Figure

(Thick Black Line) Estimate of N_{e }from equation (2)

**(Thick Black Line) Estimate of N _{e }from equation (2).** (Dashed Thin Lines) Census population in domain. The squares (□) are for a domain 1024 km in size, the circles (○) are for a domain 4096 km in size, and the diamonds (◇) for a domain 16384 km in size. (Solid Lines) Estimates of N

Discussion

The concept of "effective population size" is typically used as a numerical trait of a population more than as a descriptor of biological reality _{e }is intended to reflect the number of individuals that contribute to the evolutionary potential of a species

The attention given to estimating N_{e }in natural populations has recently been focused on a number of demographic causes for reduced N_{e}/N ratios _{e}μ (or N_{e}μ for haploid markers), in an advective environment most of these are more quickly lost due to advection (a time in generations of about L_{domain}/L_{adv}) than to stochastic genetic drift (Figure _{adv }increases, this source region becomes smaller, and with uniform density of individuals reduces N_{e }concomitantly.

Thus, different species with distinct larval dispersal traits can have distinct N_{e}/N ratios in the same region, all else being equal. This mechanism does not hinge on the reproductive "sweepstakes" between individuals at the same location – instead, it is an effect of the differential reproductive success of individuals from different regions, and the effects of mean larval transport. Mean larval transport, L_{adv}, can change from generation to generation _{e }from generation to generation, and years of especially strong mean advection could reduce net diversity. Thus N_{e }can be reduced not only by year-to-year variation in reproductive success, but also by inter-annual changes in the physical environment that affect larval dispersal.

A growing body of literature attempts to link patterns of genetic diversity with patterns of biodiversity, for the purposes of elucidating the mechanisms underlying broad-scale biogeographic structure and for conservation-focused predictions _{e }will be more disassociated from actual census size than for populations less affected by advection.

To test this prediction, one might imagine comparing species with very different dispersal strategies, or comparing the same or similar species in two locations with different dispersal conditions. However, while classical population genetics predicts elevated diversity _{e}. Metapopulation structure in general may bias the measurement of N_{e }and gene flow measures

Conclusion

Overall, some of the lowest N_{e}/N ratios observed are for species with broad dispersal potential in regions where ocean currents would be expected to generate a large L_{adv }_{e }and actual census size. In the end, any mechanism that increases the variance in reproductive success among individuals, whether due to stochastic, biological, or spatial processes, will reduce genetic variation in a species _{e}/N ratios (_{reten }it will also be protecting 'sink' regions; if it is smaller than this length, reserve size will be a limiting factor on total genetic diversity.

Authors' contributions

This work was developed as equal-authorship collaboration between JPW, who conceived of the study and drafted the manuscript, and JMP who developed the numerical simulations and scalings and coordinated the results, and helped to draft the manuscript. All authors read and approved the final manuscript.

Acknowledgements

Many thanks to John Wakeley, Scott Small, Mike Hickerson, Robin Waples, Bob Holt, and two anonymous reviewers for comments and discussions during the writing of this manuscript. This work was supported by University of Georgia Research Foundation funds to JPW and NSF grant OCE-0453792 to JMP.